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A. Clare, Zacharias Psaradakis, Stephen Thomas (1995)
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Seasonality in the Risk-Return Relationship: Some International EvidenceJournal of Finance, 42
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Abstract This paper examines the relationship between seasonality, stock returns and the macroeconomy using a multifactor model of stocks returns. Observed seasonal patterns in excess returns are found to be a result of seasonality in excess expected returns. By utilising a multifactor model of stock returns, these higher returns are found to be a compensation for risk associated with a number of macroeconomic factors at certain times of the year. It is well known that stock returns exhibit seasonal patterns.1 What is less well known is why these seasonal patterns exist. In the United States, the January effect has been attributed to tax loss selling (see, for example, Roll (1983) and Reinganum (1983)). However, this explanation has been questioned by Chan (1986) who shows that it is not consistent with a model that ascribes the January seasonal to optimal tax trading. In addition, given that the end of the tax year in the United Kingdom is April, the tax loss selling explanation for the January seasonal pattern observed in the United Kingdom is at best weak, relying on contagion from the US market. An alternative explanation is that seasonal patterns are due to the risk‐return structure. However, Clare et al. (1995) (henceforth CPT) find that dummy variables in the months of January, April, September and December all add to the explanatory power of a conditional CAPM. On the other hand, Tinic and West (1984) find that when an unconditional CAPM is used to model the risk‐return structure in the United States, the only month with a statistically significant risk premium is January. These findings present a puzzle to economists since there is no theoretical justification within the CAPM for the existence of seasonal patterns in stock returns, whether the effect is manifest in the risk‐return structure or alternatively it lies outside of it. Some authors have analysed the issue of seasonality using multifactor models of the risk‐return relationship. Attempts have been made to relate the January seasonal in the United States to observed risk factors. Chang and Pinegar (1990) and Kramer (1994) both estimate multifactor models that allow for different risk premia in January and the rest of the year. While the results are encouraging in the sense that they show the estimated risk premia are different in seasonal and non‐seasonal periods, they offer little scope in interpreting the results. For example, Chang and Pinegar (1990) argue that interpretation is difficult since considerable problems of multicollinearity in the factors and a lack of any theoretical rationale makes interpretations difficult. Consequently, no reason has emerged for this finding.23 The contribution of this paper is to estimate a multifactor model of stock returns using observed factors with the explicit aim of examining whether seasonal patterns in UK stock returns are due to seasonality in expected returns. This issue is important for at least three reasons. First, it is important, not least from the Efficient Market Hypothesis (EMH) standpoint, that we can rationalise observed seasonal patterns. Secondly, if seasonal patterns are related to expected returns through observed risk factors this will have important implications for the specification and inferences of asset pricing models that do not explicitly take account of seasonal patterns in expected returns. Thirdly, understanding seasonal patterns in expected returns will foster our understanding of financial markets and the economy in general. To anticipate the results, employing a multifactor model of returns, January, April and December, as opposed to the rest of the year, are found to have significantly positive expected returns. We attempt to rationalise this finding using evidence regarding a seasonal, yearly, business cycle (see Barsky and Miron (1989)). The remainder of the paper is organised as follows. Section I discusses issues that arise in the specification and estimation of the multifactor model. The factors and their construction are discussed in Section II. Empirical results regarding the presence of seasonal patterns in returns and the existence of seasonality in the risk‐return relationship are presented in Section III. A conclusion is provided in Section IV. I. A Seasonal Multifactor Model of Stock Returns We begin by assuming that the return generating process of assets is governed by a multifactor model of the form (see Ross (1976)) (1) where rt is a N‐vector of returns, ft is a