Economic-State Variation in Uncertainty-Yield Dynamics
Economic-State Variation in Uncertainty-Yield Dynamics
Connolly, Robert A; Dubofsky, David; Stivers, Chris
2021-02-15 00:00:00
Abstract We show there is a much stronger negative, dynamic relation between changes in economic uncertainty and Treasury yields over weaker economic times since at least 1990. We document this economic-state variation in uncertainty-yield dynamics for weekly and monthly change horizons, for nominal yields and real-yield proxies, for multiple economic-state identification methods, and for different economic uncertainty metrics. We present additional findings that suggest short-term fluctuations in precautionary-savings and consumption-smoothing forces are more impactful on interest rate dynamics during weaker economic times, especially relying on surveys of expected economic growth and inflation. Received February 8, 2019; editorial decision August 24, 2020 by Editor Nikolai Roussanov. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online. Standard economic theory generally predicts a negative relation between economic uncertainty and interest rates. In this paper, our empirical objective is to analyze the sign and magnitude of the dynamic uncertainty-yield relation for Treasury bonds over the 1- to 10-year horizons in the term structure, with a special focus on differences between weaker economic times and baseline times. By “dynamic,” we refer to the relation between changes in economic uncertainty and changes in interest rates over weekly to monthly change horizons. By “changes in economic uncertainty,” we refer to proxies for changes in the expected conditional volatility of consumption growth and/or economic growth. Our principal contribution is to expose a much stronger negative, dynamic relation between economic uncertainty and Treasury yields over weaker economic times. Prior literature has analyzed how interest rate levels vary with the level of economic uncertainty or vary across the business cycle, typically evaluating long samples of quarterly or annual data. See, for example, Ang, Bekaert, and Wei (2008), Bekaert, Engstrom, and Xing (2009), Bansal and Shaliastovich (2013), Drechsler (2013), Hartzmark (2016), and Bekaert and Engstrom (2017). Our work is a complement to this literature, in that we study the relation between changes in economic uncertainty and changes in interest rates over weekly and monthly periods. Economic cycles are slow moving, so the economic state is relatively stable over most weeks and months. Thus, our empirical investigation is focused on the dynamic relation between higher-frequency uncertainty fluctuations and yield changes when within an economic state, and whether the within-state relation is different for weaker economic states. We primarily rely on the VIX, the CBOE’s 30-day S&P 500 implied volatility index, to identify changes in economic uncertainty over weekly to monthly horizons, while also controlling for changes in the Treasury-bond implied yield volatility. As an example of our main results, we find that the negative partial relation between monthly VIX changes and the monthly changes in 5- and 10-year nominal Treasury yields is about four times stronger during our primary weaker economic (WE) segments, relative to the baseline segments, over our featured 1997:10 to 2017:12 sample period.1 The average R2 for our yield uncertainty regressions over the WE segments is also about four times that for the baseline segments, at 38.9% versus 9.3% (see Table 2). In Section 1, building on the literature, we discuss what short-horizon VIX changes might imply about changes in the core economic determinants of interest rates. We argue that VIX increases are likely to imply both greater economic growth uncertainty and a decline in expected economic growth, with both these VIX-growth relations likely being stronger over weaker economic (WE) times. If so, interest rates should decline more with a given VIX spike during WE times, because of the relatively larger increase in economic uncertainty (implying an elevated negative precautionary savings influence) and/or a relatively larger decline in expected economic growth (implying a declining consumption-smoothing influence). Further, risk aversion is also likely higher during WE times, which could amplify these VIX-related influences on interest rates. We focus on the 1990–2017 period, based on VIX availability. Most of our empirical work investigates separately the 20.25 years over 1997:10 to 2017:12 (our “featured period”) and the 7.75 years over 1990:01 to 1997:09 (our short “earlier period”). In our main analysis, we subdivide the sample because (1) our primary interest is whether the uncertainty-yield dynamics are different for weaker economic times versus more normal environments, and (2) we identify a striking shift in the predominant relation between yield changes and VIX changes in late 1997, from positive to strongly negative.2 Additionally, in light of recent economic stress related to the COVID-19 health crisis, we perform an out-of-sample exercise over 2018:01–2020:06 to further analyze the validity of our findings. Consistent with our primary results, we find that the dynamic VIX-yield relation is also more negative over weaker economic times for this out-of-sample period. Section 2 explains our data choices and methodology, and Section 3 establishes our main findings. We find consistent results across eight different methods for identifying weaker economic (WE) segments. Our principal WE method employs Bai and Perron (1998, 2003) analysis to identify reliable shifts in the relation between yield changes and implied volatility changes. This Bai-Perron method indicates WE segments of substantial length (2.2 to 3.2 years) that commence later in recessions and persist well into the uncertain recovery. The Bai-Perron WE segments fit as WE times on other dimensions too, including having a higher forecasted likelihood of a real gross domestic product (GDP) decline, near proximity to a prior bear stock market, a low and/or declining targeted Fed Funds rate, a higher equity variance risk premium, and a compelling mapping to weaker economic times identified in related studies.3 Our other WE identification methods are based on recession timing, survey expectations on the real GDP growth, the trend in the targeted Federal Funds rate, the level of the equity variance risk premium, or the economic uncertainty index and risk aversion index of Bekaert, Engstrom, and Xu (2019). Thus, by “weaker economic” times, we broadly refer to economic states characterized by some combination of lower expected economic growth, higher economic uncertainty, and higher risk aversion, relative to long-run averages. Section 4 investigates alternative uncertainty-related measures (in lieu of the VIX) and alternative asset pricing dynamics (in addition to yield dynamics). When substituting the VIX with the daily economic policy uncertainty index of Baker, Bloom, and Davis (2016), we find a similar dynamic uncertainty-yield relation. Further, we find similar VIX dynamics for both a long/short risk-differential equity position (the price of volatile stocks, following from Pflueger, Siriwardane, and Sunderam 2020) and a long/short risk-differential bond market position (the default yield spread). Section 4 also shows that our primary findings for the VIX-yield are more attributable to the conditional volatility component of the VIX (which the literature suggests is more aligned with economic uncertainty) rather than to the volatility risk premium component of the VIX (which the literature suggests is more aligned with risk aversion).4 Thus, our VIX decomposition analysis favors a time-varying economic uncertainty interpretation of our main findings. Section 5 presents evidence on economic channels that likely contribute to the VIX-yield dynamics. Our analysis there features survey expectations of real GDP growth and inflation. We show that a higher VIX predicts higher volatility in expected economic growth and that VIX changes are negatively related to concurrent changes in expected economic growth; both of these VIX-growth relations are stronger during weaker economic (WE) times. From the context of the precautionary savings and consumption-smoothing channels discussed in Section 1, these findings fit with our state-contingent VIX-yield dynamic findings. Further, we evaluate an augmented yield change specification that adds a “concurrent change in expected economic growth” explanatory term, in addition to the implied volatility changes. With this economic growth term serving as a control for changes in the consumption-smoothing influence, we find (1) that the partial VIX-yield relations over WE times decline by around 30%, indicating a material consumption-smoothing influence behind our main VIX-yield results and (2) our coefficient estimates on the economic growth term suggest an intertemporal elasticity of substitution of about 2.5, plausibly close to the 2.2 value estimated in Bansal, Kiku, and Yaron (2016). In Section 5, we also show that our main VIX-yield dynamic findings remain similarly evident over periods without changes in the targeted Federal Funds rate and with stable monetary policy, inconsistent with the idea that monetary-policy implementation is substantially driving our main results. Sections 5 to 7 also relate our findings to the literature; especially Bansal and Yaron (2004), Bansal and Shaliastovich (2013), Drechsler (2013), David and Veronesi (2013), Bansal et al. (2014), Hartzmark (2016), and Bekaert and Engstrom (2017). Our main analysis also includes changes in the Treasury-bond yield implied volatility (TIV) as an additional explanatory term for yield changes, along with the VIX changes. Section 6 discusses our TIV results, drawing two primary conclusions. First, we generally find a significantly positive, partial, dynamic TIV-yield relation, especially for longer-maturity bonds. The combined dynamic VIX-yield and TIV-yield relations align with two implications for nominal yields discussed in Bansal and Shaliastovich (2013): (1) economic uncertainty (more aligned with VIX) should be negatively related to yields, and (2) inflation uncertainty (more aligned with TIV) should be positively related to yields. Second, including TIV mitigates omitted-variable bias, leading to better estimates of the partial VIX-yield relation. Section 8 concludes with a summary of our results and a discussion of their implications. 1. Interest Rate Determinants and VIX Movements In fundamental economic models, real risk-free interest rates are generally jointly determined by the motives of precautionary savings and consumption smoothing. Risk aversion levels influence the strength of these economic motives. An important question for our study is what the short-horizon VIX movements imply about movements in these interest rate determinants. Section 1.1 offers a simple framework from which to discuss consumption smoothing, precautionary savings, and risk aversion in a model-free fashion. By “model-free,” we mean a qualitative discussion of how these determinants might influence interest rates, without explicitly enforcing restrictions from any specific theoretical model.5 From this framework, Section 1.2 discusses likely linkages between VIX movements and changes in these fundamental economic determinants of interest rates. In this section, we discuss determinants of the real risk-free interest rate (rather than nominal). By initially discussing the “real risk-free rate,” we adopt a simple perspective of the linkage between economic uncertainty and yields, while sidestepping the influences of inflation uncertainty. In our empirical implementation, we evaluate both nominal yields and real-yield proxies, out to the 10-year point in the term structure. After establishing our main findings, we will later discuss our results from richer frameworks that include inflation uncertainty and bonds of varying maturity (see especially Section 6). 1.1 Model-free framework for discussing interest rate determinants Economic uncertainty can affect interest rates through a precautionary savings channel in many theoretical specifications. For example, in the well-known stochastic discount factor framework with log consumption growth and power utility, the log real risk-free interest rate is represented as follows, with γ representing the relative-risk aversion: rtf=δ+γ Et(Δ ln ct+1)−γ22 σt2(Δ ln ct+1).(1) See Cochrane (2005, pp. 11–12) for a detailed development. This equation shows the interest rate determinants of (1) consumption smoothing in the second term with the “expected consumption growth rate,” which imparts a positive influence if the expected growth rate is positive; (2) precautionary savings in the third term with the “expected volatility of the consumption growth rate,” which imparts a negative influence; and (3) risk aversion through the γ multipliers for the second and third terms whose impact is ambiguous, depending on the model’s parameterization. However, our empirical investigation is not tied to a specific framework. Rather, following from Hartzmark (2016), we consider the following general model-free representation: rtf=β0+β1,t Et(Δ ln ct+1)+β2,t σt2(Δ ln ct+1).(2) Here, β0, β1,t , and β2,t are instead generic unconstrained parameters that do not impose restrictions from any specific model. Prior literature and intuition suggest a generally positive β1 with consumption smoothing and a negative β2 with precautionary savings. The time subscripts on β1 and β2 reflect the possibility of state-contingent variation in the consumption-smoothing and precautionary-savings influences. Further, if time-varying economic uncertainty has a negative feedback effect on expected economic growth, then increasing economic uncertainty could also have an indirect negative influence on interest rates through the consumption-smoothing channel.6 From Equation (2), time variation in the expected economic-growth and its expected volatility and/or time-variation in risk aversion (or other influences on β1,t and/or β2,t ) also imply a change in the interest rate. We focus here on what VIX movements might imply about changes in these interest rate determinants in a comparative-statics sense; finer details of the transition dynamics are beyond the scope of our model-free empirical study. With our model-free approach, we make the simplifying assertion that variation in expected consumption growth and expected economic growth, and their expected volatilities, are significantly positively related. Consistent with this assertion, Bansal and Shaliastovich (2013) provide evidence of a strong relation between survey-based expected economic growth and future consumption growth. Thus, by “economic uncertainty” in our narrative, we refer to the conditional volatility of both the consumption growth rate and the economic growth rate. Similarly, by “expected economic growth,” we refer to both expected real economic growth and expected consumption growth. 1.2 Potential economic channels contributing to our VIX-yield findings 1.2.1 Relating VIX movements to changes in economic uncertainty. Foremost, the VIX is the expected risk-neutral S&P 500 implied volatility. Past literature has also used VIX as a measure of economic uncertainty. Jurado, Ludvigson, and Ng (2015) and Hartzmark (2016) both regard VIX as one of the most common forward-looking measures of economic uncertainty. Hartzmark’s findings include that the VIX level is negatively related to interest rate levels over 1990–2010. Bloom (2009) shows that VIX is related to cross-sectional measures of economic uncertainty, including firms’ profit growth, total factor productivity, and dispersion in forecasters’ GDP predictions. Similarly, Bekaert and Hoerova (2014) suggest that expected stock-market volatility can be viewed as a market-based measure of economic uncertainty. VIX is an excellent measure of the expected equity volatility (Blair, Poon, and Taylor 2010), so this view also suggests a VIX/economic uncertainty linkage. Thus, we assert that VIX increases should indicate higher economic uncertainty (and, perhaps, a lower future economic growth rate; see Section 1.2.2). In David and Veronesi (2013), beliefs about the unobservable economic state are more volatile and responsive to news in more uncertain times; see additional discussion in Section 7. Thus, higher economic-state uncertainty over atypical weaker economic (WE) times suggests that investors’ economic-state beliefs should be more sensitive to VIX changes over WE times. If so, this suggests a more negative, dynamic VIX-yield relation in WE times. We provide survey-based evidence supportive of this channel in Sections 5.1 and 5.2. 1.2.2 Relating VIX movements to changes in expected economic growth. The literature suggests a negative relation between changes in economic uncertainty and changes in the expected consumption-growth rate and/or economic growth rate. For example, in the long-run risks framework of Bansal and Yaron (2004), time-varying economic uncertainty has an associated feedback effect, where return news and volatility news are negatively correlated. Similarly, in Bansal et al. (2014), an increase in economic uncertainty is typically accompanied by a decline in expected consumption growth. Thus, if VIX increases indicate elevated economic uncertainty, then a VIX increase is also likely to indicate a decline in expected economic growth. Further, during weaker economic (WE) times, it seems plausible that the negative relation (or feedback) between changes in economic uncertainty and changes in expected economic growth would be relatively stronger. If so, this uncertainty-growth feedback channel suggests a more negative VIX-yield relation in WE times, operating through the consumption-smoothing channel. We develop this channel further and provide supportive survey-based evidence in Section 5.2. 1.2.3 Higher risk aversion in weaker economic times. Going back at least to Fama and French (1989), the literature has argued that aggregate risk aversion is countercylical, suggesting higher aggregate risk aversion during our weaker economic (WE) segments. In Equation (2), higher risk aversion implies a likely larger β1,t and β2,t . Thus, higher risk aversion is likely to amplify the channels suggested in Sections 1.2.1 and 1.2.2 above, implying that VIX changes would be especially negatively related to interest rate movements in WE times. In Section 5.3, we present evidence that risk aversion is likely higher over our WE segments, including the time-series behavior of the equity “volatility risk premium.” 2. Data Description and Empirical Implementation 2.1 Choice of weekly and monthly changes Our primary empirical interest is the partial relation between economic uncertainty changes and yield changes over weekly to monthly change horizons. As discussed in our introduction, prior literature has analyzed how interest rate levels vary with the economic growth state and/or economic uncertainty state, focusing principally on quarterly or annual data. Economic cycles are slow moving, so economic states are relatively stable over most weeks and months. Thus, our empirical analysis principally focuses on the dynamic relation between higher-frequency uncertainty fluctuations and yield changes within an economic state and whether the within-state relation differs for weaker economic states relative to more typical baseline times. Further, monthly changes (1) match the availability of expected-inflation data, allowing us to evaluate the dynamics of inflation-adjusted real-yield proxies; (2) are suggested by the approximate 1-month decision interval estimated in the long-run risk analysis of Bansal, Kiku, and Yaron (2016); (3) fit with many monthly macro-news release cycles; and (4) mitigate microstructure measurement concerns that would be more influential in a higher-frequency analysis (e.g., daily). Weekly changes provide more nonoverlapping observations, which is useful for analyzing dynamic time-varying relations over weaker economic periods of modest length. 