Stocks hedge against inflation in the long run: evidence from a cointegration analysis for Denmark

Olesen, Jan Overgaard Working Paper Stocks hedge against inflation in the long run: evidence from a cointegration analysis for Denmark Working paper, No. 6-2000 Provided in Cooperation with: Department of Economics, Copenhagen Business School (CBS) Suggested Citation: Olesen, Jan Overgaard (2000) : Stocks hedge against inflation in the long run: evidence from a cointegration analysis for Denmark, Working paper, No. 6-2000, Copenhagen Business School (CBS), Department of Economics, Frederiksberg, https://hdl.handle.net/10398/7557 This Version is available at: https://hdl.handle.net/10419/208428 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc-nd/3.0/ Institut for Nationaløkonomi Handelshøjskolen i København Working paper 6-2000 STOCKS HEDGE AGAINST INFLATION IN THE LONG RUN: EVIDENCE FROM A COINTEGRATION ANALYSIS FOR DENMARK Jan Overgaard Olesen Department of Economics - Copenhagen Business School Solbjerg Plads 3, DK-2000 Frederiksberg This version, March 2000 First draft, August 1998 Stocks Hedge against Inflation in the Long Run: Evidence from a Cointegration Analysis for Denmark * by Jan Overgaard Olesen Department of Economics and EPRU§ Copenhagen Business School Denmark Abstract We suggest an alternative approach to testing whether stocks provide a hedge against inflation in the long run. Based on a simple structural model, we test the hedge hypothesis in terms of the long-run linkage between stock prices and the general price level, as estimated by cointegration analysis. Using data for the Danish stock market over the post-World War II-period, results give strong support for the hedge property, defined in the narrow sense of a perfect hedge. This contrasts with the weak support found in the literature and also represents stronger support than produced by standard methods. We argue that our approach has the advantage of allowing for a clear distinction between shortand long-run dynamics of stock prices which adjust slowly to long-run equilibrium. Postal address: Department of Economics, Copenhagen Business School, Solbjerg Plads 3 (5th), DK-2000 Frederiksberg, DENMARK. Phone: +45 3815 2575. As of March 20 2000: Danmarks Nationalbank, Havnegade 5, DK-1093 Copenhagen K, DENMARK. Phone: +45 3363 6363. * We have benefited from comments by participants at the EPRU workshop “The Stock Market and The Macroeconomy”, Copenhagen Business School, May 1998. In particular, we thank Ole Risager for comments. § The Economic Policy Research Unit (EPRU) is financed by a grant from The Danish National Research Foundation. 2 1. Introduction Stocks are said to provide a hedge against inflation if they compensate investors completely (and not by more) for increases in the general price level through corresponding increases in nominal stock returns, thereby leaving real returns unaffected. That is, stocks hedge against inflation if their real value or purchasing power is immune to changes in the general price level. Whether or not stocks hedge against inflation is relevant to any rational investor who cares about real wealth. The above definition is one of a perfect hedge as it demands a one-for-one compensation for inflation. This contrasts with the weaker notion of an imperfect (or partial) hedge, as often encountered in the literature, which requires the relation between nominal stock returns and inflation (or equivalently, between nominal stock prices and the general price level) to be significant and positive but it may be less or larger than one-for-one. However, with an imperfect hedge, the real value of a portfolio of stocks is subject to uncertainty due to the uncertainty about future inflation. This is not the case when the hedge is perfect. As we interpret an inflation hedge as a device of eliminating the uncertainty deriving from inflation uncertainty, we shall throughout use the term in its most restrictive sense of a perfect hedge. Apriori it can be argued that stocks should provide a hedge against inflation, at least in the long run where firms’ profit margins can reasonably be assumed to be fixed. The argument is that stocks are claims on current and future profit opportunities which in the long run (with profit margins being fixed) increase with the general price level in relation one-for-one, that is, in the long run stocks are basically claims on real profit opportunities. As a result, we should expect the real value of stocks to remain unaffected by inflation and, hence, stocks should hedge against inflation in the long run. What happens in the short run is, on the other hand, more ambiguous because slow adjustment in output prices and real production imply that profit margins may be significantly affected by inflation. Whether stocks also provide a hedge against inflation empirically has been studied extensively in the literature, see e.g. Fama and Schwert (1977), Gultekin (1983), Boudoukh and 3 Richardson (1993), Ely and Robinson (1997) and Barnes et al. (1999). With the only exception of Ely and Robinson (1997), cf. below, the literature has based its inference on return regressions where nominal stock returns are regressed on inflation and possibly further explanatory variables such as real production growth and changes in a relevant discount rate measure. The inflation hedge hypothesis is then put to a test by testing whether the coefficient to inflation is significant and equal to 11. Results of the literature are fairly mixed, but a general conclusion is that stocks do not hedge against inflation in the short run (investment horizons less than 1-2 years), where inflation usually turns out to have an insignificant effect on stock returns. In fact, at short horizons the estimated relation between nominal stock returns and inflation may even be negative, see e.g. Fama and Schwert (1977) and Gultekin (1983). There is some evidence of a significant positive relationship on longer horizons (more than 2 years) but often with a coefficient different from 1 so that the inflation hedge is not perfect, cf. Boudoukh and Richardson (1993). Hence, the hedge hypothesis comes closer to receiving support at longer horizons but the evidence is still weak. On balance it therefore seems that the empirical evidence tends to reject the hypothesis of stocks providing a (perfect) hedge against inflation. This paper tests the inflation hedge hypothesis for stocks by taking a different approach to that used in the literature. We test the hypothesis by focusing on the long-run relation between stock prices and the general price level rather than the relation between stock returns and inflation. Most importantly, this shift of focus allows us to take account of slow adjustment in stock prices in the event of inflation. The latter is from the outset precluded in the standard return regressions approach which (implicitly) assumes that stock prices adjust completely to inflationary shocks over the prespecified, fixed investment horizon, see section 5 below for a further discussion. We focus explicitly on the long-run horizon where the fixedprofit-margin assumption underlying the hedge hypothesis apriori seems most relevant. We 1 Some studies frame the test in terms of real rather than nominal stock returns, testing whether inflation has a significant influence on real stock returns, see for instance Fama (1981) and Kaul (1987). A survey of the literature including a detailed account of the empirical results is provided by Frennberg and Hansson (1993). The latter study at the same time represents an exception in the literature as the authors conclude that Swedish stocks provide a hedge against inflation even at fairly short horizons (down to one month). Another survey of the literature can be found in Sellin (1998). He concludes that “Stocks seem to be a good hedge against both expected and unexpected inflation at longer horizons” (Sellin 1998, p. 25). However, this 4 proceed as follows. Motivated by a simple theoretical framework, we formulate a structural model for stock prices which includes the general price level, real production and stock investors’ discount rate as explanatory variables. We identify the long-run relationships between the variables by cointegration analysis, using the cointegrated VAR-model, see e.g. Johansen (1996). We estimate a cointegrating relation for stock prices and, finally, test the inflation hedge hypothesis by testing whether this relation implies a one-for-one relationship between stock prices and the general price level. We test the hypothesis for the market portfolio of Danish stocks, using annual data from 1948 to 1996. While the sample may be considered small in terms of the number of observations, the sample period spans many years which is crucial for the analysis of “the long run”. In the empirical analysis, we use small sample versions of tests whenever possible. Moreover, we check the robustness of results from the cointegrated VAR model by also using single-equation-cointegration-methods to test the hedge hypothesis. Our approach has similarities with that of Ely and Robinson (1997) who also differ from the standard literature by focusing on the relation between stock prices and the general