K‐vector of factors, B is a N × K matrix of sensitivities to the factors, et is a N‐vector of idiosyncratic returns and E(r) is a N‐vector of expected returns. In the absence of arbitrage opportunities (see Ross (1976)), the following holds: (2) where λ0 is the risk free return, ιN is a N vector of ones and λ is a K‐vector of prices of risk associated with the factors. The traditional use of this model in empirical tests assumes that expected returns are constant across all months. An alternative representation of the risk‐return relationship can be achieved by estimating two separate models in order to arrive at a model which explicitly allows for the presence of non‐constant seasonal elements of B and λ. For example, assume that there are two seasonal months in actual excess returns4 and the remaining ten months are non‐seasonal. Also, assume that the two seasonal months are a product of the risk‐return relationship. In this case the return generating process (1) can be split into two parts, first a non‐seasonal model which contains data on the returns and factors in non‐seasonal months and a second seasonal model that contains data on returns and factors in seasonal months. Given there is more than one seasonal month, the separate effects of each one on B and λ in the seasonal model can be obtained by premultiplying the seasonal model data with dummy variables for each seasonal month. Consequently, from the first model we will obtain an estimate of B and λ for the non‐seasonal months and from the second model we obtain estimates of B and λ for each individual seasonal month from the seasonal data. Previous empirical studies which have examined multifactor models of the form of (1) usually employ a two‐step estimation procedure where the matrix B is estimated in one period and the associated vector λ is estimated in a subsequent period. This leads to an Errors In Variables (EIV) problem which can severely effect the standard errors (see Shanken (1992)). An alternative methodology is developed in McElroy et al. (1985) and involves the joint estimation of B and λ and hence avoids the EIV problem. Specifically, the model can be written as a system of nonlinear seemingly unrelated regressions (NLSUR) by substituting (2) into (1) and stacking the equations for the N securities, to give (3) where r is an NT vector of returns, λ is a K vector of prices of risk, F is a T × K matrix of observations on the K factors, b is a NK vector of sensitivities, IN is a N × N identity matrix and ⊗ is the Kronecker product operator. The nonlinearity arises from the restrictions the multifactor model specification places on a more general unrestricted linear factor model. The unrestricted linear factor model is (1) with E(r) replaced by a vector of constants, a. The pricing restrictions imposed by the multifactor model hold when the nonlinear, cross equation restrictions a = E(rt) are valid.5 II. The Factors Following Chen et al. (1986) henceforth (CRR) most studies specify the following potential risk factors: unexpected industrial production, shocks to the term structure of interest rates and default risk, the change in expected inflation and unexpected inflation. In addition, we include a measure of unexpected money supply which Beenstock and Chan (1988) find important in the United Kingdom, and shocks in the exchange rate to measure the relative open nature of the UK economy (all data were collected from Datastream). An important issue is the generation of the unexpected components.6 This issue is especially important when analysing seasonality in returns because failure to generate unexpected components that do not have seasonal means will lead to the factors appearing to explain the seasonal variation in returns when in fact they may not. Therefore, in order to arrive at reliable results regarding the seasonality in expected returns the expectations generating process must take account of both seasonal and non‐seasonal patterns in the variables.7 Recent evidence regarding the different possible forms of seasonality can complicate this issue. For example, seasonality in a series can be deterministic, stochastic or slowly changing over time (trigonometric) (see, for example, Harvey and Scott (1994)). Incorrectly specifying the form of seasonality can lead to dynamic misspecification and may result, in our case, in the expectations generating process being misspecified and consequently effecting the form of the unexpected component and thus biasing the results from the multifactor model. To avoid the aforementioned problems we use general time series models which use the Kalman filter to provide an updating scheme for the type of seasonality. An advantage of this approach is that other forms of seasonality such as deterministic seasonality are a