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Is the US Phillips curve stable? Evidence from Bayesian vector autoregressions*

Is the US Phillips curve stable? Evidence from Bayesian vector autoregressions* IntroductionIn association with the Great Recession, the US unemployment rate rose from 4.8 percent in the fourth quarter of 2007 to 9.3 percent in the second quarter of 2009. Such a dramatic increase in unemployment could be expected to generate substantial downward pressure on inflation. But while the US inflation rate certainly fell during that period, the fall was smaller than expected by many economists. Similar developments were also found in many other countries, giving rise to a general discussion about a “missing disinflation” (see, e.g., International Monetary Fund, 2013). One explanation for this development is that the Phillips curve might have become flatter in recent years; see, for example, Bean (2006), Gaiotti (2010), Ihrig et al. (2010), and Kuttner and Robinson (2010).1Globalization is commonly suggested as an important reason for such a development. This explanation has, however, been questioned by Ball (2006).The suggestion that the Phillips curve has become flatter is not undisputed though. For example, Blanchard et al. (2015) suggest that the Phillips curve has largely been stable since the early 1990s.2Berger et al. (2016) go even further and conclude that the Phillips curve was stable between 1959 and 2014.,3Further criticism can be found in Gordon (2013), Coibion and Gorodnichenko (2015), and Blanchard (2016).In addition, other explanations have also been suggested. Among these, Clark (2014) and Watson (2014) argue that a well‐anchored trend level of inflation contributed to keeping inflation fairly high around the Great Recession; Ball and Mazumder (2019) propose that these inflation outcomes were due to anchored inflation expectations and a short‐term unemployment rate, which rose less than the total unemployment rate, whereas Lindé (2019) point to the importance of allowing for non‐linearities in price‐ and wage‐setting.While the question of the Phillips curve's stability has generated an intense academic debate, it also matters to policymakers. For decades, different versions of the Phillips curve have been – and are still – widely used by central banks when forecasting inflation; its properties are accordingly of interest to a wide audience.The purpose of this paper is to add empirical evidence concerning the properties of the US Phillips curve by assessing its stability over time. This is done in a Bayesian vector autoregression (BVAR) framework where we estimate bivariate models with stochastic volatility using quarterly data on personal consumption expenditures (PCE) inflation and the unemployment rate ranging from 1990Q1 to 2019Q4. More specifically, we rely on the hybrid time‐varying parameter BVAR framework of Chan and Eisenstat (2018), which allows us to assess whether none, one, or both equations in the VAR are subject to time‐varying parameters.In conducting this analysis, we contribute to the existing literature in several ways. First, the hybrid time‐varying parameter BVAR framework has previously not been employed to study the US Phillips curve. One benefit of this framework is that we can perform formalized model selection in a Bayesian manner (using marginal likelihoods) to distinguish between different models; our focus when it comes to model selection is on establishing whether the VAR has time‐varying parameters. While formal model comparison between different BVAR models is not completely novel,4For example, Koop et al. (2009) and Karlsson and Österholm (2020a, 2020b) performed model selection based on marginal likelihoods in similar settings and studies such as D'Agostino et al. (2013) and Barnett et al. (2014) have relied on out‐of‐sample predictive criteria for the same purpose.our analysis nevertheless stands in contrast to a large part of the previous literature using time‐varying parameters and/or stochastic volatilities. In general, such modelling choices have been assumed rather than supported through statistical testing procedures (see, e.g., Cogley and Sargent, 2005; Bianchi and Civelli, 2015). Second, by analysing a low‐dimensional system, we move in the direction of the original observation of Phillips (1958) where simple correlations were in focus; we accordingly believe that we offer information on this relevant topic in a fairly intuitive framework. In addition, we argue that it is beneficial that both variables in the system are observable. While we do not deny the usefulness of analysis based on the output gap or unemployment gap (such as Ball and Mazumder, 2011; Leduc and Wilson, 2017), we think that it can be useful for the discussion to avoid the, at times, not very fruitful debate of what the “correct” measure of the relevant gap is.5That said, it can be noted that we conduct sensitivity analysis using the output gap instead of the unemployment rate in Section 4.Third, and from a policy perspective, we provide evidence concerning the environment in which monetary policy is acting today. For example, a flat Phillips curve means that the unemployment rate might have to be quite low in order to build up substantial inflationary pressure. Such information should prove useful to the Federal Reserve.Briefly mentioning our results, we note that they indicate that time‐varying parameters appears to be a relevant feature. We accordingly conclude that the US Phillips curve has not been stable. There are indications that the Phillips curve might have been somewhat flatter between 2005 and 2013 than in the decade preceding that period. In addition, we find that trend inflation was also reasonably high around the Great Recession. Both of these findings might have contributed to high inflation around the Great Recession. However, looking at conditional forecasts from the model, it can be questioned whether inflation actually was surprisingly high in association with the Great Recession.The remainder of this paper is organized as follows. In Section 2, we describe the methodological framework that we rely on; that is, the specification of the different BVAR models, our choice of prior distributions and parametrization as well as model selection is discussed. We present data, conduct the empirical analysis, and present our results in Section 3. Section 4 provides a sensitivity analysis, showing that the results are largely robust to both prior specification and variations in the data used. Finally, we conclude in Section 5.Methodological frameworkThe Phillips curve is an analytical tool that has reached widespread fame since its introduction in 1958, and it is commonly used in both theoretical and empirical work. However, as well as being popular, it also comes in many different forms, where both the specification and variables included can differ.6See Phillips (1958) for the seminal work. Phelps (1967) and Friedman (1968) provided important early contributions. For an example of the Phillips curve in the New Keynesian literature, see Galí and Gertler (1999). Recent empirical studies – taking a less structural approach and relying on time‐series econometrics for analysis – include Svensson (2015), Chan et al. (2016), Knotek and Zaman (2017), and Alexius et al. (2020). It is accordingly not completely transparent what one means when referring to the “Phillips curve”.7Using the words of Clark and McCracken (2006, p. 1127), the Phillips curve can be “[…] broadly defined […] as a model relating inflation to the unemployment rate, output gap, or capacity utilization”.In this paper, we rely on bivariate BVARs with the unemployment rate and inflation for our analysis. One important reason behind this choice is that the inflation equation of these BVARs can be seen as what King and Watson (1994, p. 172), referred to as the “dynamic generalization of the Phillips curve”. However, in general, we focus on the system as a whole and not just the inflation equation. By doing this, we reduce the risk that important dynamic effects between the two variables are omitted. Reflecting upon our modelling choice at a more general level, using sparse specifications is of course not something new in the literature related to the Phillips curve; see, for example, parts of the analysis in Stock and Watson (1999) and Faust and Wright (2013), or the contributions by Clark and McCracken (2006), Chan et al. (2016), Dotsey et al. (2018), Knotek and Zaman (2017), Alexius et al. (2020), and Karlsson and Österholm (2020b) in which only small models are used. It does, however, stand in contrast to literature relying on larger models, such as Laséen and Taheri Sanjani (2016). We do believe that a drawback with VAR models with a large number of variables in a Phillips curve framework is that they become more difficult to interpret and accordingly prefer our bivariate setting.The Bayesian VAR modelWe initially define yt=(utπt)′, where ut is the unemployment rate and πt is PCE inflation. We then follow Chan and Eisenstat (2018) when specifying the most general model that will be used – the BVAR with time‐varying parameters and stochastic volatility. The model is given as1B0tyt=μt+B1tyt−1+⋯+Bptyt−p+εt,where B0t is a 2×2 lower triangular matrix with ones on the diagonal;8In line with, for example, Cogley and Sargent (2005) and Primiceri (2005), we accordingly rely on a recursive structure in order to identify the orthogonal disturbances to the system. While this assumption can be questioned, it is nevertheless a benchmark assumption in empirical macroeconomic analysis using VARs.μt is a 2×1 vector of time‐varying intercepts; B1t,…,Bpt are 2×2 matrices with the parameters describing the dynamics of the BVAR; and εt is a 2×1 vector of disturbances, εt∼N(0,Σt), where Σt=diag(exp(h1t),exp(h2t)).We gather μt and the free parameters of the matrices B0t,…,Bpt in the vector θt=(θ1t′θ2t′)′, where θ1t contains the parameters of the equation for the unemployment rate and θ2t contains the parameters of the inflation equation. The processes for the time‐varying parameters and log‐volatilities are then specified as random walks,2θt=θt−1+ηt,3ht=ht−1+ζt,where ηt∼N(0,Σθ) and ζt∼N(0,Σh); Σθ and Σh are both diagonal matrices.The model described above – which has time‐varying parameters and stochastic volatility – is the most general model we consider in our analysis. We largely treat stochastic volatility as a given feature of the model. This is based on both a growing empirical literature pointing to the importance of allowing for time‐varying shock volatility in macroeconomics9See, for example, Cogley and Sargent (2005), Sims and Zha (2006), Clark (2011), Stock and Watson (2012), Franta et al. (2014), Akram and Mumtaz (2019), Koop and Korobilis (2019), and Karlsson and Österholm (2020a, 2020b).and empirical analysis that we conduct ourselves (see Table B1 in the Online Appendix and the discussion in footnote 12). Considering time‐varying parameters, we note that this is an appealing way to allow for a potentially unstable Phillips curve, something which has found some support in earlier research. For example, King and Watson (1994, p. 209), found “important evidence of econometric instability over subsamples”. Also Stock and Watson (1999) found evidence of an unstable Phillips curve.10In addition, Gallegati et al. (2011) found evidence of an unstable (wage) Phillips curve in the United States. Interestingly, their analysis pointed to the Phillips curve having become steeper since the mid‐1990s.By imposing various restrictions on the most general model, we can assess how important different features are. For example, if θ1t is constant, we obtain a BVAR with stochastic volatility where the equation for the unemployment rate has constant parameters but the equation for inflation has time‐varying parameters. If we instead make θt and ht constant, we obtain a traditional BVAR with time‐invariant parameters and a covariance matrix. As pointed out above, we largely take stochastic volatility as a given feature. Under the assumption of stochastic volatility, we accordingly compare four different BVAR specifications: (i) the parameters of both equations are constant (θt constant); (ii) time variation in the parameters of the equation for the unemployment rate (time variation in θ1t and θ2t constant); (iii) time variation in the parameters of the equation for inflation (θ1t constant and time variation in θ2t); and (iv) time variation in the parameters of both equations (time variation in θt). Here it can be noted that our definition of stability of the Phillips curve refers to all parameters in the inflation equation (i.e., the entire vector θ2t); instability can, accordingly, be due to the intercept, the coefficients on lags of inflation, or the coefficients on the (contemporaneous and lagged) unemployment rate.11An alternative to this would be to focus on the coefficients on the (contemporaneous and lagged) unemployment rate in the inflation equation. In the terminology