price level in testing the inflation hedge hypothesis. Ely and Robinson (1997) test the hypothesis for 16 OECD countries, based on impulse response analysis in a cointegrated VAR model with 4 variables - stock prices, the general price level, real production and money supply. They find for almost all countries that stocks overcompensate for inflation and conclude, using an imperfect hedge definition, that stocks hedge against inflation. However, using the more restrictive definition of a perfect hedge, the evidence in Ely and Robinson (1997) does not give support to the hedge hypothesis. Our approach differs from Ely and Robinson (1997) in several ways. First of all, we differ in the definition of an inflation hedge. In addition to the use of a perfect rather than an imperfect hedge definition, we define an inflation hedge in terms of the ‘partial’ sensitivity of stock prices wrt. the general price level within the context of a structural model for the former. Thus, we address the question: What happens to stock prices in the event of shocks to the conclusion is based on an imperfect hedge definition, which allows stock prices (or returns) to respond more 5 price level, all other factors (real production and the discount rate) kept constant ? Ely and Robinson (1997), on the other hand, examine the response in stock prices within a VAR model which we interpret as a reduced form model for stock prices and the price level where real production and the money stock are the ‘driving’ (exogenous) variables2. Hence, they address the question: What happens to stock prices in the event of shocks to the price level, when other factors (e.g. real production and the discount rate) are allowed to vary ? Our ceteris paribus definition of an inflation hedge resembles that used in the literature of return regressions. Second, we test the hedge hypothesis in terms of a cointegrating relation for stock prices and, hence, do not rely on impulse response analysis as in Ely and Robinson (1997). This may be viewed as an advantage, given the critique raised by e.g. Faust and Leeper (1997), who show that results from impulse response analysis depend crucially on the assumptions needed to identify the underlying structural shocks of the VAR model. This may question the robustness of results derived from impulse response analysis. Moreover, by focusing on the cointegrating relation, we can perform an explicit parametric test of the hedge hypothesis instead of the ‘qualitative’ test criteria used in Ely and Robinson (1997)3. Finally, we can test whether the underlying framework of our approach - the structural model for stock prices - is reasonable empirically by testing whether it is validated as a cointegrating relation. This turns out to be the case, implying that we can have (some) confidence in the framework underlying the test of the hedge hypothesis. For instance, the evidence of cointegration suggests that we do not lack an important variable in modeling the long-run linkages between stock prices and the general price level. Such a validity test of the underlying framework is not (directly) possible in the approach of Ely and Robinson (1997). than proportionately to shocks to the general price level (or to inflation). 2 Ely and Robinson (1997) do not provide a theoretical foundation for their VAR model. 3 Based on the impulse response analysis, Ely and Robinson (1997) test the hedge hypothesis at a qualitative level, concluding that “In those cases where the impact on stock prices is significantly positive (negative) and/or where the impact on goods prices is significantly negative (positive), stocks offer (do not offer) a hedge against inflation in the sense that the relative value of stock prices to goods prices rises (falls)” and “Stocks can also be said to offer a hedge in those cases where neither stock price nor goods price innovations are statistically significant”, Ely and Robinson (1997, page 151). 6 Compared to the existing literature, the contribution of the paper is three-fold. First of all, we suggest an alternative approach to testing the inflation hedge hypothesis. Second, it turns out that results give strong support to the hypothesis which contrasts with the weak support found in the literature. Third, the paper provides results for Denmark, a case which to our knowledge has not been examined thoroughly before4. The paper is organized as follows. In section 2 an operational empirical model for the longrun is formulated. Section 3 reviews the data and section 4 reports the empirical results. Section 5 concludes the paper with a summary and a comparison of our approach with that used in the literature. 2. An Empirical Model for the Long Run We formulate an empirical structural model for stock prices based on a simple theoretical framework that links stock prices to the general price level. The framework is ad hoc and rests on a set of assumptions which are restrictive but facilitate the formulation of an empirically tractable model. We focus on the long-run horizon with the objective of a model that can act as a good approximation to the long-run movements in stock prices. This provides us with a sound empirical (and a theoretical) foundation for testing the inflation hedge hypothesis in the long run. Whether the model actually is a good approximation, is tested as part of the empirical analysis by testing whether it can be validated as a cointegrating relation for stock prices. The starting point is the usual 1-period no-arbitrage relation between stocks and bonds under the assumption of perfect capital markets. Excluding risk premia, this relation demands that the expected 1-period holding return on stocks, consisting of a capital gain and a dividend yield, is equal to the 1-period return (yield-to-maturity) on bonds: 4 Bonnichsen (1983) is an informal study of the relationship between Danish stock returns and inflation in the period 1900-1982. He examines whether the nominal stock return exceeds inflation at long investment horizons, that is, whether the real return at long horizons is positive, and concludes this to be the case. However, this evidence does not address the basic issue whether stocks hedge against inflation. The latter requires an analysis of how stock returns (or stock prices) respond to changes in the inflation rate (or the general price level). Thus, apriori the real return on stocks may still be positive in a situation where the nominal stock return does not respond to changes in the inflation rate, that is, in a situation where stocks do not hedge against inflation. 7 (1) Q Q Q D QB tet t te t t + + −+ = 1 1 where Qt is the (ex dividend) stock price per share at time t, Dt+1 is the dividend payment per share during period t+1 and Bt is the 1-period bond return as of time t. Superscript “e” denotes expectations on unknown future variables. The stock is assumed to be a claim on a representative firm (in our case representative for all firms listed at the Copenhagen Stock Exchange). We shall assume that investors only form point expectations on future variables, i.e., that ‘Certainty Equivalence’ applies, and that investors, furthermore, expect bond returns to be constant over time. This, and the exclusion of rational bubbles, gives the forward-looking stock price solution5: (2) QBD t t i t i e i =+      + + + = ∞ ∑1 1 1 1 0 which determines the stock price as the expected discounted value of all future dividend payments. Now make the following assumptions: (A1) Constant profit margin π*, i.e., profits Πt t t PY=π* (A2) Output price Pt and real production Yt are expected to grow at constant growth rates gp and gy, respectively, i.e. P P g and Y Y gTt T etpTtTetyTt = + = + ≥ − − ( ) ( ) ( ) ( ) ( ) 1 1 (A3) All profits are paid out as dividends each period, i.e. Dt t = Π 14 As the initial step in the estimation, the appropriate lag length (k) of the VAR model has to be determined11. Various procedures can be used, including the explicit testing on lag coefficients in a “general-to-specific” procedure and the use of information criteria. Using the “general-to-specific” procedure, we start out with 6 lags which is sufficient to ensure that the white noise requirements on the disturbance term are fulfilled. We then successively remove insignificant lags from the top, performing a Likelihood Ratio test of the hypothesis that all coefficients at the largest lag are zero12. This procedure results in a lag length of k=4, using conventional significance levels. The test for removing all variables at lag 4 leads to a clear rejection (critical significance level of 0.2%), while the hypothesis of reducing the lag length from 5 to 4 is firmly accepted (critical significance level of 58%). A lag length of 4 is supported by the Hannan-Quinn and Akaike information criteria while the Schwarz criterion suggests a shorter lag length of 2. Table 2 reports both univariate and multivariate specification tests of the VAR model with 4 lags. Diagnostics for each equation in the model, including fitted values for the endogenous variables, are furthermore graphed in Figure 2. The specification tests test whether the residuals from the VAR model fulfill the white noise requirements of being serially uncorrelated, homoskedastic and normally distributed. According to the univariate test, the hypothesis of normally distributed residuals is rejected for the discount rate equation, using conventional significance levels. For the price level equation, the normality hypothesis is close to a rejection. However, the normality assumption is not crucial to the cointegrated VAR model, see Johansen (1996, Part II), who shows that it is a sufficient condition for using this method that the disturbance terms are identically distributed over time. The violation of the normality hypothesis is therefore not a problem for the inference to be drawn. There are no signs of misspecification according to the other, more critical specification tests for serial correlation and heteroskedasticity. Hence, we conclude that the VAR model with 4 lags is well specified and proceed with this specification. 