special case. For each variable we start by estimating a simple unobserved components models with trigonometric seasonal dummies. Evidence of serial correlation in the residuals results in a more general autoregressive model being estimated. Analysis of the statistical significance of the variance of the trigonometric seasonal dummies determines the form of the seasonal dummies. In terms of estimation, the variance parameters of the model are estimated by maximum likelihood and then the trend and seasonal parameters are extracted by a smoothing algorithm (see Harvey and Scott (1994) and Harvey (1989) for specification and estimation details). The results from estimating the expectations models are available on request. Briefly, the two interest rate variables and the change in expected inflation have slowly changing seasonal patterns while industrial production and unexpected inflation have stochastic seasonal patterns. The money supply has deterministic seasonal patterns. Box‐Ljung Q‐statistics revealed no evidence of autocorrelation of the residuals.8 The implication of these results is that if we find there is seasonality in expected returns this is due to the innovations in the factors and not spurious seasonality caused by the failure to remove observed seasonality adequately in the time series of the factors. The estimated correlation matrix of the unexpected components reveals that the highest correlation is − 0.1886 (p‐value = 0.01) between the term structure and the exchange rate. The only remaining significant correlation is between the term structure and industrial production (– 0.1798, p‐value = 0.02).9 Given the low and almost insignificant correlations, it should be possible to interpret the estimates of the prices of risk independently of each other. III. Risk, Return and Seasonality: Empirical Results Data on fifty‐nine randomly selected, individual stock returns (total returns in excess of the UK 3 month T‐Bill rate) traded on the London Stock Exchange over the period October 1968 to December 1993 were collected. Including an equation for the stock market index gives a system of 60 equations to be estimated. As a consistency check of our data with the results of CPT we regressed the returns of the FT All Share Index on zero‐one dummy variables for January, April, September and December. The results confirm the presence of seasonal patterns in the months of January, April and December, however, we found no evidence of a September seasonal (this result could be a product of the difference in sample periods).10 In order to simplify the estimation of the multifactor model a preliminary analysis of the stability of the factor betas, where the hypothesis tested is that the factor betas are the same in seasonal months as in the non‐seasonal months, is undertaken. Evidence of non‐stable betas would indicate that there is seasonality in the multifactor pricing relationship and hence seasonality in expected returns. Wald tests of the null hypothesis that the betas in non‐seasonal months are the same as the betas in the seasonal months are reported in Table 1. The results indicate that eight of the factors have unstable betas across different months. This provides the first indication that the seasonality in returns may be due to seasonality in the risk‐return relationship. Thus in the seasonal model there are eight factors corresponding to the unstable betas and in the non‐seasonal model there are the seven original factors. Table 1 A Wald Test for the Restriction that the Factor Betas in a Seasonal Month (βm, s) are the Same as the Factor Betas in the Non‐Seasonal Month (βm, ns) Factor . Ho: βm, ns = βm, jan . Ho: βm, ns = βm, apr . Ho: βm, ns = βm, dec . Default risk 33·79 (0·00) 0·91 (0·34) 0·10 (0·75) Money supply 2·37 (0·12) 0·71 (0·39) 13·9 (0·00) Industrial production 23·01 (0·00) 3·99 (0·04) 6·47 (0·01) Term structure of IR 31·67 (0·00) 2·10 (0·14) 0·36 (0·55) Unexpected inflation 5·578 (0·01) 9·67 (0·00) 0·56 (0·61) Change in expected inflation 2·16 (0·141) 0·73 (0·40) 0·43 (0·51) Exchange rate 0·75 (0·38) 0·09 (0·76) 2·09 (0·15) Factor . Ho: βm, ns = βm, jan . Ho: βm, ns = βm, apr . Ho: βm, ns = βm, dec . Default risk 33·79 (0·00) 0·91 (0·34) 0·10 (0·75) Money supply 2·37 (0·12) 0·71 (0·39) 13·9 (0·00) Industrial production 23·01 (0·00) 3·99 (0·04) 6·47 (0·01) Term structure of IR 31·67 (0·00) 2·10 (0·14) 0·36 (0·55) Unexpected inflation 5·578 (0·01) 9·67 (0·00) 0·56 (0·61) Change in expected inflation 2·16 (0·141) 0·73 (0·40) 0·43 (0·51) Exchange rate 0·75 (0·38) 0·09 (0·76) 2·09 (0·15) p values in parentheses. Open in new tab Table 1 A Wald Test for the Restriction that the Factor Betas in a Seasonal Month (βm, s) are the Same as the Factor Betas in the Non‐Seasonal Month (βm, ns) Factor . Ho: βm, ns = βm, jan . Ho: βm, ns = βm, apr . Ho: βm, ns = βm, dec . Default risk 33·79 (0·00) 0·91 (0·34) 0·10 (0·75) Money supply 2·37 (0·12) 0·71 (0·39) 13·9 (0·00) Industrial production 23·01 (0·00) 3·99 (0·04) 6·47 (0·01) Term structure of IR 31·67 (0·00) 2·10 (0·14) 0·36 (0·55) Unexpected inflation 5·578 (0·01) 9·67 (0·00) 0·56 (0·61) Change in expected inflation 2·16 (0·141) 0·73 (0·40) 0·43 (0·51) Exchange rate 0·75 (0·38) 0·09 (0·76) 2·09 (0·15) Factor . Ho: βm, ns = βm, jan . Ho: βm, ns = βm, apr . Ho: βm, ns = βm, dec . Default risk 33·79 (0·00) 0·91 (0·34) 0·10 (0·75) Money supply 2·37 (0·12) 0·71 (0·39) 13·9 (0·00) Industrial production 23·01 (0·00) 3·99 (0·04) 6·47 (0·01) Term structure of IR 31·67 (0·00) 2·10 (0·14) 0·36 (0·55) Unexpected inflation 5·578 (0·01) 9·67 (0·00) 0·56 (0·61) Change in expected inflation 2·16 (0·141) 0·73 (0·40) 0·43 (0·51) Exchange rate 0·75 (0·38) 0·09 (0·76) 2·09 (0·15) p values in parentheses. Open in new tab The estimates of the prices of risk for the seasonal and the non‐seasonal multifactor models are reported in Table 2, panels a and b respectively.11 It is clear that the seasonal months are priced differently from the non‐seasonal months. In January shocks to industrial production and shocks to the default risk are important. In April shocks to inflation are important and in December shocks to industrial production and shocks to the money supply are important. In the non‐seasonal model all factors except industrial production are important. The interesting question that remains is whether or not the expected returns are different in the seasonal months and the non‐seasonal months. Table 3 reports the calculated expected return from the multifactor model in the three seasonal months and the non‐seasonal period along with Fama‐MacBeth t‐ratios. The three seasonal months have positive and statistically significant expected returns. Importantly, these calculated expected returns are similar to the estimated coefficients on the dummy variable in the regression of the returns on seasonal dummies (the coefficients are 4.64 (t = 3.38), 2.71 (t = 1.97) and 2.53 (t = 1.87) for January, April and December). Thus, the seasonal factor model appears to describe the seasonality in UK equity returns well.12 The non‐seasonal months have both a zero expected return and a zero actual return. Table 2 Estimates of the Prices of Risk Factor . Estimate . S.E. . t‐ratio . (a) Seasonal factor model Default risk (Jan) 0·4679 0·2546 1·84 Industrial production (Jan) – 0·3946 0·1972 – 2·00 Industrial production (Dec) – 2·1405 1·2700 1·69 Money supply (Dec) – 2·7140 0·1542 1·76 Unexpected inflation (Apr) – 1·5596 0·8729 – 1·79 (b) Non‐seasonal factor model Default risk – 1·0478 0·4930 – 2·13 Money supply – 1·2985 0·6199 – 2·09 Term structure of IR – 1·5436 0·7722 – 2·00 Unexpected inflation 1·4963 0·7230 2·07 Exchange rate 0·9300 0·5024 1·85 Change in expected inflation – 1·5436 0·7641 – 1·98 Factor . Estimate . S.E. . t‐ratio . (a) Seasonal factor model Default risk (Jan) 0·4679 0·2546 1·84 Industrial production (Jan) – 0·3946 0·1972 – 2·00 Industrial production (Dec) – 2·1405 1·2700 1·69 Money supply (Dec) – 2·7140 0·1542 1·76 Unexpected inflation (Apr) – 1·5596 0·8729 – 1·79 (b) Non‐seasonal factor model Default risk – 1·0478 0·4930 – 2·13 Money supply – 1·2985 0·6199 – 2·09 Term structure of IR – 1·5436 0·7722 – 2·00 Unexpected inflation 1·4963 0·7230 2·07 Exchange rate 0·9300 0·5024 1·85 Change in expected inflation – 1·5436 0·7641 – 1·98 Open in new tab Table 2 Estimates of the Prices of Risk Factor . Estimate . S.E. . t‐ratio . (a) Seasonal factor model Default risk (Jan) 0·4679 0·2546 1·84 Industrial production (Jan) – 0·3946 0·1972 – 2·00 Industrial production (Dec) – 2·1405 1·2700 1·69 Money supply (Dec) – 2·7140 0·1542 1·76 Unexpected inflation (Apr) – 1·5596 0·8729 – 1·79 (b) Non‐seasonal factor model Default risk – 1·0478 0·4930 – 2·13 Money supply – 1·2985 0·6199 – 2·09 Term structure of IR – 1·5436 0·7722 – 2·00 Unexpected inflation 1·4963 0·7230 2·07 Exchange rate 0·9300 0·5024 1·85 Change in expected inflation – 1·5436 0·7641 – 1·98 Factor . Estimate . S.E. . t‐ratio . (a) Seasonal factor model Default risk (Jan) 0·4679 0·2546 1·84 Industrial production (Jan) – 0·3946 0·1972 – 2·00 Industrial production (Dec) – 2·1405 1·2700 1·69 Money supply (Dec) – 2·7140 0·1542 1·76 Unexpected inflation (Apr) – 1·5596 0·8729 – 1·79 (b) Non‐seasonal factor model Default risk – 1·0478 0·4930 – 2·13 Money supply – 1·2985 0·6199 – 2·09 Term structure of IR – 1·5436 0·7722 – 2·00 Unexpected inflation 1·4963 0·7230 2·07 Exchange rate 0·9300 0·5024 1·85 Change in expected inflation – 1·5436 0·7641 – 1·98 Open in new tab Table 3 Estimates of the Expected Returns from the Non‐Seasonal and the Seasonal Multifactor Model . January . April . December . Non‐seasonal . Mean 3·3658 2·8523 2·6131 – 0·2285 t‐ratio 3·249 3·050 2·766 – 0·61 . January . April . December . Non‐seasonal . Mean 3·3658 2·8523 2·6131 – 0·2285 t‐ratio 3·249 3·050 2·766 – 0·61 Open in new tab Table 3 Estimates of the Expected Returns from the Non‐Seasonal and the Seasonal Multifactor Model . January . April . December . Non‐seasonal . Mean 3·3658 2·8523 2·6131 – 0·2285 t‐ratio 3·249 3·050 2·766 – 0·61 . January . April . December . Non‐seasonal . Mean 3·3658 2·8523 2·6131 – 0·2285 t‐ratio 3·249 3·050 2·766 – 0·61 Open in new tab Overall the results suggest that rather than the presence of seasonal patterns being at odds with the EMH, seasonality is caused by higher risk at particular times in the year. While there exists no theoretical rationale within the APT for the existence of such patterns in expected returns, the question as to why they exist needs addressing. Barsky and Miron (1989) provide evidence that the business cycle in the United States is mirrored by a seasonal cycle with increases in the second and fourth quarters and falls in the first quarter. It is possible that the information contained in this seasonal cycle is important for subsequent periods and thus gives rise to higher levels of risk. For example, sales and outputs are generally higher in the holiday period around Christmas. Announcements of the level of macroeconomic activity (through, for example, industrial production, money supply and inflation) around this period provide important information regarding the performance of the economy and subsequent levels of activity in the coming year. This has implications for the general health of the economy and specifically for firms cash flows. Therefore, the risk shareholders face is that of worse than expected figures for these macroeconomic variables and the consequent implications this has for the economy. Given that agents make an expectation of the level of the variables, the unexpected component derives a higher risk premium because of the higher risk in these periods. That is, investors are rewarded more in these periods because the information contained in the uncertain macroeconomic variables is more important than at other times of the year. In our model industrial production, the money supply and inflation are certainly related to output and consumption patterns in this important quarter. With regard to the April seasonal effect there exists two main informational effects. First, March is the month the UK government announces the yearly Budget containing the government's income plans and information on fiscal and monetary policy and targets. Second, April is the end of the UK tax year. It is possible that information released in the Budget will cause investors to reassess the future performance of the economy and thus this period can also be viewed as a riskier period in the UK economy. It is quite possible that changes in the March budget could effect a firms investment and output plans (through, say, tax effects or changes in macroeconomic policy) and consequently, any unexpected changes will cause investors to reassess the future cash flows of firms. This is the risk investors hold and hence they require a higher expected return. V. Conclusion In this paper, we have sought to examine the nature of seasonality in UK stock returns. By specifying a set of candidate variables for systematic risk factors and taking particular care in generating unexpected components in them, we estimate a multifactor model of stock returns. The first finding is that seasonalities in UK stock returns are caused by seasonalities in expected returns. We find that the seasonality in expected returns is a direct result of particular risk factors in seasonal months. The evidence suggests that the seasonality in stock returns is due to the high risks involved in holding stocks, first in January and December because this is an important period in the yearly business cycle and has implications for current and subsequent levels of economic activity. Second, the April seasonal may be related to the risk of changes in government policy that may come about due to the annual Government Budget and the end of the tax year, both of which may effect future economic activity. From these results we are able to make two important inferences. First, the once regarded puzzle of seasonal patterns in stock returns appears to be solved and second the relationship between stock returns and the macroeconomy is more complex than first thought. Consequently, the specification of asset pricing models that do not explicitly take account of seasonality in expected returns must be treated with caution. Footnotes 1 US stock returns exhibit significantly higher returns in January (see, amongst others, Keim (1983)). UK stock returns exhibit significant seasonal patterns in January, April, September and December (see Clare et al. (1995)). Corhay et al. (1987) and Gultekin and Gultekin (1983) present further international evidence regarding stock return seasonality. 