Inflation and real stock returns revisited.

Lin, Shu-Chin

I. INTRODUCTION

This article empirically reinvestigates the relationship between real stock returns and inflation. According to Fisher (1930), (1) in an efficient market, real stock returns should reflect expectations only about real factors such as growth in output and time preference of savers such that real stock returns and inflation should vary independently. In other words, on average, investors are fully compensated for erosions in purchasing power over time. However, this so-called generalized Fisher hypothesis has received little empirical support. While Kaul (1990), Boudoukh and Richardson (1993), Solnik and Solnik (1997), Madsen (2005), and Ryan (2006) find positive or weakly positive correlation between stock returns and inflation, most empirical studies, including Nelson (1976), Fama (1981, 1983), Geske and Roll (1983), Gultekin (1983), Kaul (1987), and Solnik (1983), uncover a negative effect of inflation (anticipated and/or unanticipated) on (real or nominal) stock returns. These apparent inconsistencies between the data and the predictions of economic theory have been termed as the stock return-inflation puzzle and cited as evidence of monetary nonsuperneutrality or irrationality (and market inefficiency). (2)

To reconcile the seemingly conflicting contributions, Danthine and Donaldson (1986) provide a rational expectations equilibrium model in which a negative relation between stock returns and inflation arises from non-monetary sources (e.g., a real output shock) and a positive relation is driven by purely monetary sources. Stulz (1986), Marshall (1992), and Bakshi and Chen (1996) also assert that stock returns may be negatively correlated with inflation, especially when the source of inflation is related to nonmonetary factors. Evidently, Hess and Lee (1999), Khil and Lee (2000), and Lee (2003) find that while monetary shocks drive a positive correlation, real (fiscal or real output) shocks result in a negative correlation between stock returns and inflation. From a different perspective, Kaul (1987, 1990) argues that the relation between these two variables depends on both the money demand and the money supply. Specifically, while the combination of money demand effects with a countercyclical monetary response accentuates a negative correlation, money demand shocks along with a procyclical monetary response lead to a neutral or positive correlation between real stock returns and inflation. Similar arguments can be found in Day (1984), Kaul and Seyhun (1990), Boudoukh, Richardson, and Whitelaw (1994), and Gallagher and Taylor (2002) and are supported by compelling empirical evidence (see, e.g., Kim 2003; Lee 1992).

As an alternative, this article models the real stock return-inflation relationship as intrinsically dynamic, explicitly distinguishing between the short run and the long run. Such a distinction is crucial, because imperfect information, rigidity, and/or contractual obligations are more prevalent in the short run than in the long run. In that sense, the response of real stock returns to inflation could differ in the short run versus long run. Moreover, we assess the long- and short-run impacts of inflation uncertainty on real stock returns. As argued, the real costs of inflation are associated with its uncertainty. Greater inflation uncertainty resulting from higher inflation may lead to economic inefficiency, lower real investment, and unemployment, thereby reducing economic growth. Thus, it is interesting to see if there are independent effects of inflation uncertainty on real stock returns both in the short run and in the long run.

In so doing, this article uses an innovative econometric technique to explore the long-and short-run relationships between real stock returns and inflation (both anticipated and unanticipated) and the relationships with inflation uncertainty. It departs from earlier work and complements recent evidence in at least three aspects. First, we conduct a panel analysis on data pooled from 16 industrialized Organization for Economic Cooperation and Development (OECD) countries for the 1957:Q1 to 2000:Q1 period. Existing studies usually build on a time series framework for a single or a group of countries. In contrast, this article recasts the issue into a dynamic panel framework to exploit both the cross-section and the time series dimensions of the data. Moreover, the dynamic panel data model allows us to control heterogeneity across countries by the inclusion of country-specific effects.

Second, to measure the effects of both expected and unexpected inflation and inflation uncertainty, we employ generalized autoregressive conditional heteroskedasticity (GARCH)-type models to obtain expected and unexpected components of inflation and conditional variance as a proxy for inflation uncertainty. The procedure thus allows us to avoid problems associated with the survey data (see Kandel, Ofer, and Sarig 1996; Woodward 1992, for the detailed discussion) as well as the problem of using volatility (or standard error) to measure inflation uncertainty (Fountas and Karanasos 2007). As argued, conditional variance is better to capture inflation risk and uncertainty.