used in this paper, this would be seen as assessing whether the slope of the Phillips curve has been stable. See Section 3.4 and footnote 14 in particular for further discussion concerning this.Priors and estimationThe choice of prior distributions and parameters of the priors can have a substantial effect on model comparisons based on marginal likelihoods, and the priors must be set up with some care. We need, in particular, to ensure that the information content is as equivalent as possible for the different models and that the prior specification does not put one of the models at a disadvantage. For example, putting a tight prior on the regression parameters (initial state in the case of time‐varying parameters) that does not agree with the data will hurt the marginal likelihood for constant parameter models more than models with time‐varying parameters.We address these concerns by specifying a proper but uninformative prior on the regression parameters or their initial states when the parameters are time‐varying. This is achieved through a normal, N(0,5I), prior on the initial states of the regression parameters, θ1,0and θ2,0. For the initial state of the log‐volatilities we use a normal prior (i.e., the variance is lognormal), hi,0∼N(μi,0.25), with μi selected to set the prior mean of exp(hi,0) equal to the residual variance of a univariate AR(p) with constant parameters. As we conduct model comparison between models with and without time‐varying parameters, the specification of the prior for the variance of the innovations to θi,t is important. In addition, stochastic volatility can to some extent compensate for a lack of time variation in the parameters and we therefore also need to be careful with the prior for the variance of the innovations to hi,t. In both cases, we use independent inverse Gamma, iG(v,S) priors for the diagonal elements of Σθ and Σh. The shape parameter, v, is fixed at 5 and the scale parameter, S, or equivalently the prior mean of the variance, S/(v−1), is selected in an empirical Bayes fashion. Starting with the prior for the variance of the innovations to the log‐volatilities, we use a grid search to find the value of S that maximizes the marginal likelihood for the model with constant parameters. This results in a prior mean for the variance of 0.1. Next, we allow the prior means of the variances of the innovations to the regression parameters to be different for the constant terms and other parameters. These are then selected in a grid search to maximize the marginal likelihood in the model with stochastic volatility and time variation in the parameters of both equations. The grid search leads to prior means of 0.01 and 0.00005 for the variance of the innovations to the constant terms and other parameters, respectively.We use the Markov chain Monte Carlo (MCMC) sampler developed by Chan and Eisenstat (2018) for posterior inference and we refer to their paper for details. Throughout we use 200,000 draws from the sampler with 50,000 draws as burn‐in and retain every tenth draw for posterior inference. Except for marginal likelihood calculations, draws with non‐stationary regression parameters are discarded.Model selectionIn order to distinguish between the models and gain insight into the stability over time of the US Phillips curve, we assess the fit of the different models using the marginal likelihood. In a Bayesian setting, the marginal likelihood is the appropriate measure of how well the model (and prior) agrees with the data.Given the triangular specification, we can write the marginal likelihood for the models with time‐varying parameters and stochastic volatility as4m(y|M)=∏i=12[∫p(yi|θi,hi,ξi,M)p(θi|ξi,M)×p(hi|ξi,M)p(ξi|M)dθidhidξi],where ξi collects the parameters θi,0, hi,0, σi,j,θ2, and σi,h2 of the state equations (2) and (3). The marginal likelihood is then estimated, equation by equation, using the method of Chan and Eisenstat (2018).Empirical analysisDataData on the seasonally adjusted PCE deflator and unemployment rate – ranging from 1990Q1 to 2019Q4 – were sourced from the FRED database of the Federal Reserve Bank of St Louis. PCE inflation is calculated as πt=100(Pt/Pt−4−1) where Pt is the PCE deflator at time t. Data are shown in Figure 1.1FigureData Notes: Both variables are measured in percent.We believe that 1990Q1 is a reasonable starting point for our analysis. By choosing a sample that is too long, we would arguably stack the analysis towards finding time variation in the parameters. For example, if the starting point for our sample was set to 1970Q1, we would include the stagflation of the 1970s, which almost surely was characterized by different time‐series properties. Even starting in 1980Q1 could be problematic in terms of giving the model with time‐invariant parameters (and covariance matrix) a fair chance, seeing that the early 1980s was when the “Volcker disinflation” took place (for a discussion, see, e.g., Goodfriend and King, 2005). There is a fairly substantial literature claiming that the inflation process changed its mean and/or dynamics in the 1970s and early 1980s (see, e.g., Kozicki and Tinsley, 2005; Cogley and Sbordone, 2008; Beechey and Österholm, 2012). Regarding the end point of the sample, we choose 2019Q4 in order to avoid the period associated with the COVID‐19 pandemic.Using the sample in this paper, we hence estimate the model over a period where it would not be unrealistic to find stability; as pointed out above, it has been suggested by, for example, Blanchard et al. (2015) that the Phillips curve largely has been stable since the beginning of this sample.Is there time variation?We next estimate and compare the four different BVARs with stochastic volatility. Lag length is in all cases set to p=4. The log marginal likelihoods from the estimation of the models are shown in Table 1.12Table B1 in the Online Appendix gives the log marginal likelihoods for the corresponding four models estimated under the assumption of a time‐invariant covariance matrix (i.e., homoscedastic errors). As can be seen, also in this set‐up, the model with time variation in the parameters of the equation of the inflation equation is judged as the best. However, it should be noted that all models have substantially lower marginal likelihoods than the models with stochastic volatility; this points to the importance of stochastic volatility in modelling the Phillips curve and we accordingly focus on models with this feature.1TableLog marginal likelihood for different BVAR specifications with stochastic volatilityModelLog marginal likelihoodTime variation in parametersi−82.4Both equations constantii−88.0Time variation in equation for unemployment rateiii−78.7Time variation in equation for inflationiv−84.2Time variation in both equationsNotes: The table gives the natural logarithm of the marginal likelihood of the models. The log marginal likelihood for the best model is shown in bold.As can be seen, the marginal likelihood is highest for the model with time variation in the parameters of the inflation equation only, that is, model iii. The marginal likelihoods clearly indicate that model iii is the preferred model. Using the scale of two times the difference in log marginal likelihood and the terminology of Kass and Raftery (1995, p. 777), the evidence in favour of model iii is “strong” or “very strong”, regardless of which model it is compared against.The fact that the parameters of the inflation equation of the VAR appear to be time‐varying indicates that the Phillips curve is not stable over time. We next illustrate what this time variation looks like by examining the properties of model iii in more detail.Impulse‐response functionsFigure 2 shows the impulse‐response function from the model with time variation in the parameters of the inflation equation only (i.e., model iii), which describes the effect of a shock to inflation on inflation itself. We initially discuss this impulse‐response function as it clearly shows why both time‐varying parameters and stochastic volatility are relevant features when modelling this bivariate system.2FigureImpulse‐response function for model iii: the effect of a shock to inflation on inflation Notes: The size of the impulse is one standard deviation. Effect on inflation in percentage points on vertical axis. Horizon in quarters and time on horizontal axes.As can be seen from the figure, the estimated standard deviation of the shocks to the inflation equation – that is, [exp(ĥ2t)]0.5 – varies a fair bit over time. Between 1996 and 1998, in what can be described as the peak of the Great Moderation, it was as low as 0.10. It then increased almost continuously until 2009Q1 when it reached a value of 0.74. Since then, the volatility of the shocks has decreased and it is by the end of the sample estimated to be approximately 0.20.Figure 2 also illustrates that the dynamic properties of the system have changed. The variation in these impulse‐response functions indicates that there is non‐negligible time variation in the coefficients of the model, which leads us to question the stability of the US Phillips curve. This is also confirmed when looking at the estimated coefficients of the inflation equation (see Figure A1 in the Appendix).We turn to the impulse‐response function that perhaps is of primary interest to us, namely the effect that shocks to the unemployment rate have on inflation; this can be found in Figure 3. As is clearly shown in the figure, a shock to the unemployment rate has a negative effect on inflation. The effect varies somewhat over time. This is because of the time‐varying parameters of the inflation equation as well as the fact that the volatility of the shock to the unemployment rate changes over time.13The standard deviation of this shock has ranged between 0.10 and 0.37 during the sample. The impulse‐response function showing the effect of a shock to the unemployment rate on the unemployment rate itself is given in Figure B1 in the Online Appendix. Figure B2 in the Online Appendix gives the impulse‐response function showing the effect of a shock to inflation on the unemployment rate.3FigureImpulse‐response function for model iii: the effect of a shock to the unemployment rate on inflation Notes: The size of the impulse is one standard deviation. Effect on inflation in percentage points on vertical axis. Horizon in quarters and time on horizontal axes.It is, however, difficult to assess whether the slope of the Phillips curve has changed markedly only through looking at the impulse‐response functions. Therefore, we now turn to a different way of addressing that issue.The slope of the Phillips curveOne measure of the slope of the Phillips curve that is used in the empirical literature is the sum of the coefficients on the lags of the unemployment rate in the inflation equation when looking at the reduced form of the VAR (see, e.g., Knotek and Zaman, 2017; Karlsson and Österholm, 2020b).The reduced form is achieved by pre‐multiplying the specification in equation (1) with B0t−1. This generates the following form of the model,5yt=δt+A1tyt−1+⋯+Aptyt−p+et,where δt=B0t−1μt, Ait=B0t−1Bit, and et=B0t−1εt. The estimated sum of the coefficients on the lags of the unemployment rate in the inflation equation is shown in Figure 4.4FigureEstimated slope of the Phillips curve for model iii Notes: Slope is measured as the sum of the coefficients on the lags of unemployment in the inflation equation in equation (5). Coloured band is 68 percent equal tail credible interval.As can be seen, the point estimate indicates a non‐negligible amount of variation within the sample. At the beginning of the sample, the slope is approximately zero but it then falls and reaches a low of −0.14 in 1998. Between 2005 and 2013, the slope is more moderate, hovering around −0.07. We should of course keep in mind that the point estimate is associated with a fair amount of uncertainty as illustrated by the 68 percent credible interval; changes should accordingly not be over‐interpreted.14Additional analysis conducted – not reported in detail but available from the authors on request – does, however, suggest that the slope of the Phillips curve has indeed changed over time. In this analysis, we estimate a version of model iii where we make the model have very close to no time variation for the coefficients on the (contemporaneous and lagged) unemployment rate in the inflation equation; this is achieved by placing very tight priors on the relevant elements in Σθ, thereby in practice making them zero. Marginal likelihood calculations for this model show that the log marginal likelihood is lower (−81.8) than that of the original version of model iii presented in Table 1.It is nevertheless our best estimate, and therefore interesting to note, that the Phillips curve might have been flatter around the financial crisis than it typically has been during this sample.15Comparing our estimates in Figure 4 to the findings from a few other recent studies, it can be noted that Blanchard et al. (2015) found a