2 Furthermore, given the construction of their risk factors, it is not at all clear whether their findings are a result of true variation in seasonal risk premia or failure to generate unexpected components of the underlying factors adequately. 3 Gultekin and Gultekin (1987) use factor analysis to estimate the APT for the US stock market. They find that while the APT can capture the January seasonal, the model is incapable of explaining the risk return structure in any other month of the year (see, also Cho and Taylor (1987) and Connor and Korajczyk (1993) for studies into the US January effect that employ statistical techniques to extract risk factors). However, using these results to rationalise the seasonal puzzle is difficult since statistical techniques cannot identify the underlying macroeconomic influences of expected returns and consequently it is not possible to attempt to justify seasonality in returns. 4 We use the term ‘seasonal’ to refer to a month which has returns significantly different from the rest of the year. For months that have returns which are not ‘seasonal’ we refer to these as non‐seasonal. 5 See McElroy (1985) for estimation details. 6 Equation (1) assumes that the elements of Ft are mean zero and serially uncorrelated. CRR simply take first differences of the factors, or for their default risk and term structure factors they simply take the difference between two interest rates. Evidence from CRR (table 3 p. 392) indicates a long, predictable time series element in the generated factors. Therefore, this methodology is unlikely to provide unexpected components which mirror agents actual unexpected components and moreover it does not satisfy the underlying assumptions of the multifactor model. 7 One way around the problem of seasonal patterns in the variables is to use seasonally adjusted data. However, Wallis (1974) notes that seasonally adjusted data can lead to distortions in the data series. Thus, to avoid any such problems the data on the macroeconomic variables are seasonally unadjusted and we take special care in obtaining unexpected components that account for any seasonality in the variables. 8 Given the use of monthly data we also performed a 12th order serial correlation test assuming all other lags are zero. We could not reject the null hypothesis of no autocorrelation in the residuals from the expectations models. A heteroscedasticity test of the residuals was also carried out, with the exception of the money supply, all the residuals were homoscedastic. The results are available on request. 9 A full correlation matrix is available on request. 10 To ascertain that the seasonal patterns in the market index are representative of the seasonal patterns in individual stock returns we estimate two regressions. The first, for the system of individual stock returns, is a regression of returns on the three, zero‐one, seasonal dummy variables, one for January, April and December respectively, and a constant. Secondly, to check for consistency, we estimate the same regression but constrain the estimates of the coefficients on the seasonal dummies to be the same across all equations. In essence, the second regression is a test of common variation in seasonal returns. The restriction on common variation is easily accepted and the dummy variables are significant with roughly the same estimated coefficients as for the market index. These results are available on request. 11 The reported results are of reduced versions of the models where the insignificant prices of risk have been omitted. The restrictions that this imposes are easily accepted: for the seasonal model the likelihood ratio test is distributed X2(186) and has a calculated value of 205 and an approximate 5% critical value of 218.55; for the non‐seasonal model the likelihood ratio test is distributed X2(62) and has a calculated value of 63 and an approximate 5% critical value of 81.29. 12 Diagnostic tests of the residuals from both the seasonal and non‐seasonal multifactor model for the presence of heteroscedasticity, serial correlation and ARCH effects, suggest that residual misspecification is not a problem. These results are available on request. References Barsky , Robert B. and Miron , Jeffrey A. ( 1989 ). ‘ The seasonal cycle and the business cycle . Journal of Political Economy , vol. 97 , pp. 503 – 34 . 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The Economic Journal – Oxford University Press
Published: Nov 1, 1997
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