2.2 Bond yield data Much of the prior literature that seeks to relate economic uncertainty to bond yields focuses on real yields (rather than nominal yields). Real yields are unobservable, but can be approximated from nominal yields with an expected-inflation adjustment or from TIPS. In our empirical work, we adopt a broad approach and examine both nominal yields and real-yield proxies. The partial relation between changes in economic uncertainty and nominal bond yields is interesting in its own right, with nominal yield movements directly translating to the actual dollar gain or loss with a bond investment. Further, for relatively short horizons, such as a week or a month, it seems reasonable to assert that changes in nominal yields are appreciably positively related to changes in real yields, in particular over our 1997:10–2017:12 featured period with its modest and stable inflation. Since nominal yields are readily observable, higher-frequency analysis (such as at the weekly horizon) is straightforward. Accordingly, we initially examine the VIX-yield relation for nominal Treasury yields. We also evaluate the partial relation between VIX changes and two proxies for real yields. Our primary proxies for real yields are inflation-adjusted yields, defined as the nominal Treasury yield less the expected inflation over the life of the bond. We rely on the monthly observations of expected-inflation data from the Haubrich, Pennacchi, and Ritchken (2012) model, made available by the Cleveland Federal Reserve Bank. Their inflation-expectations model provides monthly estimates of expected inflation out to the 30-year horizon. Their model relies on both market data and survey data, including inflation swap rates since April 2003. We examine TIPS yields as a second real-yield proxy. We consider our TIPS analysis in a secondary role for two reasons: (1) the TIPS data are not available until 1999, and (2) TIPS yields also reflect time-varying inflation risk compensation and liquidity (see Gurkaynak, Sack, and Wright 2010). Regarding bond maturities, the literature indicates that longer-maturity Treasury bonds have taken on an elevated hedging role since the late 1990s. This view suggests that we include mid- to long maturities in our analysis.7 Moreover, at the very short end of the yield curve, Cieslak and Povala (2016) point out that the yield dynamics are confounded by money market noise, Treasury-bill factors not shared by longer-maturity Treasuries, and institutional effects. Accordingly, their analysis focuses on Treasury maturities of 2 years and greater. For these reasons and supported by our analysis discussed in the next paragraph, we evaluate Treasury zero coupon bond (ZCB) yields at the 1-, 2-, 5-, and 10-year points in the term structure over our full 1990–2017 sample and 5- and 10-year TIPS ZCB yields since 1999, using the yield data described in Gurkaynak, Sack, and Wright (2007, 2010). All of the yields exhibit an appreciable downward trend since 1990. Figure 1 charts the weekly volatility of the four nominal yields. We see that the 10- and 5-year weekly yield volatilities are roughly similar across our entire sample. However, the volatility for the 2- and 1-year yields are depressed over December 2008 to December 2015, a period when the targeted Federal Funds rate (FFR) was at the zero lower bound. This depressed volatility suggests a distortion linked to the near-zero FFR, further suggesting that our analysis should exclude the very short end of the yield curve. Figure 1 Open in new tabDownload slide Weekly variability in Treasury yields, 1990:01–2017:12 This figure shows weekly changes in Treasury yields, Friday-to-Friday, over 1990:01 to 2017:12 in basis points. Panels A to D report 10-, 5-, 2-, and 1-year Treasury ZCB yields. Figure 1 Open in new tabDownload slide Weekly variability in Treasury yields, 1990:01–2017:12 This figure shows weekly changes in Treasury yields, Friday-to-Friday, over 1990:01 to 2017:12 in basis points. Panels A to D report 10-, 5-, 2-, and 1-year Treasury ZCB yields. 2.3 The CBOE’s volatility index As previously discussed in Section 1.2, our empirical work features the CBOE’s 30-day volatility index (VIX). Figure 2 illustrates the substantial variability of the VIX over our 1990 to 2017 sample. Panel A shows the VIX level, and panel C shows the week-to-week log(VIX) changes. We note that the VIX variability over weekly to monthly horizons is appreciable over our full sample, and over both the indicated Bai-Perron weaker economic (WE) segments and baseline segments. Using the absolute value of the log change in VIX over 5 weekdays as a VIX variability metric, we note that the average VIX variability is not statistically significantly different over the WE segments. These observations suggest that time variation in the VIX variability is not an important factor behind our results. Figure 2 Open in new tabDownload slide Equity and Treasury implied volatility, 1990–2017 This figure displays the time series of the equity market’s implied volatility, as indicated by the VIX in panel A and the Treasury market’s implied yield volatility as indicated by the MOVE index in panel B. Panels C and D show the weekly log changes in the equity and Treasury implied volatility series, respectively ( Δlog(IVt−5,t) ), for Friday-to-Friday weeks. The upper-shaded areas indicate NBER recessions, and the lower-shaded areas indicate our primary Bai-Perron weaker economic segments. The overall correlation between these VIX and TIV weekly changes is 0.34, with a ΔVIX (ΔTIV) standard deviation of 12.5% (7.7%). For the Bai-Perron weaker economic segments, the comparable correlation is 0.20, with a ΔVIX (ΔTIV) standard deviation of 11.5% (7.3%). For the other times, the comparable correlation is 0.39, with a ΔVIX (ΔTIV) standard deviation of 12.9% (7.9%). Figure 2 Open in new tabDownload slide Equity and Treasury implied volatility, 1990–2017 This figure displays the time series of the equity market’s implied volatility, as indicated by the VIX in panel A and the Treasury market’s implied yield volatility as indicated by the MOVE index in panel B. Panels C and D show the weekly log changes in the equity and Treasury implied volatility series, respectively ( Δlog(IVt−5,t) ), for Friday-to-Friday weeks. The upper-shaded areas indicate NBER recessions, and the lower-shaded areas indicate our primary Bai-Perron weaker economic segments. The overall correlation between these VIX and TIV weekly changes is 0.34, with a ΔVIX (ΔTIV) standard deviation of 12.5% (7.7%). For the Bai-Perron weaker economic segments, the comparable correlation is 0.20, with a ΔVIX (ΔTIV) standard deviation of 11.5% (7.3%). For the other times, the comparable correlation is 0.39, with a ΔVIX (ΔTIV) standard deviation of 12.9% (7.9%). 2.4 Other data Our study also relies on the following other data in several supporting roles: (1) the Merrill Lynch 1-month horizon MOVE series, as an option-derived implied volatility of Treasury yields (referred to as the Treasury implied volatility, or TIV, in our study); (2) a conditional volatility (CV) and equity volatility risk premium (VRP) based on Bekaert and Hoerova’s (2014) favored model; (3) inflation expectations from the Haubrich, Pennacchi, and Ritchken (2012) model; (4) the targeted Federal Funds rate; (5) survey data from the Survey of Professional Forecasters on the expected real GDP growth; (6) the risk aversion index and economic uncertainty index of Bekaert, Engstrom, and Xu (2019); (7) the daily economic policy uncertainty index of Baker, Bloom, and Davis (2016); (8) aggregate stock market returns and decile portfolio returns based on sorting by a stock’s standard deviation, from CRSP; (9) corporate bond yields; and (10) Federal Reserve holdings of debt securities. For brevity, these data are described in Appendix A. 2.5 Econometric methods Our main empirical work focuses on daily observations of rolling 1-week, 2-week, and 1-month (approximate) changes in yields and other variables, constructed from rolling periods of 5, 10, and 22 weekdays. A rolling analysis is attractive for measuring dynamic relations (see Richardson and Smith 1991); it also provides more flexibility for identifying shifts in the VIX-yield relation as economic conditions change. In robustness checks, we find that our main results remain evident when evaluating nonoverlapping 1-week and 1-month changes. In our main tables, we report ordinary least squares (OLS) coefficients with t-statistics that are calculated with autocorrelation and heteroscedastic-consistent standard errors using the Newey and West (1987) method. In our main analysis, we evaluate changes in financial market variables, so there is little autocorrelation beyond that induced when using overlapping observations. To be conservative in our inferences, we set the Newey-West lag length to equal two times the number of days that are cumulated to calculate the longer-horizon change. Internet Appendix B.1 confirms the robustness of our main VIX-yield results in a maximum likelihood approach that simultaneously estimates the conditional mean and conditional variance of the yield changes in a conditional normal-density specification; the conditional variance is modeled as a function of the lagged Treasury-yield implied volatility. Given this robustness, we report our main tabular results with OLS and Newey-West-based t-statistics, due to this method’s familiarity, comparability across the literature, and simple intuition of OLS R2 values. 3. Economic-State Variation in the VIX-Yield Dynamics Our primary interest is the dynamic uncertainty-yield relation over weaker economic times, as compared to more typical or baseline times. This section provides compelling evidence using VIX to measure economic uncertainty. Later, Section 4.1 evaluates alternative uncertainty metrics. Our main analysis also includes the changes in the Treasury implied yield volatility (TIV) as a control explanatory variable. This choice (1) enables us to interpret the coefficient for the VIX change term as a partial relation, while controlling for changes in yield uncertainty and (2) provides us interesting TIV-yield results that also contribute in their own right. To focus on the VIX-yield results here, we postpone discussion of our TIV-yield findings until Section 6. As discussed in our Introduction, we find that the unconditional partial dynamic VIX-yield relation shifts to strongly negative for our featured 1997:10–2017:12 period from marginally positive for the earlier 1990:01–1997:09 period.8 For brevity, we relegate estimation details to Internet Appendix B.2. Since our focus is on whether the VIX-yield relation is different over weaker economic times versus other baseline times, we evaluate these two periods separately. Recall that our footnote 2 summarizes related literature that also suggests this late 1997 shift. This section is organized as follows. Section 3.1 provides our initial results using the Bai-Perron method to identify weaker economic (WE) times. Then, Section 3.2 confirms our main results using seven alternative methods for categorizing WE times, based on economic indicators suggested by the literature. Finally, Section 3.3 reports additional results that both assist in our interpretation and confirm robustness. 3.1 VIX-yield dynamics over Bai-Perron weaker economic segments. In this section, we evaluate VIX-yield dynamics featuring weaker economic segments identified by our Bai-Perron (1998, 2003) analysis. The Bai-Perron approach is attractive because the null is that the dynamic uncertainty-yield relation does not vary, but then “the data speak” in a one-step estimation regarding the existence of relational breaks. In our Bai-Perron estimations, we analyze daily observations of rolling weekly changes to provide more power in identifying shifts in dynamic relations (as compared to monthly changes). We analyze the relation between the yield changes, as the dependent variable, and both the VIX changes and the TIV changes as the explanatory terms, allowing both the VIX-yield and TIV-yield relations to vary in the Bai-Perron estimation. Our Bai-Perron analysis suggests weaker economic (WE) segments over January 2, 1991, to March 9, 1993; November 13, 2001, to May 6, 2004; and January 12, 2009, to April 12, 2012. These Bai-Perron WE segments: (1) commence after the onset of a recession, (2) extend well into the uncertain post-recession recovery period, (3) are the best performers amongst the eight WE-identification methods compared in Section 3.2, and (4) exhibit substantial overlap with our alternative WE-identification methods.9 In Section 5, we connect our Bai-Perron WE segments to stressful economic periods identified in other studies and evidence suggestive of higher risk aversion. For these reasons, we feature the Bai-Perron WE segments as our principal WE segments. 3.1.1 Nominal yields, VIX, and Bai-Perron weaker economic segments. We begin by estimating the following equation to show how Treasury yields move differently with VIX changes over weaker economic times: ΔZCBnYt−j,t=α0+(λ1+λ2DtWE)Δlog(VIXt−j,t)+(γ1+γ2DtWE)Δlog(TIVt−j,t)+ϵt−j,t,(3) where ΔZCBnY indicates the change in Treasury n-year nominal ZCB yields over weekdays t – j to t; n indicates 1-, 2-, 5-, or 10-year ZCB yields; VIX and TIV are the equity and Treasury-bond implied volatilities with the Δlog and subscripts t−j,t indicating the difference in the natural log of each variable between weekdays t – j and t; DtWE is an indicator variable that equals one if the change observation over t – j to t is in a Bai-Perron weaker economic segment and zero otherwise; ϵt−j,t is the residual; and the α, λs, and γs are coefficients to be estimated. Table 1, panel A, reports the results over 1997:10–2017:12, for both the weekly (j=5) and monthly (j=22) change horizons and for the 10-, 5- and 2-year maturities. The VIX-yield relations are always strongly more negative over the weaker economic (WE) segments. Across the three yield maturities and two change horizons, the average baseline VIX-yield relation is -0.22 (λ1) and the average WE VIX-yield relation is −0.67 ( λ1+λ2 ). The changes in the VIX-yield relation (λ2) over the WE segments are all statistically significant. Table 1 VIX, Treasury yields, and weaker economic states, 1 . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A.1. Change in 10-year ZCB yields, 1997:10–2017:12 1. ΔZCB10Y –0.22 –0.48 –0.71 0.24 0.70 0.94 20.5% (1 week, j=5) (–7.92) (–8.69) (–14.71) (4.18) (6.98) (11.30) 2. ΔZCB10Y –0.19 –0.70 –0.89 0.19 0.76 0.95 17.6% (1 month, j=22) (–2.27) (–5.46) (–8.77) (1.41) (3.26) (4.91) A.2. Change in 5-year ZCB yields, 1997:10–2017:12 1. ΔZCB05Y –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (1 week, j=5) (–9.48) (–8.01) (–15.22) (2.76) (7.25) (11.02) 2. ΔZCB05Y –0.25 –0.60 –0.84 0.01 0.95 0.96 17.6% (1 month, j=22) (–3.89) (–5.91) (–10.80) (0.08) (3.93) (5.06) A.3. Change in 2-year ZCB yields, 1997:10–2017:12 1. ΔZCB02Y –0.19 –0.21 –0.41 –0.01 0.55 0.55 11.6% (1 week, j=5) (–7.92) (–4.73) (–10.69) (–0.03) (6.39) (7.92) 2. ΔZCB02Y –0.20 –0.34 –0.54 –0.25 0.87 0.63 14.5% (1 month, j=22) (–3.09) (–3.18) (–6.57) (–1.97) (4.63) (4.41) B.1. Change in 5-year ZCB yields, 1990:01–1997:09 1. ΔZCB05Y 0.33 –0.50 –0.17 0.63 –0.06 0.57 15.7% (1 week, j=5) (6.70) (–5.19) (–2.03) (6.49) (–0.24) (2.66) 2. ΔZCB05Y 0.34 –0.57 –0.23 0.69 –0.11 0.59 15.5% (2 weeks, j=10) (4.55) (–3.81) (–1.75) (4.66) (–0.36) (2.36) B.2. Change in 2-year ZCB yields, 1990:01–1997:09 1. ΔZCB02Y 0.22 –0.41 –0.20 0.63 –0.13 0.50 12.5% (1 week, j=5) (4.17) (–4.16) (–2.32) (6.49) (–0.58) (2.52) 2. ΔZCB02Y 0.21 –0.46 –0.26 0.71 –0.20 0.51 12.0% (2 weeks, j=10) (2.41) (–3.13) (–2.09) (4.59) (–0.71) (2.12) B.3. Change in 1-year ZCB yields, 1990:01–1997:09 1. ΔZCB01Y 0.14 –0.30 –0.17 0.51 –0.14 0.37 9.9% (1 week, j=5) (2.77) (–3.29) (–2.13) (6.41) (–0.75) (2.18) 2. ΔZCB01Y 0.12 –0.33 –0.21 0.60 –0.20 0.40 9.7% (2 weeks, j=10) (1.45) (–2.41) (–1.91) (4.45) (–0.84) (1.93) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A.1. Change in 10-year ZCB yields, 1997:10–2017:12 1. ΔZCB10Y –0.22 –0.48 –0.71 0.24 0.70 0.94 20.5% (1 week, j=5) (–7.92) (–8.69) (–14.71) (4.18) (6.98) (11.30) 2. ΔZCB10Y –0.19 –0.70 –0.89 0.19 0.76 0.95 17.6% (1 month, j=22) (–2.27) (–5.46) (–8.77) (1.41) (3.26) (4.91) A.2. Change in 5-year ZCB yields, 1997:10–2017:12 1. ΔZCB05Y –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (1 week, j=5) (–9.48) (–8.01) (–15.22) (2.76) (7.25) (11.02) 2. ΔZCB05Y –0.25 –0.60 –0.84 0.01 0.95 0.96 17.6% (1 month, j=22) (–3.89) (–5.91) (–10.80) (0.08) (3.93) (5.06) A.3. Change in 2-year ZCB yields, 1997:10–2017:12 1. ΔZCB02Y –0.19 –0.21 –0.41 –0.01 0.55 0.55 11.6% (1 week, j=5) (–7.92) (–4.73) (–10.69) (–0.03) (6.39) (7.92) 2. ΔZCB02Y –0.20 –0.34 –0.54 –0.25 0.87 0.63 14.5% (1 month, j=22) (–3.09) (–3.18) (–6.57) (–1.97) (4.63) (4.41) B.1. Change in 5-year ZCB yields, 1990:01–1997:09 1. ΔZCB05Y 0.33 –0.50 –0.17 0.63 –0.06 0.57 15.7% (1 week, j=5) (6.70) (–5.19) (–2.03) (6.49) (–0.24) (2.66) 2. ΔZCB05Y 0.34 –0.57 –0.23 0.69 –0.11 0.59 15.5% (2 weeks, j=10) (4.55) (–3.81) (–1.75) (4.66) (–0.36) (2.36) B.2. Change in 2-year ZCB yields, 1990:01–1997:09 1. ΔZCB02Y 0.22 –0.41 –0.20 0.63 –0.13 0.50 12.5% (1 week, j=5) (4.17) (–4.16) (–2.32) (6.49) (–0.58) (2.52) 2. ΔZCB02Y 0.21 –0.46 –0.26 0.71 –0.20 0.51 12.0% (2 weeks, j=10) (2.41) (–3.13) (–2.09) (4.59) (–0.71) (2.12) B.3. Change in 1-year ZCB yields, 1990:01–1997:09 1. ΔZCB01Y 0.14 –0.30 –0.17 0.51 –0.14 0.37 9.9% (1 week, j=5) (2.77) (–3.29) (–2.13) (6.41) (–0.75) (2.18) 2. ΔZCB01Y 0.12 –0.33 –0.21 0.60 –0.20 0.40 9.7% (2 weeks, j=10) (1.45) (–2.41) (–1.91) (4.45) (–0.84) (1.93) This table shows how Treasury yields move differently with VIX changes over weaker economic (WE) times. We report on the following regression for 1-week, 2-week, or 1-month changes (j=5, 10, or 22): ΔZCBnYt−j,t=α0+(λ1+λ2DtWE)Δlog(VIXt−j,t)+(γ1+γ2DtWE)Δlog(TIVt−j,t)+ϵt−j,t, where ΔZCBnY indicates the change in the n-year nominal ZCB yield over weekdays t – j to t, n is 1, 2, 5, or 10 years; VIX and TIV are the equity and Treasury-bond implied volatilities with the Δlog and subscripts t−j,t indicating the difference in the natural log of each variable between weekdays t – j and t; DtWE is an indicator variable that equals one if in a Bai-Perron WE segment; ϵt−j,t is the residual; and α, λs, and γs are the coefficients to be estimated. The sample dates are provided in each subpanel heading. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 1 VIX, Treasury yields, and weaker economic states, 1 . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A.1. Change in 10-year ZCB yields, 1997:10–2017:12 1. ΔZCB10Y –0.22 –0.48 –0.71 0.24 0.70 0.94 20.5% (1 week, j=5) (–7.92) (–8.69) (–14.71) (4.18) (6.98) (11.30) 2. ΔZCB10Y –0.19 –0.70 –0.89 0.19 0.76 0.95 17.6% (1 month, j=22) (–2.27) (–5.46) (–8.77) (1.41) (3.26) (4.91) A.2. Change in 5-year ZCB yields, 1997:10–2017:12 1. ΔZCB05Y –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (1 week, j=5) (–9.48) (–8.01) (–15.22) (2.76) (7.25) (11.02) 2. ΔZCB05Y –0.25 –0.60 –0.84 0.01 0.95 0.96 17.6% (1 month, j=22) (–3.89) (–5.91) (–10.80) (0.08) (3.93) (5.06) A.3. Change in 2-year ZCB yields, 1997:10–2017:12 1. ΔZCB02Y –0.19 –0.21 –0.41 –0.01 0.55 0.55 11.6% (1 week, j=5) (–7.92) (–4.73) (–10.69) (–0.03) (6.39) (7.92) 2. ΔZCB02Y –0.20 –0.34 –0.54 –0.25 0.87 0.63 14.5% (1 month, j=22) (–3.09) (–3.18) (–6.57) (–1.97) (4.63) (4.41) B.1. Change in 5-year ZCB yields, 1990:01–1997:09 1. ΔZCB05Y 0.33 –0.50 –0.17 0.63 –0.06 0.57 15.7% (1 week, j=5) (6.70) (–5.19) (–2.03) (6.49) (–0.24) (2.66) 2. ΔZCB05Y 0.34 –0.57 –0.23 0.69 –0.11 0.59 15.5% (2 weeks, j=10) (4.55) (–3.81) (–1.75) (4.66) (–0.36) (2.36) B.2. Change in 2-year ZCB yields, 1990:01–1997:09 1. ΔZCB02Y 0.22 –0.41 –0.20 0.63 –0.13 0.50 12.5% (1 week, j=5) (4.17) (–4.16) (–2.32) (6.49) (–0.58) (2.52) 2. ΔZCB02Y 0.21 –0.46 –0.26 0.71 –0.20 0.51 12.0% (2 weeks, j=10) (2.41) (–3.13) (–2.09) (4.59) (–0.71) (2.12) B.3. Change in 1-year ZCB yields, 1990:01–1997:09 1. ΔZCB01Y 0.14 –0.30 –0.17 0.51 –0.14 0.37 9.9% (1 week, j=5) (2.77) (–3.29) (–2.13) (6.41) (–0.75) (2.18) 2. ΔZCB01Y 0.12 –0.33 –0.21 0.60 –0.20 0.40 9.7% (2 weeks, j=10) (1.45) (–2.41) (–1.91) (4.45) (–0.84) (1.93) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A.1. Change in 10-year ZCB yields, 1997:10–2017:12 1. ΔZCB10Y –0.22 –0.48 –0.71 0.24 0.70 0.94 20.5% (1 week, j=5) (–7.92) (–8.69) (–14.71) (4.18) (6.98) (11.30) 2. ΔZCB10Y –0.19 –0.70 –0.89 0.19 0.76 0.95 17.6% (1 month, j=22) (–2.27) (–5.46) (–8.77) (1.41) (3.26) (4.91) A.2. Change in 5-year ZCB yields, 1997:10–2017:12 1. ΔZCB05Y –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (1 week, j=5) (–9.48) (–8.01) (–15.22) (2.76) (7.25) (11.02) 2. ΔZCB05Y –0.25 –0.60 –0.84 0.01 0.95 0.96 17.6% (1 month, j=22) (–3.89) (–5.91) (–10.80) (0.08) (3.93) (5.06) A.3. Change in 2-year ZCB yields, 1997:10–2017:12 1. ΔZCB02Y –0.19 –0.21 –0.41 –0.01 0.55 0.55 11.6% (1 week, j=5) (–7.92) (–4.73) (–10.69) (–0.03) (6.39) (7.92) 2. ΔZCB02Y –0.20 –0.34 –0.54 –0.25 0.87 0.63 14.5% (1 month, j=22) (–3.09) (–3.18) (–6.57) (–1.97) (4.63) (4.41) B.1. Change in 5-year ZCB yields, 1990:01–1997:09 1. ΔZCB05Y 0.33 –0.50 –0.17 0.63 –0.06 0.57 15.7% (1 week, j=5) (6.70) (–5.19) (–2.03) (6.49) (–0.24) (2.66) 2. ΔZCB05Y 0.34 –0.57 –0.23 0.69 –0.11 0.59 15.5% (2 weeks, j=10) (4.55) (–3.81) (–1.75) (4.66) (–0.36) (2.36) B.2. Change in 2-year ZCB yields, 1990:01–1997:09 1. ΔZCB02Y 0.22 –0.41 –0.20 0.63 –0.13 0.50 12.5% (1 week, j=5) (4.17) (–4.16) (–2.32) (6.49) (–0.58) (2.52) 2. ΔZCB02Y 0.21 –0.46 –0.26 0.71 –0.20 0.51 12.0% (2 weeks, j=10) (2.41) (–3.13) (–2.09) (4.59) (–0.71) (2.12) B.3. Change in 1-year ZCB yields, 1990:01–1997:09 1. ΔZCB01Y 0.14 –0.30 –0.17 0.51 –0.14 0.37 9.9% (1 week, j=5) (2.77) (–3.29) (–2.13) (6.41) (–0.75) (2.18) 2. ΔZCB01Y 0.12 –0.33 –0.21 0.60 –0.20 0.40 9.7% (2 weeks, j=10) (1.45) (–2.41) (–1.91) (4.45) (–0.84) (1.93) This table shows how Treasury yields move differently with VIX changes over weaker economic (WE) times. We report on the following regression for 1-week, 2-week, or 1-month changes (j=5, 10, or 22): ΔZCBnYt−j,t=α0+(λ1+λ2DtWE)Δlog(VIXt−j,t)+(γ1+γ2DtWE)Δlog(TIVt−j,t)+ϵt−j,t, where ΔZCBnY indicates the change in the n-year nominal ZCB yield over weekdays t – j to t, n is 1, 2, 5, or 10 years; VIX and TIV are the equity and Treasury-bond implied volatilities with the Δlog and subscripts t−j,t indicating the difference in the natural log of each variable between weekdays t – j and t; DtWE is an indicator variable that equals one if in a Bai-Perron WE segment; ϵt−j,t is the residual; and α, λs, and γs are the coefficients to be estimated. The sample dates are provided in each subpanel heading. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 1, panel B, reports the results over 1990:01–1997:09, for both the weekly (j=5) and 10-weekday (j=10) change horizons and for 5-, 2-, and 1-year maturities. In panel B, we evaluate the 2-week horizon (rather than the monthly horizon as in panel A) to provide more statistical power over this short 93-month period with a single weaker economic segment of about 26 months. Again, the VIX-yield relations are always appreciably more negative over the weaker economic segments. Across the three yield maturities and two change horizons, the average baseline VIX-yield relation is +0.23 and the average weaker economic VIX-yield relation is -0.21 ( λ1+λ2 ). The state-contingent changes in the VIX-yield relation are again all statistically significant. Our featured 1997:10–2017:12 period comprises five subperiods: two weaker economic (WE) segments and three baseline segments. Rather than pooling the baseline segments together and the WE segments together (as in Table 1), we next estimate Equation (3) (without the λ2 and γ2 terms) over each of the five subperiods separately.10 Table 2 reports the results. We find a striking contrast between each WE segment (rows 2 and 4), and the three baseline segments (rows 1, 3, and 5), in terms of both the estimated VIX-yield relation and the R2 values. For example, for the monthly regressions with the 5-year yield changes, the λ1 coefficients for the ΔVIX term are -1.07 and -0.74 for the WE segments and -0.24, -0.16, and -0.29 for the three baseline segments, respectively. Moreover, the R2 values are 34.7% and 44.0% for the WE segments versus 9.6%, 4.2%, and 17.7% for the three baseline segments. Table 2 VIX, Treasury yields, and weaker economic states, 2 . Weekly horizon, j = 5 . Monthly horizon, j = 22 . 1. Segment: . 2. λ1,ΔVIX . 3. γ1,ΔTIV . 4. R2 . 5. λ1,ΔVIX . 6. γ1,ΔTIV . 7. R2 . A. Change in 10-year ZCB yields, baseline versus WE segment results 1. Baseline –0.17 (−2.59) 0.22 (2.00) 3.3% –0.20 (−2.29) –0.10 (−0.46) 4.3% 2. WE –0.66 (−7.58) 0.92 (6.34) 33.0% –0.88 (−7.12) 1.01 (3.09) 33.7% 3. Baseline –0.19 (−3.65) 0.03 (0.30) 2.9% 0.09 (0.41) –0.03 (−0.15) 0.3% 4. WE –0.73 (−13.69) 0.95 (9.95) 46.7% –0.89 (−6.62) 0.91 (3.89) 43.1% 5. Baseline –0.28 (−7.62) 0.47 (5.72) 19.3% –0.36 (−4.96) 0.58 (3.23) 19.9% B. Change in 5-year ZCB yields, baseline versus WE segment results 1. Baseline –0.23 (−3.65) 0.12 (0.98) 3.7% –0.24 (−2.24) –0.32 (−1.24) 9.6% 2. WE –0.76 (−9.14) 0.94 (5.89) 32.8% –1.07 (−7.53) 0.96 (2.59) 34.7% 3. Baseline –0.29 (−5.78) 0.01 (0.01) 6.9% –0.16 (−1.17) –0.20 (−1.01) 4.2% 4. WE –0.59 (−12.76) 0.84 (11.17) 41.7% –0.74 (−8.34) 0.93 (4.77) 44.0% 5. Baseline –0.24 (−8.06) 0.38 (4.91) 17.0% –0.29 (−5.33) 0.49 (3.13) 17.7% C. Change in 2-year ZCB yields, baseline versus WE segment results 1. Baseline –0.24 (−4.27) –0.01 (−0.06) 5.1% –0.21 (−1.65) –0.46 (−1.94) 13.1% 2. WE –0.66 (−8.87) 0.69 (5.37) 28.7% –0.95 (−5.35) 0.73 (2.09) 35.1% 3. Baseline –0.35 (−6.01) –0.06 (−0.70) 10.8% –0.35 (−2.89) –0.35 (−1.92) 15.5% 4. WE –0.27 (−9.05) 0.41 (8.79) 24.9% –0.37 (−8.23) 0.60 (6.35) 43.0% 5. Baseline –0.11 (−6.95) 0.12 (3.36) 8.9% –0.10 (−2.74) 0.12 (1.60) 4.4% . Weekly horizon, j = 5 . Monthly horizon, j = 22 . 1. Segment: . 2. λ1,ΔVIX . 3. γ1,ΔTIV . 4. R2 . 5. λ1,ΔVIX . 6. γ1,ΔTIV . 7. R2 . A. Change in 10-year ZCB yields, baseline versus WE segment results 1. Baseline –0.17 (−2.59) 0.22 (2.00) 3.3% –0.20 (−2.29) –0.10 (−0.46) 4.3% 2. WE –0.66 (−7.58) 0.92 (6.34) 33.0% –0.88 (−7.12) 1.01 (3.09) 33.7% 3. Baseline –0.19 (−3.65) 0.03 (0.30) 2.9% 0.09 (0.41) –0.03 (−0.15) 0.3% 4. WE –0.73 (−13.69) 0.95 (9.95) 46.7% –0.89 (−6.62) 0.91 (3.89) 43.1% 5. Baseline –0.28 (−7.62) 0.47 (5.72) 19.3% –0.36 (−4.96) 0.58 (3.23) 19.9% B. Change in 5-year ZCB yields, baseline versus WE segment results 1. Baseline –0.23 (−3.65) 0.12 (0.98) 3.7% –0.24 (−2.24) –0.32 (−1.24) 9.6% 2. WE –0.76 (−9.14) 0.94 (5.89) 32.8% –1.07 (−7.53) 0.96 (2.59) 34.7% 3. Baseline –0.29 (−5.78) 0.01 (0.01) 6.9% –0.16 (−1.17) –0.20 (−1.01) 4.2% 4. WE –0.59 (−12.76) 0.84 (11.17) 41.7% –0.74 (−8.34) 0.93 (4.77) 44.0% 5. Baseline –0.24 (−8.06) 0.38 (4.91) 17.0% –0.29 (−5.33) 0.49 (3.13) 17.7% C. Change in 2-year ZCB yields, baseline versus WE segment results 1. Baseline –0.24 (−4.27) –0.01 (−0.06) 5.1% –0.21 (−1.65) –0.46 (−1.94) 13.1% 2. WE –0.66 (−8.87) 0.69 (5.37) 28.7% –0.95 (−5.35) 0.73 (2.09) 35.1% 3. Baseline –0.35 (−6.01) –0.06 (−0.70) 10.8% –0.35 (−2.89) –0.35 (−1.92) 15.5% 4. WE –0.27 (−9.05) 0.41 (8.79) 24.9% –0.37 (−8.23) 0.60 (6.35) 43.0% 5. Baseline –0.11 (−6.95) 0.12 (3.36) 8.9% –0.10 (−2.74) 0.12 (1.60) 4.4% This table presents subperiod analysis that shows how Treasury yields move differently with VIX changes over weaker economic (WE) times. We report on the following regression: ΔZCBnYt−j,t=α0+λ1Δlog(VIXt−j,t)+γ1Δlog(TIVt−j,t)+ϵt−j,t, where ΔZCBnY indicates the change in the n-year nominal ZCB yield over weekdays t – j to t, n is 10, 5, or 2 years, and other terms are defined in Table 1. We report estimations for both the weekly and monthly change horizons, using rolling 5 weekday periods (j = 5) or rolling 22 weekday periods (j = 22). Columns 2 to 4 report the weekly change horizon and columns 5 to 7 the monthly change horizon. The overall sample period is 1997:10 to 2017:12, but here rows 1 to 5 report five different subperiod estimations to contrast between the baseline segments (October 1, 1997, to November 12, 2001; May 7, 2004, to January 9, 2009; and April 13, 2012, to December 29, 2017) and the two WE segments (November 13, 2001, to May 6, 2004; and January 12, 2009, to April 12, 2012). t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 2 VIX, Treasury yields, and weaker economic states, 2 . Weekly horizon, j = 5 . Monthly horizon, j = 22 . 1. Segment: . 2. λ1,ΔVIX . 3. γ1,ΔTIV . 4. R2 . 5. λ1,ΔVIX . 6. γ1,ΔTIV . 7. R2 . A. Change in 10-year ZCB yields, baseline versus WE segment results 1. Baseline –0.17 (−2.59) 0.22 (2.00) 3.3% –0.20 (−2.29) –0.10 (−0.46) 4.3% 2. WE –0.66 (−7.58) 0.92 (6.34) 33.0% –0.88 (−7.12) 1.01 (3.09) 33.7% 3. Baseline –0.19 (−3.65) 0.03 (0.30) 2.9% 0.09 (0.41) –0.03 (−0.15) 0.3% 4. WE –0.73 (−13.69) 0.95 (9.95) 46.7% –0.89 (−6.62) 0.91 (3.89) 43.1% 5. Baseline –0.28 (−7.62) 0.47 (5.72) 19.3% –0.36 (−4.96) 0.58 (3.23) 19.9% B. Change in 5-year ZCB yields, baseline versus WE segment results 1. Baseline –0.23 (−3.65) 0.12 (0.98) 3.7% –0.24 (−2.24) –0.32 (−1.24) 9.6% 2. WE –0.76 (−9.14) 0.94 (5.89) 32.8% –1.07 (−7.53) 0.96 (2.59) 34.7% 3. Baseline –0.29 (−5.78) 0.01 (0.01) 6.9% –0.16 (−1.17) –0.20 (−1.01) 4.2% 4. WE –0.59 (−12.76) 0.84 (11.17) 41.7% –0.74 (−8.34) 0.93 (4.77) 44.0% 5. Baseline –0.24 (−8.06) 0.38 (4.91) 17.0% –0.29 (−5.33) 0.49 (3.13) 17.7% C. Change in 2-year ZCB yields, baseline versus WE segment results 1. Baseline –0.24 (−4.27) –0.01 (−0.06) 5.1% –0.21 (−1.65) –0.46 (−1.94) 13.1% 2. WE –0.66 (−8.87) 0.69 (5.37) 28.7% –0.95 (−5.35) 0.73 (2.09) 35.1% 3. Baseline –0.35 (−6.01) –0.06 (−0.70) 10.8% –0.35 (−2.89) –0.35 (−1.92) 15.5% 4. WE –0.27 (−9.05) 0.41 (8.79) 24.9% –0.37 (−8.23) 0.60 (6.35) 43.0% 5. Baseline –0.11 (−6.95) 0.12 (3.36) 8.9% –0.10 (−2.74) 0.12 (1.60) 4.4% . Weekly horizon, j = 5 . Monthly horizon, j = 22 . 1. Segment: . 2. λ1,ΔVIX . 3. γ1,ΔTIV . 4. R2 . 5. λ1,ΔVIX . 6. γ1,ΔTIV . 7. R2 . A. Change in 10-year ZCB yields, baseline versus WE segment results 1. Baseline –0.17 (−2.59) 0.22 (2.00) 3.3% –0.20 (−2.29) –0.10 (−0.46) 4.3% 2. WE –0.66 (−7.58) 0.92 (6.34) 33.0% –0.88 (−7.12) 1.01 (3.09) 33.7% 3. Baseline –0.19 (−3.65) 0.03 (0.30) 2.9% 0.09 (0.41) –0.03 (−0.15) 0.3% 4. WE –0.73 (−13.69) 0.95 (9.95) 46.7% –0.89 (−6.62) 0.91 (3.89) 43.1% 5. Baseline –0.28 (−7.62) 0.47 (5.72) 19.3% –0.36 (−4.96) 0.58 (3.23) 19.9% B. Change in 5-year ZCB yields, baseline versus WE segment results 1. Baseline –0.23 (−3.65) 0.12 (0.98) 3.7% –0.24 (−2.24) –0.32 (−1.24) 9.6% 2. WE –0.76 (−9.14) 0.94 (5.89) 32.8% –1.07 (−7.53) 0.96 (2.59) 34.7% 3. Baseline –0.29 (−5.78) 0.01 (0.01) 6.9% –0.16 (−1.17) –0.20 (−1.01) 4.2% 4. WE –0.59 (−12.76) 0.84 (11.17) 41.7% –0.74 (−8.34) 0.93 (4.77) 44.0% 5. Baseline –0.24 (−8.06) 0.38 (4.91) 17.0% –0.29 (−5.33) 0.49 (3.13) 17.7% C. Change in 2-year ZCB yields, baseline versus WE segment results 1. Baseline –0.24 (−4.27) –0.01 (−0.06) 5.1% –0.21 (−1.65) –0.46 (−1.94) 13.1% 2. WE –0.66 (−8.87) 0.69 (5.37) 28.7% –0.95 (−5.35) 0.73 (2.09) 35.1% 3. Baseline –0.35 (−6.01) –0.06 (−0.70) 10.8% –0.35 (−2.89) –0.35 (−1.92) 15.5% 4. WE –0.27 (−9.05) 0.41 (8.79) 24.9% –0.37 (−8.23) 0.60 (6.35) 43.0% 5. Baseline –0.11 (−6.95) 0.12 (3.36) 8.9% –0.10 (−2.74) 0.12 (1.60) 4.4% This table presents subperiod analysis that shows how Treasury yields move differently with VIX changes over weaker economic (WE) times. We report on the following regression: ΔZCBnYt−j,t=α0+λ1Δlog(VIXt−j,t)+γ1Δlog(TIVt−j,t)+ϵt−j,t, where ΔZCBnY indicates the change in the n-year nominal ZCB yield over weekdays t – j to t, n is 10, 5, or 2 years, and other terms are defined in Table 1. We report estimations for both the weekly and monthly change horizons, using rolling 5 weekday periods (j = 5) or rolling 22 weekday periods (j = 22). Columns 2 to 4 report the weekly change horizon and columns 5 to 7 the monthly change horizon. The overall sample period is 1997:10 to 2017:12, but here rows 1 to 5 report five different subperiod estimations to contrast between the baseline segments (October 1, 1997, to November 12, 2001; May 7, 2004, to January 9, 2009; and April 13, 2012, to December 29, 2017) and the two WE segments (November 13, 2001, to May 6, 2004; and January 12, 2009, to April 12, 2012). t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Next, we note that our main VIX-yield dynamic findings are reliably evident with or without the inclusion of the ΔTIV explanatory term. Inclusion of the ΔTIV explanatory term (1) results in an estimated VIX-yield relation over the weaker economic segments that is more negative and more statistically reliable, a finding discussed in more detail in Section 6, and (2) appreciably increases the R2 values for the regressions. Internet Appendix B.3.1 reports the details. Finally, we estimate an augmented Equation (3) over 1990 to 2017 that also allows the VIX-yield relation to shift in October 1997. We again see that the dynamic VIX-yield relation is appreciably more negative over the weaker economy segments for all four yield maturities, with or without the inclusion of the ΔTIV term. See Internet Appendix B.3.2 for the tabular details. 3.1.2 Real yields, the VIX, and Bai-Perron weaker economic segments Next, we shift to real-yield proxies. First, we examine nonoverlapping monthly changes in inflation-adjusted yields, defined as the nominal yield minus the expected annualized inflation over the bond’s life. In Section 6, we discuss limitations on interpreting these inflation-adjusted yields as true “real yields.” Second, we examine overlapping weekly and monthly changes in TIPS yields. Table 3 reports the results for the nonoverlapping monthly changes for both inflation-adjusted yields (panels B and D) and comparable nominal yields (panels A and C).11 Thus, our results here also serve to show that our main dynamic findings between VIX and nominal yields (as in Table 1) are also evident for nonoverlapping monthly changes, strongly for our featured 1997:10 to 2017:12 period and marginally for the earlier 1990:01–1997:09 period. Table 3 VIX, nominal and inflation-adjusted yields, and weaker economic times . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . Variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. Nominal yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10Y –0.44 0.46 10.8% (–3.91) (2.72) 2. ΔZCB05Y –0.42 0.30 8.6% (–4.38) (1.74) 3. ΔZCB02Y –0.26 –0.02 5.1% (–3.24) (–0.18) 4. ΔZCB10Y –0.24 –0.80 –1.04 0.18 0.84 1.02 17.7% (–2.36) (–4.25) (–6.19) (1.05) (2.69) (3.91) 5. ΔZCB05Y –0.24 –0.69 –0.94 –0.03 0.98 0.95 15.9% (–2.75) (–3.50) (–5.25) (–0.17) (3.36) (4.22) 6. ΔZCB02Y –0.15 –0.41 –0.56 –0.32 0.86 0.54 11.2% (–1.82) (–2.18) (–3.26) (–2.07) (3.61) (2.95) B. Inflation-adjusted yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10YRl –0.24 0.38 9.2% (–2.80) (3.17) 2. ΔZCB05YRl –0.20 0.24 5.3% (–3.21) (2.06) 3. ΔZCB02YRl –0.01 0.04 0.1% (–0.06) (0.32) 4. ΔZCB10YRl –0.11 –0.54 –0.65 0.21 0.50 0.71 14.8% (–1.35) (–3.72) (–4.90) (1.81) (2.16) (3.56) 5. ΔZCB05YRl –0.10 –0.39 –0.49 0.03 0.63 0.66 11.3% (–1.59) (–2.93) (–4.00) (0.22) (3.27) (4.49) 6. ΔZCB02YRl 0.03 –0.06 –0.03 –0.17 0.62 0.44 3.2% (0.33) (–0.29) (–0.17) (–1.31) (2.77) (2.46) C. Nominal yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05Y 0.31 0.80 11.5% (1.57) (1.97) 2. ΔZCB02Y 0.17 0.65 6.4% (0.81) (1.69) 3. ΔZCB01Y 0.13 0.41 3.6% (0.66) (1.24) 4. ΔZCB05Y 0.45 –0.91 –0.47 1.05 –0.43 0.62 15.6% (2.08) (–1.66) (–0.93) (1.81) (–0.51) (1.09) 5. ΔZCB02Y 0.27 –0.72 –0.45 0.92 –0.48 0.43 9.5% (1.14) (–1.22) (–0.84) (1.52) (–0.59) (0.90) 6. ΔZCB01Y 0.21 –0.53 –0.33 0.59 –0.33 0.27 5.7% (0.93) (–0.95) (–0.64) (1.12) (–0.45) (0.64) D. Inflation-adjusted yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05YRl 0.08 0.58 9.5% (0.53) (2.07) 2. ΔZCB02YRl –0.09 0.56 5.5% (–0.54) (2.06) 3. ΔZCB01YRl –0.20 0.45 3.4% (–1.05) (1.71) 4. ΔZCB05YRl 0.16 –0.57 –0.41 0.74 –0.25 0.49 13.1% (0.96) (–1.54) (–1.25) (1.76) (–0.45) (1.32) 5. ΔZCB02YRl –0.02 –0.55 –0.57 0.76 –0.37 0.40 9.0% (–0.08) (–1.26) (–1.46) (1.86) (–0.71) (1.27) 6. ΔZCB01YRl –0.11 –0.64 –0.74 0.61 –0.27 0.35 6.6% (–0.49) (–1.36) (–1.81) (1.59) (–0.58) (1.23) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . Variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. Nominal yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10Y –0.44 0.46 10.8% (–3.91) (2.72) 2. ΔZCB05Y –0.42 0.30 8.6% (–4.38) (1.74) 3. ΔZCB02Y –0.26 –0.02 5.1% (–3.24) (–0.18) 4. ΔZCB10Y –0.24 –0.80 –1.04 0.18 0.84 1.02 17.7% (–2.36) (–4.25) (–6.19) (1.05) (2.69) (3.91) 5. ΔZCB05Y –0.24 –0.69 –0.94 –0.03 0.98 0.95 15.9% (–2.75) (–3.50) (–5.25) (–0.17) (3.36) (4.22) 6. ΔZCB02Y –0.15 –0.41 –0.56 –0.32 0.86 0.54 11.2% (–1.82) (–2.18) (–3.26) (–2.07) (3.61) (2.95) B. Inflation-adjusted yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10YRl –0.24 0.38 9.2% (–2.80) (3.17) 2. ΔZCB05YRl –0.20 0.24 5.3% (–3.21) (2.06) 3. ΔZCB02YRl –0.01 0.04 0.1% (–0.06) (0.32) 4. ΔZCB10YRl –0.11 –0.54 –0.65 0.21 0.50 0.71 14.8% (–1.35) (–3.72) (–4.90) (1.81) (2.16) (3.56) 5. ΔZCB05YRl –0.10 –0.39 –0.49 0.03 0.63 0.66 11.3% (–1.59) (–2.93) (–4.00) (0.22) (3.27) (4.49) 6. ΔZCB02YRl 0.03 –0.06 –0.03 –0.17 0.62 0.44 3.2% (0.33) (–0.29) (–0.17) (–1.31) (2.77) (2.46) C. Nominal yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05Y 0.31 0.80 11.5% (1.57) (1.97) 2. ΔZCB02Y 0.17 0.65 6.4% (0.81) (1.69) 3. ΔZCB01Y 0.13 0.41 3.6% (0.66) (1.24) 4. ΔZCB05Y 0.45 –0.91 –0.47 1.05 –0.43 0.62 15.6% (2.08) (–1.66) (–0.93) (1.81) (–0.51) (1.09) 5. ΔZCB02Y 0.27 –0.72 –0.45 0.92 –0.48 0.43 9.5% (1.14) (–1.22) (–0.84) (1.52) (–0.59) (0.90) 6. ΔZCB01Y 0.21 –0.53 –0.33 0.59 –0.33 0.27 5.7% (0.93) (–0.95) (–0.64) (1.12) (–0.45) (0.64) D. Inflation-adjusted yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05YRl 0.08 0.58 9.5% (0.53) (2.07) 2. ΔZCB02YRl –0.09 0.56 5.5% (–0.54) (2.06) 3. ΔZCB01YRl –0.20 0.45 3.4% (–1.05) (1.71) 4. ΔZCB05YRl 0.16 –0.57 –0.41 0.74 –0.25 0.49 13.1% (0.96) (–1.54) (–1.25) (1.76) (–0.45) (1.32) 5. ΔZCB02YRl –0.02 –0.55 –0.57 0.76 –0.37 0.40 9.0% (–0.08) (–1.26) (–1.46) (1.86) (–0.71) (1.27) 6. ΔZCB01YRl –0.11 –0.64 –0.74 0.61 –0.27 0.35 6.6% (–0.49) (–1.36) (–1.81) (1.59) (–0.58) (1.23) This table shows how monthly changes in both nominal Treasury yields and inflation-adjusted Treasury yields (as a measure of real yields) move with VIX changes for nonoverlapping monthly changes. We allow for different risk-yield relations over weaker economic (WE) times using our Bai-Perron primary WE identification method. Panels A and C report on the following specification: ΔZCBnYt−1,t=α0+(λ1+λ2DtWE)Δlog(VIXt−1,t)+(γ1+γ2DtWE)Δlog(TIVt−1,t)+ϵt−1,t, where ΔZCBnYt−1,t is the change in the Treasury nominal n-year yield, n is 1, 2, 5, or 10 years as indicated in column one; DtWE is an indicator variable that equals one if in a Bai-Perron WE segment; and the other terms are as defined in Table 1. Here, we report nonoverlapping monthly changes (j=1, with the time units in months), with the inflation expectations from the Haubrich, Pennacchi, and Ritchken (2012) model. Panels B and D report on the same regression and timing, but for the comparable change in inflation-adjusted yields, ΔZCBnYt−1,tRl with the superscript Rl denoting a real-yield proxy. The inflation-adjusted yield is the difference between the nominal yield and the expected annualized inflation over the life of the bond. The sample periods are indicated in each panel heading. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 3 VIX, nominal and inflation-adjusted yields, and weaker economic times . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . Variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. Nominal yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10Y –0.44 0.46 10.8% (–3.91) (2.72) 2. ΔZCB05Y –0.42 0.30 8.6% (–4.38) (1.74) 3. ΔZCB02Y –0.26 –0.02 5.1% (–3.24) (–0.18) 4. ΔZCB10Y –0.24 –0.80 –1.04 0.18 0.84 1.02 17.7% (–2.36) (–4.25) (–6.19) (1.05) (2.69) (3.91) 5. ΔZCB05Y –0.24 –0.69 –0.94 –0.03 0.98 0.95 15.9% (–2.75) (–3.50) (–5.25) (–0.17) (3.36) (4.22) 6. ΔZCB02Y –0.15 –0.41 –0.56 –0.32 0.86 0.54 11.2% (–1.82) (–2.18) (–3.26) (–2.07) (3.61) (2.95) B. Inflation-adjusted yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10YRl –0.24 0.38 9.2% (–2.80) (3.17) 2. ΔZCB05YRl –0.20 0.24 5.3% (–3.21) (2.06) 3. ΔZCB02YRl –0.01 0.04 0.1% (–0.06) (0.32) 4. ΔZCB10YRl –0.11 –0.54 –0.65 0.21 0.50 0.71 14.8% (–1.35) (–3.72) (–4.90) (1.81) (2.16) (3.56) 5. ΔZCB05YRl –0.10 –0.39 –0.49 0.03 0.63 0.66 11.3% (–1.59) (–2.93) (–4.00) (0.22) (3.27) (4.49) 6. ΔZCB02YRl 0.03 –0.06 –0.03 –0.17 0.62 0.44 3.2% (0.33) (–0.29) (–0.17) (–1.31) (2.77) (2.46) C. Nominal yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05Y 0.31 0.80 11.5% (1.57) (1.97) 2. ΔZCB02Y 0.17 0.65 6.4% (0.81) (1.69) 3. ΔZCB01Y 0.13 0.41 3.6% (0.66) (1.24) 4. ΔZCB05Y 0.45 –0.91 –0.47 1.05 –0.43 0.62 15.6% (2.08) (–1.66) (–0.93) (1.81) (–0.51) (1.09) 5. ΔZCB02Y 0.27 –0.72 –0.45 0.92 –0.48 0.43 9.5% (1.14) (–1.22) (–0.84) (1.52) (–0.59) (0.90) 6. ΔZCB01Y 0.21 –0.53 –0.33 0.59 –0.33 0.27 5.7% (0.93) (–0.95) (–0.64) (1.12) (–0.45) (0.64) D. Inflation-adjusted yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05YRl 0.08 0.58 9.5% (0.53) (2.07) 2. ΔZCB02YRl –0.09 0.56 5.5% (–0.54) (2.06) 3. ΔZCB01YRl –0.20 0.45 3.4% (–1.05) (1.71) 4. ΔZCB05YRl 0.16 –0.57 –0.41 0.74 –0.25 0.49 13.1% (0.96) (–1.54) (–1.25) (1.76) (–0.45) (1.32) 5. ΔZCB02YRl –0.02 –0.55 –0.57 0.76 –0.37 0.40 9.0% (–0.08) (–1.26) (–1.46) (1.86) (–0.71) (1.27) 6. ΔZCB01YRl –0.11 –0.64 –0.74 0.61 –0.27 0.35 6.6% (–0.49) (–1.36) (–1.81) (1.59) (–0.58) (1.23) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Dependent . Base . WE diff. . WE total . Base . WE diff. . WE total . . Variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. Nominal yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10Y –0.44 0.46 10.8% (–3.91) (2.72) 2. ΔZCB05Y –0.42 0.30 8.6% (–4.38) (1.74) 3. ΔZCB02Y –0.26 –0.02 5.1% (–3.24) (–0.18) 4. ΔZCB10Y –0.24 –0.80 –1.04 0.18 0.84 1.02 17.7% (–2.36) (–4.25) (–6.19) (1.05) (2.69) (3.91) 5. ΔZCB05Y –0.24 –0.69 –0.94 –0.03 0.98 0.95 15.9% (–2.75) (–3.50) (–5.25) (–0.17) (3.36) (4.22) 6. ΔZCB02Y –0.15 –0.41 –0.56 –0.32 0.86 0.54 11.2% (–1.82) (–2.18) (–3.26) (–2.07) (3.61) (2.95) B. Inflation-adjusted yields, primary 1997:10–2017:12 subperiod 1. ΔZCB10YRl –0.24 0.38 9.2% (–2.80) (3.17) 2. ΔZCB05YRl –0.20 0.24 5.3% (–3.21) (2.06) 3. ΔZCB02YRl –0.01 0.04 0.1% (–0.06) (0.32) 4. ΔZCB10YRl –0.11 –0.54 –0.65 0.21 0.50 0.71 14.8% (–1.35) (–3.72) (–4.90) (1.81) (2.16) (3.56) 5. ΔZCB05YRl –0.10 –0.39 –0.49 0.03 0.63 0.66 11.3% (–1.59) (–2.93) (–4.00) (0.22) (3.27) (4.49) 6. ΔZCB02YRl 0.03 –0.06 –0.03 –0.17 0.62 0.44 3.2% (0.33) (–0.29) (–0.17) (–1.31) (2.77) (2.46) C. Nominal yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05Y 0.31 0.80 11.5% (1.57) (1.97) 2. ΔZCB02Y 0.17 0.65 6.4% (0.81) (1.69) 3. ΔZCB01Y 0.13 0.41 3.6% (0.66) (1.24) 4. ΔZCB05Y 0.45 –0.91 –0.47 1.05 –0.43 0.62 15.6% (2.08) (–1.66) (–0.93) (1.81) (–0.51) (1.09) 5. ΔZCB02Y 0.27 –0.72 –0.45 0.92 –0.48 0.43 9.5% (1.14) (–1.22) (–0.84) (1.52) (–0.59) (0.90) 6. ΔZCB01Y 0.21 –0.53 –0.33 0.59 –0.33 0.27 5.7% (0.93) (–0.95) (–0.64) (1.12) (–0.45) (0.64) D. Inflation-adjusted yields, earlier 1990:01–1997:09 subperiod 1. ΔZCB05YRl 0.08 0.58 9.5% (0.53) (2.07) 2. ΔZCB02YRl –0.09 0.56 5.5% (–0.54) (2.06) 3. ΔZCB01YRl –0.20 0.45 3.4% (–1.05) (1.71) 4. ΔZCB05YRl 0.16 –0.57 –0.41 0.74 –0.25 0.49 13.1% (0.96) (–1.54) (–1.25) (1.76) (–0.45) (1.32) 5. ΔZCB02YRl –0.02 –0.55 –0.57 0.76 –0.37 0.40 9.0% (–0.08) (–1.26) (–1.46) (1.86) (–0.71) (1.27) 6. ΔZCB01YRl –0.11 –0.64 –0.74 0.61 –0.27 0.35 6.6% (–0.49) (–1.36) (–1.81) (1.59) (–0.58) (1.23) This table shows how monthly changes in both nominal Treasury yields and inflation-adjusted Treasury yields (as a measure of real yields) move with VIX changes for nonoverlapping monthly changes. We allow for different risk-yield relations over weaker economic (WE) times using our Bai-Perron primary WE identification method. Panels A and C report on the following specification: ΔZCBnYt−1,t=α0+(λ1+λ2DtWE)Δlog(VIXt−1,t)+(γ1+γ2DtWE)Δlog(TIVt−1,t)+ϵt−1,t, where ΔZCBnYt−1,t is the change in the Treasury nominal n-year yield, n is 1, 2, 5, or 10 years as indicated in column one; DtWE is an indicator variable that equals one if in a Bai-Perron WE segment; and the other terms are as defined in Table 1. Here, we report nonoverlapping monthly changes (j=1, with the time units in months), with the inflation expectations from the Haubrich, Pennacchi, and Ritchken (2012) model. Panels B and D report on the same regression and timing, but for the comparable change in inflation-adjusted yields, ΔZCBnYt−1,tRl with the superscript Rl denoting a real-yield proxy. The inflation-adjusted yield is the difference between the nominal yield and the expected annualized inflation over the life of the bond. The sample periods are indicated in each panel heading. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab The regressions in rows 1 to 3 of Table 3 show the unconditional relations between the yield changes and implied volatility changes. The results show the generally more negative VIX-yield relations over 1997:10–2017:12, relative to 1990:01–1997:09, as previously discussed. We later revisit these unconditional results in Section 6 when discussing across-maturity differences. Table 3, panel B, reports the inflation-adjusted (IA) yields over 1997:10–2017:12. We see the same strong and reliable VIX-yield dynamic patterns for the 5- and 10-year IA yields, but the weaker economy difference is not meaningfully evident for the 2-year IA yields. Table 3, panel D, reports the inflation-adjusted yields over 1990:01–1997:09. The estimated VIX-yield coefficients remain sizably more negative for the weaker economic segment, but the differences lack statistical significance for this short subperiod. Shifting to the TIPS yields, we find that the stronger VIX-yield relations over weaker economic (WE) segments remain sizable and reliably strong over both weekly and monthly change horizons for both the 5- and 10-year TIPS. However, the negative, dynamic VIX-yield relation over the WE segments is appreciably lower than that observed for the nominal yields, by roughly one-half in magnitude. See Internet Appendix B.4 for the details. In sum, changes in these real-yield proxies indicate the same qualitative pattern; the dynamic VIX-yield relation is more negative over weaker economic segments. However, the economic-state differences in the dynamic VIX-yield relation (the estimated λ2s) are generally appreciably more modest for our real-yield proxies as compared to the same maturity nominal yields.12 3.2 Alternative methods for identifying weaker economic times So far, we have relied on Bai-Perron methods to identify periods with a more negative partial relation between yield changes and VIX changes. In this subsection, we examine seven other identification schemes to assess whether our primary VIX-yield results are robust to alternative methods for describing weaker economic times. We cast a wide net for this investigation, using economic data or metrics from other studies that indicate lower economic growth, heightened economic uncertainty, and/or elevated risk aversion. Table 4 presents the estimation results over 1997:10–2017:12, estimating Equation (3) with different weaker economic (WE) segments identified by one of the following eight methods: Bai-Perron analysis, as previously described. NBER recession months. High anxious index (AI): The AI is the probability of a real GDP decline for quarter t, based on the response in quarter t – 1 from the Survey of Professional Forecasters. A threshold-regression is estimated to identify the high AI threshold, where the dynamic VIX-yield and TIV-yield relations differ from the baseline case yield. For our 1997:10–2017:12 period, the AI threshold was estimated at an 11.6% probability. Top-quartile AI with an associated recession: A WE segment commences after a recession starts, but not before the AI also exceeds a top-quartile value. The WE segment lasts as long as there are AI values in the top quartile following the recession and there are no increases in the targeted Federal Funds rate. Declining targeted Federal Funds rate (TFFR): An observation is categorized as being in a WE segment if the TFFR either declined that day or the last change was a decline. This method is predicated on the view that the Fed generally lowers the TFFR in response to economic weakness. Top-quartile equity volatility risk premium (VRP): Periods following a top-quartile equity VRP, as measured over the prior rolling month using the VRP from Bekaert and Hoerova (2014). The literature suggests that a higher VRP is associated with higher aggregate risk aversion. We use a rolling monthly average here, because the series has occasional extreme day-to-day variation. Top-quartile risk aversion: Periods following a top-quartile value of risk aversion, using the rolling 5-weekday average of the daily risk aversion index from Bekaert, Engstrom, and Xu (2019). Top-quartile economic uncertainty: Periods following a top-quartile value of economic uncertainty, using the rolling 5-weekday average of the daily uncertainty index from Bekaert, Engstrom, and Xu (2019). Table 4 VIX and yield dynamics with alternative weaker economic segments, 1997:10–2017:12 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . WE . Base . WE diff. . WE total . Base . WE diff. . WE total . . method . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . 1. Bai-Perron –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (28.4% / 100% ovlp) (–9.48) (–8.02) (–15.21) (2.76) (7.26) (11.02) 2. Recession –0.33 –0.29 –0.62 0.37 0.03 0.40 13.8% (11.5% / 22.3% ovlp) (–13.03) (–2.96) (–6.58) (6.80) (0.16) (2.48) 3. Hi anx. ind. –0.21 –0.28 –0.49 0.16 0.36 0.52 15.2% (51.8% / 41.7% ovlp) (–6.71) (–5.92) (–13.96) (2.20) (3.59) (7.35) 4. Top-quartile AI & rec. –0.23 –0.32 –0.56 0.22 0.35 0.57 15.6% (38.2% / 63.1% ovlp) (–8.00) (–6.56) (–14.03) (3.35) (3.22) (6.61) 5. Target FFR –0.16 –0.30 –0.46 0.06 0.44 0.51 15.5% (61.7% / 45.9% ovlp) (–3.97) (–5.74) (–13.71) (0.79) (4.30) (7.68) 6. Hi VRP –0.33 –0.16 –0.49 0.35 0.09 0.43 13.5% (25.0% / 53.9% ovlp) (–12.21) (–2.58) (–8.50) (6.40) (0.67) (3.66) 7. Hi RA index –0.30 –0.24 –0.55 0.38 –0.01 0.37 14.2% (25.0% / 54.7% ovlp) (–11.39) (–4.21) (–10.45) (7.04) (–0.07) (3.31) 8. Hi EU index –0.32 –0.14 –0.46 0.33 0.15 0.48 13.5% (25.0% / 42.3% ovlp) (–11.09) (–2.23) (–8.42) (5.52) (1.30) (4.73) 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . WE . Base . WE diff. . WE total . Base . WE diff. . WE total . . method . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . 1. Bai-Perron –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (28.4% / 100% ovlp) (–9.48) (–8.02) (–15.21) (2.76) (7.26) (11.02) 2. Recession –0.33 –0.29 –0.62 0.37 0.03 0.40 13.8% (11.5% / 22.3% ovlp) (–13.03) (–2.96) (–6.58) (6.80) (0.16) (2.48) 3. Hi anx. ind. –0.21 –0.28 –0.49 0.16 0.36 0.52 15.2% (51.8% / 41.7% ovlp) (–6.71) (–5.92) (–13.96) (2.20) (3.59) (7.35) 4. Top-quartile AI & rec. –0.23 –0.32 –0.56 0.22 0.35 0.57 15.6% (38.2% / 63.1% ovlp) (–8.00) (–6.56) (–14.03) (3.35) (3.22) (6.61) 5. Target FFR –0.16 –0.30 –0.46 0.06 0.44 0.51 15.5% (61.7% / 45.9% ovlp) (–3.97) (–5.74) (–13.71) (0.79) (4.30) (7.68) 6. Hi VRP –0.33 –0.16 –0.49 0.35 0.09 0.43 13.5% (25.0% / 53.9% ovlp) (–12.21) (–2.58) (–8.50) (6.40) (0.67) (3.66) 7. Hi RA index –0.30 –0.24 –0.55 0.38 –0.01 0.37 14.2% (25.0% / 54.7% ovlp) (–11.39) (–4.21) (–10.45) (7.04) (–0.07) (3.31) 8. Hi EU index –0.32 –0.14 –0.46 0.33 0.15 0.48 13.5% (25.0% / 42.3% ovlp) (–11.09) (–2.23) (–8.42) (5.52) (1.30) (4.73) This table reports the partial relation between changes in Treasury yields and VIX changes, while allowing for a different VIX-yield relation over weaker economic (WE) times. We report on our Bai-Perron primary WE identification method and seven alternative WE classification methods as described below. We report on regressions of the following form for the 5-year ZCB yield over our featured 1997:10–2017:12 period: ΔZCB05Yt−5,t=α0+(λ1+λ2DtWE)Δlog(VIXt−5,t)+(γ1+γ2DtWE)Δlog(TIVt−5,t)+ϵt−5,t, where DtWE is an indicator variable that equals one if in a WE segment using one of eight different WE classification methods; and the other terms are as defined in Table 1. We report on the weekly change horizon, based on rolling 5 weekday periods. The specification in row 1 reports the WE segments identified by our primary Bai-Perron estimation. For the specification in row 2, NBER recessions are used to identify WE segments. For the specification in row 3, a WE segment is in place when the estimated probability of a real GDP decline for the current quarter exceeds an estimated threshold, above which we observe a more negative VIX-yield relation. This threshold is determined from a one-step threshold regression estimation. We use the Survey of Professional Forecasters’ response from the preceding quarter’s survey, referred to as the anxious index (AI), as our measure of the probability of a real GDP decline. For the specification in row 4, a WE segment commences when there is both a recession in place and the estimated AI first exceeds its top-quartile value after the commencement of the recession; then the WE persists as long as there are occurrences of an AI greater than its top quartile following the recent recession and there is no targeted federal funds rate (TFFR) increase. For the specification in row 5, a WE segment is in place for weekday t if either there was a concurrent decrease in the TFFR on that day or the last change in the TFFR was a decline. For the specification in row 6, a WE segment is in place if the lagged monthly moving average of the equity variance risk premium (VRP) (over weekdays t – 6 to t – 27) is a top quartile observation. For the specification in row 7, a WE segment is in place if the lagged weekly moving average of the risk aversion index of Bekaert, Engstrom, and Xu (2019) (over weekdays t – 6 to t – 10) is a top quartile observation. For the specification in row 8, a WE segment is in place if the lagged weekly moving average of the uncertainty index of Bekaert, Engstrom, and Xu (2019) (over weekdays t – 6 to t – 10) is a top quartile observation. Column 1 reports (a) the WE method, (b) the proportion of observations that the method classifies as a WE observation, with the first percentage in parentheses, and (c) the overlap for that WE method with our primary Bai-Perron WE method, with the second percentage in parentheses denoted by “ovlp.” t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 4 VIX and yield dynamics with alternative weaker economic segments, 1997:10–2017:12 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . WE . Base . WE diff. . WE total . Base . WE diff. . WE total . . method . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . 1. Bai-Perron –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (28.4% / 100% ovlp) (–9.48) (–8.02) (–15.21) (2.76) (7.26) (11.02) 2. Recession –0.33 –0.29 –0.62 0.37 0.03 0.40 13.8% (11.5% / 22.3% ovlp) (–13.03) (–2.96) (–6.58) (6.80) (0.16) (2.48) 3. Hi anx. ind. –0.21 –0.28 –0.49 0.16 0.36 0.52 15.2% (51.8% / 41.7% ovlp) (–6.71) (–5.92) (–13.96) (2.20) (3.59) (7.35) 4. Top-quartile AI & rec. –0.23 –0.32 –0.56 0.22 0.35 0.57 15.6% (38.2% / 63.1% ovlp) (–8.00) (–6.56) (–14.03) (3.35) (3.22) (6.61) 5. Target FFR –0.16 –0.30 –0.46 0.06 0.44 0.51 15.5% (61.7% / 45.9% ovlp) (–3.97) (–5.74) (–13.71) (0.79) (4.30) (7.68) 6. Hi VRP –0.33 –0.16 –0.49 0.35 0.09 0.43 13.5% (25.0% / 53.9% ovlp) (–12.21) (–2.58) (–8.50) (6.40) (0.67) (3.66) 7. Hi RA index –0.30 –0.24 –0.55 0.38 –0.01 0.37 14.2% (25.0% / 54.7% ovlp) (–11.39) (–4.21) (–10.45) (7.04) (–0.07) (3.31) 8. Hi EU index –0.32 –0.14 –0.46 0.33 0.15 0.48 13.5% (25.0% / 42.3% ovlp) (–11.09) (–2.23) (–8.42) (5.52) (1.30) (4.73) 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . WE . Base . WE diff. . WE total . Base . WE diff. . WE total . . method . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . 1. Bai-Perron –0.25 –0.40 –0.64 0.16 0.72 0.88 18.4% (28.4% / 100% ovlp) (–9.48) (–8.02) (–15.21) (2.76) (7.26) (11.02) 2. Recession –0.33 –0.29 –0.62 0.37 0.03 0.40 13.8% (11.5% / 22.3% ovlp) (–13.03) (–2.96) (–6.58) (6.80) (0.16) (2.48) 3. Hi anx. ind. –0.21 –0.28 –0.49 0.16 0.36 0.52 15.2% (51.8% / 41.7% ovlp) (–6.71) (–5.92) (–13.96) (2.20) (3.59) (7.35) 4. Top-quartile AI & rec. –0.23 –0.32 –0.56 0.22 0.35 0.57 15.6% (38.2% / 63.1% ovlp) (–8.00) (–6.56) (–14.03) (3.35) (3.22) (6.61) 5. Target FFR –0.16 –0.30 –0.46 0.06 0.44 0.51 15.5% (61.7% / 45.9% ovlp) (–3.97) (–5.74) (–13.71) (0.79) (4.30) (7.68) 6. Hi VRP –0.33 –0.16 –0.49 0.35 0.09 0.43 13.5% (25.0% / 53.9% ovlp) (–12.21) (–2.58) (–8.50) (6.40) (0.67) (3.66) 7. Hi RA index –0.30 –0.24 –0.55 0.38 –0.01 0.37 14.2% (25.0% / 54.7% ovlp) (–11.39) (–4.21) (–10.45) (7.04) (–0.07) (3.31) 8. Hi EU index –0.32 –0.14 –0.46 0.33 0.15 0.48 13.5% (25.0% / 42.3% ovlp) (–11.09) (–2.23) (–8.42) (5.52) (1.30) (4.73) This table reports the partial relation between changes in Treasury yields and VIX changes, while allowing for a different VIX-yield relation over weaker economic (WE) times. We report on our Bai-Perron primary WE identification method and seven alternative WE classification methods as described below. We report on regressions of the following form for the 5-year ZCB yield over our featured 1997:10–2017:12 period: ΔZCB05Yt−5,t=α0+(λ1+λ2DtWE)Δlog(VIXt−5,t)+(γ1+γ2DtWE)Δlog(TIVt−5,t)+ϵt−5,t, where DtWE is an indicator variable that equals one if in a WE segment using one of eight different WE classification methods; and the other terms are as defined in Table 1. We report on the weekly change horizon, based on rolling 5 weekday periods. The specification in row 1 reports the WE segments identified by our primary Bai-Perron estimation. For the specification in row 2, NBER recessions are used to identify WE segments. For the specification in row 3, a WE segment is in place when the estimated probability of a real GDP decline for the current quarter exceeds an estimated threshold, above which we observe a more negative VIX-yield relation. This threshold is determined from a one-step threshold regression estimation. We use the Survey of Professional Forecasters’ response from the preceding quarter’s survey, referred to as the anxious index (AI), as our measure of the probability of a real GDP decline. For the specification in row 4, a WE segment commences when there is both a recession in place and the estimated AI first exceeds its top-quartile value after the commencement of the recession; then the WE persists as long as there are occurrences of an AI greater than its top quartile following the recent recession and there is no targeted federal funds rate (TFFR) increase. For the specification in row 5, a WE segment is in place for weekday t if either there was a concurrent decrease in the TFFR on that day or the last change in the TFFR was a decline. For the specification in row 6, a WE segment is in place if the lagged monthly moving average of the equity variance risk premium (VRP) (over weekdays t – 6 to t – 27) is a top quartile observation. For the specification in row 7, a WE segment is in place if the lagged weekly moving average of the risk aversion index of Bekaert, Engstrom, and Xu (2019) (over weekdays t – 6 to t – 10) is a top quartile observation. For the specification in row 8, a WE segment is in place if the lagged weekly moving average of the uncertainty index of Bekaert, Engstrom, and Xu (2019) (over weekdays t – 6 to t – 10) is a top quartile observation. Column 1 reports (a) the WE method, (b) the proportion of observations that the method classifies as a WE observation, with the first percentage in parentheses, and (c) the overlap for that WE method with our primary Bai-Perron WE method, with the second percentage in parentheses denoted by “ovlp.” t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab While these eight methods share common observations, there are also substantial differences in the proportion of time that each method characterizes as being in a weaker economic (WE) segment. Over our 1997:10–2017:12 period, the proportion of WE observations (as reported with the first percentage in parentheses in column 1 in the table) is as low as 11.5% for the NBER-recession method and as high as 61.7% for the TFFR method. Figure 3 visualizes the timing of the WE segments for each method. Figure 3 Open in new tabDownload slide Weaker economic times across eight different methods This figure shows weaker economic (WE) times for each of our eight classification methods. Panel A shows the WE times by NBER recessions (upper-shaded areas) and by the Bai-Perron estimation (lower-shaded areas). Panel B plots the anxious index (AI) for the quarter and shows the WE times for the high AI method (upper-shaded area, high value estimated by a threshold regression) and the combination top-quintile AI/recession method (lower-shaded area). Panel C plots the targeted Fed Funds rate, and the shaded area represents WE times for the Fed Funds trend method. Panel D plots the monthly moving average of the equity variance risk premium (VRP), with the shaded areas representing the WE times for our top-quartile VRP method. We use the equity VRP, as estimated in Bekaert and Hoerova’s (2014) favored model (8). Panel E plots the weekly moving average of the daily risk aversion (RA) index of Bekaert, Engstrom, and Xu (2019), with the shaded areas representing the WE times for our top-quartile RA method. The RA series briefly exceeds 10 in October 2008, but the scale is truncated at 10 in the figure. Panel F plots the weekly moving average of the daily economic uncertainty (EU) index of Bekaert, Engstrom, and Xu (2019) scaled up by 10,000 for ease of review, with the shaded areas representing the WE times for our top-quartile EU method. Figure 3 Open in new tabDownload slide Weaker economic times across eight different methods This figure shows weaker economic (WE) times for each of our eight classification methods. Panel A shows the WE times by NBER recessions (upper-shaded areas) and by the Bai-Perron estimation (lower-shaded areas). Panel B plots the anxious index (AI) for the quarter and shows the WE times for the high AI method (upper-shaded area, high value estimated by a threshold regression) and the combination top-quintile AI/recession method (lower-shaded area). Panel C plots the targeted Fed Funds rate, and the shaded area represents WE times for the Fed Funds trend method. Panel D plots the monthly moving average of the equity variance risk premium (VRP), with the shaded areas representing the WE times for our top-quartile VRP method. We use the equity VRP, as estimated in Bekaert and Hoerova’s (2014) favored model (8). Panel E plots the weekly moving average of the daily risk aversion (RA) index of Bekaert, Engstrom, and Xu (2019), with the shaded areas representing the WE times for our top-quartile RA method. The RA series briefly exceeds 10 in October 2008, but the scale is truncated at 10 in the figure. Panel F plots the weekly moving average of the daily economic uncertainty (EU) index of Bekaert, Engstrom, and Xu (2019) scaled up by 10,000 for ease of review, with the shaded areas representing the WE times for our top-quartile EU method. Figure 3 Open in new tabDownload slide (Continued) Figure 3 Open in new tabDownload slide (Continued) Over our featured 1997:10–2017:12 period, Table 4 shows that the partial VIX-yield relation is appreciably more negative over weaker economic (WE) segments for each classification method. The estimated λ2 coefficients, which indicate the difference in the VIX-yield relation over the WE segments, have a p-value of better than 1% for every method except method 8 (which has a 5% p-value). Further, the total VIX-yield relation over weaker economic segments ( λ1+λ2 ) is sizably negative and highly statistically significant with p-values of less than 0.1% in every case. As reported in column 1, we note that the overlap between the Bai-Perron segments and the alternative weaker economic segments is moderate, six of the seven alternative methods have an overlap of between 42% and 63%.13 Over the earlier 1990:01–1997:09 period, we also find that the partial VIX-yield relation is always more negative over weaker economic times for each classification method, but the differences lack statistical significance for methods 2 and 6. To save space, we relegate tabular details to Internet Appendix B.5. In sum, our results indicate that the partial VIX-yield relation is more negative over weaker economic (WE) times, relative to the baseline segments, for a wide variety of different WE classification methods. Across our two periods and eight methods, the λ2 coefficients are always negative with statistical significance for 14 of the 16 cases. Further, the substantial overlap of the Bai-Perron segments with the alternative WE segments indicates that the Bai-Perron segments capture multiple dimensions of a WE environment. 