somewhat steeper Phillips curve for the United States; estimating a single‐equation regression model with constant parameters employing quarterly data from 1990 to 2014 – and a shorter subsample ranging from 2007 to 2014 – the estimated slope was around −0.25. Knotek and Zaman (2017) report estimates from just two points in time from their BVAR with time‐varying parameters – 1999Q3 and 2017Q3 – both of which are associated with a quite modest slope (−0.07 and −0.06, respectively). Their numbers are fairly close to that found by Murphy (2018). Based on a single‐equation regression model with constant parameters and a sample of quarterly data ranging from 1990 to 2014, he found a slope of −0.07. Karlsson and Österholm (2020b) report a time‐varying slope of the Swedish Phillips curve that, while generally somewhat steeper, shows a similar pattern to Figure 4. One should of course be careful when making comparisons seeing that the specifications, estimation methods, and data used vary between studies.It hence seems that a flatter Phillips curve might have contributed to the “surprisingly high” inflation outcomes in the aftermath of the financial crisis.Before leaving the discussion of the slope of the Phillips curve behind, it is also important to be point out that the Phillips curve has not been flatter than usual during the most recent years. Since 2015, the slope has been between −0.10 and −0.13, which is actually quite steep relative to the rest of the sample; this steepness is on a par with that of the late 1990s. This appears, at least to some extent, to contradict a flatter Phillips curve as an explanation for the weak inflation outcomes that the Federal Reserve has been struggling with during this period.Trend inflationVarious properties of the model have been illustrated above and provided information concerning the stability of the Phillips curve. However, in a model with time‐varying parameters, there is information based on the change in parameters that occurs over time, which is not communicated through the above tools. As this can be of importance when it comes to the analysis of the Phillips curve, we now address the issue of the estimated trend inflation from the model.We define “trend inflation” as the level to which the inflation forecasts from the model will converge; this definition is in line with, for example, that employed by Faust and Wright (2013) and Clark and Doh (2014).16In addition, what we denote as trend inflation has the same interpretation as what Cogley and Sargent (2001, 2005) define as core inflation.Perhaps more intuitively, it can be thought of as the model's long‐run expectation of inflation. Employing the reduced form of the VAR in equation (5), we rewrite it in companion form as6y˜t=δ˜t+Aty˜t−1+e˜t,with y˜t=(yt′,yt−1′,…,yt−p+1′)′, δ˜t=(δt′,0′,…,0′)′,7At=A1tA2t⋯Ap−1,tAptI0⋯00⋱⋮⋱⋮0⋯0I0,and e˜t=(et′,0′,…,0′)′, and solve for trend inflation as the second element of ϕt=(I−At)−1δ˜t.The median estimated trend inflation at each point in time from the model is shown in Figure 5. This shows that trend inflation was fairly high when the Great Recession hit the US economy in the second quarter of 2007, namely 2.5 percent; it had reached this value after drifting up from 1.3 percent in 1998. Trend inflation also stayed reasonably high until 2008Q3 – when it was 2.5 percent – but then took on a clear downward trajectory, reaching a minimum of 1.2 percent in 2015. It hence seems that part of the high inflation during the Great Recession can be explained by high trend inflation.17This result can be seen as being in the same spirit as the findings presented in Clark (2014) and Watson (2014), who suggested that inflation was kept reasonably high around the Great Recession partly as a result of a well‐anchored trend level of inflation.5FigureEstimated trend inflation from model iii Notes: Trend inflation and inflation are measured in percent. Coloured band is 68 percent equal tail credible interval.Figure 5 also illustrates the stubbornly low inflation during the last few years of our sample. As can be seen from the figure, trend inflation has at the same time been low. The low inflationary pressure that we have seen over the last years has hence been reflected in the estimated parameters of the model and caused estimated trend inflation to decrease noticeably. For the Federal Reserve, this means that while shocks to the unemployment rate might have roughly the same effect now as it has historically, there should be scope to allow for additional inflationary pressures to build up because such pressures could be needed in order to bring trend inflation back up to the target level of 2 percent.18It can be noted that the formal inflation target that the Federal Reserve introduced in 2012 can be seen as being at conflict with models with time‐varying parameters of the type used in this paper. Providing a clear focal point for future inflation, it can be argued that inflation forecasts should converge to the target level of 2 percent and that, for example, models with steady‐state priors of the type suggested by Villani (2009) or Clark (2011) should be considered. However, we believe that the fact that the Federal Reserve's formal target has been active during only a reasonably small part of our sample is a good reason to employ our chosen framework.Summing up the empirical analysis so far, we note that the time‐varying parameters help explain the surprisingly high inflation around the financial crisis. There is an indication that the Phillips curve might have been somewhat flatter around the financial crisis and also that trend inflation was high.Conditional forecastsOne common usage of econometric models of the type investigated above is forecasting. In many cases, the models are simply employed to generate an endogenous forecast. It is, however, also common – particularly at policy institutions – to use the models for conditional forecasting.19For contributions in the field of conditional forecasting – dealing with methodological issues and/or empirical applications – see, for example, Waggoner and Zha (1999), Hamilton and Herrera (2004), Österholm (2009), Baumeister and Kilian (2013), Clark and McCracken (2014), and Giannone et al. (2014).Concerning the US Phillips curve as studied in this paper, we believe that a conditional forecasting exercise is of certain interest as it can provide us with information regarding the missing disinflation associated with the Great Recession. In line with the analysis presented above, we conduct this exercise using model iii.Forecasts of inflation are generated for the period 2008Q1–2010Q4 conditional upon an assumed path for the unemployment rate; more specifically, this path is given by the actual unemployment rate. This will show us how the model would have predicted future inflation given perfect foresight regarding the unemployment rate. The conditional forecasts are generated using a method that we believe represents common current practice, namely that of Waggoner and Zha (1999).We generate the conditional forecasts based on three sets of parameters. First, to obtain more precise estimates of the parameter values as of 2007Q4, we estimate the model on the full sample rather than data up to 2007Q4 where the estimates would suffer from end‐of‐sample uncertainty. These conditional forecasts are given by the red dashed line in Figure 6. Generating the forecasts in this manner obviously does not correspond to how a conditional forecasting exercise would have been conducted in real time; rather, it is based on today's best estimate of the parameters of the model at the beginning of the financial crisis.20It should be noted that when the conditional forecasts are generated, the parameters and volatilities are kept fixed at their 2007Q4 values.Second, we construct forecasts using data only up until 2007Q4 in order to see if this affects our conclusion. Using this shorter sample, we provide a better approximation to what the model would have suggested in real time.21We do not use real‐time data though, so this does not show exactly what the model would have predicted in real time. However, given the purpose of the exercise, we do not believe that this is strictly necessary.The conditional forecasts from this exercise are given by the blue dashed line with asterisks in Figure 6.22We also generate conditional forecasts using 2008Q4 rather than 2007Q4 as the date from which the forecasts originate; forecasts are accordingly generated for the period 2009Q1–2010Q4 (using the parameter estimates associated with 2008Q4). Regardless of whether we use parameter estimates based on the full sample or only on data up until 2008Q4, the results are similar. In neither case do we find evidence of missing disinflation. (Detailed results are not reported here but are available from the authors on request.)The third set of parameters we employ comes from the estimation of the model using the full sample; however, rather than using the parameter estimates associated with 2007Q4, we use the estimates associated with 1997Q4. This can be seen as providing further information on the stability of the Phillips curve. The conditional forecasts from this exercise are given by the blue dashed line with asterisks in Figure 7; for comparison, the forecasts from the model using the parameter estimates associated with 2007Q4 (estimated using the full sample) are also shown.6FigureModel‐based inflation forecasts using model iii Notes: Median forecasts. Parameter estimates for long sample forecast use full sample and short sample forecasts use data up to 2007Q4. Parameter estimates used for forecasting are dated 2007Q4.7FigureModel‐based inflation forecasts using model iii: alternative date of parameters Notes: Median forecasts. Parameter estimates use full sample. Parameter estimates used for forecasting are dated 1997Q4 and 2007Q4.Note that in order to see the effect that conditioning on the unemployment rate has, we also provide the model's unconditional forecasts. These forecasts – given by the dot‐dashed lines (red, and blue with asterisks) – are also shown in Figures 6 and 7.Looking at the conditional forecasts based on parameter estimates from 2007Q4 using the full sample (i.e., the red dashed line in Figures 6 and 7), it can be seen that the model suggests a fall in inflation in response to the increasing unemployment rate. However, the predicted fall in inflation is actually less deep than the outcome. While actual inflation hit its lowest point in 2009Q3 at −1 percent, the conditional forecast for the same quarter was 0.7 percent. Comparing the full path of the forecasts to the outcomes, it is clear that the model did not want substantially lower inflation at any time point than what was actually the case (even though it can be noted that the predictions for 2009Q4 and 2010Q1 are somewhat lower than the outcomes). We accordingly conclude that the model does not suggest that there was a missing disinflation associated with the financial crisis. As can be seen from Figure 6, this conclusion does not change if the shorter sample is used for parameter estimation.23We also generate conditional forecasts from the BVAR with time‐invariant θt – that is, model i – in order to assess the importance of the time‐varying parameters. These results are presented in Figure B3 in the Online Appendix. As can be seen, the story told is very much the same as when model iii was used. The model does not suggest a more sizeable drop in inflation in response to the increasing unemployment rate than what was actually the case.A somewhat different picture is, however, painted when looking at the conditional forecasts from the model when the parameter estimates associated with 1997Q4 are employed. As can be seen from Figure 7, these forecasts are substantially lower than those based on the parameter estimates associated with 2007Q4; we also note that they consistently lie below the actual values. This shows that the time variation in the parameters of the model that we have established is quantitatively meaningful. In addition, if one assumes that these parameter estimates also reflect views held by economists, these results can be interpreted as favouring a claim of missing disinflation; if someone believed that the economy in 2007 had the structure of 1997 (as given by the VAR), a more substantial fall in inflation than what actually materialized would have been expected.Sensitivity analysisHaving conducted the above analysis, we finally want to assess how sensitive our results are with respect to some key concepts, namely our choice of priors and the data used. In this section, accordingly, we first vary some of our prior parameters (using the data employed in the main analysis). We then conduct analysis using alternative measures of inflation and, finally, consider an alternative measure of capacity utilization by replacing the unemployment rate with the output gap.24In addition, we have also conducted some sensitivity analysis using a trivariate system, which we