Last, and most importantly, we apply the pooled mean group (PMG) estimator of Pesaran, Shin, and Smith (1999) to estimate short- and long-run relationships between real stock returns and inflation. Specifically, the estimator can be applied to either I(1) and/ or I(O) variables and does not require the pre-testing of unit roots. (3) By focusing on different time spans, we set the basis for an explanation of the contradictory effects of inflation on real stock returns. This methodology can be summarized as a panel error-correction model, where short- and long-run effects are estimated jointly from a general autoregressive distributed lag (ARDL) model. The PMG estimator has been recently applied to measure the effect of exchange rate uncertainty on investment (Byrne and Davis 2005a, 2005b), to assess the trade effect of real effective exchange rates by Catao and Solomou (2005), to estimate the impacts of fiscal deficits on inflation (Catao and Terrones, 2005), and to estimate the relationship between financial development and economic growth (Loayza and Ranciere, 2006).

Our empirical findings are summarized as follows. First, while anticipated inflation has insignificant short-run impacts on real stock returns, it tends to have significantly negative long-run impacts on real stock returns. Second, regarding unanticipated inflation, we find that a negative long-run effect coexists with a positive short-run effect on real stock returns. Finally, inflation uncertainty is found to have independent impacts and tends to exert a significantly negative long-run effect but an insignificant short-run effect on real stock returns. These findings are robust to alternative measures of expected inflation, unexpected inflation, and inflation volatility.

The remainder of this article is organized as follows. Section II introduces the PMG estimator proposed by Pesaran, Shin, and Smith (1999) and elaborates alternative measures of expected and unexpected inflation and inflation uncertainty. Section III describes the data and reports empirical results and robustness tests. Section IV concludes the analysis.

II. INFLATION AND REAL STOCK RETURNS

A. The Generalized Fisher Hypothesis

This section tests the generalized Fisher hypothesis, which states that the real return on common stocks and (both expected and unexpected components of) inflation should vary independently of each other. This conjecture makes sense to many economists since, as claims against real physical capital, common stocks ought to act as a hedge against inflation. (4) This generalized hypothesis is commonly tested by a regression of the form using time series data:

(1) [r.sub.t] = [mu] + [theta][[pi].sup.e.sub.t] + [[epsilon].sub.t],

where [r.sub.t] is the real stock returns and [[pi].sup.e.sub.t] is the expected inflation rate. According to the generalized Fisher hypothesis, the real rates of return on common stocks and expected inflation rates are independent. Thus, the coefficient [theta] should not be significantly different from 0 if the generalized Fisher hypothesis holds. The same logic applies for the unexpected inflation cases.

B. The ARDL Approach

In this article, as an alternative, we assess the effect of expected inflation on real stock return using panel econometric techniques. In particular, Equation (1) is extended to a panel data specification. Moreover, following conventional wisdom, we assume that there exists a long-run relationship between real stock return ([r.sub.it]) and expected inflation rate ([[pi].sup.e.sub.it]), which can be written as:

(2) [r.sub.it] = [[mu].sub.i] + [theta][[pi].sup.e.sub.it] + [[epsilon].sub.it]

where [[mu].sub.i] is the fixed (country-specific) effect, i = 1, 2, ..., N is the country indicator, and t = 1, 2, ..., T is the time index. In the time series framework, Pesaran and Shin (1998) and Pesaran, Shin, and Smith (2001) propose the ARDL models to estimate the long-run cointegrating relationship among variables of interest. In a panel data specification, we follow the suggestion of Pesaran, Shin, and Smith (1999) to nest Equation (2) in an ARDL specification to allow for rich dynamics in the way that real stock return adjusts to changes in the anticipated inflation and to other explanatory variables, if any. The ARDL(p, q) model, that is, the dependent and independent variables enter the right-hand side with lags of order p and q, respectively, can be written as:

(3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [x.sub.it], is a k x 1 vector of independent variables, which includes only the expected inflation [[pi].sup.e.sub.it] (in some cases, the unexpected inflation [[pi].sup.u.sub.t]), in our main application. In addition to [[pi].sup.e.sub.it], however, we also consider the inflation uncertainty ([h.sub.it]) as an additional explanatory variable in our later application.