11 Estimations are performed in PCFIML, cf. Doornik and Hendry (1997). 12 We use the approximate F-form of the Likelihood Ratio test suggested by Rao, cf. Doornik and Hendry (1997). This F-form which corrects for degrees of freedom is generally considered to have better small sample properties than the uncorrected c2-form. 15 < Table 2> < Figure 2 > The cointegration part of the VAR model (Ι and ϑ*) is estimated by Maximum Likelihood, using the Johansen procedure, cf. e.g. Johansen (1996). Table 3 shows the (standardized) estimates of α and β* together with estimated eigenvalues. Table 3 also reports statistics from trace tests on the rank of Π. Two trace test statistics are shown. The first statistic which is the one used in Johansen (1996) is the outcome of an asymptotic test. The evidence in Reimers (1992) suggests that this test is “over-sized” in small samples, implying that when using this test we tend to accept too many cointegrating relations compared to the significance level which we are actually willing to use. Based on this evidence and the fact that we have to deal with a small sample, we have more faith in the second trace test which adjusts the former test for degrees of freedom in the way discussed by Reimers (1992). This test is reported to have significantly better small sample properties in the sense that the actual significance levels of the test come close (closer) to the nominal levels in small samples. < Table 3 > Both rank tests lead to the conclusion that there is at least one cointegrating relation as both tests firmly reject the hypothesis of no cointegration (r=0) at conventional significance levels. The first (asymptotic) trace test also rejects the hypothesis of 1 cointegrating relation in favor of the alternative of more than 1 cointegrating relation. However, this hypothesis cannot be rejected according to the second (degrees-of-freedom-adjusted) trace test. Based on the latter test, we conclude that there is one and only one cointegrating relation between the variables (r=1). The second trace test gives a clear rejection of the hypothesis of no cointegration (the critical significance level is 1.9% by linear interpolation). Hence, the evidence of cointegration is strong. The econometric identification of the cointegrating relations is relatively straightforward with only 1 cointegrating relation because normalizing on one of the variables suffices. Motivated by the modeling framework of section 2 (and the lack of an obvious alternative), we interpret 16 the cointegrating relation as a model for stock prices and normalize on this variable. The resulting estimates of the normalized cointegrating vector and the corresponding adjustment coefficients appear in Table 3 as the first column of β* (i.e., β1*), respectively, the first column of α (i.e., the adjustment coefficients wrt. ϑ1*’Xt*). The assumption that the cointegrating relation is a model for stock prices is actually supported by the estimates of the Ι-coefficients, because the error-correction in the short-run dynamics is strong in the direction of stock prices, whereas the corrections in the directions of the price level and the discount rate are very small in magnitude and can actually be shown to be insignificant, cf. below. The estimation gives the following long-run model for stock prices (indicative standard errors of the parameter estimates in parenthesis)13: (9) q p tr t t t = + + −096 104 0011 542 013 0009 2 4 . . . . (.) ( .) ( .) All coefficients have signs consistent with theory. The trend may appear to be insignificant, using the indicative standard error, but we proceed with (9) because our interest lies with the price level coefficient and we do not want to condition the inference on the coefficients to the remaining variables. We take the estimated cointegrating relation as evidence in favor of the modeling framework of section 2, hence establishing a firm empirical framework within which to test the inflation hedge hypothesis. The hedge hypothesis is tested by Likelihood Ratio (LR) tests on the coefficient to the price level in the cointegrating relation. These tests compare the likelihood of the unrestricted VAR model (where the price level coefficient can vary freely) with the likelihood of the restricted VAR model (where the price level coefficient is restricted). Testing, first, the null hypothesis that the price level has an insignificant effect on stock prices (ϑ1=0), the outcome is a LR test statistic of 11.8 which has to be compared with a χ2(1)- distribution. The critical significance level is for all practical purposes zero, leading to a strong rejection of the null. Hence, the price level has a significant effect on stock prices in 13 The constant term in (9) is calculated from the formula ρα α α µ 010 =− (')' where µ0 is the unrestricted constant term, cf. (8), and ρ0 is the component of this constant term which enters the cointegrating relation, see Johansen (1996, p. 81). Ι here denotes the first column of the estimated Ι-matrix in Table 3. 17 the long run. Next, testing the null hypothesis that stock prices and the price level move onefor-one (ϑ1=1) gives a test statistic of 0.04 which, again, has to be compared to a χ2(1)- distribution. The critical significance level is 83% which leads to the unambiguous test result that the null can not be rejected. The conclusion is strong support for the long-run inflation hedge hypothesis. < Figures 3 and 4 > To check the robustness of this conclusion, we have examined whether results are stable over time by estimating the cointegrated VAR model recursively. Figures 3 and 4 provide the results, showing the recursive estimates of the three eigenvalues and of the coefficients of the (one) cointegrating vector, respectively. The eigenvalues are fairly stable over the sample period, so the conclusion of one and only one cointegrating relation in the data is robust over time. Figure 4 shows that the long-run coefficients are reasonably stable, maybe with the exception of a slight instability of the coefficient to the discount rate in the late part of the sample. Most importantly, the coefficient to the price level is very stable. We take these results as evidence that the conclusion in favor of the inflation hedge hypothesis is robust over time. The cointegrated VAR model approach has the advantages, compared to single-equationcointegration methods, that it allows for more than one cointegrating relation in the data and, in general, leads to consistent and asymptotically efficient estimates of the long-run parameters (ϑ*). However, as noted by e.g. Gonzalo and Lee (1998), Johansen (1999) and Juselius (1999), the cointegrated VAR model is sensitive to the number of observations. Thus, evidence based on Monte Carlo simulations suggests that the test of cointegration and the tests of hypotheses on the long-run coefficients may suffer from poor small sample performance (size distortions and low power). Moreover, inference from the model is based on the condition that the VAR specification gives the correct model not only for the variable of interest (stock prices) but also for the remaining variables (the general price level and the discount rate). As a further check on the robustness of conclusions, we have therefore reestimated (5) by single-equation-cointegration methods. These give valid and efficient 18 inference in our case because we only have one cointegrating relation and because there is only error-correction in the direction of stock prices, implying that the price level and the discount rate are weakly exogenous for the parameters of the cointegrating vector, cf. Johansen (1996, Chp. 8). The latter can be shown by formal testing14. Given the evidence of cointegration, OLS estimation of (5) produces consistent estimates of the coefficients15. However, testing coefficient hypotheses based on these estimates is in general difficult due to a (possible) correlation between the error term in the cointegrating relation and the innovations in the regressors, cf. Hamilton (1994). In particular, usual t-test statistics calculated from the OLS coefficients and the OLS standard errors do not have standard (known) distributions. Therefore, we have to refine the estimation of the cointegrating relation. Several approaches have been suggested for this purpose, cf. e.g. Phillips and Loretan (1991), Stock and Watson (1993) and Phillips and