only briefly report in this footnote. (Further details are available from the authors on request.) Modelling the unemployment rate, PCE inflation, and the three‐month Treasury bill rate, we find that a specification with time variation only in the inflation equation is ranked first (of the eight assessed models); the conditional forecasts from this model do not suggest that there was any missing disinflation.PriorsThe choice of prior parameters affects the marginal likelihood and can influence the ranking of the models when comparing marginal likelihoods. In this sensitivity analysis, we focus on the prior parameters that govern time‐varying parameters and time‐varying volatilities as they are the key ones in the present application. More specifically, we modify the prior means of the variance of the changes in the parameters and log‐volatilities by halving or doubling Sθ and Sh and re‐estimate the models using the same data as we employed in our main analysis (i.e., the data described in Section 3).The results from this exercise can be summarized very briefly. The finding from our main analysis is confirmed as the model with time variation in the parameters in the inflation equation only (model iii) is ranked first in all cases (see Table B2 in the Online Appendix). We thus conclude that this is a robust finding with respect to the choice of priors.Quarter‐on‐quarter inflationWhen modelling inflation, it is not obvious that it is the annual change in the PCE deflator that we want to use as our measure. Another relevant alternative is to use the (annualized) quarterly change; that is, we define PCE inflation as πt=400(Pt/Pt−1−1). Figure A2 in the Appendix presents the data on quarter‐on‐quarter PCE inflation. Marginal likelihoods from the four different models estimated using this inflation measure can be found in Table A1 in the Appendix.As can be seen from Table A1, the preferred model is that with constant parameters in both equations (model i). This specification has a somewhat higher marginal likelihood than the specification with time variation in the inflation equation (model iii). While the evidence in favour of model i against model iii – again using the scale and terminology of Kass and Raftery (1995, p. 777) – is “not worth more than a bare mention”, it is nevertheless the best performing model, and we conclude that the Phillips curve in this case is judged to have been stable.In terms of the model's conditional forecasts, this model does not want a substantial decrease in inflation in response to the rising unemployment rate. In fact, the decrease in inflation is very modest (see Figure B4 in the Online Appendix). This is because the Phillips curve is extremely flat; the point estimate of the slope – based on the full sample – is only −0.01. Given that the 68 percent credible interval for the slope is (−0.08, 0.05), one can even question the existence of the Phillips curve in this case.25The model's impulse‐response function of inflation with respect to shocks to the unemployment rate is shown in Figure B5 in the Online Appendix.Core PCE inflationWhen discussing monetary policy, it is common to relate central banks' actions to “core” measures of inflation – that is, inflation measures that have had some volatile components removed. As we have used PCE inflation in our analysis above, we here investigate if our results are robust to using core PCE inflation instead. In constructing this measure, food and energy prices have been removed from the PCE deflator. Inflation is calculated as πtcore=100(Ptc/Pt−4c−1), where Ptc is the core PCE deflator at time t. Data on core PCE inflation are shown in Figure A2 in the Appendix and we again present marginal likelihoods from the four different models in Table A1 in the Appendix.The preferred model in this setting is again model iii (i.e., the model with time variation in the parameters of the inflation equation); the evidence in favour of it ranges from “strong” to “very strong” (Kass and Raftery, 1995, p. 777).We find that the picture painted is similar to that in our main analysis. The impulse‐response function of core PCE inflation with respect to a shock to the unemployment rate shows a qualitatively similar picture to what we saw in Figure 3 when PCE inflation was used (see Figure B6 in the Online Appendix). Turning to the slope of the Phillips curve (Figure B7), we find that using core PCE inflation does not change our main findings from above, though it can be noted that the slope is now somewhat steeper in general (judging by the point estimates). Finally, the conditional forecasts (Figure B8) show that the model predicts a decrease in inflation that was smaller than what materialized in this setting as well. We again therefore conclude that there was no missing disinflation.GDP deflator inflationThe last inflation measure that we analyse is based on the seasonally adjusted GDP deflator and is also used fairly frequently in the empirical literature (see, e.g., Rudebusch and Svensson, 1999; Lindé, 2018). Following Rudebusch and Svensson (1999), we define inflation as πtGDP=400ln(PtGDP/Pt−1GDP) where PtGDP is the GDP deflator at time t. The data are shown in Figure A2 in the Appendix and marginal likelihoods from the four different models are again presented in Table A1 in the Appendix. As can be seen, we also here find that model iii is preferred by the data; the evidence in favour of it ranges from “positive” to “very strong” (Kass and Raftery, 1995, p. 777).The key properties of the model are also in this case similar to what we have found previously. The impulse‐response function of inflation with respect to a shock to the unemployment rate is again similar to that shown in Figure 3 (see Figure B9 in the Online Appendix). Further, we note that the slope of the Phillips curve (Figure B10) resembles that of the benchmark model (Figure 4), though it is somewhat less steep in general. Concerning the conditional forecasts from the model (Figure B11), these largely confirm our previous findings; even if the model – when using the parameters from the shorter sample – wanted a somewhat lower inflation than the actual outcome during 2009 and 2010, it does not signal that there was an obvious missing disinflation.26We also conducted some additional analysis – not reported in detail but available from the authors on request – relating our work more closely to that of Rudebusch and Svensson (1999). This was done by using data on the output gap (as described in Section 4.5) and GDP deflator inflation; in line with Rudebusch and Svensson, both series were demeaned. We restricted the inflation equation so that the only coefficients that were allowed to be non‐zero were those on lags of inflation and the first lag of the unemployment rate; all other coefficients (i.e., the intercept and lags 0, 2, 3, and 4 on the output gap) were forcefully shrunk to zero using very tight priors. In this setting, the marginal likelihoods from the four models were extremely similar, pointing to the data being unable to distinguish between the models. All four models had similar key properties though. A shock to the output gap (with sign reversed) generated lower inflation. The slope of the Phillips curve was somewhat flatter than in our main analysis, approximately −0.04. The conditional forecasts from the models did not indicate that there was any missing disinflation.Output gapThe Phillips curve is often interpreted as a general relation between the level of real activity in the economy and the inflation rate. Accordingly, the unemployment rate is not the only relevant measure and we investigate how robust the results are to a change of measure of real activity. We thus replace the unemployment rate in the model with the output gap (and return to inflation being measured as the annual change in the PCE deflator – that is, πt=100(Pt/Pt−4−1)). Data on the output gap are based on the Congressional Budget Office's estimate of potential GDP and are shown in Figure A2 in the Appendix. Note that the output gap is given with reversed sign in order to make comparisons with the models that relied on the unemployment rate easier.We present marginal likelihoods from the four different models in Table A1 in the Appendix. The preferred model is once again model iii (i.e., the model with time variation in the parameters of the inflation equation). In this case, the second‐best model is that with time variation in the parameters of both equations (i.e., model iv). The evidence in favour of model iii against model iv is only just “positive” (Kass and Raftery, 1995, p. 777); against the other two models, the evidence is “strong”.Turning to the properties of the model, the impulse‐response function of inflation with respect to a shock to the output gap is qualitatively similar to that presented in Figure 3 (see Figure B12 in the Online Appendix). The slope of the Phillips curve is fairly stable over time (Figure B13). Finally, the conditional forecasts (Figure B14) largely confirm our previous findings. While the model wanted a somewhat lower inflation than actual between 2009Q4 and 2010Q2, the general message is that the model wanted a substantially higher inflation than the outcome. The overall picture is therefore hardly one of missing disinflation. We accordingly conclude that it seems to be a fairly robust finding that our BVAR models do not support a claim of a missing disinflation in association with the Great Recession.ConclusionsThe Phillips curve is a popular tool in macroeconomics, which, among other things, is used to assess inflationary pressure in the economy. In this paper, we have added empirical evidence concerning the properties of the US Phillips curve by analysing different specifications of BVAR models.Conducting Bayesian model selection using recently developed methods for this purpose, we find that the inflation equation of the VAR appears to be subject to time‐varying parameters. Our results also indicate that the Phillips curve might have been somewhat flatter between 2005 and 2013 than in the decade preceding that period. This fact – together with high trend inflation – can help explain why inflation was high around the Great Recession.However, our findings from the conditional forecasting exercises conducted suggest that as far as the models are concerned, there might not have been that much of a missing disinflation associated with the Great Recession. Conditioning on the increase in the unemployment rate that actually took place, different model specifications yield somewhat different results. In general though, the models did not predict that inflation should have been lower than what turned out to be the case.From a policy perspective, our results imply two things. First, policy institutions and forecasters need to consider the fact that both dynamic relations and the volatility of shocks might be changing over time when building econometric models. Because methods for model selection have been improved, it is also the case that the econometric modelling choice can stand on firmer ground nowadays; where previously an econometrician largely had to make an assumption regarding the relevance of different specifications, they can now be formally compared. Second, while inflation in the United States recently has increased quite rapidly, it was stubbornly low the years before that. This low inflation appears, at least to some extent, to be due to low inflationary pressures that have affected the estimated trend inflation. Looking at the present unemployment rate, it is still somewhat elevated due to the COVID‐19 pandemic, but it has come down fairly rapidly from its peak in April 2020. Our findings indicate that there could be scope for the federal funds rate to be quite low even in light of a low unemployment rate, in order to let inflationary pressures build up in the economy and reliably have inflation close to the target level.AppendixA1FigureEstimated coefficients of the inflation equation of model iii Notes: Time on horizontal axes.A2FigureData on variables used for sensitivity analysis Notes: Core PCE inflation and annualized quarter‐on‐quarter PCE inflation are measured in percent. GDP deflator inflation is 400 times the log difference of the series. The output gap is based on the Congressional Budget Office's estimate of potential GDP and is given as percent of potential GDP.A1TableLog marginal likelihood for different BVAR specifications: sensitivity analysis with different dataModelLog marginal likelihoodTime variation in parametersQ‐on‐QCore PCEGDP deflatorOutputinflationinflationinflationgapi−205.714.0−148.7 −201.5Both equations constantii−211.6 9.2−156.3 −202.6Time variation in equationfor unemployment rateiii−206.620.5−146.4 −197.8Time variation in equation for inflationiv−211.715.6−153.7−198.9Time variation in both equationsNotes: The table gives the natural logarithm of the marginal likelihood of the models. The log marginal likelihood for the best model is shown in bold.ReferencesAkram, F. and Mumtaz, H. (2019), Time‐varying dynamics of the Norwegian economy, Scandinavian Journal of Economics 121, 407–434.Alexius, A., Lundholm, M., and Nielsen, L. (2020), Is the Phillips curve dead? 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F. and Zha, T. (1999), Conditional forecasts in dynamic multivariate models, Review of Economics and Statistics 81, 639–651.Watson, M. W. (2014), Inflation persistence, the NAIRU, and the Great Recession, American Economic Review 104 (5), 31–36. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png The Scandinavian Journal of Economics Wiley