3.3 Additional investigation and robustness analysis Next, we summarize four additional results that confirm robustness and assist in our interpretation. 3.3.1 Linear changes in the VIX and TIV. We find that the VIX-yield dynamics are quite similar and reliably evident when evaluating the simple linear changes in VIX and TIV as the explanatory terms. So, our main results are not limited to the “change in log” functional form. See Internet Appendix B.6.1 for the details. 3.3.2 Nonoverlapping weekly changes. Our prior empirical results for the weekly horizon used overlapping observations constructed from rolling 5-weekday changes. We confirm that our results are quite similar when evaluating nonoverlapping Friday-to-Friday weekly observations. See Internet Appendix B.6.2 for the details. 3.3.3 Unexpected VIX and TIV changes. Since uncertainty tends to mean revert, we expect that implied volatility (IV) changes would be modestly predictable in an autoregressive framework. Accordingly, we use a vector autoregressive (VAR) estimation to decompose the total IV changes, into an “unexpected IV change” (the residuals from the VAR) and the “predictable IV changes” (the fitted values from the VAR). Consistent with an interpretation that the economic uncertainty innovation is the relevant factor for understanding yield movements, we find that the strong state-contingent differences in the VIX-yield relations (as in Table 1) are essentially entirely attributable to the unpredictable component of the VIX changes. See Internet Appendix B.6.3 for the details. 3.3.4 Out-of-sample investigation over 2018:01–2020:06. The sample period in our initial empirical work ended in December 2017. Here, we discuss results for the 2018:01–2020:6 period as an out-of-sample evaluation. The last 12 months of this 30-month period can be considered a relatively weaker economic (WE) segment based on a consistently elevated anxious index (AI) and the COVID-19-related recession. We find the same qualitative patterns over this 30-month period. The dynamic VIX-yield relation is appreciably and statistically significantly more negative over the WE segment, with or without the inclusion of the TIV explanatory term. For brevity, we report details in Internet Appendix B.6.4. 4. Other Uncertainty Evidence in lieu of VIX and Treasury Yields Our main results in Section 3 indicate a more negative, dynamic relation between economic uncertainty and Treasury-bond yields in weaker economic times, at least when using VIX movements to measure changes in uncertainty. This section extends our investigation along two dimensions. Section 4.1 evaluates how three other measures of time-varying economic uncertainty and/or risk aversion are related to the Treasury yield movements, in lieu of VIX. We start by evaluating the two components of VIX separately, the conditional-volatility (CV) component and the volatility-risk-premium (VRP) component. This analysis is intended to assess whether our main VIX-yield dynamic results are more closely related to the equity market’s CV or the equity market’s VRP. The literature suggests that CV is more aligned with economic uncertainty and the VRP is more aligned with risk aversion; see the literature in our footnote 4. We also evaluate the daily economic policy uncertainty (EPU) index of Baker, Bloom, and Davis (2016), in place of VIX. We perform the EPU analysis to test for a similar weaker economy contingency between yields and uncertainty when using a direct news-driven uncertainty metric. Section 4.2 evaluates alternative dependent variables for Equation (3), in lieu of Treasury yields. If our main VIX-yield findings are substantially attributed to an uncertainty/precautionary savings channel, then we expect VIX changes to be similarly related to the relative pricing dynamics in comparisons of safer versus riskier financial assets. To evaluate this implication in an alternative bond setting, we examine how VIX movements are related to the default yield spread. To evaluate this implication in stock pricing, we examine how VIX movements are related to the returns of a long/short position that is long higher-volatility stocks and short lower-volatility stocks. 4.1 Relating alternative risk measures to yield dynamics over WE times 4.1.1 The equity market’s conditional volatility and volatility risk premium. This section evaluates both the conditional volatility (CV) and the volatility risk premium (VRP) from a VIX decomposition, in lieu of VIX, in Equation (3). Since we previously showed that the weaker economic (WE) variation in the VIX-yield relation is consistently evident for a variety of alternative ways to categorize WE conditions, this section shows results only for our primary Bai-Perron WE segments. For this evaluation, it is an empirical issue whether to use (1) a standard deviation form of CV, with a VRP defined as the difference between the standard deviation form of VIX and the conditional standard deviation or (2) a variance form of CV, with a VRP defined as the difference between the squared VIX and the conditional variance. Theory generally takes on a variance functional form, but empirically the variance form can lead to explanatory terms with high kurtosis that may be undesirable econometrically. Hence, following French, Schwert, and Stambaugh (1987), we embrace a practical approach and evaluate both functional forms. We focus our discussion on the standard-deviation-based CV and VRP, as the better performer in our setting.14 Table 5 reports the specification and estimation results. First, we find that the dynamic relations between the yield changes and both the ΔCV and ΔVRP terms are always more negative over our weaker economic (WE) segments; see the estimated λ2 coefficients. The λ2 estimates are always sizably negative and highly statistically significant for the ΔCV terms in the regressions in row 1. The estimated λ2s are negative for the ΔVRP terms in the regressions in row 2, but they generally lack statistical significance. The total WE yield relation ( λ1+λ2 ) is also always negative for both ΔCV and ΔVRP , but again the ΔCV relation is reliably stronger. Table 5 Treasury yield dynamics with the conditional volatility and volatility risk premium 1. Dependent . 2. Base . 3. WE diff. . 4. WE total . 5. Base . 6. WE diff. . 7. WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. 1997:10–2017:12, ΔZCB05Yt−j,t 1. ΔCV –0.69 –1.38 –2.07 0.10 0.65 0.74 12.4% (1 week, j=5) (–2.62) (–3.94) (–8.93) (1.44) (6.63) (10.37) 2. ΔVRP –0.60 –0.37 –0.97 0.01 0.60 0.60 7.9% (1 week, j=5) (–3.18) (–1.27) (–4.39) (0.13) (5.49) (6.83) 3. ΔVIX –1.09 –1.09 –2.18 0.15 0.60 0.75 18.0% (1 week, j=5) (–7.34) (–4.95) (–13.20) (2.25) (6.27) (10.70) 1. ΔCV –0.73 –1.96 –2.69 –0.06 0.82 0.76 12.8% (1 month, j=22) (–1.15) (–2.65) (–6.17) (–0.36) (3.54) (4.29) 2. ΔVRP –0.49 –1.39 –1.88 –0.19 0.80 0.61 7.1% (1 month, j=22) (–0.82) (–1.94) (–5.19) (–1.55) (3.39) (3.04) 3. ΔVIX –0.83 –2.05 –2.88 –0.05 0.81 0.77 15.7% (1 month, j=22) (–2.01) (–4.06) (–8.97) (–0.31) (3.69) (4.33) B. 1990:01–1997:09, ΔZCB02Yt−j,t 1. ΔCV 1.32 –4.75 –3.43 0.64 0.03 0.67 13.8% (1 week, j=5) (2.99) (–5.04) (–4.08) (7.03) (0.16) (3.57) 2. ΔVRP 0.58 –0.63 –0.05 0.68 –0.30 0.38 11.0% (1 week, j=5) (1.66) (–1.04) (–0.10) (7.39) (–1.42) (2.02) 3. ΔVIX 0.98 –2.23 –1.24 0.61 –0.04 0.57 12.7% (1 week, j=5) (3.27) (–4.08) (–2.71) (6.72) (–0.18) (2.91) 1. ΔCV 1.64 –5.85 –4.21 0.68 –0.05 0.63 13.7% (2 weeks, j=10) (2.57) (–4.37) (–3.54) (4.84) (–0.19) (2.61) 2. ΔVRP 0.26 –0.61 –0.35 0.75 –0.38 0.37 10.2% (2 weeks, j=10 (0.47) (–0.60) (–0.39) (5.22) (–1.39) (1.58) 3. ΔVIX 0.97 –2.54 –1.57 0.68 –0.11 0.57 12.0% (2 weeks, j=10 (1.85) (–3.06) (–2.40) (4.61) (–0.38) (2.21) 1. Dependent . 2. Base . 3. WE diff. . 4. WE total . 5. Base . 6. WE diff. . 7. WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. 1997:10–2017:12, ΔZCB05Yt−j,t 1. ΔCV –0.69 –1.38 –2.07 0.10 0.65 0.74 12.4% (1 week, j=5) (–2.62) (–3.94) (–8.93) (1.44) (6.63) (10.37) 2. ΔVRP –0.60 –0.37 –0.97 0.01 0.60 0.60 7.9% (1 week, j=5) (–3.18) (–1.27) (–4.39) (0.13) (5.49) (6.83) 3. ΔVIX –1.09 –1.09 –2.18 0.15 0.60 0.75 18.0% (1 week, j=5) (–7.34) (–4.95) (–13.20) (2.25) (6.27) (10.70) 1. ΔCV –0.73 –1.96 –2.69 –0.06 0.82 0.76 12.8% (1 month, j=22) (–1.15) (–2.65) (–6.17) (–0.36) (3.54) (4.29) 2. ΔVRP –0.49 –1.39 –1.88 –0.19 0.80 0.61 7.1% (1 month, j=22) (–0.82) (–1.94) (–5.19) (–1.55) (3.39) (3.04) 3. ΔVIX –0.83 –2.05 –2.88 –0.05 0.81 0.77 15.7% (1 month, j=22) (–2.01) (–4.06) (–8.97) (–0.31) (3.69) (4.33) B. 1990:01–1997:09, ΔZCB02Yt−j,t 1. ΔCV 1.32 –4.75 –3.43 0.64 0.03 0.67 13.8% (1 week, j=5) (2.99) (–5.04) (–4.08) (7.03) (0.16) (3.57) 2. ΔVRP 0.58 –0.63 –0.05 0.68 –0.30 0.38 11.0% (1 week, j=5) (1.66) (–1.04) (–0.10) (7.39) (–1.42) (2.02) 3. ΔVIX 0.98 –2.23 –1.24 0.61 –0.04 0.57 12.7% (1 week, j=5) (3.27) (–4.08) (–2.71) (6.72) (–0.18) (2.91) 1. ΔCV 1.64 –5.85 –4.21 0.68 –0.05 0.63 13.7% (2 weeks, j=10) (2.57) (–4.37) (–3.54) (4.84) (–0.19) (2.61) 2. ΔVRP 0.26 –0.61 –0.35 0.75 –0.38 0.37 10.2% (2 weeks, j=10 (0.47) (–0.60) (–0.39) (5.22) (–1.39) (1.58) 3. ΔVIX 0.97 –2.54 –1.57 0.68 –0.11 0.57 12.0% (2 weeks, j=10 (1.85) (–3.06) (–2.40) (4.61) (–0.38) (2.21) This table evaluates yield relations to both the conditional volatility (CV) and the volatility risk premium (VRP), based on a VIX decomposition. Row 1 reports on the following regression form: ΔZCBnYt−j,t=α0+(λ1+λ2DtWE)ΔCVt−j,t+(γ1+γ2DtWE)ΔTIVt−j,t+ϵt−j,t, where ΔCVt−j,t and ΔTIVt−j,t are the simple linear change over weekday t – j to t for the conditional volatility and TIV; and the other terms and tabular details are as in Table 1. Row 2 replaces Δ(CV) with ΔVRP . Row 3 replaces ΔCV with ΔVIX . Our CV and VRP are based on Bekaert-Hoerova’s (2014) favored model 8, but we use an annualized standard deviation function form for the CV and VRP (as explained in Section 4.1.1). The VIX, TIV, CV, and VRP are in annualized decimal units. The panel headings indicate the sample and yield horizon, with the 5-year yield (2-year yield) representing our featured 1997:10–2017:12 period (earlier 1990:01–1997:09 period). Open in new tab Table 5 Treasury yield dynamics with the conditional volatility and volatility risk premium 1. Dependent . 2. Base . 3. WE diff. . 4. WE total . 5. Base . 6. WE diff. . 7. WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. 1997:10–2017:12, ΔZCB05Yt−j,t 1. ΔCV –0.69 –1.38 –2.07 0.10 0.65 0.74 12.4% (1 week, j=5) (–2.62) (–3.94) (–8.93) (1.44) (6.63) (10.37) 2. ΔVRP –0.60 –0.37 –0.97 0.01 0.60 0.60 7.9% (1 week, j=5) (–3.18) (–1.27) (–4.39) (0.13) (5.49) (6.83) 3. ΔVIX –1.09 –1.09 –2.18 0.15 0.60 0.75 18.0% (1 week, j=5) (–7.34) (–4.95) (–13.20) (2.25) (6.27) (10.70) 1. ΔCV –0.73 –1.96 –2.69 –0.06 0.82 0.76 12.8% (1 month, j=22) (–1.15) (–2.65) (–6.17) (–0.36) (3.54) (4.29) 2. ΔVRP –0.49 –1.39 –1.88 –0.19 0.80 0.61 7.1% (1 month, j=22) (–0.82) (–1.94) (–5.19) (–1.55) (3.39) (3.04) 3. ΔVIX –0.83 –2.05 –2.88 –0.05 0.81 0.77 15.7% (1 month, j=22) (–2.01) (–4.06) (–8.97) (–0.31) (3.69) (4.33) B. 1990:01–1997:09, ΔZCB02Yt−j,t 1. ΔCV 1.32 –4.75 –3.43 0.64 0.03 0.67 13.8% (1 week, j=5) (2.99) (–5.04) (–4.08) (7.03) (0.16) (3.57) 2. ΔVRP 0.58 –0.63 –0.05 0.68 –0.30 0.38 11.0% (1 week, j=5) (1.66) (–1.04) (–0.10) (7.39) (–1.42) (2.02) 3. ΔVIX 0.98 –2.23 –1.24 0.61 –0.04 0.57 12.7% (1 week, j=5) (3.27) (–4.08) (–2.71) (6.72) (–0.18) (2.91) 1. ΔCV 1.64 –5.85 –4.21 0.68 –0.05 0.63 13.7% (2 weeks, j=10) (2.57) (–4.37) (–3.54) (4.84) (–0.19) (2.61) 2. ΔVRP 0.26 –0.61 –0.35 0.75 –0.38 0.37 10.2% (2 weeks, j=10 (0.47) (–0.60) (–0.39) (5.22) (–1.39) (1.58) 3. ΔVIX 0.97 –2.54 –1.57 0.68 –0.11 0.57 12.0% (2 weeks, j=10 (1.85) (–3.06) (–2.40) (4.61) (–0.38) (2.21) 1. Dependent . 2. Base . 3. WE diff. . 4. WE total . 5. Base . 6. WE diff. . 7. WE total . . variable: . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. 1997:10–2017:12, ΔZCB05Yt−j,t 1. ΔCV –0.69 –1.38 –2.07 0.10 0.65 0.74 12.4% (1 week, j=5) (–2.62) (–3.94) (–8.93) (1.44) (6.63) (10.37) 2. ΔVRP –0.60 –0.37 –0.97 0.01 0.60 0.60 7.9% (1 week, j=5) (–3.18) (–1.27) (–4.39) (0.13) (5.49) (6.83) 3. ΔVIX –1.09 –1.09 –2.18 0.15 0.60 0.75 18.0% (1 week, j=5) (–7.34) (–4.95) (–13.20) (2.25) (6.27) (10.70) 1. ΔCV –0.73 –1.96 –2.69 –0.06 0.82 0.76 12.8% (1 month, j=22) (–1.15) (–2.65) (–6.17) (–0.36) (3.54) (4.29) 2. ΔVRP –0.49 –1.39 –1.88 –0.19 0.80 0.61 7.1% (1 month, j=22) (–0.82) (–1.94) (–5.19) (–1.55) (3.39) (3.04) 3. ΔVIX –0.83 –2.05 –2.88 –0.05 0.81 0.77 15.7% (1 month, j=22) (–2.01) (–4.06) (–8.97) (–0.31) (3.69) (4.33) B. 1990:01–1997:09, ΔZCB02Yt−j,t 1. ΔCV 1.32 –4.75 –3.43 0.64 0.03 0.67 13.8% (1 week, j=5) (2.99) (–5.04) (–4.08) (7.03) (0.16) (3.57) 2. ΔVRP 0.58 –0.63 –0.05 0.68 –0.30 0.38 11.0% (1 week, j=5) (1.66) (–1.04) (–0.10) (7.39) (–1.42) (2.02) 3. ΔVIX 0.98 –2.23 –1.24 0.61 –0.04 0.57 12.7% (1 week, j=5) (3.27) (–4.08) (–2.71) (6.72) (–0.18) (2.91) 1. ΔCV 1.64 –5.85 –4.21 0.68 –0.05 0.63 13.7% (2 weeks, j=10) (2.57) (–4.37) (–3.54) (4.84) (–0.19) (2.61) 2. ΔVRP 0.26 –0.61 –0.35 0.75 –0.38 0.37 10.2% (2 weeks, j=10 (0.47) (–0.60) (–0.39) (5.22) (–1.39) (1.58) 3. ΔVIX 0.97 –2.54 –1.57 0.68 –0.11 0.57 12.0% (2 weeks, j=10 (1.85) (–3.06) (–2.40) (4.61) (–0.38) (2.21) This table evaluates yield relations to both the conditional volatility (CV) and the volatility risk premium (VRP), based on a VIX decomposition. Row 1 reports on the following regression form: ΔZCBnYt−j,t=α0+(λ1+λ2DtWE)ΔCVt−j,t+(γ1+γ2DtWE)ΔTIVt−j,t+ϵt−j,t, where ΔCVt−j,t and ΔTIVt−j,t are the simple linear change over weekday t – j to t for the conditional volatility and TIV; and the other terms and tabular details are as in Table 1. Row 2 replaces Δ(CV) with ΔVRP . Row 3 replaces ΔCV with ΔVIX . Our CV and VRP are based on Bekaert-Hoerova’s (2014) favored model 8, but we use an annualized standard deviation function form for the CV and VRP (as explained in Section 4.1.1). The VIX, TIV, CV, and VRP are in annualized decimal units. The panel headings indicate the sample and yield horizon, with the 5-year yield (2-year yield) representing our featured 1997:10–2017:12 period (earlier 1990:01–1997:09 period). Open in new tab We find that the total ΔVIX term has considerably more information about the yield movements than does either of the two VIX components (the ΔCV and ΔVRP terms) over our featured 1997:10–2017:12 period (compare the regressions in row 3 to the regressions in rows 1 and 2). However, ΔCV and ΔVIX contain similar information for yield changes over the earlier 1990:01–1997:09 period. In sum, for the more negative dynamic VIX-yield relation over weaker economic times, the estimated coefficients for the ΔCV and ΔVRP terms indicate that each component contributes to our strong VIX-yield findings. However, the ΔCV results are appreciably stronger, supporting more of a time-varying economic uncertainty interpretation for our main results. 4.1.2 Economic policy uncertainty index. The EPU index is not the same concept as economic uncertainty, but a high EPU is one factor that could contribute to a heightened economic uncertainty. Given that our results in Section 4.1.1 favor an economic uncertainty interpretation of our main VIX-yield results, we next evaluate how changes in the daily EPU index are related to Treasury yield movements. Consistent with our main VIX-yield findings, we find that the dynamic EPU-yield relation is also more negative over weaker economic times. See Internet Appendix B.7 for details. 4.2 Other asset pricing dynamics and VIX changes If time-varying economic uncertainty has an elevated influence on yields in weaker economic times through the precautionary savings channel discussed in Section 1, then it seems likely that a related pricing dynamic would be evident in the pricing of other lower-risk securities relative to comparable higher-risk securities. Accordingly, this section evaluates how VIX changes are related to the relative pricing of lower-risk bonds versus higher-risk bonds and lower-risk stocks versus higher-risk stocks. By taking a within-asset-class approach, we focus on risk differentials, holding the security-type constant. To examine bond-risk differentials, we evaluate changes in a default yield spread (DYS) defined as the difference between the Moodys Baa yield and the 10-year Treasury yield. Table 6, panel A, reports on a regression similar to Equation (3), but with the DYS change as the dependent variable in place of the yield changes. Consistent with this subsection’s premise, we find that the DYS widens relatively more with VIX increases over our weaker economic (WE) segments (compare the WE relation in column 4 to the baseline relation in column 2). Table 6 Default yield spread, the price of volatile stocks, and VIX changes . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Sample period . Base . WE diff. . WE total . Base . WE diff. . WE total . . (change horizon) . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. ΔDYS 1. 1997:10–2017:12 0.12 0.09 0.21 0.09 –0.40 –0.28 8.5% (1 week, j=5) (5.93) (2.25) (6.37) (2.04) (–4.71) (–4.27) 2. 1997:10–2017:12 0.29 0.26 0.56 0.29 –0.75 –0.45 20.7% (1 month, j=22) (3.02) (1.92) (5.46) (3.24) (–3.17) (–2.05) 3. 1990:01–1997:09 –0.10 0.16 0.06 –0.16 –0.06 –0.23 7.0% (1 week, j=5) (–4.54) (2.93) (1.19) (–5.20) (–0.50) (–1.89) 4. 1990:01–1997:09 –0.09 0.21 0.12 –0.16 –0.04 –0.19 6.1% (2 weeks, j=10) (–2.86) (2.61) (1.61) (–3.26) (–0.26) (–1.48) B. PVS return 5. 1997:10–2017:12 –9.38 –9.25 –18.63 –0.45 1.87 1.42 24.0% (1 week, j=5) (–12.85) (–6.40) (–15.19) (–0.38) (0.66) (0.57) 6. 1997:10–2017:12 –14.15 –10.94 –25.09 1.43 –2.33 –0.91 19.9% (1 month, j=22) (–7.01) (–2.23) (–5.61) (0.47) (–0.30) (–0.13) 7. 1990:01–1997:09 –2.16 –3.33 –5.49 –0.20 7.83 7.63 3.0% (1 week, j=5) (–2.53) (–1.47) (–2.59) (–0.16) (0.98) (0.97) 8. 1990:01–1997:09 –1.44 –6.78 –8.21 –0.67 13.28 12.61 3.1% (2 weeks, j=10) (–0.98) (–1.58) (–2.00) (–0.35) (0.83) (0.80) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Sample period . Base . WE diff. . WE total . Base . WE diff. . WE total . . (change horizon) . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. ΔDYS 1. 1997:10–2017:12 0.12 0.09 0.21 0.09 –0.40 –0.28 8.5% (1 week, j=5) (5.93) (2.25) (6.37) (2.04) (–4.71) (–4.27) 2. 1997:10–2017:12 0.29 0.26 0.56 0.29 –0.75 –0.45 20.7% (1 month, j=22) (3.02) (1.92) (5.46) (3.24) (–3.17) (–2.05) 3. 1990:01–1997:09 –0.10 0.16 0.06 –0.16 –0.06 –0.23 7.0% (1 week, j=5) (–4.54) (2.93) (1.19) (–5.20) (–0.50) (–1.89) 4. 1990:01–1997:09 –0.09 0.21 0.12 –0.16 –0.04 –0.19 6.1% (2 weeks, j=10) (–2.86) (2.61) (1.61) (–3.26) (–0.26) (–1.48) B. PVS return 5. 1997:10–2017:12 –9.38 –9.25 –18.63 –0.45 1.87 1.42 24.0% (1 week, j=5) (–12.85) (–6.40) (–15.19) (–0.38) (0.66) (0.57) 6. 1997:10–2017:12 –14.15 –10.94 –25.09 1.43 –2.33 –0.91 19.9% (1 month, j=22) (–7.01) (–2.23) (–5.61) (0.47) (–0.30) (–0.13) 7. 1990:01–1997:09 –2.16 –3.33 –5.49 –0.20 7.83 7.63 3.0% (1 week, j=5) (–2.53) (–1.47) (–2.59) (–0.16) (0.98) (0.97) 8. 1990:01–1997:09 –1.44 –6.78 –8.21 –0.67 13.28 12.61 3.1% (2 weeks, j=10) (–0.98) (–1.58) (–2.00) (–0.35) (0.83) (0.80) This table shows how the default yield spread (DYS) and the price of volatile stocks (PVS) move concurrently with changes in the implied volatility of both the equity and Treasury-bond markets, while allowing for a different relation over weaker economic (WE) times. We estimate variations of the regression specification in Table 1 but with different dependent variables. In panel A, we report the DYS change ( ΔDYS ) over weekdays t – j to t, where the DYS is the difference between Moody’s Baa yield and the 10-year Treasury Constant Maturity yield. In panel B, we report the PVS return, defined as the equally weighted percentage return over weekdays t – j to t of a long/short position that is long the highest volatility-quintile of stocks and short the lowest volatility-quintile of stocks. We use the CRSP standard-deviation-sorted portfolios; the volatility sort is on the standard deviation over the prior calendar year. We report both the weekly change horizon and a longer horizon, as indicated by the number of weekdays j for each row. The WE segments are our featured Bai-Perron segments, as in Table 1. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 6 Default yield spread, the price of volatile stocks, and VIX changes . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Sample period . Base . WE diff. . WE total . Base . WE diff. . WE total . . (change horizon) . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. ΔDYS 1. 1997:10–2017:12 0.12 0.09 0.21 0.09 –0.40 –0.28 8.5% (1 week, j=5) (5.93) (2.25) (6.37) (2.04) (–4.71) (–4.27) 2. 1997:10–2017:12 0.29 0.26 0.56 0.29 –0.75 –0.45 20.7% (1 month, j=22) (3.02) (1.92) (5.46) (3.24) (–3.17) (–2.05) 3. 1990:01–1997:09 –0.10 0.16 0.06 –0.16 –0.06 –0.23 7.0% (1 week, j=5) (–4.54) (2.93) (1.19) (–5.20) (–0.50) (–1.89) 4. 1990:01–1997:09 –0.09 0.21 0.12 –0.16 –0.04 –0.19 6.1% (2 weeks, j=10) (–2.86) (2.61) (1.61) (–3.26) (–0.26) (–1.48) B. PVS return 5. 1997:10–2017:12 –9.38 –9.25 –18.63 –0.45 1.87 1.42 24.0% (1 week, j=5) (–12.85) (–6.40) (–15.19) (–0.38) (0.66) (0.57) 6. 1997:10–2017:12 –14.15 –10.94 –25.09 1.43 –2.33 –0.91 19.9% (1 month, j=22) (–7.01) (–2.23) (–5.61) (0.47) (–0.30) (–0.13) 7. 