Hansen (1990). Hamilton (1994) and Mills (1993) provide surveys. We employ two of these procedures, both suggested by Phillips and Loretan (1991); the Phillips-Loretan OLS procedure (PLOLS) and the Phillips-Loretan Non-linear least squares procedure (PLNLS). In both approaches, the static regression of (5) is augmented by stationary terms which capture the short-run dynamics of the explanatory variables. The PLOLS procedure augments (5) with current, lagged and leaded first differences of the explanatory variables (the price level and the discount rate), leading to the dynamic regression: (10) q p trpru t t t i t i iN N i t i iN N t = + + + + + + − =− − =− ∑ ∑ β β β β γ γ 0 1 2 3 1 2 1 1 2 2 ∆ ∆ t denotes as before the deterministic time trend (replacing yt in (5)) and ut is the new residual term. N1 and N2 which determine the number of lags and leads in the regression have to be 14 We have weak exogeneity if the equilibrium error does not affect the short-run dynamics of the price level and the discount rate, i.e., if the corresponding adjustment coefficients in α (see first column, second and third entry of α in Table 3) are both zero. This hypothesis can be tested formally by a LR test. The LR test statistic is 1.04 which has to be compared with a χ2(2)-distribution. The critical significance level is 59%, leading to the conclusion that weak exogeneity can not be rejected. 19 specified prior to estimation. We use different specifications, cf. below, in order to check the sensitivity of coefficient estimates. (10) is estimated by OLS. The PLNLS procedure augments (5) further by adding lagged levels of the error correction term, i.e., the difference between stock prices and their long-run equilibrium level as determined by (5), [q p tr t t t − + + +( )β β β β 0 1 2 3 ] : (11) q p trprq p t i rv t t t i t i iN N i t i iN N i t i t i t i i N t = + + + + + + − − − − − + − =− − =− − − − = ∑ ∑ ∑ β β β β γ γ φβ β β β 0 1 2 3 1 2 0 1 2 3 1 1 1 2 2 3 ∆ ∆ ( ( ) ) The error correction terms are included in order to eliminate serial correlation in the disturbance term (vt) and increase the efficiency of the coefficient estimates, cf. Hamilton (1994). Because the coefficients of the cointegrating relation enter the lagged error correction terms, (11) is estimated by Non-linear least squares (NLS). < Table 4 > Results including t-tests on the price level coefficient are reported in Table 4. In the first entry, results from estimating (5) by OLS (no augmentation) are shown together with OLS standard errors which are indicative only. The PLOLS procedure is used in three regression specifications which differ according to the included first differences of the price level and the discount rate (entries 2 trough 4). In the first application (entry 2), current first differences and first differences at lead 1 and lag 1, respectively, are included. The disturbance term shows serial correlation up to lag 5 so standard errors and t-statistics have to be corrected. We use the adjustment method suggested by Hamilton (1994), based on an AR(5)-model fitted to the residuals of the PLOLS regression16. In the second application (entry 3), first differences of up to 2 leads and 2 lags are included. This further augmentation only has a minor effect on the estimated price level coefficient. It turns out that the disturbance term 15 Using the two-step procedure of Engle and Granger (1987), we can, at the 10% significance level, confirm (5) as a cointegrating relation, see Appendix A (the alternative based on CPI and a trend). 16 The adjusted t-statistics reported in Table 4 (entry 1) are calculated as the ordinary OLS t-statistics multiplied by the ratio (s/l), where s is the ordinary standard error of the residual in (10) while l is calculated from an AR(5)-model fitted to the residual, see Hamilton (1994, p. 610). l can, heuristically, be interpreted as an estimate of the residual standard error in ‘long-run equilibrium’ of the AR(5)-model. The reported standard errors of the coefficient estimates are adjusted accordingly. 20 shows no sign of misspecification in this formulation (no serial correlation) so usual OLS standard errors can be used. Finally, in the third application (entry 4), we use a “specific-togeneral” procedure and augment (5) with current, lagged and leaded first differences until the disturbance term fulfills the white noise requirements. The resulting regression is just a reduced version of the second PLOLS regression (entry 3) where insignificant first difference terms have been omitted. Again, the estimated price level coefficient is only mildly affected. PLOLS regressions have also been carried out with more leads and lags but the coefficients and, in particular, the price level coefficient are stable wrt. this further augmentation. The PLNLS regression in entry 5 has the augmenting terms shown in the first column of the table, including one lag of the error correction term. The augmenting terms are chosen in a “specific-to-general” manner in order to ensure a white noise disturbance. The reported standard errors are NLS calculated standard errors. The results show that while the coefficient estimates for the trend and especially the discount rate are sensitive to the estimation procedure used, the estimate of the price level coefficient is fairly robust (and also comes close to the estimate obtained from the cointegrated VAR model). Turning to the inflation hedge hypothesis, the t-tests show that the price level coefficient is significant in all four cases. Moreover, in none of the cases we can reject the hypothesis that the price level coefficient is 1. The evidence in terms of critical significance levels is very strong. Hence, we conclude that single-equation-cointegration methods confirm the strong evidence in favor of the hedge hypothesis. 5. Conclusion and Discussion We have examined whether Danish stocks provide a hedge against inflation, focusing explicitly on the long-run horizon. We have tested the hypothesis based on the long-run relation between stock prices and the general price level, estimated by cointegration analysis. Using the Consumer Price Index as the relevant price measure, results give strong support to the hedge hypothesis. The evidence supports the hedge property in its most restrictive sense of a perfect hedge. The conclusion is confirmed by both multivariate and univariate cointegration methods and is robust over time. The inflation hedge hypothesis is tested within 21 a firm modeling framework which is validated by the data as a cointegrating relation for stock prices. The inflation hedge property of stocks (defined as a perfect hedge) only receives weak support, if any, in the literature. The strong support in this paper is therefore not a standard result. We do not believe that the Danish stock market has unique characteristics compared to other stock markets but rather attribute the difference to the literature to other factors. First of all, the use of different investment horizons is one possible explanation. We test the inflation hedge hypothesis in a long-run framework whereas others, e.g. Fama and Schwert (1977) and Gultekin (1983), examine relatively short investment horizons (less than 6 months). A plausible and reconciling interpretation of this evidence is that stocks hedge against inflation in the long run, but not in the short run. Second, the use of different sample periods may be important. In this paper, we include observations until 1996, while other studies, e.g. Fama (1981) and Gultekin (1983), use samples that only cover the period until the end of the 1970s. As well-known, the 1970s were in almost all OECD countries a period of very high and increasing inflation due to the 1973 and 1979 oil price shocks. The use of a sample ending shortly after the oil price shocks ignores the subsequent and major adjustment in stock prices and may have triggered the (false) conclusion that stocks do not hedge against inflation. In this context, it may in particular be important that real oil prices, while increasing substantially during the oil crises with a deteriorating effect on profit margins, have by the beginning of the 1990s returned to the pre-oil crises level, hence allowing for a restoration of “normal” profit margins. Our study differs from the older literature by including the important adjustment period after the 1970s. Finally, we have taken a different approach compared to the literature where it has been standard to test the inflation hedge hypothesis based on return regressions. We use cointegration methods to disentangle the short-run dynamics of and the long-run linkages between stock prices and the general price level, explicitly allowing for slow adjustment in stock prices to long-run equilibrium in the event of shocks to (not least) the general price level. This approach has the advantage of allowing for a clear identification of long-run stock 22 price behavior. The return regressions approach, on the other hand, does not distinguish between short-run dynamics and long-run linkages and the identification of long-run stock price behavior is conducted merely by investigating a sufficiently ‘long’ investment horizon. However, this muddles short-run dynamics and long-run