Is the US Phillips curve stable? Evidence from Bayesian vector autoregressions*

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Abstract

IntroductionIn association with the Great Recession, the US unemployment rate rose from 4.8 percent in the fourth quarter of 2007 to 9.3 percent in the second quarter of 2009. Such a dramatic increase in unemployment could be expected to generate substantial downward pressure on inflation. But while the US inflation rate certainly fell during that period, the fall was smaller than expected by many economists. Similar developments were also found in many other countries, giving rise to a general discussion about a “missing disinflation” (see, e.g., International Monetary Fund, 2013). One explanation for this development is that the Phillips curve might have become flatter in recent years; see, for example, Bean (2006), Gaiotti (2010), Ihrig et al. (2010), and Kuttner and Robinson (2010).1Globalization is commonly suggested as an important reason for such a development. This explanation has, however, been questioned by Ball (2006).The suggestion that the Phillips curve has become flatter is not undisputed though. For example, Blanchard et al. (2015) suggest that the Phillips curve has largely been stable since the early 1990s.2Berger et al. (2016) go even further and conclude that the Phillips curve was stable between 1959 and 2014.,3Further criticism can be found in Gordon (2013), Coibion and Gorodnichenko (2015), and Blanchard (2016).In addition, other explanations have also been suggested. Among these, Clark (2014) and Watson (2014) argue that a well‐anchored trend level of inflation contributed to keeping inflation fairly high around the Great Recession; Ball and Mazumder (2019) propose that these inflation outcomes were due to anchored inflation expectations and a short‐term unemployment rate, which rose less than the total unemployment rate, whereas Lindé (2019) point to the importance of allowing for non‐linearities in price‐ and wage‐setting.While the question of the Phillips curve's stability has generated an intense academic debate, it also matters to policymakers. For decades, different versions of the Phillips curve have been – and are still – widely used by central banks when forecasting inflation; its properties are accordingly of interest to a wide audience.The purpose of this paper is to add empirical evidence concerning the properties of the US Phillips curve by assessing its stability over time. This is done in a Bayesian vector autoregression (BVAR) framework where we estimate bivariate models with stochastic volatility using quarterly data on personal consumption expenditures (PCE) inflation and the unemployment rate ranging from 1990Q1 to 2019Q4. More specifically, we rely on the hybrid time‐varying parameter BVAR framework of Chan and Eisenstat (2018), which allows us to assess whether none, one, or both equations in the VAR are subject to time‐varying parameters.In conducting this analysis, we contribute to the existing literature in several ways. First, the hybrid time‐varying parameter BVAR framework has previously not been employed to study the US Phillips curve. One benefit of this framework is that we can perform formalized model selection in a Bayesian manner (using marginal likelihoods) to distinguish between different models; our focus when it comes to model selection is on establishing whether the VAR has time‐varying parameters. While formal model comparison between different BVAR models is not completely novel,4For example, Koop et al. (2009) and Karlsson and Österholm (2020a, 2020b) performed model selection based on marginal likelihoods in similar settings and studies such as D'Agostino et al. (2013) and Barnett et al. (2014) have relied on out‐of‐sample predictive criteria for the same purpose.our analysis nevertheless stands in contrast to a large part of the previous literature using time‐varying parameters and/or stochastic volatilities. In general, such modelling choices have been assumed rather than supported through statistical testing procedures (see, e.g., Cogley and Sargent, 2005; Bianchi and Civelli, 2015). Second, by analysing a low‐dimensional system, we move in the direction of the original observation of Phillips (1958) where simple correlations were in focus; we accordingly believe that we offer information on this relevant topic in a fairly intuitive framework. In addition, we argue that it is beneficial that both variables in the system are observable. While we do not deny the usefulness of analysis based on the output gap or unemployment gap (such as Ball and Mazumder, 2011; Leduc and Wilson, 2017), we think that it can be useful for the discussion to avoid the, at times, not very fruitful debate of what the “correct” measure of the relevant gap is.5That said, it can be noted that we conduct sensitivity analysis using the output gap instead of the unemployment rate in Section 4.Third, and from a policy perspective, we provide evidence concerning the environment in which monetary policy is acting today. For example, a flat Phillips curve means that the unemployment rate might have to be quite low in order to build up substantial inflationary pressure. Such information should prove useful to the Federal Reserve.Briefly mentioning our results, we note that they indicate that time‐varying parameters appears to be a relevant feature. We accordingly conclude that the US Phillips curve has not been stable. There are indications that the Phillips curve might have been somewhat flatter between 2005 and 2013 than in the decade preceding that period. In addition, we find that trend inflation was also reasonably high around the Great Recession. Both of these findings might have contributed to high inflation around the Great Recession. However, looking at conditional forecasts from the model, it can be questioned whether inflation actually was surprisingly high in association with the Great Recession.The remainder of this paper is organized as follows. In Section 2, we describe the methodological framework that we rely on; that is, the specification of the different BVAR models, our choice of prior distributions and parametrization as well as model selection is discussed. We present data, conduct the empirical analysis, and present our results in Section 3. Section 4 provides a sensitivity analysis, showing that the results are largely robust to both prior specification and variations in the data used. Finally, we conclude in Section 5.Methodological frameworkThe Phillips curve is an analytical tool that has reached widespread fame since its introduction in 1958, and it is commonly used in both theoretical and empirical work. However, as well as being popular, it also comes in many different forms, where both the specification and variables included can differ.6See Phillips (1958) for the seminal work. Phelps (1967) and Friedman (1968) provided important early contributions. For an example of the Phillips curve in the New Keynesian literature, see Galí and Gertler (1999). Recent empirical studies – taking a less structural approach and relying on time‐series econometrics for analysis – include Svensson (2015), Chan et al. (2016), Knotek and Zaman (2017), and Alexius et al. (2020). It is accordingly not completely transparent what one means when referring to the “Phillips curve”.7Using the words of Clark and McCracken (2006, p. 1127), the Phillips curve can be “[…] broadly defined […] as a model relating inflation to the unemployment rate, output gap, or capacity utilization”.In this paper, we rely on bivariate BVARs with the unemployment rate and inflation for our analysis. One important reason behind this choice is that the inflation equation of these BVARs can be seen as what King and Watson (1994, p. 172), referred to as the “dynamic generalization of the Phillips curve”. However, in general, we focus on the system as a whole and not just the inflation equation. By doing this, we reduce the risk that important dynamic effects between the two variables are omitted. Reflecting upon our modelling choice at a more general level, using sparse specifications is of course not something new in the literature related to the Phillips curve; see, for example, parts of the analysis in Stock and Watson (1999) and Faust and Wright (2013), or the contributions by Clark and McCracken (2006), Chan et al. (2016), Dotsey et al. (2018), Knotek and Zaman (2017), Alexius et al. (2020), and Karlsson and Österholm (2020b) in which only small models are used. It does, however, stand in contrast to literature relying on larger models, such as Laséen and Taheri Sanjani (2016). We do believe that a drawback with VAR models with a large number of variables in a Phillips curve framework is that they become more difficult to interpret and accordingly prefer our bivariate setting.The Bayesian VAR modelWe initially define yt=(utπt)′, where ut is the unemployment rate and πt is PCE inflation. We then follow Chan and Eisenstat (2018) when specifying the most general model that will be used – the BVAR with time‐varying parameters and stochastic volatility. The model is given as1B0tyt=μt+B1tyt−1+⋯+Bptyt−p+εt,where B0t is a 2×2 lower triangular matrix with ones on the diagonal;8In line with, for example, Cogley and Sargent (2005) and Primiceri (2005), we accordingly rely on a recursive structure in order to identify the orthogonal disturbances to the system. While this assumption can be questioned, it is nevertheless a benchmark assumption in empirical macroeconomic analysis using VARs.μt is a 2×1 vector of time‐varying intercepts; B1t,…,Bpt are 2×2 matrices with the parameters describing the dynamics of the BVAR; and εt is a 2×1 vector of disturbances, εt∼N(0,Σt), where Σt=diag(exp(h1t),exp(h2t)).We gather μt and the free parameters of the matrices B0t,…,Bpt in the vector θt=(θ1t′θ2t′)′, where θ1t contains the parameters of the equation for the unemployment rate and θ2t contains the parameters of the inflation equation. The processes for the time‐varying parameters and log‐volatilities are then specified as random walks,2θt=θt−1+ηt,3ht=ht−1+ζt,where ηt∼N(0,Σθ) and ζt∼N(0,Σh); Σθ and Σh are both diagonal matrices.The model described above – which has time‐varying parameters and stochastic volatility – is the most general model we consider in our analysis. We largely treat stochastic volatility as a given feature of the model. This is based on both a growing empirical literature pointing to the importance of allowing for time‐varying shock volatility in macroeconomics9See, for example, Cogley and Sargent (2005), Sims and Zha (2006), Clark (2011), Stock and Watson (2012), Franta et al. (2014), Akram and Mumtaz (2019), Koop and Korobilis (2019), and Karlsson and Österholm (2020a, 2020b).and empirical analysis that we conduct ourselves (see Table B1 in the Online Appendix and the discussion in footnote 12). Considering time‐varying parameters, we note that this is an appealing way to allow for a potentially unstable Phillips curve, something which has found some support in earlier research. For example, King and Watson (1994, p. 209), found “important evidence of econometric instability over subsamples”. Also Stock and Watson (1999) found evidence of an unstable Phillips curve.10In addition, Gallegati et al. (2011) found evidence of an unstable (wage) Phillips curve in the United States. Interestingly, their analysis pointed to the Phillips curve having become steeper since the mid‐1990s.By imposing various restrictions on the most general model, we can assess how important different features are. For example, if θ1t is constant, we obtain a BVAR with stochastic volatility where the equation for the unemployment rate has constant parameters but the equation for inflation has time‐varying parameters. If we instead make θt and ht constant, we obtain a traditional BVAR with time‐invariant parameters and a covariance matrix. As pointed out above, we largely take stochastic volatility as a given feature. Under the assumption of stochastic volatility, we accordingly compare four different BVAR specifications: (i) the parameters of both equations are constant (θt constant); (ii) time variation in the parameters of the equation for the unemployment rate (time variation in θ1t and θ2t constant); (iii) time variation in the parameters of the equation for inflation (θ1t constant and time variation in θ2t); and (iv) time variation in the parameters of both equations (time