1990:01–1997:09 –2.16 –3.33 –5.49 –0.20 7.83 7.63 3.0% (1 week, j=5) (–2.53) (–1.47) (–2.59) (–0.16) (0.98) (0.97) 8. 1990:01–1997:09 –1.44 –6.78 –8.21 –0.67 13.28 12.61 3.1% (2 weeks, j=10) (–0.98) (–1.58) (–2.00) (–0.35) (0.83) (0.80) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . 1. Sample period . Base . WE diff. . WE total . Base . WE diff. . WE total . . (change horizon) . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . 8. R2 . A. ΔDYS 1. 1997:10–2017:12 0.12 0.09 0.21 0.09 –0.40 –0.28 8.5% (1 week, j=5) (5.93) (2.25) (6.37) (2.04) (–4.71) (–4.27) 2. 1997:10–2017:12 0.29 0.26 0.56 0.29 –0.75 –0.45 20.7% (1 month, j=22) (3.02) (1.92) (5.46) (3.24) (–3.17) (–2.05) 3. 1990:01–1997:09 –0.10 0.16 0.06 –0.16 –0.06 –0.23 7.0% (1 week, j=5) (–4.54) (2.93) (1.19) (–5.20) (–0.50) (–1.89) 4. 1990:01–1997:09 –0.09 0.21 0.12 –0.16 –0.04 –0.19 6.1% (2 weeks, j=10) (–2.86) (2.61) (1.61) (–3.26) (–0.26) (–1.48) B. PVS return 5. 1997:10–2017:12 –9.38 –9.25 –18.63 –0.45 1.87 1.42 24.0% (1 week, j=5) (–12.85) (–6.40) (–15.19) (–0.38) (0.66) (0.57) 6. 1997:10–2017:12 –14.15 –10.94 –25.09 1.43 –2.33 –0.91 19.9% (1 month, j=22) (–7.01) (–2.23) (–5.61) (0.47) (–0.30) (–0.13) 7. 1990:01–1997:09 –2.16 –3.33 –5.49 –0.20 7.83 7.63 3.0% (1 week, j=5) (–2.53) (–1.47) (–2.59) (–0.16) (0.98) (0.97) 8. 1990:01–1997:09 –1.44 –6.78 –8.21 –0.67 13.28 12.61 3.1% (2 weeks, j=10) (–0.98) (–1.58) (–2.00) (–0.35) (0.83) (0.80) This table shows how the default yield spread (DYS) and the price of volatile stocks (PVS) move concurrently with changes in the implied volatility of both the equity and Treasury-bond markets, while allowing for a different relation over weaker economic (WE) times. We estimate variations of the regression specification in Table 1 but with different dependent variables. In panel A, we report the DYS change ( ΔDYS ) over weekdays t – j to t, where the DYS is the difference between Moody’s Baa yield and the 10-year Treasury Constant Maturity yield. In panel B, we report the PVS return, defined as the equally weighted percentage return over weekdays t – j to t of a long/short position that is long the highest volatility-quintile of stocks and short the lowest volatility-quintile of stocks. We use the CRSP standard-deviation-sorted portfolios; the volatility sort is on the standard deviation over the prior calendar year. We report both the weekly change horizon and a longer horizon, as indicated by the number of weekdays j for each row. The WE segments are our featured Bai-Perron segments, as in Table 1. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab To examine stock-risk differentials, we contrast the returns of high-volatility stocks versus low-volatility stocks. Specifically, we analyze returns of an equity position that is long the top volatility-quintile of stocks and short the bottom volatility-quintile of stocks, which we refer to as the “price of volatile stocks” (PVS), following Pflueger, Siriwardane, and Sunderam (2020).15 Table 6, panel B, reports on a regression similar to Equation (3), but with the PVS as the dependent variable in place of the yield changes. Consistent with this subsection’s premise, we find an appreciably larger negative relation between VIX changes and the PVS returns over weaker economic segments for our primary 1997:10–2017:12 period. Over the earlier 1990:01–1997:09 period, the results are qualitatively similar but the estimated λ2 coefficients fall short of statistical significance. 5. Evidence on Possible Channels behind VIX-Yield Dynamics Having established our main findings, we now revisit the three VIX-yield economic channels considered in Section 1. Sections 5.1, 5.2, and 5.3 further investigate the precautionary savings channel, the consumption smoothing channel, and time-varying risk aversion, respectively. Section 5.4 summarizes evidence that is at odds with a monetary policy implementation channel. To assess changes in expected economic growth, Sections 5.1 and 5.2 feature the expected real GDP growth from the quarterly Survey of Professional Forecasters (SPF). Bansal and Shaliastovich (2013) find that the SPF forecasts of real GDP growth contain significant information about future real consumption growth, prompting our SPF use. Here, we also use the SPF expected CPI inflation to comparably evaluate changes in expected inflation. Ang, Bekaert, and Wei (2007) find that survey-based expected inflation provides more accurate inflation forecasts than time-series models or term-structure models with macroeconomic variables. Thus, our work here evaluates quarterly changes, rather than the weekly and monthly changes in Sections 3 and 4. This section reports results only for our featured 1997:10–2017:12 period, because our short earlier 1990:01 to 1997:09 period: (1) has relatively few quarterly observations, and (2) does not have a meaningful relation between changes in the SPF’s expected economic growth and yield changes. Internet Appendix B.8.1 summarizes results for this earlier period. 5.1 Predicting Volatility in Expected Economic Growth with the VIX Section 1.2.1 discusses a precautionary-saving channel where VIX movements are positively related to changes in economic growth uncertainty. Our findings of a more negative, dynamic VIX-yield relation in weaker economic (WE) times fit if VIX movements are more related to expected economic growth uncertainty in WE times. Here, we investigate the following two empirical questions suggested by this channel. Does the VIX level predict the subsequent realized volatility of expected economic growth? And, if so, is the VIX relation to the subsequent economic growth volatility stronger over WE times? We use the absolute change in the quarter-to-quarter SPF estimates of a quarter’s expected economic growth rate as a realized-volatility measure. Table 7, panel A, investigates these questions regarding economic growth uncertainty; see the table for specification details.16 We find that the lagged VIX level contains appreciable information about the subsequent volatility in expected economic growth. Further, using weaker economic (WE) segments identified by our primary Bai-Perron method, the VIX-volatility relation is reliably stronger over the WE segments: panel A, row 4 indicates that the relation between VIX and subsequent economic growth volatility is reliably positive for WE times and about 83% greater than the relation in baseline times. Internet Appendix B.8.2 reports qualitatively similar findings when using the alternative “anxious index/recession” method for categorizing WE times. Thus, our evidence suggests an affirmative answer to both empirical questions in the preceding paragraph, consistent with a precautionary savings channel. Table 7, panel B, reports on a similar investigation with inflation uncertainty, using the absolute change in the quarter-to-quarter SPF expected inflation as a realized inflation-volatility measure. Consistent with our intuition and later discussion about the different information in VIX and TIV, we note that VIX is relatively more informative about subsequent economic growth volatility and TIV about subsequent inflation volatility. Table 7 Predicting expected growth and inflation volatility from the lagged VIX and TIV A. Volatility of the SPF’s expected real GDP growth rate . . 2. VIX . 3. VIX . 4. VIX . 5. VIX . 6. TIV . 7. . 1. . base . WE diff. . WE total . oth. rec. . Base . . Model: . α1 . α2 . α1+α2 . α3 . α4 . R 2 . 1. 4.78 (3.40) – – – – 19.3% 2. – – – – 1.25 (3.17) 15.3% 3. 2.28 (1.57) 2.07 (1.83) 4.35 (3.48) 2.25 (0.87) – 24.2% 4. 2.61 (1.33) 2.16 (1.72) 4.77 (2.10) 2.46 (0.92) –0.16 (-0.33) 24.3% B. Volatility of the SPF’s expected CPI inflation rate 2. VIX 3. TIV 4. TIV 5. TIV 6. TIV 7. 1. base base WE diff. WE total oth. rec. Model: α1 α4 α5 α4+α5 α6 R2 1. 0.26 (0.97) – – – – 0.5% 2. – 0.24 (2.08) – – – 5.1% 3. – 0.02 (0.18) 0.08 (1.03) 0.10 (1.06) 0.38 (1.58) 16.6% 4. –0.66 (-1.02) 0.17 (0.95) 0.08 (1.07) 0.26 (1.34) 0.36 (1.51) 17.7% A. Volatility of the SPF’s expected real GDP growth rate . . 2. VIX . 3. VIX . 4. VIX . 5. VIX . 6. TIV . 7. . 1. . base . WE diff. . WE total . oth. rec. . Base . . Model: . α1 . α2 . α1+α2 . α3 . α4 . R 2 . 1. 4.78 (3.40) – – – – 19.3% 2. – – – – 1.25 (3.17) 15.3% 3. 2.28 (1.57) 2.07 (1.83) 4.35 (3.48) 2.25 (0.87) – 24.2% 4. 2.61 (1.33) 2.16 (1.72) 4.77 (2.10) 2.46 (0.92) –0.16 (-0.33) 24.3% B. Volatility of the SPF’s expected CPI inflation rate 2. VIX 3. TIV 4. TIV 5. TIV 6. TIV 7. 1. base base WE diff. WE total oth. rec. Model: α1 α4 α5 α4+α5 α6 R2 1. 0.26 (0.97) – – – – 0.5% 2. – 0.24 (2.08) – – – 5.1% 3. – 0.02 (0.18) 0.08 (1.03) 0.10 (1.06) 0.38 (1.58) 16.6% 4. –0.66 (-1.02) 0.17 (0.95) 0.08 (1.07) 0.26 (1.34) 0.36 (1.51) 17.7% This table reports how the VIX and TIV levels are related to the subsequent realized volatility of the SPF’s expected real GDP growth (panel A) and the subsequent realized volatility of the SPF’s CPI inflation growth (panel B). Panel A reports on the following regression: abs(ΔRGDPt−1,tGr1Q)=α0+(α1+α2 DtWE+α3 DtOthRec) VIXt−1+(α4+α5 DtWE+α6 DtOthRec) TIVt−1+ϵt−1,t, where abs(ΔRGDPt−1,tGr1Q) is the absolute quarter-to-quarter change in the SPF’s expected annualized real-GDP growth rate over the next quarter in annualized percentage units, as a realized volatility measure; DtWE is an indicator variable that equals one when in a Bai-Perron weaker economic (WE) segment; DtOthRec is an “Other Recession” indicator variable that equals one when in a recession that is not also a WE quarter, included so the α1 and α4 baselines include only expansionary quarters; VIXt−1 is the lagged 5-day moving average VIX level in decimal units at the close of the first trading day of the second month in calendar quarter t – 1 (to just precede the survey responses for quarter t – 1); TIVt−1 is the lagged 5-day moving average MOVE Treasury yield implied volatility metric divided by 100 with the timing aligned with VIX; the α’s are coefficients to be estimated; and ϵt−1,t is the residual. Panel B reports on the same regression, except the SPF’s expected CPI inflation over the next quarter replaces the expected growth term. We report on our featured 1997:10–2017:12 period. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 7 Predicting expected growth and inflation volatility from the lagged VIX and TIV A. Volatility of the SPF’s expected real GDP growth rate . . 2. VIX . 3. VIX . 4. VIX . 5. VIX . 6. TIV . 7. . 1. . base . WE diff. . WE total . oth. rec. . Base . . Model: . α1 . α2 . α1+α2 . α3 . α4 . R 2 . 1. 4.78 (3.40) – – – – 19.3% 2. – – – – 1.25 (3.17) 15.3% 3. 2.28 (1.57) 2.07 (1.83) 4.35 (3.48) 2.25 (0.87) – 24.2% 4. 2.61 (1.33) 2.16 (1.72) 4.77 (2.10) 2.46 (0.92) –0.16 (-0.33) 24.3% B. Volatility of the SPF’s expected CPI inflation rate 2. VIX 3. TIV 4. TIV 5. TIV 6. TIV 7. 1. base base WE diff. WE total oth. rec. Model: α1 α4 α5 α4+α5 α6 R2 1. 0.26 (0.97) – – – – 0.5% 2. – 0.24 (2.08) – – – 5.1% 3. – 0.02 (0.18) 0.08 (1.03) 0.10 (1.06) 0.38 (1.58) 16.6% 4. –0.66 (-1.02) 0.17 (0.95) 0.08 (1.07) 0.26 (1.34) 0.36 (1.51) 17.7% A. Volatility of the SPF’s expected real GDP growth rate . . 2. VIX . 3. VIX . 4. VIX . 5. VIX . 6. TIV . 7. . 1. . base . WE diff. . WE total . oth. rec. . Base . . Model: . α1 . α2 . α1+α2 . α3 . α4 . R 2 . 1. 4.78 (3.40) – – – – 19.3% 2. – – – – 1.25 (3.17) 15.3% 3. 2.28 (1.57) 2.07 (1.83) 4.35 (3.48) 2.25 (0.87) – 24.2% 4. 2.61 (1.33) 2.16 (1.72) 4.77 (2.10) 2.46 (0.92) –0.16 (-0.33) 24.3% B. Volatility of the SPF’s expected CPI inflation rate 2. VIX 3. TIV 4. TIV 5. TIV 6. TIV 7. 1. base base WE diff. WE total oth. rec. Model: α1 α4 α5 α4+α5 α6 R2 1. 0.26 (0.97) – – – – 0.5% 2. – 0.24 (2.08) – – – 5.1% 3. – 0.02 (0.18) 0.08 (1.03) 0.10 (1.06) 0.38 (1.58) 16.6% 4. –0.66 (-1.02) 0.17 (0.95) 0.08 (1.07) 0.26 (1.34) 0.36 (1.51) 17.7% This table reports how the VIX and TIV levels are related to the subsequent realized volatility of the SPF’s expected real GDP growth (panel A) and the subsequent realized volatility of the SPF’s CPI inflation growth (panel B). Panel A reports on the following regression: abs(ΔRGDPt−1,tGr1Q)=α0+(α1+α2 DtWE+α3 DtOthRec) VIXt−1+(α4+α5 DtWE+α6 DtOthRec) TIVt−1+ϵt−1,t, where abs(ΔRGDPt−1,tGr1Q) is the absolute quarter-to-quarter change in the SPF’s expected annualized real-GDP growth rate over the next quarter in annualized percentage units, as a realized volatility measure; DtWE is an indicator variable that equals one when in a Bai-Perron weaker economic (WE) segment; DtOthRec is an “Other Recession” indicator variable that equals one when in a recession that is not also a WE quarter, included so the α1 and α4 baselines include only expansionary quarters; VIXt−1 is the lagged 5-day moving average VIX level in decimal units at the close of the first trading day of the second month in calendar quarter t – 1 (to just precede the survey responses for quarter t – 1); TIVt−1 is the lagged 5-day moving average MOVE Treasury yield implied volatility metric divided by 100 with the timing aligned with VIX; the α’s are coefficients to be estimated; and ϵt−1,t is the residual. Panel B reports on the same regression, except the SPF’s expected CPI inflation over the next quarter replaces the expected growth term. We report on our featured 1997:10–2017:12 period. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab 5.2 VIX movements and concurrent changes in expected economic growth 5.2.1 Relating VIX movements to changes in expected economic growth. As discussed in Section 1.2.2, prior literature indicates a negative relation (or feedback) between economic uncertainty and future economic growth. Regarding understanding our main VIX-yield findings, this negative relation implies a possible consumption-smoothing influence through the uncertainty-to-growth feedback channel. If upward VIX movements are associated with relatively larger declines in expected economic growth over weaker economic times, this could help explain our primary findings for the state-contingent VIX-yield relation. Here, we investigate this issue, using the quarterly time series of the SPF’s “expected real-GDP growth rate.” We estimate the concurrent relation between changes in the expected economic growth rate and VIX changes, while allowing for a different relation over weaker economic (WE) times. Table 8 provides the specification and results, again using WE segments identified by our primary Bai-Perron method. Here, we use the 1-year-ahead expected real GDP growth, following from Bansal and Shaliastovich (2013). We find that VIX changes are negatively related to the changes in expected economic growth, with a much stronger relation over WE times (see Table 8, columns 3 and 4). Internet Appendix B.8.3 reports qualitatively similar findings for the alternative “anxious index/recession” method used to categorize WE times. These findings are supportive of the feedback consumption-smoothing channel proposed in Section 1.2.2. Results in Table 8 also indicate that the VIX changes are more negatively related to coincident changes in the expected real GDP growth rate, relative to TIV changes (compare rows 1 and 2). This finding helps distinguish between the economic information content of the VIX and the TIV, a topic discussed further in Section 6. Table 8 Relating changes in the “expected real GDP growth rate” to VIX movements . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔVIX . . . 1. . base . WE diff. . WE total . oth. rec. . 6. ΔTIV . . Model: . α1 . α2 . α1+α2 . α3 . α4 . 7. R2 . 1. –0.57 – – – – 26.2% (–3.94) 2. – – – – –0.46 11.9% (–2.59) 3. –0.18 –0.82 –1.01 –0.65 – 37.4% (–2.08) (–3.29) (–4.36) (–2.87) 4. –0.15 –0.82 –0.97 –0.62 –0.09 37.7% (–1.84) (–3.15) (–3.83) (–2.60) (–0.46) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔVIX . . . 1. . base . WE diff. . WE total . oth. rec. . 6. ΔTIV . . Model: . α1 . α2 . α1+α2 . α3 . α4 . 7. R2 . 1. –0.57 – – – – 26.2% (–3.94) 2. – – – – –0.46 11.9% (–2.59) 3. –0.18 –0.82 –1.01 –0.65 – 37.4% (–2.08) (–3.29) (–4.36) (–2.87) 4. –0.15 –0.82 –0.97 –0.62 –0.09 37.7% (–1.84) (–3.15) (–3.83) (–2.60) (–0.46) This table reports how a quarter’s change in the estimated real GDP growth moves with the concurrent change in log(VIX), while allowing for the relation to be different for weaker economic (WE) segments. We report on the following regression: ΔRGDPt−1,tGr4Q=α0+(α1+α2 DtWE+α3 DtOthRec)Δlog(VIXt−1,t)+α4Δlog(TIVt−1,t)+ϵt−1,t, where ΔRGDPt−1,tGr4Q is the change in the four-quarters-ahead (or 1-year) SPF survey’s expected annualized real-GDP growth rate; DtWE is an indicator variable that equals one when in a Bai-Perron WE segment; DtOthRec is an “Other Recession” indicator variable that equals one when in a recession that is not also a WE quarter, included so the α1 baseline includes only expansionary quarters; Δlog(VIXt−1,t) ( Δlog(TIVt−1,t) ) is the concurrent change in log(VIX) (log(TIV)); α’s are the coefficients to be estimated; and ϵt−1,t is the residual. To align with the survey timing, we measure the quarterly change in VIX (TIV) using the 5-day moving average VIX (5-day moving average TIV) from the first trading-day of the second month of successive calendar quarters. ΔRGDPt−1,tGr4Q refers to the difference between quarter t – 1’s and quarter t’s survey for the four-quarter-ahead growth estimates expected over the subsequent four quarters (over quarters t to t + 3 for the survey in quarter t – 1 and over quarters t + 1 to t + 4 for the survey in quarter t). The sample period is 1997:10–2017:12. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 8 Relating changes in the “expected real GDP growth rate” to VIX movements . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔVIX . . . 1. . base . WE diff. . WE total . oth. rec. . 6. ΔTIV . . Model: . α1 . α2 . α1+α2 . α3 . α4 . 7. R2 . 1. –0.57 – – – – 26.2% (–3.94) 2. – – – – –0.46 11.9% (–2.59) 3. –0.18 –0.82 –1.01 –0.65 – 37.4% (–2.08) (–3.29) (–4.36) (–2.87) 4. –0.15 –0.82 –0.97 –0.62 –0.09 37.7% (–1.84) (–3.15) (–3.83) (–2.60) (–0.46) . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔVIX . . . 1. . base . WE diff. . WE total . oth. rec. . 6. ΔTIV . . Model: . α1 . α2 . α1+α2 . α3 . α4 . 7. R2 . 1. –0.57 – – – – 26.2% (–3.94) 2. – – – – –0.46 11.9% (–2.59) 3. –0.18 –0.82 –1.01 –0.65 – 37.4% (–2.08) (–3.29) (–4.36) (–2.87) 4. –0.15 –0.82 –0.97 –0.62 –0.09 37.7% (–1.84) (–3.15) (–3.83) (–2.60) (–0.46) This table reports how a quarter’s change in the estimated real GDP growth moves with the concurrent change in log(VIX), while allowing for the relation to be different for weaker economic (WE) segments. We report on the following regression: ΔRGDPt−1,tGr4Q=α0+(α1+α2 DtWE+α3 DtOthRec)Δlog(VIXt−1,t)+α4Δlog(TIVt−1,t)+ϵt−1,t, where ΔRGDPt−1,tGr4Q is the change in the four-quarters-ahead (or 1-year) SPF survey’s expected annualized real-GDP growth rate; DtWE is an indicator variable that equals one when in a Bai-Perron WE segment; DtOthRec is an “Other Recession” indicator variable that equals one when in a recession that is not also a WE quarter, included so the α1 baseline includes only expansionary quarters; Δlog(VIXt−1,t) ( Δlog(TIVt−1,t) ) is the concurrent change in log(VIX) (log(TIV)); α’s are the coefficients to be estimated; and ϵt−1,t is the residual. To align with the survey timing, we measure the quarterly change in VIX (TIV) using the 5-day moving average VIX (5-day moving average TIV) from the first trading-day of the second month of successive calendar quarters. ΔRGDPt−1,tGr4Q refers to the difference between quarter t – 1’s and quarter t’s survey for the four-quarter-ahead growth estimates expected over the subsequent four quarters (over quarters t to t + 3 for the survey in quarter t – 1 and over quarters t + 1 to t + 4 for the survey in quarter t). The sample period is 1997:10–2017:12. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab 5.2.2 Yield changes, VIX changes, and changes in expected economic growth. The results in Section 5.2.1 are suggestive of a feedback channel, where VIX spikes predict lower subsequent real growth and an associated decline in yields because of a changing consumption-smoothing influence. Such an economic channel presumes a positive β1 in our Equation (2), which is generally implied by theory but has not always been supported by empirical estimates of the intertemporal elasticity of substitution (IES).17 In this section, we revisit the relation between yield changes and both VIX changes and TIV changes as in our Equation (3), but when also adding the coincident change in the SPF expected 1-year-ahead real GDP growth rate as an additional explanatory term ( ΔRGDPGr ). Our premise is that the ΔRGDPGr term should serve as a consumption-smoothing control variable that comoves positively with changes in expected consumption growth. If so, our estimation results should be informative about the sign of β1 (and the IES) over our sample. If the feedback consumption-smoothing channel from Section 1.2.2 is relevant, we would expect to find: (a) a positive coefficient for the ΔRGDPGr term (implying a positive β1 from our Equation (2)), and (b) a decline in the negative, partial, dynamic VIX-yield relation, as compared to the case that does not include the ΔRGDPGr explanatory term (implying that a portion of the partial, negative, dynamic VIX-yield relation is attributable to the feedback consumption-smoothing channel). Given the SPF’s quarterly data availability, our investigation here also expands our earlier investigation by assessing the dynamic VIX-yield relation for nonoverlapping quarterly changes. Table 9 provides the specification and estimation results. We focus on four different quarterly yield changes, both nominal and inflation-adjusted 5- and 2-year Treasury ZCB yields. We highlight three findings. Table 9 Treasury yields, VIX, and expected GDP growth 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . . Model . Base . WE diff. . WE total . Base . WE diff. . WE total . 