linkages. Moreover, by linking stock returns to contemporaneous inflation, return regressions from the outset preclude slow adjustment in stock prices. In principle, this could trigger a false conclusion that stocks do not hedge against inflation in the long run. That is, stocks may be a perfect hedge against inflation with a lagged response in stock prices, but return regressions may fail to establish this because they do not explicitly take account of the lagged adjustment. As an illustration, assume that stocks hedge against inflation after a lagged adjustment over (say) 3 years, i.e., stock prices adjust completely to current inflation after 3 years. A return regression for even a long investment horizon of e.g. 5 years may not be able to detect this because stock returns do not reflect (completely) inflation in the last 3 years of each horizon, while at the same time, stock returns in the first 3 years are a result of adjustment to inflation in the preceding years17. To highlight the difference between the standard return regressions approach and our approach (the cointegration approach) more formally, consider the cointegrated VAR model of (8) with a lag length of (for simplicity) k=1 and let us assume that this is the ´true´ reduced form model. The structural form of the cointegrated VAR model is formally derived by premultiplying this reduced form by a non-singular matrix, cf. Johansen (1996). The resulting dynamic equation for stock prices can be written as (ignoring the disturbance term): (12) ∆ ∆ ∆q a a p a raX t t t t = + + + −0 1 2 3 1 β*' * where the ai’s are structural coefficients. The term β*’Xt-1* denotes as before the errorcorrection term from the long-run stock price relation (as of period t-1). Now notice, that the 17 The possibility of a slow adjustment in stock prices or rather stock returns to a change in the inflation rate has also been noted by Barnes et al. (1999). They test the inflation hedge hypothesis on a large sample of countries using the standard return regressions approach. To take account of the possible slow adjustment, they include both the contemporaneous and the lagged inflation rate in the return regressions. However, this does not alter the evidence significantly. The general result in Barnes et al. (1999) is a rejection of the inflation hedge hypothesis for stocks. 23 first differences part of (12) resembles a return regression for an investment horizon of 1 year, by regressing the first differences of log-to-stock prices (Dqt), which is a proxy for the 1-year stock return, on 1-year inflation (Dpt) and the 1-year change in the discount rate (Drt). The 1year return regression, therefore, can be viewed as a special case of (12) where the level term (the cointegration term) has been excluded. In terms of (12), what distinguishes the cointegration approach from the standard approach is that the former is concerned with the long-run coefficients to stock prices and the general price level, i.e., the cointegrating vector ϑ*. The return regressions approach, on the other hand, is concerned with the dynamic coefficient to the price level, i.e., the coefficient a1 which captures the short-run or contemporaneous response in stock prices to inflation. This difference reflects our explicit focus on the long-run horizon whereas the existing literature has mainly examined the inflation hedge hypothesis over relatively short horizons. (12) also suggests a possible shortcoming of the standard approach. In standard return regressions, the level term of (12) is excluded which (implicitly) assumes either that stock prices adjust immediately to their long-run equilibrium level as determined by the cointegrating relation (i.e., the equilibrium error ϑ*’X*t-1 is always zero), or that there is no cointegration between the level variables (i.e., the cointegrating rank is zero and no cointegrating vectors ϑ* exist). In our case, both possibilities are rejected by the data. Therefore, the 1-year return regression must be misspecified because it omits a significant regressor (the level term of (12)), which reflects the slow adjustment in stock prices. In general, the result is inconsistent coefficient estimates, which affects the inference on the coefficient to inflation and, hence, the inflation hedge hypothesis. In the cointegration approach, we explicitly allow for slow adjustment18. For the purpose of comparison, we have also tested the inflation hedge hypothesis for Danish stocks using standard return regressions over the same sample period as considered above. We have run a return regression where nominal stock returns are regressed (OLS) on 18 The cointegrated VAR model is more general than implied by (12) because it allows for more short-run dynamic terms, i.e., lagged first differences, when the lag length k is larger than 1. Moreover, in the case of a cointegrated VAR model with an explicit measure for real production, we would also have included the 1-year real growth rate (Dyt) as a first-differences regressor in (12). Note that (12) focuses on the 1-year investment horizon. For longer horizons, the implied model for returns will be more complicated but the fundamental insight remains that return regressions omit significant level terms. 30 Engle, R.F. and C.W.J. Granger (editors) (1991), Long-Run Economic Relationships: Readings in Cointegration, Oxford University Press. Fama, E.F. (1981), “Stock Returns, Real Activity, Inflation and Money”, American Economic Review 71 (4), 545-565. Fama, E.F. and G.W. Schwert (1977), “Asset Returns and Inflation”, Journal of Financial Economics 5, 115-146. Faust, J. and E.M. Leeper (1997), “When Do Long-Run Identifying Restrictions Give Reliable Results ?”, Journal of Business and Economic Statistics 15 (3), 345-353. Frennberg, P. and B. Hansson (1993), “Stock Returns and Inflation in Sweden. Do Stocks Hedge against Inflation”, Working Paper #25, Department of Economics, Lund University. Gonzalo, J. and T.H. Lee (1998), “Pitfalls in Testing for Long Run Relationships”, Journal of Econometrics 86, 129-154. Grandmont, J.-M. (1988), Money and Value, Cambridge University Press, New York. Gultekin, N.B. (1983), “Stock Market Returns and Inflation: Evidence from Other Countries”, Journal of Finance XXXVIII (1), 49-65. Hamilton, J.D. (1994), Time Series Analysis, Princeton University Press. Hansen, H. and S. Johansen (1996), “Recursive Analysis of Eigenvalues in Cointegrated VAR-Models”, revised version of Discussion Paper 92-13, Institute of Economics, University of Copenhagen. Johansen, S. (1996), Likelihood-Based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press. 31 Johansen, S. (1999), “A Bartlett Correction Factor for Tests on the Cointegrating Relations”, European University Institute Working Paper ECO 99/10. Juselius, K. (1999), “Models and Relations in Economics and Econometrics”, Working Paper, University of Copenhagen. Kaul, G. (1987), “Stock Returns and Inflation: The Role of the Monetary Sector”, Journal of Financial Economics 18, 253-276. Kwiatkowski, D., P.C.B. Phillips, P. Schmidt and Y. Shin (1992), “Testing the Null Hypothesis of Stationarity Against the Alternative of a Unit Root”, Journal of Econometrics 54, 159-178. MacKinnon, J.G. (1991), “Critical Values for Cointegration Tests”, Chp. 13 in Engle and Granger (eds.) (1991). Mills, T.C. (1993), The Econometric Modelling of Financial Time Series, Cambridge University Press. Newey, W.K. and K.D. West (1987), “A Simple Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix”, Econometrica 55, 703-708. Nielsen, S., J.O. Olesen and O. Risager (1997), Danish Stock Market and Macroeconomic Database, Department of Economics, Copenhagen Business School. Phillips, P.C.B. and B.E. Hansen (1990), “Statistical Inference in Instrumental Variables Regression with I(1) Processes”, Review of Economic Studies 57, 99-125. Phillips, P.C.B. and M. Loretan (1991), “Estimating Long-Run Economic Equilibria”, Review of Economic Studies 58, 407-436. 32 Phillips, P.C.B. and P. Perron (1988), “Testing for a Unit Root in Time Series Regression”, Biometrika 75, 335-346. Reimers, H.E. (1992), “Comparisons of Tests For Multivariate Cointegration”, Statistical Papers 33, 335-359. Sellin, P. (1998), “Monetary Policy and the Stock Market: Theory and Empirical Evidence”, Working Paper #72, Sveriges Riksbank. Stock, J.H. and M.W. Watson (1993), “A Simple Estimator of Cointegrating Vectors in Higher Order Integrated Systems”, Econometrica 61, 783-820. 