variation in θt). Here it can be noted that our definition of stability of the Phillips curve refers to all parameters in the inflation equation (i.e., the entire vector θ2t); instability can, accordingly, be due to the intercept, the coefficients on lags of inflation, or the coefficients on the (contemporaneous and lagged) unemployment rate.11An alternative to this would be to focus on the coefficients on the (contemporaneous and lagged) unemployment rate in the inflation equation. In the terminology used in this paper, this would be seen as assessing whether the slope of the Phillips curve has been stable. See Section 3.4 and footnote 14 in particular for further discussion concerning this.Priors and estimationThe choice of prior distributions and parameters of the priors can have a substantial effect on model comparisons based on marginal likelihoods, and the priors must be set up with some care. We need, in particular, to ensure that the information content is as equivalent as possible for the different models and that the prior specification does not put one of the models at a disadvantage. For example, putting a tight prior on the regression parameters (initial state in the case of time‐varying parameters) that does not agree with the data will hurt the marginal likelihood for constant parameter models more than models with time‐varying parameters.We address these concerns by specifying a proper but uninformative prior on the regression parameters or their initial states when the parameters are time‐varying. This is achieved through a normal, N(0,5I), prior on the initial states of the regression parameters, θ1,0and θ2,0. For the initial state of the log‐volatilities we use a normal prior (i.e., the variance is lognormal), hi,0∼N(μi,0.25), with μi selected to set the prior mean of exp(hi,0) equal to the residual variance of a univariate AR(p) with constant parameters. As we conduct model comparison between models with and without time‐varying parameters, the specification of the prior for the variance of the innovations to θi,t is important. In addition, stochastic volatility can to some extent compensate for a lack of time variation in the parameters and we therefore also need to be careful with the prior for the variance of the innovations to hi,t. In both cases, we use independent inverse Gamma, iG(v,S) priors for the diagonal elements of Σθ and Σh. The shape parameter, v, is fixed at 5 and the scale parameter, S, or equivalently the prior mean of the variance, S/(v−1), is selected in an empirical Bayes fashion. Starting with the prior for the variance of the innovations to the log‐volatilities, we use a grid search to find the value of S that maximizes the marginal likelihood for the model with constant parameters. This results in a prior mean for the variance of 0.1. Next, we allow the prior means of the variances of the innovations to the regression parameters to be different for the constant terms and other parameters. These are then selected in a grid search to maximize the marginal likelihood in the model with stochastic volatility and time variation in the parameters of both equations. The grid search leads to prior means of 0.01 and 0.00005 for the variance of the innovations to the constant terms and other parameters, respectively.We use the Markov chain Monte Carlo (MCMC) sampler developed by Chan and Eisenstat (2018) for posterior inference and we refer to their paper for details. Throughout we use 200,000 draws from the sampler with 50,000 draws as burn‐in and retain every tenth draw for posterior inference. Except for marginal likelihood calculations, draws with non‐stationary regression parameters are discarded.Model selectionIn order to distinguish between the models and gain insight into the stability over time of the US Phillips curve, we assess the fit of the different models using the marginal likelihood. In a Bayesian setting, the marginal likelihood is the appropriate measure of how well the model (and prior) agrees with the data.Given the triangular specification, we can write the marginal likelihood for the models with time‐varying parameters and stochastic volatility as4m(y|M)=∏i=12[∫p(yi|θi,hi,ξi,M)p(θi|ξi,M)×p(hi|ξi,M)p(ξi|M)dθidhidξi],where ξi collects the parameters θi,0, hi,0, σi,j,θ2, and σi,h2 of the state equations (2) and (3). The marginal likelihood is then estimated, equation by equation, using the method of Chan and Eisenstat (2018).Empirical analysisDataData on the seasonally adjusted PCE deflator and unemployment rate – ranging from 1990Q1 to 2019Q4 – were sourced from the FRED database of the Federal Reserve Bank of St Louis. PCE inflation is calculated as πt=100(Pt/Pt−4−1) where Pt is the PCE deflator at time t. Data are shown in Figure 1.1FigureData Notes: Both variables are measured in percent.We believe that 1990Q1 is a reasonable starting point for our analysis. By choosing a sample that is too long, we would arguably stack the analysis towards finding time variation in the parameters. For example, if the starting point for our sample was set to 1970Q1, we would include the stagflation of the 1970s, which almost surely was characterized by different time‐series properties. Even starting in 1980Q1 could be problematic in terms of giving the model with time‐invariant parameters (and covariance matrix) a fair chance, seeing that the early 1980s was when the “Volcker disinflation” took place (for a discussion, see, e.g., Goodfriend and King, 2005). There is a fairly substantial literature claiming that the inflation process changed its mean and/or dynamics in the 1970s and early 1980s (see, e.g., Kozicki and Tinsley, 2005; Cogley and Sbordone, 2008; Beechey and Österholm, 2012). Regarding the end point of the sample, we choose 2019Q4 in order to avoid the period associated with the COVID‐19 pandemic.Using the sample in this paper, we hence estimate the model over a period where it would not be unrealistic to find stability; as pointed out above, it has been suggested by, for example, Blanchard et al. (2015) that the Phillips curve largely has been stable since the beginning of this sample.Is there time variation?We next estimate and compare the four different BVARs with stochastic volatility. Lag length is in all cases set to p=4. The log marginal likelihoods from the estimation of the models are shown in Table 1.12Table B1 in the Online Appendix gives the log marginal likelihoods for the corresponding four models estimated under the assumption of a time‐invariant covariance matrix (i.e., homoscedastic errors). As can be seen, also in this set‐up, the model with time variation in the parameters of the equation of the inflation equation is judged as the best. However, it should be noted that all models have substantially lower marginal likelihoods than the models with stochastic volatility; this points to the importance of stochastic volatility in modelling the Phillips curve and we accordingly focus on models with this feature.1TableLog marginal likelihood for different BVAR specifications with stochastic volatilityModelLog marginal likelihoodTime variation in parametersi−82.4Both equations constantii−88.0Time variation in equation for unemployment rateiii−78.7Time variation in equation for inflationiv−84.2Time variation in both equationsNotes: The table gives the natural logarithm of the marginal likelihood of the models. The log marginal likelihood for the best model is shown in bold.As can be seen, the marginal likelihood is highest for the model with time variation in the parameters of the inflation equation only, that is, model iii. The marginal likelihoods clearly indicate that model iii is the preferred model. Using the scale of two times the difference in log marginal likelihood and the terminology of Kass and Raftery (1995, p. 777), the evidence in favour of model iii is “strong” or “very strong”, regardless of which model it is compared against.The fact that the parameters of the inflation equation of the VAR appear to be time‐varying indicates that the Phillips curve is not stable over time. We next illustrate what this time variation looks like by examining the properties of model iii in more detail.Impulse‐response functionsFigure 2 shows the impulse‐response function from the model with time variation in the parameters of the inflation equation only (i.e., model iii), which describes the effect of a shock to inflation on inflation itself. We initially discuss this impulse‐response function as it clearly shows why both time‐varying parameters and stochastic volatility are relevant features when modelling this bivariate system.2FigureImpulse‐response function for model iii: the effect of a shock to inflation on inflation Notes: The size of the impulse is one standard deviation. Effect on inflation in percentage points on vertical axis. Horizon in quarters and time on horizontal axes.As can be seen from the figure, the estimated standard deviation of the shocks to the inflation equation – that is, [exp(ĥ2t)]0.5 – varies a fair bit over time. Between 1996 and 1998, in what can be described as the peak of the Great Moderation, it was as low as 0.10. It then increased almost continuously until 2009Q1 when it reached a value of 0.74. Since then, the volatility of the shocks has decreased and it is by the end of the sample estimated to be approximately 0.20.Figure 2 also illustrates that the dynamic properties of the system have changed. The variation in these impulse‐response functions indicates that there is non‐negligible time variation in the coefficients of the model, which leads us to question the stability of the US Phillips curve. This is also confirmed when looking at the estimated coefficients of the inflation equation (see Figure A1 in the Appendix).We turn to the impulse‐response function that perhaps is of primary interest to us, namely the effect that shocks to the unemployment rate have on inflation; this can be found in Figure 3. As is clearly shown in the figure, a shock to the unemployment rate has a negative effect on inflation. The effect varies somewhat over time. This is because of the time‐varying parameters of the inflation equation as well as the fact that the volatility of the shock to the unemployment rate changes over time.13The standard deviation of this shock has ranged between 0.10 and 0.37 during the sample. The impulse‐response function showing the effect of a shock to the unemployment rate on the unemployment rate itself is given in Figure B1 in the Online Appendix. Figure B2 in the Online Appendix gives the impulse‐response function showing the effect of a shock to inflation on the unemployment rate.3FigureImpulse‐response function for model iii: the effect of a shock to the unemployment rate on inflation Notes: The size of the impulse is one standard deviation. Effect on inflation in percentage points on vertical axis. Horizon in quarters and time on horizontal axes.It is, however, difficult to assess whether the slope of the Phillips curve has changed markedly only through looking at the impulse‐response functions. Therefore, we now turn to a different way of addressing that issue.The slope of the Phillips curveOne measure of the slope of the Phillips curve that is used in the empirical literature is the sum of the coefficients on the lags of the unemployment rate in the inflation equation when looking at the reduced form of the VAR (see, e.g., Knotek and Zaman, 2017; Karlsson and Österholm, 2020b).The reduced form is achieved by pre‐multiplying the specification in equation (1) with B0t−1. This generates the following form of the model,5yt=δt+A1tyt−1+⋯+Aptyt−p+et,where δt=B0t−1μt, Ait=B0t−1Bit, and et=B0t−1εt. The estimated sum of the coefficients on the lags of the unemployment rate in the inflation equation is shown in Figure 4.4FigureEstimated slope of the Phillips curve for model iii Notes: Slope is measured as the sum of the coefficients on the lags of unemployment in the inflation equation in equation (5). Coloured band is 68 percent equal tail credible interval.As can be seen, the point estimate indicates a non‐negligible amount of variation within the sample. At the beginning of the sample, the slope is approximately zero but it then falls and reaches a low of −0.14 in 1998. Between 2005 and 2013, the slope is more moderate, hovering around −0.07. We should of course keep in mind that the point estimate is associated with a fair amount of uncertainty as illustrated by the 68 percent credible interval; changes should accordingly not be over‐interpreted.14Additional analysis conducted – not reported in detail but available from the authors on request – does, however, suggest that the slope of the Phillips curve has indeed changed over time. In this analysis, we estimate a version of model iii where we make the model have very close to no time variation for the coefficients on the (contemporaneous and lagged) unemployment rate in the inflation equation; this is achieved by placing very tight priors on the relevant elements in Σθ, thereby in practice making them zero. Marginal