8. ΔRGDP . . . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . ψ1 . 9. R2 . A. Nominal ΔZCB05Y 1. –0.19 –1.22 –1.41 –0.31 1.23 0.93 21.9% (–0.99) (–3.22) (–4.50) (–0.71) (2.39) (3.23) 2. –0.10 –0.90 –1.00 –0.13 0.90 0.77 0.40 26.3% (–0.60) (–2.24) (–2.93) (–0.30) (1.56) (2.65) (2.07) B. Nominal ΔZCB02Y 1. –0.08 –1.02 –1.11 –0.84 1.43 0.59 24.7% (–0.42) (–2.67) (–3.52) (–2.34) (3.45) (3.08) 2. –0.01 –0.78 –0.79 –0.71 1.18 0.47 0.31 27.2% (–0.08) (–2.02) (–2.47) (–1.79) (2.46) (2.55) (2.08) C. Inflation-adjusted ΔZCB05YIA 1. –0.02 –1.18 –1.19 –0.01 1.17 1.16 15.7% (–0.09) (–3.28) (–4.30) (–0.02) (2.23) (4.03) 2. 0.07 –0.88 –0.81 0.15 0.86 1.01 0.38 20.1% (0.34) (–2.02) (–2.37) (0.34) (1.55) (3.56) (1.72) D. Inflation-adjusted ΔZCB02YIA 1. 0.09 –0.98 –0.89 –0.54 1.36 0.82 14.5% (0.53) (–2.82) (–3.09) (–1.51) (3.30) (3.91) 2. 0.15 –0.76 –0.60 –0.42 1.13 0.71 0.29 17.1% (0.96) (–2.03) (–1.89) (–1.09) (2.51) (3.76) (1.94) 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . . Model . Base . WE diff. . WE total . Base . WE diff. . WE total . 8. ΔRGDP . . . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . ψ1 . 9. R2 . A. Nominal ΔZCB05Y 1. –0.19 –1.22 –1.41 –0.31 1.23 0.93 21.9% (–0.99) (–3.22) (–4.50) (–0.71) (2.39) (3.23) 2. –0.10 –0.90 –1.00 –0.13 0.90 0.77 0.40 26.3% (–0.60) (–2.24) (–2.93) (–0.30) (1.56) (2.65) (2.07) B. Nominal ΔZCB02Y 1. –0.08 –1.02 –1.11 –0.84 1.43 0.59 24.7% (–0.42) (–2.67) (–3.52) (–2.34) (3.45) (3.08) 2. –0.01 –0.78 –0.79 –0.71 1.18 0.47 0.31 27.2% (–0.08) (–2.02) (–2.47) (–1.79) (2.46) (2.55) (2.08) C. Inflation-adjusted ΔZCB05YIA 1. –0.02 –1.18 –1.19 –0.01 1.17 1.16 15.7% (–0.09) (–3.28) (–4.30) (–0.02) (2.23) (4.03) 2. 0.07 –0.88 –0.81 0.15 0.86 1.01 0.38 20.1% (0.34) (–2.02) (–2.37) (0.34) (1.55) (3.56) (1.72) D. Inflation-adjusted ΔZCB02YIA 1. 0.09 –0.98 –0.89 –0.54 1.36 0.82 14.5% (0.53) (–2.82) (–3.09) (–1.51) (3.30) (3.91) 2. 0.15 –0.76 –0.60 –0.42 1.13 0.71 0.29 17.1% (0.96) (–2.03) (–1.89) (–1.09) (2.51) (3.76) (1.94) This table reports how Treasury yields move differently with VIX changes over weaker economic (WE) times at the quarterly change horizon, when also controlling for TIV changes and changes in the expected real GDP growth. Panels A and B report on the following regression for nonoverlapping quarterly changes: ΔZCBnYt−1,t=α0+(λ1+λ2DtWE)Δlog(VIXt−1,t)+(γ1+γ2DtWE)Δlog(TIVt−1,t)+ψ1ΔRGDPt−1,tGr4Q+ϵt−1,t, where ΔZCBnY indicates the change in the Treasury nominal n-year ZCB yield over quarters t – 1 to t, n is either 5 or 2 years as indicated in the panel heading; ΔRGDPt−1,tGr4Q is the quarterly change in the 1-year-ahead expected real GDP growth from the SPF; the α, λs, γs, and ψ1 are coefficients to be estimated; and the other terms are as defined in Table 1. Here, to precisely match the yield changes to the VIX and TIV changes and align approximately to the survey, the quarterly changes use the yield, VIX, and TIV from the close of the first trading-day of the second month of calendar quarters. Panels C and D report on the same model, except with inflation-adjusted ZCB yields; the inflation-adjustment is the nominal yield less the expected 1-year-ahead CPI inflation from the SPF survey. The WE segments are as identified by our primary Bai-Perron-method. The sample period is 1997:10–2017:12. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab Table 9 Treasury yields, VIX, and expected GDP growth 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . . Model . Base . WE diff. . WE total . Base . WE diff. . WE total . 8. ΔRGDP . . . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . ψ1 . 9. R2 . A. Nominal ΔZCB05Y 1. –0.19 –1.22 –1.41 –0.31 1.23 0.93 21.9% (–0.99) (–3.22) (–4.50) (–0.71) (2.39) (3.23) 2. –0.10 –0.90 –1.00 –0.13 0.90 0.77 0.40 26.3% (–0.60) (–2.24) (–2.93) (–0.30) (1.56) (2.65) (2.07) B. Nominal ΔZCB02Y 1. –0.08 –1.02 –1.11 –0.84 1.43 0.59 24.7% (–0.42) (–2.67) (–3.52) (–2.34) (3.45) (3.08) 2. –0.01 –0.78 –0.79 –0.71 1.18 0.47 0.31 27.2% (–0.08) (–2.02) (–2.47) (–1.79) (2.46) (2.55) (2.08) C. Inflation-adjusted ΔZCB05YIA 1. –0.02 –1.18 –1.19 –0.01 1.17 1.16 15.7% (–0.09) (–3.28) (–4.30) (–0.02) (2.23) (4.03) 2. 0.07 –0.88 –0.81 0.15 0.86 1.01 0.38 20.1% (0.34) (–2.02) (–2.37) (0.34) (1.55) (3.56) (1.72) D. Inflation-adjusted ΔZCB02YIA 1. 0.09 –0.98 –0.89 –0.54 1.36 0.82 14.5% (0.53) (–2.82) (–3.09) (–1.51) (3.30) (3.91) 2. 0.15 –0.76 –0.60 –0.42 1.13 0.71 0.29 17.1% (0.96) (–2.03) (–1.89) (–1.09) (2.51) (3.76) (1.94) 1. . 2. ΔVIX . 3. ΔVIX . 4. ΔVIX . 5. ΔTIV . 6. ΔTIV . 7. ΔTIV . . . Model . Base . WE diff. . WE total . Base . WE diff. . WE total . 8. ΔRGDP . . . λ1 . λ2 . ( λ1+λ2 ) . γ1 . γ2 . ( γ1+γ2 ) . ψ1 . 9. R2 . A. Nominal ΔZCB05Y 1. –0.19 –1.22 –1.41 –0.31 1.23 0.93 21.9% (–0.99) (–3.22) (–4.50) (–0.71) (2.39) (3.23) 2. –0.10 –0.90 –1.00 –0.13 0.90 0.77 0.40 26.3% (–0.60) (–2.24) (–2.93) (–0.30) (1.56) (2.65) (2.07) B. Nominal ΔZCB02Y 1. –0.08 –1.02 –1.11 –0.84 1.43 0.59 24.7% (–0.42) (–2.67) (–3.52) (–2.34) (3.45) (3.08) 2. –0.01 –0.78 –0.79 –0.71 1.18 0.47 0.31 27.2% (–0.08) (–2.02) (–2.47) (–1.79) (2.46) (2.55) (2.08) C. Inflation-adjusted ΔZCB05YIA 1. –0.02 –1.18 –1.19 –0.01 1.17 1.16 15.7% (–0.09) (–3.28) (–4.30) (–0.02) (2.23) (4.03) 2. 0.07 –0.88 –0.81 0.15 0.86 1.01 0.38 20.1% (0.34) (–2.02) (–2.37) (0.34) (1.55) (3.56) (1.72) D. Inflation-adjusted ΔZCB02YIA 1. 0.09 –0.98 –0.89 –0.54 1.36 0.82 14.5% (0.53) (–2.82) (–3.09) (–1.51) (3.30) (3.91) 2. 0.15 –0.76 –0.60 –0.42 1.13 0.71 0.29 17.1% (0.96) (–2.03) (–1.89) (–1.09) (2.51) (3.76) (1.94) This table reports how Treasury yields move differently with VIX changes over weaker economic (WE) times at the quarterly change horizon, when also controlling for TIV changes and changes in the expected real GDP growth. Panels A and B report on the following regression for nonoverlapping quarterly changes: ΔZCBnYt−1,t=α0+(λ1+λ2DtWE)Δlog(VIXt−1,t)+(γ1+γ2DtWE)Δlog(TIVt−1,t)+ψ1ΔRGDPt−1,tGr4Q+ϵt−1,t, where ΔZCBnY indicates the change in the Treasury nominal n-year ZCB yield over quarters t – 1 to t, n is either 5 or 2 years as indicated in the panel heading; ΔRGDPt−1,tGr4Q is the quarterly change in the 1-year-ahead expected real GDP growth from the SPF; the α, λs, γs, and ψ1 are coefficients to be estimated; and the other terms are as defined in Table 1. Here, to precisely match the yield changes to the VIX and TIV changes and align approximately to the survey, the quarterly changes use the yield, VIX, and TIV from the close of the first trading-day of the second month of calendar quarters. Panels C and D report on the same model, except with inflation-adjusted ZCB yields; the inflation-adjustment is the nominal yield less the expected 1-year-ahead CPI inflation from the SPF survey. The WE segments are as identified by our primary Bai-Perron-method. The sample period is 1997:10–2017:12. t-statistics, calculated with heteroscedastic and autocorrelation consistent standard errors, appear in parentheses. Open in new tab First, for all four yield changes, we find that our primary VIX-yield dynamic findings (as in Section 3) are also evident with the nonoverlapping quarterly changes; the partial negative VIX-yield relation is stronger over weaker economic times. See the regressions in row 1 of Table 9 and note the coefficient estimates in columns 3 and 4.18 Second, when adding the ΔRGDPGr explanatory term to Equation (3), we find that the estimated ψ1 coefficient for the ΔRGDPGr term is positive and statistically significant for all four yield changes, consistently estimated in the 0.3 to 0.4 range. If we interpret our ψ1 estimates as β1 values in our Equation (2) and under the common interpretation that β1 can be viewed as the IES inverse, then our results imply an IES roughly in the 2.5 to 3 range. This seems plausible, given the IES estimate of 2.2 in Bansal, Kiku, and Yaron (2016). Third, we find that the partial negative VIX-yield relations in weaker economic (WE) times (the relations in column 4) decline by about 30% when adding the ΔRGDPGr explanatory term; compare the coefficient from column 4 in the regressions from row 2 to that in the regressions from row 1. For example, for the 5-year nominal yield, the WE VIX-yield relation falls to -1.00 in row 2 from -1.41 in row 1, a decline of 29%. Internet Appendix B.8.4 reports on this evaluation with the alternative “anxious index/recession” method for categorizing weaker economic (WE) times. We find generally qualitatively similar results, but the WE VIX-yield relation decline is more appreciable in this case when adding the ΔRGDPGr term. In sum, our findings align with the empirical predictions from a feedback consumption-smoothing channel, as laid out in the beginning of this subsection. Our collective findings in Tables 8 and 9 suggest that the feedback consumption-smoothing channel (the β1 term in our Equation (2)) is a material factor in understanding our primary VIX-yield findings. However, a sizable component of the dynamic VIX-yield relation remains to be explained, presumably attributed to other channels. 5.3 Weaker economic segments and risk aversion Section 1.2.3 suggests that higher risk aversion over weaker economic (WE) times might also be a contributor to our main VIX-yield findings. In our Equation (2), a higher risk aversion over WE times also suggests a more negative β2,t coefficient over WE times, which could act to amplify the negative precautionary savings influence on interest rates. Secondarily, this could also suggest a more positive β1,t coefficient over WE times, which could act to amplify the feedback consumption-smoothing influence. Here, appealing to related literature, we discuss evidence that suggests higher risk aversion over our Bai-Perron WE segments. First, recall that the equity variance risk premium (VRP) has been proposed as a measure that moves positively with risk aversion; see Section 4. Using the VRP estimates from Bekaert and Hoerova’s (2014) favored model (8), we find that the average VRP is reliably higher over our Bai-Perron weaker economic (WE) segments. Over our 1997:10–2017:12 period, the VRP average over the WE segments is about 80% higher than the VRP average over the baseline segments. Over the earlier 1990:01–1997:09 period, the average VRP for the WE segment is about 39% higher than the average VRP over the expansionary baseline segment from March 10, 1993, to September 30, 1997. These WE VRP differences are statistically significant at 1% and 10% p-values, respectively. Second, when the stock market has suffered an appreciable decline coincident with an economic recession, then one would expect a higher marginal utility of consumption in the aggregate, with an associated relatively lower surplus consumption ratio and higher risk aversion.19 Thus, we analyze stock market performance around our Bai-Perron weaker economic (WE) segments. The stock market declined by: (1) about -20% over July 16, 1990, to October 11, 1990, prior to the onset of our 1991–1993 WE segment; (2) about -30% prior to the onset of our 2001–2004 WE segment, with a prior peak on March 24, 2000; and (3) about -43% prior to the onset of our 2009–2012 WE segment, with a prior peak on October 9, 2007. Accordingly, heightened risk aversion over our WE segments plausibly could have a role in accounting for our VIX-yield dynamic findings. 5.4 Possible monetary policy influence Monetary policy implementation could help explain our main VIX-yield findings if (1) both VIX and Treasury yields were responding to policy actions or (2) policy actions were responsive to VIX movements and then the policy actions influenced bond yields.20 Here, we briefly discuss evidence inconsistent with this channel being a material contributor to our primary VIX-yield dynamic findings. First, we find that our primary VIX-yield dynamic findings remain reliably evident when estimated: (1) over reduced samples that exclude observations that have changes in the targeted federal funds rate (TFFR); and (2) over the December 2008 to December 2015 subperiod, which had no TFFR changes. Second, during our 2009:01 to 2012:04 Bai-Perron weaker economic segment, the Federal Reserve undertook unprecedented asset purchases in a quantitative-easing (QE) effort. However, the QE purchases are largely consistent week-to-week and unrelated to VIX movements, so QE behavior seems incapable of explaining our primary VIX-yield findings. See Internet Appendix B.9 for details. 6. Role of Treasury-Bond Implied Yield Volatility in our Analysis Our main empirical analysis in Section 3 estimated specifications with the change in both the equity implied volatility ( ΔVIX ) and Treasury-bond implied yield volatility ( ΔTIV ) as explanatory variables. In this section, we discuss the TIV-yield results and their implications. This discussion also includes the possible role of inflation uncertainty in our results, an important consideration since we evaluate nominal yields out to the 10-year maturity. Cieslak and Povala (CP, 2016) show that the “constant yield volatility” assumption commonly made in the term structure literature is not valid because yield volatility exhibits substantial time variation. Our Figure 2 depicts the appreciable time variation in the implied yield volatility. In Internet Appendix B.10, we show that our featured TIV measure (the 1-month MOVE series) contains appreciable forward-looking information for the subsequent yield volatility. CP (2016) and Collin-Dufresne and Goldstein (2002) present evidence of generally little comovement between yield levels and yield volatility shocks. However, our main results indicate a strongly positive partial TIV-yield relation over weaker economic times in our featured 1997:10–2017:12 period and a generally positive partial TIV-yield relation over the earlier 1990:01–1997:09 period (see Tables 1 to 4). Thus, our TIV-yield results show that shocks to the yield volatility have a strong positive partial relation to yield movements in some economic states, at least for weekly to monthly change horizons over 1990–2017. What could explain this positive partial relation between TIV movements and the yield changes? Bansal and Shaliastovich (2013) develop a promising framework for explaining our TIV-yield results. Building on the long-run risks framework of Bansal and Yaron (2004), they find that higher inflation uncertainty raises nominal bond premiums and increases the realized term premium. Conversely, higher economic uncertainty lowers nominal bond premiums and the realized term premium. Their analysis evaluates bond returns and term premiums using quarterly data and long-run forecasting regressions. In the framework of Bansal and Shaliastovich (2013) and others, changes in economic uncertainty can contribute to yield volatility. This view suggests that both VIX and TIV should positively comove with economic uncertainty. Here, we review our evidence that indicates VIX and TIV movements do share a modest commonality. First, the simple correlations between our featured Δlog(VIX) and Δlog(TIV) metrics are modest at (1) +0.39 (+0.35) for the monthly (weekly) changes over our featured 1997:10 to 2017:12 period, and (2) +0.41 (+0.36) for the monthly (weekly) changes over the earlier 1990:01–1997:09 period. Second, our results in Sections 5.1 and 5.2 indicate that the TIV level is positively related to future economic growth volatility and that TIV changes are negatively related to concurrent changes in expected economic growth.21 However, comparatively speaking, our results indicate that (1) the TIV (VIX) is more predictive about the future volatility of expected inflation (expected real GDP growth), relative to VIX (TIV), and (2) ΔVIX is more related to coincident changes in expected real GDP growth than is ΔTIV . These observations suggest that including ΔTIV can be important for identifying the partial, dynamic VIX-yield relation, by helping to mitigate omitted variable bias. For our interest in weekly to quarterly yield changes, the Bansal-Shaliastovich framework suggests the following interpretation. Movements in TIV, as an option-derived implied volatility, should capture the change in total expected yield volatility, including the components attributed to time-varying economic uncertainty and time-varying inflation uncertainty. VIX movements should be more aligned with time-varying economic uncertainty. Thus, the component of the TIV movement that is distinct from the VIX movement should be more related to inflation uncertainty (rather than economic uncertainty). In this way, the positive partial relation between TIV movements and nominal yields, along with the negative partial relation between VIX movements and nominal yields, is consistent with Bansal-Shaliastovich’s conclusions that inflation uncertainty and economic uncertainty affect nominal yields differently. This interpretation of the positive partial, dynamic relation between TIV and the nominal yields implies that (1) the simple, dynamic TIV-yield relation, when excluding the ΔVIX explanatory term, should be smaller in magnitude (less positive) than the partial TIV-yield relations when including the ΔVIX explanatory term; and (2) the positive, partial, dynamic TIV-yield relation should be increasing with a bond’s maturity, because the influence of inflation uncertainty on nominal bond yields should rise with a bond’s maturity.22 This is generally what we find. An exception is that there is little difference between the simple TIV-yield relations and the partial TIV-yield relations for the earlier 1990:01–1997:09 period. See Internet Appendix B.11 for details. At first glance, the positive TIV-yield relations for the inflation-adjusted yields in Table 3 appear to be inconsistent with the prediction for real yields in Bansal and Shaliastovich (2013). In their framework with inflation non-neutrality, inflation volatility can also induce a “flight-to-quality” effect in real yields, suggesting a decline in real yields with increasing inflation volatility. Thus, since TIV should comove positively with inflation volatility, their framework suggests a partial negative relation between TIV changes and real-yield changes. While an exhaustive investigation into this issue is beyond the scope of our paper, we make the following observations. First, our inflation-adjusted yields in Table 3 are only approximations of real yields. The nominal yield is commonly represented as the real yield plus expected inflation plus an inflation risk premium (Haubrich, Pennacchi, and Ritchken 2012). Since our inflation-adjusted yields equal the nominal yield minus the expected inflation, this means that our inflation-adjusted yields should still have an embedded inflation-risk premium. Since the inflation risk-premium should increase with a bond’s maturity, this suggests that the partial TIV-yield relation also might be increasing with maturity for our inflation-adjusted yields (as we find). Relatedly, over our featured 1997:10–2017:12 period, we note that the unconditional, dynamic TIV-yield relations are only meaningfully positive for the longer 5- and 10-year yields (Table 3, panels A and B, rows 1 to 3). Second, Bansal-Shaliastovich’s framework does not model bond risks related to time-varying illiquidity risk or uncertainty about potential Federal Reserve action (which might be especially influential in weaker economic times). Thus, in sum, we feel that our TIV-yield results for the inflation-adjusted yields in Table 3 are not necessarily a meaningful deviation from the Bansal-Shaliastovich real-yield prediction. 