33 Appendix A: Alternative Candidates for a Cointegrating Relation This appendix reports the results from estimating alternative candidates for a cointegrating relation for stock prices, cf. (5), while using alternative measures of the general price level and real production, respectively. For the general price level, we have examined three different measures, the official Consumer Price Index (CPI) (denoted by pt in the following), the implicit price deflator for total GDP in factor prices (pyft) and, finally, the implicit price deflator for GDP in factor prices in the sector of manufacturing (pyfit). For real production, we use data on total GDP (denoted by yft), respectively, GDP in manufacturing (yfit), both in fixed 1980-factor prices. The price deflators and production measures are taken from National Accounts and all series are in log-levels. For the growth-adjusted real discount rate (rt in (5)), we, throughout, use the same proxy as in the main text, cf. section 3. The estimations and tests are performed by a single-equation cointegration method, that is, the Engle and Granger (1987) two-step procedure (EG2). To begin with, we have to test for the stationarity properties of the data series (stock prices, all price and production measures and the discount rate proxy). Unit root tests have been performed following the same approach as in section 3 and the conclusion is that all series are integrated of order 1, i.e., non-stationary in levels but stationary in first differences (tests not reported). Hence, the regression in (5) is balanced which is a prerequisite for using the EG2 procedure for estimation purposes. < Table A.1 > Table A.1 reports the alternative estimates of (5) and the corresponding tests for cointegration. The measures used for the general price level and real production are indicated in the first column. For example, the regression of the first entry uses the price deflator for total GDP and, correspondingly, total GDP in fixed prices as the relevant measures. The second column shows the OLS estimates of the coefficients of (5) (stated as a cointegrating vector which is normalized on stock prices), together with indicative OLS standard errors. For instance, the price level coefficient in the regression of the first entry (the estimated ϑ1) is 1.38 with an indicative OLS standard error of 0.14. A residual-based test for cointegration is 34 performed by testing the null hypothesis of a unit root in the process for the OLS residuals. If the null is rejected, the residuals are stationary and the regression (5) is concluded to be a cointegrating relation. The test results and conclusions on cointegration are reported in the remaining columns of the table, using the cointegrating regression Dickey-Fuller test (CRDF), cf. Engle and Granger (1987) or Hamilton (1994). The number of augmenting lags (of the first differences of the residuals) in the CRDF test is chosen according to a “specific-togeneral” procedure, taking the simple Dickey-Fuller regression without augmentation as the starting point and - in case this regression shows signs of being misspecified (serial correlation in the disturbance term) - including lags until diagnostic tests are passed. A maximum of 1 augmenting lag suffices in the tests reported in Table A.122. The choice of production measure is important for whether or not (5) is a cointegrating relation. The first two regressions in Table A.1 which both use an explicit measure of production show no cointegration at the 10% significance level. Because the OLS estimates indicate that the production measures are insignificant, a regression is run (third entry) where prices and production are combined in nominal production (using total GDP in current prices as the relevant measure, denoted by yft* in the table) to check whether this enhances the presence of cointegration. This is not the case23. In the last three regressions in Table A.1, we have replaced the explicit production measure by a deterministic trend (denoted by t), which can be interpreted as a proxy for the trend growth in production. Results show that the inclusion of a deterministic trend leads to cointegration at the 10% significance level when measuring the general price level by CPI (entry 4) or the factor price deflator for manufacturing (entry 6). Cointegration is most evident in the latter case with cointegration being accepted also at the 5% significance level. 22 The level of augmentation used in Table A.1 is the same as one would get from a “general-to-specific” procedure, starting out with a Dickey-Fuller regression with 5 augmenting lags and then, successively, removing insignificant lags from the highest order. 23 We have examined alternative measures of real production including GDP for the private sector and GDP for the private sector excluding farming and housing, but without any further success. The lack of cointegration and the apparent insignificance of the production measures, basically, suggests that we have not been able to find a good proxy for the production of goods and services by the representative firm on the Copenhagen Stock Exchange. 35 Cointegration is just rejected when using the factor price deflator for total GDP as the price measure (entry 5). Despite the fact that the evidence of cointegration is strongest when using the price deflator for manufacturing as the price measure, we prefer to test the hedge hypothesis in terms of CPI inflation for the reasons stated in section 324. 24 While CPI seems most relevant to stock investors, a deflator for GDP in factor prices (or a net price index) may actually be more relevant to firm profits and, hence, more adequate for the theoretical framework of section 2. Thus, CPI includes indirect taxes paid by the consumers. Furthermore, CPI measures the prices of (domestically consumed) consumer goods and services and, thereby, ignores (say) the prices of investment goods. A factor price deflator captures the prices of all goods and services produced. However, whatever price measure used, it is just a proxy for what we really want to measure, and that is the prices of goods and services produced by the representative firm at the Copenhagen Stock Exchange. In particular, what we need is a good proxy for the long-run movements in the ‘true’ prices and, in this respect, CPI may do as well as e.g. a factor price deflator. It should also be recalled that the use of a proxy for the price level does not undermine the asymptotic consistency of the coefficient estimates in a cointegrating relation, cf. Hamilton (1994), provided cointegration is preserved. 36 Appendix B: Evidence from Return Regressions In the literature, the inflation hedge hypothesis is tested by examining the link between stock returns and contemporaneous inflation. This appendix provides comparable results for Denmark, focusing on the three investment horizons of 1, 5 and 10 years. The Empirical Model Based on the theoretical framework of section 2, we use the following empirical model for stock returns over the k-year investment horizon25: (B1) Sk Ik GYk GRk kand years t t t t t = + + + + = β β β β ε 0 1 2 3 1 5 10, , Skt denotes the annualized total stock return over the k-year investment horizon, including both capital gains and dividend yield. Ik Pk tkt ≡ (ln )/ ∆ and GYk Yk tkt ≡ (ln )/ ∆ are, respectively, the (continuously compounded) annual inflation rate and the (continuously compounded) annual growth rate in real production over the same k year horizon. GRk Rk tkt ≡ (ln )/ ∆ is the per annum relative change in the discount rate over the investment horizon while εt, finally, denotes the usual disturbance term. According to (B1), stock returns should be regressed on a constant term and contemporaneous values of the inflation rate, the real growth rate and the relative change in the discount rate. Whether or not stocks provide a hedge against inflation is captured by the coefficient to inflation, β1, measuring the direct or partial effect from inflation to stock returns. A formal test of the hedge hypothesis is performed in two steps, by testing (i) whether inflation has a significant effect on stock returns (β1¹0), and, if the inflation effect is significant, (ii) whether the relationship between (changes in) stock returns and inflation, furthermore, is one-to-one (β1=1). The Data 25 The theoretical counterpart of (B1) is obtained by taking k-year differences in the logarithmic analog to (3) and dividing through by k to obtain per annum continuous growth rates. We substitute total stock returns (including dividend yields) for capital gains as the endogenous variable to allow for a comparison with the literature. It can be shown that this does not affect the empirical results significantly. The latter reflects the fact that the variation in dividend yields have played only a minor role for the variation in Danish stock returns over the sample period. 