likelihood calculations for this model show that the log marginal likelihood is lower (−81.8) than that of the original version of model iii presented in Table 1.It is nevertheless our best estimate, and therefore interesting to note, that the Phillips curve might have been flatter around the financial crisis than it typically has been during this sample.15Comparing our estimates in Figure 4 to the findings from a few other recent studies, it can be noted that Blanchard et al. (2015) found a somewhat steeper Phillips curve for the United States; estimating a single‐equation regression model with constant parameters employing quarterly data from 1990 to 2014 – and a shorter subsample ranging from 2007 to 2014 – the estimated slope was around −0.25. Knotek and Zaman (2017) report estimates from just two points in time from their BVAR with time‐varying parameters – 1999Q3 and 2017Q3 – both of which are associated with a quite modest slope (−0.07 and −0.06, respectively). Their numbers are fairly close to that found by Murphy (2018). Based on a single‐equation regression model with constant parameters and a sample of quarterly data ranging from 1990 to 2014, he found a slope of −0.07. Karlsson and Österholm (2020b) report a time‐varying slope of the Swedish Phillips curve that, while generally somewhat steeper, shows a similar pattern to Figure 4. One should of course be careful when making comparisons seeing that the specifications, estimation methods, and data used vary between studies.It hence seems that a flatter Phillips curve might have contributed to the “surprisingly high” inflation outcomes in the aftermath of the financial crisis.Before leaving the discussion of the slope of the Phillips curve behind, it is also important to be point out that the Phillips curve has not been flatter than usual during the most recent years. Since 2015, the slope has been between −0.10 and −0.13, which is actually quite steep relative to the rest of the sample; this steepness is on a par with that of the late 1990s. This appears, at least to some extent, to contradict a flatter Phillips curve as an explanation for the weak inflation outcomes that the Federal Reserve has been struggling with during this period.Trend inflationVarious properties of the model have been illustrated above and provided information concerning the stability of the Phillips curve. However, in a model with time‐varying parameters, there is information based on the change in parameters that occurs over time, which is not communicated through the above tools. As this can be of importance when it comes to the analysis of the Phillips curve, we now address the issue of the estimated trend inflation from the model.We define “trend inflation” as the level to which the inflation forecasts from the model will converge; this definition is in line with, for example, that employed by Faust and Wright (2013) and Clark and Doh (2014).16In addition, what we denote as trend inflation has the same interpretation as what Cogley and Sargent (2001, 2005) define as core inflation.Perhaps more intuitively, it can be thought of as the model's long‐run expectation of inflation. Employing the reduced form of the VAR in equation (5), we rewrite it in companion form as6y˜t=δ˜t+Aty˜t−1+e˜t,with y˜t=(yt′,yt−1′,…,yt−p+1′)′, δ˜t=(δt′,0′,…,0′)′,7At=A1tA2t⋯Ap−1,tAptI0⋯00⋱⋮⋱⋮0⋯0I0,and e˜t=(et′,0′,…,0′)′, and solve for trend inflation as the second element of ϕt=(I−At)−1δ˜t.The median estimated trend inflation at each point in time from the model is shown in Figure 5. This shows that trend inflation was fairly high when the Great Recession hit the US economy in the second quarter of 2007, namely 2.5 percent; it had reached this value after drifting up from 1.3 percent in 1998. Trend inflation also stayed reasonably high until 2008Q3 – when it was 2.5 percent – but then took on a clear downward trajectory, reaching a minimum of 1.2 percent in 2015. It hence seems that part of the high inflation during the Great Recession can be explained by high trend inflation.17This result can be seen as being in the same spirit as the findings presented in Clark (2014) and Watson (2014), who suggested that inflation was kept reasonably high around the Great Recession partly as a result of a well‐anchored trend level of inflation.5FigureEstimated trend inflation from model iii Notes: Trend inflation and inflation are measured in percent. Coloured band is 68 percent equal tail credible interval.Figure 5 also illustrates the stubbornly low inflation during the last few years of our sample. As can be seen from the figure, trend inflation has at the same time been low. The low inflationary pressure that we have seen over the last years has hence been reflected in the estimated parameters of the model and caused estimated trend inflation to decrease noticeably. For the Federal Reserve, this means that while shocks to the unemployment rate might have roughly the same effect now as it has historically, there should be scope to allow for additional inflationary pressures to build up because such pressures could be needed in order to bring trend inflation back up to the target level of 2 percent.18It can be noted that the formal inflation target that the Federal Reserve introduced in 2012 can be seen as being at conflict with models with time‐varying parameters of the type used in this paper. Providing a clear focal point for future inflation, it can be argued that inflation forecasts should converge to the target level of 2 percent and that, for example, models with steady‐state priors of the type suggested by Villani (2009) or Clark (2011) should be considered. However, we believe that the fact that the Federal Reserve's formal target has been active during only a reasonably small part of our sample is a good reason to employ our chosen framework.Summing up the empirical analysis so far, we note that the time‐varying parameters help explain the surprisingly high inflation around the financial crisis. There is an indication that the Phillips curve might have been somewhat flatter around the financial crisis and also that trend inflation was high.Conditional forecastsOne common usage of econometric models of the type investigated above is forecasting. In many cases, the models are simply employed to generate an endogenous forecast. It is, however, also common – particularly at policy institutions – to use the models for conditional forecasting.19For contributions in the field of conditional forecasting – dealing with methodological issues and/or empirical applications – see, for example, Waggoner and Zha (1999), Hamilton and Herrera (2004), Österholm (2009), Baumeister and Kilian (2013), Clark and McCracken (2014), and Giannone et al. (2014).Concerning the US Phillips curve as studied in this paper, we believe that a conditional forecasting exercise is of certain interest as it can provide us with information regarding the missing disinflation associated with the Great Recession. In line with the analysis presented above, we conduct this exercise using model iii.Forecasts of inflation are generated for the period 2008Q1–2010Q4 conditional upon an assumed path for the unemployment rate; more specifically, this path is given by the actual unemployment rate. This will show us how the model would have predicted future inflation given perfect foresight regarding the unemployment rate. The conditional forecasts are generated using a method that we believe represents common current practice, namely that of Waggoner and Zha (1999).We generate the conditional forecasts based on three sets of parameters. First, to obtain more precise estimates of the parameter values as of 2007Q4, we estimate the model on the full sample rather than data up to 2007Q4 where the estimates would suffer from end‐of‐sample uncertainty. These conditional forecasts are given by the red dashed line in Figure 6. Generating the forecasts in this manner obviously does not correspond to how a conditional forecasting exercise would have been conducted in real time; rather, it is based on today's best estimate of the parameters of the model at the beginning of the financial crisis.20It should be noted that when the conditional forecasts are generated, the parameters and volatilities are kept fixed at their 2007Q4 values.Second, we construct forecasts using data only up until 2007Q4 in order to see if this affects our conclusion. Using this shorter sample, we provide a better approximation to what the model would have suggested in real time.21We do not use real‐time data though, so this does not show exactly what the model would have predicted in real time. However, given the purpose of the exercise, we do not believe that this is strictly necessary.The conditional forecasts from this exercise are given by the blue dashed line with asterisks in Figure 6.22We also generate conditional forecasts using 2008Q4 rather than 2007Q4 as the date from which the forecasts originate; forecasts are accordingly generated for the period 2009Q1–2010Q4 (using the parameter estimates associated with 2008Q4). Regardless of whether we use parameter estimates based on the full sample or only on data up until 2008Q4, the results are similar. In neither case do we find evidence of missing disinflation. (Detailed results are not reported here but are available from the authors on request.)The third set of parameters we employ comes from the estimation of the model using the full sample; however, rather than using the parameter estimates associated with 2007Q4, we use the estimates associated with 1997Q4. This can be seen as providing further information on the stability of the Phillips curve. The conditional forecasts from this exercise are given by the blue dashed line with asterisks in Figure 7; for comparison, the forecasts from the model using the parameter estimates associated with 2007Q4 (estimated using the full sample) are also shown.6FigureModel‐based inflation forecasts using model iii Notes: Median forecasts. Parameter estimates for long sample forecast use full sample and short sample forecasts use data up to 2007Q4. Parameter estimates used for forecasting are dated 2007Q4.7FigureModel‐based inflation forecasts using model iii: alternative date of parameters Notes: Median forecasts. Parameter estimates use full sample. Parameter estimates used for forecasting are dated 1997Q4 and 2007Q4.Note that in order to see the effect that conditioning on the unemployment rate has, we also provide the model's unconditional forecasts. These forecasts – given by the dot‐dashed lines (red, and blue with asterisks) – are also shown in Figures 6 and 7.Looking at the conditional forecasts based on parameter estimates from 2007Q4 using the full sample (i.e., the red dashed line in Figures 6 and 7), it can be seen that the model suggests a fall in inflation in response to the increasing unemployment rate. However, the predicted fall in inflation is actually less deep than the outcome. While actual inflation hit its lowest point in 2009Q3 at −1 percent, the conditional forecast for the same quarter was 0.7 percent. Comparing the full path of the forecasts to the outcomes, it is clear that the model did not want substantially lower inflation at any time point than what was actually the case (even though it can be noted that the predictions for 2009Q4 and 2010Q1 are somewhat lower than the outcomes). We accordingly conclude that the model does not suggest that there was a missing disinflation associated with the financial crisis. As can be seen from Figure 6, this conclusion does not change if the shorter sample is used for parameter estimation.23We also generate conditional forecasts from the BVAR with time‐invariant θt – that is, model i – in order to assess the importance of the time‐varying parameters. These results are presented in Figure B3 in the Online Appendix. As can be seen, the story told is very much the same as when model iii was used. The model does not suggest a more sizeable drop in inflation in response to the increasing unemployment rate than what was actually the case.A somewhat different picture is, however, painted when looking at the conditional forecasts from the model when the parameter estimates associated with 1997Q4 are employed. As can be seen from Figure 7, these forecasts are substantially lower than those based on the parameter estimates associated with 2007Q4; we also note that they consistently lie below the actual values. This shows that the time variation in the parameters of the model that we have established is quantitatively meaningful. In addition, if one assumes that these parameter estimates also reflect views held by economists, these results can be interpreted as favouring a claim of missing disinflation; if someone believed that the economy in 2007 had the structure of 1997 (as given by the VAR), a more substantial fall in inflation than what actually materialized would have been expected.Sensitivity analysisHaving conducted the above analysis, we finally want to assess how sensitive our results