7. Implications of our Findings for Other Related Literature Next, we discuss the implications of our findings in the context of the macro-finance and asset pricing literature more broadly. Our focus is on the determinants of risk-free interest rates and the “information in” or the “signal from” the short-horizon VIX changes. For brevity, we limit our discussion here to representative papers that seem particularly relevant to our findings. First, evaluating movements in yield levels and economic uncertainty going back to 1871, Hartzmark (2016) concludes that higher economic uncertainty is associated with lower interest rates, as suggested by a precautionary savings motive. Consistently, under the view that economic uncertainty is higher during recessions, Ang, Bekaert, and Wei (2008) find a procyclical real rate in their analysis of quarterly yield and inflation data over 1952 to 2004. Our dynamic findings with weekly and monthly changes complement this evidence, by also indicating an important negative precautionary saving influence on interest rates that is amplified over weaker economic times. Second, David and Veronesi (2013) consider an economy where the growth rates of real earnings, consumption, and inflation follow a regime-switching process. Economic-state uncertainty is present because the economic regime is not observable; rather, agents learn about the economic state in a Bayesian-updating sense by observing real growth, inflation, and other signals. In their framework, news affects beliefs about the underlying economic regime more substantially under high economic-state uncertainty. This intuition suggests that the signaling intensity from VIX movements might be stronger in weaker economic (WE) times under heightened economic-state uncertainty. Further, their figure 4 indicates that low-growth, low-inflation regimes were more likely around the onset of our 2001 and 2009 Bai-Perron WE segments. These are times when longer-term bonds would be more attractive as hedging instruments, fitting with our VIX-yield findings for the 5- and 10-year bonds over these WE segments. Third, aspects of our findings are consistent with the literature on long-run risks (Bansal and Yaron 2004; Bansal and Shaliastovich 2013; Bansal et al. 2014). Consistent with uncertainty affecting expected economic growth, we find a negative relation between VIX changes and survey-based estimates of expected economic growth that is stronger over weaker economic (WE) times. This uncertainty-growth dynamic suggests that VIX increases can also negatively affect the consumption-smoothing influence on interest rates, especially in WE times. In Section 6, we appealed to Bansal and Shaliastovich (2013) for understanding our findings of both a negative, partial, dynamic VIX-yield relation and a positive, partial, dynamic TIV-yield relation. For nominal yields, their framework indicates economic uncertainty should be negatively related to yields and inflation uncertainty should be positively related to yields. Fourth, Bekaert and Engstrom (2017) adopt the habit preference structure of Campbell and Cochrane (1999), but add the innovation of modeling consumption growth with both bad and good environment state variables. Their bad environments feature a lower consumption surplus ratio, higher risk aversion, and both a greater conditional consumption-growth volatility and a more negative skewness in the consumption-growth distribution. Their bad environments also feature a stronger precautionary savings motive, with risk-free rates that are depressed below what would be predicted by the usual precautionary savings effects attributed to volatility. Figure 1 in Bekaert and Engstrom (2017) plots the time series of their estimated consumption-growth skewness. Their estimated skewness is particularly negative later in recessions and tends to remain more negative for months; this timing quite closely coincides with our Bai-Perron weaker economic (WE) segments. An elevated precautionary savings motive during our WE segments fits with our state-contingent VIX-yield dynamic findings. However, Xu’s (2020) analysis indicates that the Bekaert-Engstrom framework cannot generate the observed time variation in the risk-free rate over 1959–2014. Such a finding casts doubt on the framework’s ability to explain the appreciable time variation in interest rates without further modification. Fifth, using a jump-diffusion dynamic model with Epstein-Zin (1989) preferences, Drechsler (2013) studies the impact of uncertainty about the underlying economic model. In his framework, greater model uncertainty both decreases the expected consumption growth and increases the precautionary savings motive due to higher expected growth volatility, imparting a negative influence on interest rates.23 He uses dispersion in the forecasted economic growth as a measure of model uncertainty. Figure 1 in his paper shows that this “forecasted growth dispersion” has local peaks near the onset of each of our Bai-Perron weaker economic (WE) segments; the dispersion remains elevated well into our WE segments. Thus, his evidence also indicates our Bai-Perron WE segments reflect “more uncertain economic times,” but this time referring to model uncertainty. If VIX increases are especially associated with increasing economic model uncertainty over such WE times, then our main VIX-yield dynamic findings are consistent with Drechsler’s intuition. Sixth, research on portfolio rebalancing and the stock-bond return relation has emphasized the cross-asset-class spillovers of volatility risk, suggestive of an “equity risk to bond pricing” avenue. See, for example, Fleming, Kirby, and Ostdiek (1998), Kodres and Pritzker (2002), Connolly, Stivers, and Sun (2005), Underwood (2009), and Bansal, Connolly, and Stivers (2014). However, none of these papers features a state-contingent uncertainty-yield dynamic comparable to ours. 8. Conclusions We study economic-state variation in uncertainty-yield dynamics over the VIX availability period since 1990. Our primary finding is uncovering an appreciably more negative, dynamic relation between economic uncertainty and Treasury yields over weaker economic (WE) times. In contrast to prior studies that relate the level of interest rates to the level of economic uncertainty over long periods, we analyze how changes in uncertainty are related to changes in interest rates over weekly to monthly change horizons. Our results imply an important negative influence of economic uncertainty on interest rates that is especially pronounced in WE times. Our principal results are evident in both nominal and real-yield proxies over the 1- to 10-year maturities in the term structure. While our focus is on uncertainty changes as measured by VIX movements, our results are robust to eight different methods for identifying weaker economic times. Further, we find consistent results for different measures of economic uncertainty (in lieu of the VIX), and for different asset pricing dynamics (in addition to Treasury yield dynamics). We also investigate likely economic channels behind these uncertainty-yield dynamics, especially relying on survey-based expected economic growth and inflation. Our findings suggest that a given upward VIX movement is associated with a relatively stronger decline in interest rates during weaker economic (WE) times because during such times the VIX increase signals: (1) a relatively larger increase in economic uncertainty, implying a stronger negative precautionary savings influence on interest rates; and/or (2) a relatively larger decline in expected economic growth, implying a more negative impact on the normally positive consumption-smoothing influence on interest rates. We also present evidence that risk aversion is higher during our WE segments, and higher risk aversion is likely to magnify these VIX-related influences on interest rates. Secondarily, we also find a generally positive, partial, dynamic relation between yield uncertainty (TIV) and Treasury yields, especially for longer-maturity bonds. The Bansal and Shaliastovich (2013) framework offers a potential explanation, whereby (1) economic growth uncertainty (more aligned with VIX) is negatively related to nominal yields, and (2) inflation uncertainty (more aligned with TIV) is positively related to nominal yields. However, while aspects of our findings are consistent with key intuition from the literature that we discuss in Sections 1, 5, 6, and 7, we acknowledge that our collective dynamic results are not a direct and unambiguous prediction of any one specific existing theoretical framework. Thus, our new stylized facts on the short-horizon dynamic relation between economic uncertainty and Treasury yields pose a challenge to future theoretical work. Professor David Dubofsky retired from the University of Louisville in 2019. We thank Ric Colacito, Connor Curry, Loan Dang, Madhu Kalimipalli, Cami Kuhnen, Ahn Le, Topaz Prawito, and Cipriana Prepeliuc and seminar participants at UNC-Chapel Hill, the University of Louisville, and the Financial Management Association meetings for helpful comments. We thank Marie Hoerova and Nancy Xu for making the data from their studies available, Bekaert and Hoerova (2014), and Bekaert, Engstrom, and Xu (2019). Finally, we thank Nikolai Roussanov (the editor) and two anonymous referees for their comments and advice that substantially improved our paper. Footnotes 1 By “partial relation between VIX changes and yield changes,” we refer to the relation in a multivariate framework that also includes changes in the Treasury-bond implied yield volatility as an explanatory term for yield changes. 2 Prior literature also suggests this late 1990s shift. First, results in David and Veronesi (2013) and Campbell, Sunderam, and Viceira (2017) indicate a heightened hedging role for Treasury bonds since the late 1990s. Second, Song (2017) suggests the U.S. economy shifted to a regime with procyclical inflation and an active monetary policy in the late 1990s, a regime where nominal bonds are hedges and safe. Third, Baele, Bekaert, and Inghelbrecht (2010) show that the stock-bond return correlation shifts from sizably positive to predominantly negative in late 1997. Fourth, Campbell, Pflueger, and Viceira (2020) demonstrate the usefulness of estimating their macroeconomic model separately over pre- and post-2000 periods. 3 See our discussion about David and Veronesi (2013), Bekaert and Engstrom (2017), and Drechsler (2013) in Section 7. 4 See, for example, Bollerslev, Tauchen, and Zhou (2009), Bekaert, Engstrom, and Xing (2009), Bollerslev, Gibson, and Zhou (2011), Bekaert and Hoerova (2014), and Bekaert and Engstrom (2017). 5 Hartzmark (2016) offers a similar model-free approach in his study on economic uncertainty and interest rates. 6 For example, the long-run risks literature (originating with Bansal and Yaron 2004) features the preference structure of Epstein and Zin (1989); their structure separates the intertemporal elasticity of substitution and risk aversion. This literature also suggests a feedback influence, whereby an increase in economic uncertainty suggests lower subsequent economic growth. 7 See, for example, David and Veronesi (2013), Campbell, Sunderam, and Viceira (2017), Song (2017), and Baele, Bekaert, and Inghelbrecht (2010). Further, practitioners tend to use the 10-year Treasury yield as a risk-free rate when applying the capital asset pricing model (CAPM) (see Brotherson et al. 2013). 8 The unconditional simple and partial VIX-yield relations are also negative over the out-of-sample 2018:01–2020:06 period that we report in Section 3.3.4. 9 These Bai-Perron segments commence later in a recession, after an economic contraction and the realization of a substantial stock market decline. Thus, under the consumption-surplus habit perspective of Campbell and Cochrane (1999), the marginal utility of consumption and risk aversion already should be elevated at the onset of each Bai-Perron weaker economy segment. See Sections 1.2.3 and 5.3 for a discussion and related evidence. 10 Our shorter 1990:01–1997:09 period contains only one WE segment and only one meaningful baseline segment. Accordingly, we do not perform an estimation comparable to Table 2 for the earlier period. 11 Since the expected inflation from Haubrich, Pennacchi, and Ritchken (2012) relies on market data from the first trading-day of the month, the changes here refer to changes over the first trading days of successive months. 12 An exception is the 1-year inflation-adjusted yield over 1990:01–1997:09, where there is little difference in the λ2s for the nominal and inflation-adjusted 1-year yields. 13 With eight different WE classification methods, we only present tabular results for the 5-year maturity to represent our results. 14 The model variation with the CV and VRP standard deviation functional form always has the highest R2 value in our setting. Further, with the greater excess kurtosis of the variance functional form, the coefficient estimates with the variance-based CV and VRP heavily rely on extreme observations. Regardless, our conclusions are consistent for both the standard deviation and the variance functional forms. 15 The PVS variable in Pflueger, Siriwardane, and Sunderam (2020) is equal to the difference in the average book-to-market equity ratio for the least-volatile stocks (the bottom quintile) and the most-volatile stocks (the top quintile), and the authors provide strong evidence that PVS reflects the market’s risk appetite. Since the book value of equity is only observable quarterly, we rely on volatility-based long/short equity returns to measure changes in the PVS over our rolling weekly and monthly horizons. 16 Here and in Section 5.2.1, we use a 5-weekday moving average of closing VIX and TIV values as our “VIX and TIV observations” (rather than the closing values from a single weekday). We choose this because of the timing ambiguity of the information set that survey respondents use when forming their response and to smooth out some of the day-to-day variability in the implied volatilities for this quarterly investigation. The results are qualitatively quite similar when using the closing implied volatility from a single day, but marginally less precise in some cases. 17 The mean reported IES estimate in a survey of over 2,700 estimates by Havranek et al. (2015) was 0.5 with substantial variation across different countries and methods. Bansal, Kiku, and Yaron (2012) conclude that the IES is likely larger than the one based on asset market data. In an empirical long-run risks model that accounts for time aggregation, Bansal, Kiku, and Yaron (2016) report an IES estimate of 2.2 with a standard error of just 0.2. 18 In Internet Appendix B.8.4, we also show that the weaker economic-state contingency in the VIX-yield relation is evident in a simpler specification, where ΔVIX is the only explanatory term. 19 This premise follows from the Campbell-Cochrane (1999) utility framework and from evidence in both Guiso, Sapienza, and Zingales (2018) and Cohn et al. (2015). 20 Krishnamurthy and Vissing-Jorgensen (2011) and Jarrow and Li (2014) present evidence that the Fed’s quantitative easing response to the 2008–2009 recession reduced interest rates overall. Piazzesi (2005) points out that bond yields respond to policy decisions by the Federal Reserve and vice versa. Yet, Fama (2013) argues that the evidence is generally inconclusive about the role of the Fed versus market forces in interest rate behavior. 21 These TIV findings are consistent with related evidence in Choi, Mueller, and Vedolin (2017) and Cremers, Fleckenstein, and Gandhi (2020). 22 In addition to Bansal and Shaliastovich (2013), evidence in Ang, Bekaert, and Wei (2008) and Haubrich, Pennacchi, and Ritchken (2012) indicate that the inflation risk premium in nominal bond yields is increasing with a bond’s maturity. 23 Hansen and Sargent (2010) show how model uncertainty can induce time-varying prices of risk, in a manner different from habit utility (Campbell and Cochrane 1999) or long-run risks (Bansal and Yaron 2004). A. Appendix This appendix provides details on the variables and data sets that serve a secondary supporting role in our analysis. The Treasury yields and equity implied volatility that compose our primary data are described in Sections 1 and 2. The Merrill Lynch MOVE series on the Option-Derived Implied Volatility of Treasury Yields The Merrill Lynch Option Volatility Estimate (MOVE) Index is an index of normalized implied volatilities from Treasury options on the 2-, 5-, 10-, and 30-year Treasury futures contracts. To match the VIX’s 1-month horizon, we use the MOVE series with a 1-month horizon also. The MOVE is normalized to have a mean value of 100. As such, the MOVE value does not directly give an implied standard deviation for any single underlying Treasury series, but the MOVE provides forward-looking information about the subsequent yield volatility. In our main text, we refer to the MOVE as the Treasury-bond implied volatility, abbreviated as TIV. Figure 2 depicts the appreciable time variation in the MOVE. In Internet Appendix B.10, we show that MOVE contains reliable and substantial information about the subsequent yield volatility of the Treasuries that we examine. The Equity Market ’s Volatility Risk Premium (VRP) and Conditional Volatility (CV) In Section 4, we consider the well-known VIX decomposition into a conditional volatility (CV) component and an equity “volatility risk premium” (VRP) component. There, our CV and VRP are based on the conditional volatility and VRP from Bekaert and Hoerova’s (2014) favored model 8, as kindly provided by Marie Hoerova. In their approach, a forecasting model estimates a projected forward-looking realized physical variance, using high-frequency 5-minute returns to measure realized volatility. Their VRP is the difference between VIX2 and the forecasted physical variance over the next rolling month. As explained in Section 4.1.1, Table 5 show results for the standard deviation functional form of CV and a VRP, defined as VIX minus the standard deviation CV. Inflation Expectations To calculate inflation-adjusted yields as real-yield proxies in Section 3.1.2, we subtract the annualized expected inflation over the life of the bond from the nominal yield of the Treasury zero coupon bond. We use the inflation expectations from the Haubrich, Pennacchi, and Ritchken (2012) model, as made available by the Cleveland Federal Reserve Bank. This inflation-expectations data are available for the subsequent 1-, 2-, 5-, and 10-year horizons (among others), matching our yield maturities. The Targeted Federal Funds Rate (TFFR) We obtain the time series of the targeted Federal Funds rate from the Federal Reserve at https://www.federalreserve.gov/monetarypolicy/openmarket.htm. Survey Data from the Survey of Professional Forecasters For our analysis in Sections 3.2, 5.1, and 5.2, we use the median survey data from the Survey of Professional Forecasters to measure: (1) the expected real GDP growth rate, (2) the likelihood of a real GDP decline next quarter, and (3) the expected 1-year-ahead CPI inflation. The “estimated likelihood of a real GDP decline next quarter” is also referred to as the anxious index (AI). These data are available quarterly, with survey responses from early in the second month of each calendar quarter. These data are available from the Philadelphia Federal Reserve Bank. The Risk Aversion Index and Economic Uncertainty Index of Bekaert, Engstrom, and Xu (2019) We also evaluate both the risk aversion (RA) index and economic uncertainty (EU) index from Bekaert, Engstrom, and Xu (2019), as kindly made available by Nancy Xu. While these index values are available daily, we evaluate rolling 5-weekday averages to remove some of the day-to-day noise. If the 5-weekday rolling average of the index over weekdays t – 10 to t – 6 is above its 75th percentile, then the yield changes over weekday t – 5 to t are considered to be in a weaker economic environment. In Section 3.2, the RA index is used for the seventh alternative weaker economic method and the EU index for the eighth. Daily Economic Policy Uncertainty (EPU) Index We use the daily EPU index of Baker, Bloom, and Davis (2016) as an alternative daily measure of uncertainty, which should be one component of our broad economic uncertainty concept. This measure is based on policy-related economic news contained in newspaper articles, with the index standardized to have an average value of 100. With its daily availability, we evaluate its changes over our rolling 22- and 10-weekday periods, corresponding to our approach in the regressions in row 2 of our main Table 1. The series is available online at www.policyuncertainty.com. CRSP Stock Market Returns and Decile Portfolio Returns Based on Standard Deviation Sorts In Section 4.2, we use the returns of the NYSE/AMEX CRSP Standard Deviation decile portfolios to measure variation in the price of volatile stocks, building on Pflueger, Siriwardane, and Sunderam (2020). For a given calendar year, these equal-weighted portfolios are formed based on sorts on realized volatility from the prior calendar year. We use the return difference between the quintile of most-volatile stocks and the quintile of least-volatile stocks to indicate short-run movement in the relative price of volatile stocks. For our evaluation in Table 6, we calculate rolling cumulative 5-, 10-, and 22-weekday returns from the daily returns of these portfolios. In our implementation, we take the average return of the two extreme higher-volatility decile portfolios and two extreme lower-volatility decile portfolio to indicate the top and bottom quintile returns, respectively. In Section 5.3, we report the cumulative stock market decline that precedes the onset of each of our Bai-Perron weaker economic segments. There, we use the CRSP value-weighted stock index to measure aggregate equity market returns. Default Yield Spread In Section 4.2, we also investigate the time-series behavior of a default yield spread, defined as the difference between Moody’s Baa series and the 10-year Treasury Constant Maturity yield. These yields are obtained from the Federal Reserve Economic Data (FRED) data service. Federal Reserve Debt Security Holdings In Section 5.4, we evaluate the weekly positions of Treasury, government agency, and mortgage-backed securities (MBS) on the Federal Reserve’s Balance Sheet. 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