37 Data for total stock returns are from the database by Nielsen, Olesen and Risager (1997) and relate to the market portfolio of all Danish stocks listed at the Copenhagen Stock Exchange. Inflation is measured by the (annualized and continuously compounded) growth in the official Consumer Price Index (CPI) by Statistics Denmark, while we as data for real growth use the (annualized and continuously compounded) growth in total GDP in fixed 1980 factor prices, taken from National Accounts. For the unobservable discount rate, we use the same proxy as in the main text, cf. section 3, and GRkt is calculated as the (annualized) change in this proxy over a k-year period. All data are annual. In order to cover the same sample period as in the main text, we consider the period 1949-1996 for the 1-year investment horizon, 1953-1996 for the 5-year horizon and 1958-1996 for the 10-year horizon. At the 5and 10-year horizons, we use overlapping observations. Figure B1 shows the stock return and the inflation at the three horizons. < Figure B1 > < Tables B1.a and B1.b > Tables B1.a and B1.b report the outcome of tests for unit roots, using the Phillips and Perron (1988) Zt-test (PP) and the test by Kwiatkowski et al. (1992) (KPSS). For the 1-year horizon, we conclude that all variables are stationary. The two tests give conflicting results for the real growth rate but the evidence seems most robust across lag lengths for the PP-test which strongly points to stationarity. For the inflation rate, the PP-test consistently concludes stationarity at the 10% significance level, while the KPSS test at the same time gives firm evidence in favor of stationarity when allowing for serial correlation in the disturbance term (lag length l³1). At the 5-year horizon, stock returns and the change in the discount rate (GR5t) are stationary. Results for inflation are ambiguous as the PP-test points to (at least one) unit root whereas the KPSS test supports stationarity. Real growth is non-stationary according to both tests. Finally, for the 10-year horizon, results for both stock returns and inflation are ambiguous. GR10t is stationary while real growth is non-stationary. To conclude, the static regression of (B1) which requires the data series to be stationary is valid for the 1year horizon whereas conclusions are less clear for the 5and 10-year horizons. At the latter horizons, the real growth should be excluded to allow for a valid regression and the 38 regression results should, in general, be interpreted with caution due to the possible nonstationary behavior of the data series. The Results < Table B2 > Results are shown in Table B2. Regressions of the type (B1) are performed for each of the three investment horizons. Furthermore, for each investment horizon three distinct regressions are examined, cf. below. For all regressions, OLS is used for estimating the parameters, producing consistent estimates. For the 5and 10-year horizons standard errors of the parameter estimates are estimated by the Newey and West (1987) method to take account of heteroskedasticity and serial correlation up to lag 5 in the disturbance term. The non-standard behavior of the disturbance term can be motivated by the use of overlapping observations. The truncation at lag 5 seems appropriate as the Newey and West (1987) standard error of the inflation coefficient becomes stable at this lag length. For the 1-year horizon, we include impulse dummies for 1972 and 1983 in order to exclude the exorbitant and exceptionally high stock returns these years (returns of 95% and 118%, respectively). These outliers can be explained by exceptional changes in the Danish economy including, in particular, the Danish favorable EEC referendum in 1972, the major shift towards a new economic policy regime in late 1982 and the introduction of a new pension fund tax on bonds in 1983. Having included these dummies, the regression residual fulfills the white noise requirements of being serially uncorrelated and homoskedastic and, hence, standard errors of the coefficients can be estimated by OLS at the 1-year horizon. The first regression for each horizon (first entry) shows the results for “the simple model”, which is the specification that has been used most extensively in the literature. This formulation is a special case of (B1) where any effects from real production and the discount rate are ignored (β2≡β3≡0) so that stock returns are explained by inflation only. The problem with this formulation is that it, according to (B1), ignores potentially relevant explanatory variables. As well known, the omission of relevant regressors leads to biased estimates for the 39 remaining regressors to the extent that the omitted and included regressors are correlated. Thus, in the simple model, the estimated coefficient to inflation could potentially be biased, as it may also capture relevant effects from real growth and a changing discount rate. Results for the simple model should, therefore, in general be interpreted with caution. The second regression for each horizon (second entry) shows results for (B1) including both real growth and the change in the discount rate (“the extended model”). The latter enters significantly and with the correct (minus) sign for all three horizons whereas real growth has the wrong sign (minus) in each case. However, the effect from real growth is also insignificant at the 5% significance level. For this reason, we exclude it from the regression (which is also preferable from unit root considerations, cf. above) and arrive at the “reduced extended model” (third entry for each horizon) which can be interpreted as the parsimonious model formulation. Comparing the reduced extended model with the simple one, we find that the inclusion of the discount rate matters at both the 5and 10-year horizons as it increases the point estimate of the inflation coefficient. In testing the inflation hedge hypothesis, we focus on the “reduced extended model” which provides the best specification in terms of included regressors. The impact of inflation on the stock return is clearly insignificant at the 1-year horizon, where the estimated inflation coefficient for all practical purposes is zero. At the 5-year horizon, the coefficient of 1.01 is very close to one, but the coefficient standard error is large (0.55) so that the inflation effect is only at the boarder of being significant, using conventional significance levels (the critical significance level of a two-sided t-test for significance is 6.6%). If the inflation effect is accepted to be significant, the hypothesis that stock returns and inflation move one-for-one is clearly accepted. At the 10-year horizon, the inflation coefficient of 1.01 is strongly significant with a t-statistic of almost 5. Moreover, the hypothesis that the inflation effect is one (ϑ1=1) receives strong support. To judge the robustness of the latter evidence, a recursive estimation of the reduced extended model at the 10-year horizon is performed, cf. Figure B2. 47 Table 2. Specification Tests of the VAR Model Estimation sample 1952-1996 Multivariate tests: Vector Autocorrelation order 2 F(18,65) = 1.05 [0.42] Vector Autocorrelation order 4 F(36,50) = 0.92 [0.60] Vector Autocorrelation order 6 F(54,33) = 0.98 [0.54] Vector Heteroskedasticity (squares) F(156,2) = 0.02 [1.00] Normality χ2(6) = 7.60 [0.27] Univariate tests: ∆qt∆pt∆rt Autocorrelation order 2, F(2,29): 0.50 [0.61]0.06 [0.95]1.80 [0.18] Autocorrelation order 4, F(4,27): 1.34 [0.28]0.71 [0.59]1.15 [0.35] Autocorrelation order 6, F(6,25): 1.14 [0.37]0.56 [0.76]0.94 [0.49] ARCH (1), F(1,29): 0.15 [0.71]1.36 [0.25]0.02 [0.89] Heteroskedast. (squares), F(26,4): 0.23 [0.99]0.07 [1.00]0.54 [0.85] Normality, χ2(2): 2.58 [0.27]5.43 [0.07]9.18 [0.01] * Goodness-of-fit: r0.98 1.00 0.81 σε0.184 0.018 0.012 Note: The VAR model has a lag length of 4 (k=4). The F-tests are small sample approximations to Lagrange Multiplier tests, being adjusted for degrees of freedom. Normality test of Doornik and Hansen (1994). For a description of the tests, see Doornik and Hendry (1997). Numbers in brackets are critical significance levels. * and ** indicate misspecification at the 5% and 1% significance level, respectively. r is the correlation between actual and fitted values for each equation (variables in levels). [Μ is the standard deviation of the residual term. 48 Table 3. Cointegration Analysis in the VAR Model Estimation sample 1952-1996 Cointegrating rank: Rank(Π) (r =) 012 Eigenvalue 0.54 0.36 0.17 Trace test 1) 63.3 *** 28.3 ** 8.2 Trace test (adj. for df.) 2) 46.4 ** 20.7 6.0 95 % critical test value 42.2 25.5 12.4 97.5 % critical test value 45.0 27.9 14.1 99 % critical test value 48.6 30.7 16.4 Standardized eigenvectors β*: β1*β2* β3* qt 1.000 0.066 0.031 pt-1.037 1.000 0.008 rt 5.423 -15.338 1.000 t-0.011 -0.053 -0.004 Standardized loadings α: 3) β1*′Xt* β2*′Xt* β3*′Xt* ∆qt-0.877 -0.136 8.049 ∆pt-0.017 -0.051 -0.565 ∆rt -0.017 0.010 -0.593 Note:Maximum Likelihood Estimation by the Johansen-method, cf. Johansen (1996). The trace tests test for each value of r the null hypothesis H0: rank(Π)≤r against the alternative HA: rank(Π)>r. The null is rejected iff the trace statistic is larger than the critical test value. Critical values from Table 15.4 in Johansen (1996). *, ** and *** indicate rejection of the null at the 5%, 2.5% and 1% significance level, respectively. The standardized eigenvectors are normalized on the diagonal wrt. the endogenous variables. Corresponding Ι-loadings. 1) The asymptotic trace test of the Johansen-method. 2) Small sample approximation to the asymptotic trace test, obtained by adjusting for degrees of freedom, cf. Reimers (1992). 3) Xt* =(qt,pt,rt,t)’ 49 Table 4. Estimation and Testing of the Cointegrating Relation: Single-Equation-Analysis The regression is: (*) q p trprecm where ecm q p tr t t t i t i iN N i t i iN N i t i t t t t t i N = + + + + + + + ≡ − − − − − =− − =− − = ∑ ∑ ∑ β β β β γ γ φ ν β β β β 0 1 2 3 1 2 0 1 2 3 1 1 1 2 2 3 ∆ ∆ , (*) is the static cointegrating regression augmented by current, leaded and lagged first differences of pt and rt and lagged error correction terms ecmt. The augmenting terms in each regression are indicated in the first column. Regression Sample (no. obs.) Coefficient Estimates t-test on price level coeff. No. of regressors (standard errors) (critical sign. level) 1) β0 β1 β2 β3H0: β1=0 H0: β1=1 1. No augmentation 1948-1996 (49) 0.372 0.898 0.024 -11.32 - - 4 (0.99) (0.22) (0.014) (1.80) 2. 