are with respect to some key concepts, namely our choice of priors and the data used. In this section, accordingly, we first vary some of our prior parameters (using the data employed in the main analysis). We then conduct analysis using alternative measures of inflation and, finally, consider an alternative measure of capacity utilization by replacing the unemployment rate with the output gap.24In addition, we have also conducted some sensitivity analysis using a trivariate system, which we only briefly report in this footnote. (Further details are available from the authors on request.) Modelling the unemployment rate, PCE inflation, and the three‐month Treasury bill rate, we find that a specification with time variation only in the inflation equation is ranked first (of the eight assessed models); the conditional forecasts from this model do not suggest that there was any missing disinflation.PriorsThe choice of prior parameters affects the marginal likelihood and can influence the ranking of the models when comparing marginal likelihoods. In this sensitivity analysis, we focus on the prior parameters that govern time‐varying parameters and time‐varying volatilities as they are the key ones in the present application. More specifically, we modify the prior means of the variance of the changes in the parameters and log‐volatilities by halving or doubling Sθ and Sh and re‐estimate the models using the same data as we employed in our main analysis (i.e., the data described in Section 3).The results from this exercise can be summarized very briefly. The finding from our main analysis is confirmed as the model with time variation in the parameters in the inflation equation only (model iii) is ranked first in all cases (see Table B2 in the Online Appendix). We thus conclude that this is a robust finding with respect to the choice of priors.Quarter‐on‐quarter inflationWhen modelling inflation, it is not obvious that it is the annual change in the PCE deflator that we want to use as our measure. Another relevant alternative is to use the (annualized) quarterly change; that is, we define PCE inflation as πt=400(Pt/Pt−1−1). Figure A2 in the Appendix presents the data on quarter‐on‐quarter PCE inflation. Marginal likelihoods from the four different models estimated using this inflation measure can be found in Table A1 in the Appendix.As can be seen from Table A1, the preferred model is that with constant parameters in both equations (model i). This specification has a somewhat higher marginal likelihood than the specification with time variation in the inflation equation (model iii). While the evidence in favour of model i against model iii – again using the scale and terminology of Kass and Raftery (1995, p. 777) – is “not worth more than a bare mention”, it is nevertheless the best performing model, and we conclude that the Phillips curve in this case is judged to have been stable.In terms of the model's conditional forecasts, this model does not want a substantial decrease in inflation in response to the rising unemployment rate. In fact, the decrease in inflation is very modest (see Figure B4 in the Online Appendix). This is because the Phillips curve is extremely flat; the point estimate of the slope – based on the full sample – is only −0.01. Given that the 68 percent credible interval for the slope is (−0.08, 0.05), one can even question the existence of the Phillips curve in this case.25The model's impulse‐response function of inflation with respect to shocks to the unemployment rate is shown in Figure B5 in the Online Appendix.Core PCE inflationWhen discussing monetary policy, it is common to relate central banks' actions to “core” measures of inflation – that is, inflation measures that have had some volatile components removed. As we have used PCE inflation in our analysis above, we here investigate if our results are robust to using core PCE inflation instead. In constructing this measure, food and energy prices have been removed from the PCE deflator. Inflation is calculated as πtcore=100(Ptc/Pt−4c−1), where Ptc is the core PCE deflator at time t. Data on core PCE inflation are shown in Figure A2 in the Appendix and we again present marginal likelihoods from the four different models in Table A1 in the Appendix.The preferred model in this setting is again model iii (i.e., the model with time variation in the parameters of the inflation equation); the evidence in favour of it ranges from “strong” to “very strong” (Kass and Raftery, 1995, p. 777).We find that the picture painted is similar to that in our main analysis. The impulse‐response function of core PCE inflation with respect to a shock to the unemployment rate shows a qualitatively similar picture to what we saw in Figure 3 when PCE inflation was used (see Figure B6 in the Online Appendix). Turning to the slope of the Phillips curve (Figure B7), we find that using core PCE inflation does not change our main findings from above, though it can be noted that the slope is now somewhat steeper in general (judging by the point estimates). Finally, the conditional forecasts (Figure B8) show that the model predicts a decrease in inflation that was smaller than what materialized in this setting as well. We again therefore conclude that there was no missing disinflation.GDP deflator inflationThe last inflation measure that we analyse is based on the seasonally adjusted GDP deflator and is also used fairly frequently in the empirical literature (see, e.g., Rudebusch and Svensson, 1999; Lindé, 2018). Following Rudebusch and Svensson (1999), we define inflation as πtGDP=400ln(PtGDP/Pt−1GDP) where PtGDP is the GDP deflator at time t. The data are shown in Figure A2 in the Appendix and marginal likelihoods from the four different models are again presented in Table A1 in the Appendix. As can be seen, we also here find that model iii is preferred by the data; the evidence in favour of it ranges from “positive” to “very strong” (Kass and Raftery, 1995, p. 777).The key properties of the model are also in this case similar to what we have found previously. The impulse‐response function of inflation with respect to a shock to the unemployment rate is again similar to that shown in Figure 3 (see Figure B9 in the Online Appendix). Further, we note that the slope of the Phillips curve (Figure B10) resembles that of the benchmark model (Figure 4), though it is somewhat less steep in general. Concerning the conditional forecasts from the model (Figure B11), these largely confirm our previous findings; even if the model – when using the parameters from the shorter sample – wanted a somewhat lower inflation than the actual outcome during 2009 and 2010, it does not signal that there was an obvious missing disinflation.26We also conducted some additional analysis – not reported in detail but available from the authors on request – relating our work more closely to that of Rudebusch and Svensson (1999). This was done by using data on the output gap (as described in Section 4.5) and GDP deflator inflation; in line with Rudebusch and Svensson, both series were demeaned. We restricted the inflation equation so that the only coefficients that were allowed to be non‐zero were those on lags of inflation and the first lag of the unemployment rate; all other coefficients (i.e., the intercept and lags 0, 2, 3, and 4 on the output gap) were forcefully shrunk to zero using very tight priors. In this setting, the marginal likelihoods from the four models were extremely similar, pointing to the data being unable to distinguish between the models. All four models had similar key properties though. A shock to the output gap (with sign reversed) generated lower inflation. The slope of the Phillips curve was somewhat flatter than in our main analysis, approximately −0.04. The conditional forecasts from the models did not indicate that there was any missing disinflation.Output gapThe Phillips curve is often interpreted as a general relation between the level of real activity in the economy and the inflation rate. Accordingly, the unemployment rate is not the only relevant measure and we investigate how robust the results are to a change of measure of real activity. We thus replace the unemployment rate in the model with the output gap (and return to inflation being measured as the annual change in the PCE deflator – that is, πt=100(Pt/Pt−4−1)). Data on the output gap are based on the Congressional Budget Office's estimate of potential GDP and are shown in Figure A2 in the Appendix. Note that the output gap is given with reversed sign in order to make comparisons with the models that relied on the unemployment rate easier.We present marginal likelihoods from the four different models in Table A1 in the Appendix. The preferred model is once again model iii (i.e., the model with time variation in the parameters of the inflation equation). In this case, the second‐best model is that with time variation in the parameters of both equations (i.e., model iv). The evidence in favour of model iii against model iv is only just “positive” (Kass and Raftery, 1995, p. 777); against the other two models, the evidence is “strong”.Turning to the properties of the model, the impulse‐response function of inflation with respect to a shock to the output gap is qualitatively similar to that presented in Figure 3 (see Figure B12 in the Online Appendix). The slope of the Phillips curve is fairly stable over time (Figure B13). Finally, the conditional forecasts (Figure B14) largely confirm our previous findings. While the model wanted a somewhat lower inflation than actual between 2009Q4 and 2010Q2, the general message is that the model wanted a substantially higher inflation than the outcome. The overall picture is therefore hardly one of missing disinflation. We accordingly conclude that it seems to be a fairly robust finding that our BVAR models do not support a claim of a missing disinflation in association with the Great Recession.ConclusionsThe Phillips curve is a popular tool in macroeconomics, which, among other things, is used to assess inflationary pressure in the economy. In this paper, we have added empirical evidence concerning the properties of the US Phillips curve by analysing different specifications of BVAR models.Conducting Bayesian model selection using recently developed methods for this purpose, we find that the inflation equation of the VAR appears to be subject to time‐varying parameters. Our results also indicate that the Phillips curve might have been somewhat flatter between 2005 and 2013 than in the decade preceding that period. This fact – together with high trend inflation – can help explain why inflation was high around the Great Recession.However, our findings from the conditional forecasting exercises conducted suggest that as far as the models are concerned, there might not have been that much of a missing disinflation associated with the Great Recession. Conditioning on the increase in the unemployment rate that actually took place, different model specifications yield somewhat different results. In general though, the models did not predict that inflation should have been lower than what turned out to be the case.From a policy perspective, our results imply two things. First, policy institutions and forecasters need to consider the fact that both dynamic relations and the volatility of shocks might be changing over time when building econometric models. Because methods for model selection have been improved, it is also the case that the econometric modelling choice can stand on firmer ground nowadays; where previously an econometrician largely had to make an assumption regarding the relevance of different specifications, they can now be formally compared. Second, while inflation in the United States recently has increased quite rapidly, it was stubbornly low the years before that. This low inflation appears, at least to some extent, to be due to low inflationary pressures that have affected the estimated trend inflation. Looking at the present unemployment rate, it is still somewhat elevated due to the COVID‐19 pandemic, but it has come down fairly rapidly from its peak in April 2020. Our findings indicate that there could be scope for the federal funds rate to be quite low even in light of a low unemployment rate, in order to let inflationary pressures build up in the economy and reliably have inflation close to the target level.AppendixA1FigureEstimated coefficients of the inflation equation of model iii Notes: Time on horizontal axes.A2FigureData on variables used for sensitivity analysis Notes: Core PCE inflation and annualized quarter‐on‐quarter PCE inflation are measured in percent. GDP deflator inflation is 400 times the log difference of the series. 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Journal

The Scandinavian Journal of Economics – Wiley

Published: Jan 1, 2023

Keywords: Inflation; model selection; stochastic volatility; time‐varying parameters; unemployment

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