1 lead, 1 lag and current first differences 1950-1995 (46) 0.283 0.948 0.015 -5.04 7.18 * 0.39 of pt and rt (N1=N2=1, N3=0 in (*)) 10 (0.60) (0.13) (0.009) (2.15) (0.000) (0.697) 3. 2 leads, 2 lags and current first differences 1951-1994 (44) 0.001 1.024 0.012 -7.20 3.94 * 0.09 of pt and rt (N1=N2=2, N3=0 in (*)) 14 (1.18) (0.26) (0.016) (4.35) (0.000) (0.928) 4. ∆pt-1, ∆rt+1 and ∆rt+2 1950-1994 (45) 0.464 0.913 0.024 -11.99 5.02 * 0.48 7 (0.83) (0.18) (0.012) (1.90) (0.000) (0.631) 5. ∆pt-1, ∆rt+2 and ecmt-1 1950-1994 (45) -0.003 1.007 0.016 -9.04 3.46 * 0.02 (NLS) 7 (1.31) (0.29) (0.019) (1.68) (0.001) (0.984) Note: Entry 1 shows the results from estimating (OLS) the static cointegrating regression (no augmentation), including indicative OLS standard errors. Entries 2 through 4 give the results from the Phillips and Loretan (1991) OLS procedure using different augmentations (as tabulated). In entry 2 the standard errors of the coefficient estimates are adjusted to take account of AR(5) serial correlation in the disturbance term (νt), using the method suggested by Hamilton (1994, p. 608f). Standard errors in entries 3 and 4 are OLS standard errors as the disturbance term fulfills the white noise requirements. Entry 5 uses the Phillips and Loretan (1991) NLS procedure. NLS standard errors, calculated from numerical derivatives of the sum of squared residuals. 1) Critical significance level for two-sided t-test, calculated from standard normal distribution (asymptotic test). A ‘*’ indicates that the null is rejected at the 5% significance level. 50 Table A.1. Cointegration Analysis: Estimates and Tests Single-equation cointegration analysis following the Engle and Granger (1987) two-step procedure. Residuals-based tests for cointegration. In the cointegrating regression, stock prices (qt) are regressed on a constant term (const), measures for the general price level and real production, and the discount rate proxy (rt), cf. (5). Sample 1948-1996 Model Candidate cointegrating vector CRDF Critical test value at Test conclusion: [OLS standard errors]1) (no. of lags) significance level 3) Cointegration at 10% 2) significance level ? 10% 5% qt,const,pyft,yft,rt( 1 ; -1.67 ; -1.38 ; 0.16 ; 11.17 ) -2.834 -3.990 -4.338 No [ 0 ; 0.65 ; 0.14 ; 0.28 ; 2.10 ] (0) (N=4,No Trend,T=48) qt,const,pyfit,yfit,rt( 1 ; 0.02 ; -1.41 ; -0.13 ; 8.78 ) -3.717 -3.990 -4.338 No [ 0 ; 0.29 ; 0.08 ; 0.13 ; 1.76 ] (0) (N=4,No Trend,T=48) qt,const,yft*,rt ( 1 ; -3.38 ; -0.87 ; 12.98 ) -3.162 -3.586 -3.927 No [ 0 ; 0.11 ; 0.04 ; 2.32 ] (1) (N=3,No Trend,T=47) qt,const,pt,t,rt( 1 ; -0.37 ; -0.90 ; -0.02 ; 11.32 ) -4.163 -4.032 -4.381 Yes [ 0 ; 0.99 ; 0.21 ; 0.01 ; 1.80 ] (1) (N=3,With Trend,T=47) qt,const,pyft,t,rt( 1 ; -2.56 ; -0.79 ; -0.03 ; 11.74 ) -3.920 -4.032 -4.381 No [ 0 ; 0.57 ; 0.23 ; 0.01 ; 1.90 ] (1) (N=3,With Trend,T=47) qt,const,pyfit,t,rt( 1 ; -1.29 ; -1.10 ; -0.02 ; 9.18 ) -4.460 -4.032 -4.381 Yes [ 0 ; 0.52 ; 0.18 ; 0.01 ; 1.59 ] (1) (N=3,With Trend,T=47) Note:For definition of variables entering the model, see text. 1) The candidate cointegrating vector is normalized on stock prices. In terms of (5), the vector is (1;-ϑ0;-ϑ1;-ϑ2;-ϑ3). OLS standard errors are indicative only. 2) CRDF is the Dickey and Fuller (1979) t-test statistic of the null of a unit root in the OLS residuals from the cointegrating regression. The null is rejected in favor of the stationary alternative (and cointegration is accepted) if the test statistic is negative and larger in absolute value than the critical test value. Number of augmenting lags of the first differences of the OLS residuals used in the unit root regression shown in parenthesis. 3) Small sample critical values from MacKinnon (1991). N=number of I(1) variables in the model; ‘No Trend’ and ’With Trend’ indicate whether a deterministic trend is included in the cointegrating regression; T=number of observations in the unit root regression, cf. Table 1 in MacKinnon (1991). 51 Figure B1. Stock Return and Inflation Annual stock return and inflation for the (by row) 1-, 5and 10-year horizon. Sample periods 1949-1996, 1953-1996 and 1958-1996, respectively. 1950 1960 1970 1980 1990 0 .5 1 1950 1960 1970 1980 1990 0 .05 .1 .15 1960 1970 1980 1990 .1 .2 .3 1960 1970 1980 1990 .05 .1 1960 1970 1980 1990 .1 .15 .2 1960 1970 1980 1990 .05 .1 S1t S5t S10t I1t I5t I10t 52 Table B1.a. Phillips and Perron (1988) Zt-Test for Unit Root Lag length (l) Series: 0 1 2 3 4 5 6 1-Year Horizon, 1949-1996: S1t-8.52*** -8.52*** -8.53*** -8.55*** -8.61*** -8.67*** -8.72*** I1t-2.79* -2.94** -2.80* -2.70* -2.73* -2.78* -2.86* GR1t-8.14*** -8.15*** -8.25*** -8.41*** -8.69*** -9.00*** -9.45*** GY1t-5.27*** -5.24*** -5.24*** -5.32*** -5.41*** -5.50*** -5.55*** 5-Year Horizon, 1953-1996: S5t-3.16** -3.11** -3.23** -3.22** -3.28** -3.15** -3.08** I5t-0.31 -0.63 -0.78 -0.91 -1.01 -1.08 -1.13 GR5t-3.25** -3.35** -3.41** -3.36** -3.31** -3.11** -2.96** GY5t-1.28 -1.41 -1.43 -1.49 -1.56 -1.54 -1.49 10-Year Horizon, 1958-1996: S10t-2.19 -2.15 -2.18 -2.19 -2.18 -2.20 -2.20 I10t-0.25 -0.57 -0.74 -0.88 -0.99 -1.08 -1.16 GR10t-2.80* -2.81* -2.80* -2.78* -2.79* -2.82* -2.82* GY10t-0.28 -0.40 -0.41 -0.49 -0.58 -0.64 -0.68 Critical test values:10 % 5 % 2.5 % 1 % Without trend -2.60 -2.93 -3.22 -3.58 Note:See note to Table 1.a. All regressions include a constant term, while no trend is allowed for. *,** and *** denote rejection of the null of a unit root at the 10%, 5% and 1% significance level, respectively. 53 Table B1.b. Kwiatkowski et al. (1992) Test for Unit Root Lag length (l) Series: 0 1 2 3 4 5 6 1-Year Horizon, 1949-1996: S1t 0.12 0.16 0.16 0.17 0.19 0.20 0.21 I1t 0.61** 0.36* 0.28 0.23 0.20 0.17 0.16 GR1t 0.05 0.06 0.07 0.09 0.11 0.12 0.15 GY1t 1.05*** 0.84*** 0.74*** 0.64** 0.56** 0.51** 0.48** 5-Year Horizon, 1953-1996: S5t 0.42* 0.26 0.20 0.18 0.16 0.16 0.16 I5t 1.01*** 0.52** 0.36* 0.28 0.23 0.20 0.18 GR5t 0.32 0.20 0.17 0.15 0.16 0.17 0.19 GY5t 2.43*** 1.28*** 0.90*** 0.70** 0.59** 0.52** 0.47** 10-Year Horizon, 1958-1996: S10t 1.31*** 0.74*** 0.54** 0.44* 0.38* 0.34* 0.31* I10t 1.19*** 0.61** 0.42* 0.33 0.27 0.24 0.21 GR10t 0.73** 0.44* 0.34 0.29 0.26 0.24 0.22 GY10t 3.04*** 1.57*** 1.07*** 0.83*** 0.68** 0.58** 0.52** Critical test values:10 % 5 % 1 % Without trend 0.35 0.46 0.74 Note: See note to Table 1.b. No trend is allowed for in the tests, i.e., the null hypothesis is mean-stationarity. *,** and *** denote rejection of the null (i.e., a unit root is present) at the 10%, 5% and 1% significance level, respectively. 54 Table B2. Return Regressions for the 1-, 5and 10-Year Investment Horizon. The model is: (*) Sk Ik GYk GRk kyears t t t t t = + + + + = β β β β ε 0 1 2 3 1 5 10, , , The ‘simple’ model in the table only includes inflation as an explanatory variable (β2≡β3≡0), while the ‘extended’ model includes all variables in (*). In the ‘reduced extended’ model, we have removed the insignificant variables from the extended model. Horizon Sample Model Coefficient estimates Goodness-of-fitt-test on inflation coeff. (years) (sample size) (standard errors) 2) (critical sign. level) 3) βoβ1 β2β3R2 σε H0: β1=0 H0: β1=1 11949-1996 Simple 1) 0.102 -0.106 - - 0.57 0.18 -0.17 - (48) (0.029) (0.61) (0.865) Extended 1) 0.152 -0.205 -1.02 -5.63 0.67 0.16 -0.30 - (0.045) (0.68) (0.88) (1.8) (0.764) Reduced extended 1) 0.111 -0.0368 - -6.01 0.66 0.16 -0.05 - (0.032) (0.73) (1.7) (0.959) 51953-1996 Simple 0.0617 0.936 - - 0.11 0.073 1.53 - (44) (0.025) (0.61) (0.126) Extended 0.0908 0.936 -0.702 -7.27 0.34 0.064 1.64 - (0.041) (0.57) (0.69) (2.8) (0.101) Reduced extended 0.0643 1.01 - -7.35 0.33 0.064 1.84 0.03 (0.025) (0.55) (2.8) (0.066) (0.979) 10 1958-1996 Simple 0.0665 0.858 - - 0.21 0.041 3.06 * -0.51 (39) (0.014) (0.28) (0.002) (0.610) Extended 0.109 0.808 -0.944 -8.85 0.59 0.030 2.89 * -0.68 (0.031) (0.28) (0.55) (2.4) (0.004) (0.497) Reduced extended 0.0668 1.01 - -9.99 0.54 0.031 4.81 * 0.04 (0.014) (0.21) (2.6) (0.000) (0.967) Note: All coefficients are estimated by OLS. Standard errors of the coefficient estimates are OLS errors for the 1-year horizon, respectively Newey and West (1987) errors for the 5and 10year horizons. The reported Newey and West (1987) errors allow for heteroskedasticity and serial correlation in the disturbance term (εt) up to lag 5. 1) Impulse dummies included for 1972 and 1983. 2) Not corrected for serial correlation or heteroskedasticity in the disturbance term. 3) Critical significance level for two-sided t-test, calculated from standard normal distribution (asymptotic test). Based on Newey and West (1987) standard errors for the 5and 10-year horizons. A ‘*’ indicates that the null is rejected at the 5% significance level. 55 Figure B2. Recursive Estimation of the 10-Year Return Regression Recursive point estimates (solid line) and 95% confidence bands for the coefficients of the 10-year return regression (reduced extended model). Recursive least squares. Full sample: 1958-1996. 1970 1975 1980 1985 1990 1995 .05 .1 .15 1970 1975 1980 1985 1990 1995 -2 -1 0 1 2 1970 1975 1980 1985 1990 1995 -10 0 10 Constant term (ϑ0) Coeff. to inflation (ϑ1) Coeff. to change in discount rate ( ϑ 3)