Equation (3) can be reparameterized and written as:

(4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [[ph].sub.i] = - (1 - [[summation].sup.p.sub.j=1] [[lambda].sup.ij], [[beta].sub.i] = [[summation].sup.q.sub.j] = [[delta].sub.ij], [[lambda].sup.*.sub.ij] = - [[summation].sup.p.sub.m=j+1] [[lambda].sub.im], and [[delta].sup.*.sub.ij] = - [[summation].sup.q.sub.m=j+1] [[delta].sub.im], j = 1, 2, ..., q - 1.

By grouping the variables in levels, Equation (4) can be rewritten as:

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where the vector [[theta].sub.i] = -[[beta].sub.i]/[[phi].sub.i] defines the long-run equilibrium relationship between [r.sub.it] and [x.sub.it] as [r.sub.it] = - ([[beta]'.sub.i]/[[phi].sub.i])[x.sub.it] + [[eta].sub.it]. In contrast, [[lambda].sup.*.sub.it] and [[delta].sup.*.sub.it] are the short-run coefficients relating real stock returns to its determinants [x.sub.it]. Finally, [[phi].sub.i] measures the speed of adjustment of [r.sub.it] toward its long-run equilibrium following a given change in [x.sub.it] and [[phi].sub.i] < 0 ensures that such a long-run relationship exists. As a result, a significant and negative value of [phi] can be treated as evidence in support of cointegration between [r.sub.it] and [x.sub.it].

As argued in Catao and Solomou (2005) and Catao and Terrones (2005), the ARDL specification in Equation (5), where all explanatory variables enter the regression with a lag, not only mitigates endogeneity issues but also accommodates the substantial persistence of inflation adjustments and captures potentially rich stock return adjustment dynamics. In addition, the model allows for heterogeneity in the relationship between stock return and expected inflation across countries since the various parameters in Equation (5) are not restricted to be the same across countries. Finally, the ARDL approach allows us to estimate an empirical model that encompasses the long- and short-run stock return effects of expected inflation using a data field composed of a relatively large sample of countries and quarterly observations.

There are a few existing procedures for estimating the above model. At one extreme, the simple pooled estimator assumes the fully homogeneous-coefficient model in which all slope and intercept parameters are restricted to be identical across countries. At the other extreme, the fully heterogeneous-coefficient model imposes no cross-country coefficients constraint and can be estimated on a country-by-country basis. This is the so-called mean group (MG) estimator introduced by Pesaran and Smith (1995). The approach amounts to estimate separate ARDL regressions for each group and obtain [theta] and [phi] as simple averages of individual group coefficient [[theta].sub.i] and [[phi].sub.i]. In particular, Pesaran and Smith (1995) show that the MG estimator will provide consistent estimates of the averages of parameters interested.

In between these two extremes, the dynamic fixed-effect method allows the intercepts to differ across groups, but it imposes homogeneity of all slope coefficients and error variances. Alternatively, Pesaran, Shin, and Smith (1999) propose the PMG estimator, which restricts the long-run parameters to be identical over the cross section, but allows the intercepts, short-run coefficients (including the speed of adjustment), and error variances to differ across groups on the cross section. If the long-run homogeneity restrictions are valid, it is known that MG estimates will be inefficient. In this case, the maximum likelihood--based PMG approach proposed by Pesaran, Shin, and Smith (1999) will yield a more efficient estimator. (5) As shown in Pesaran, Shin, and Smith (1999), the validity of a cross-sectional, long-run homogeneity restriction of the form [[theta].sub.i] = [theta], i = l, 2 , ... *, N--and hence the suitability of the PMG estimator--can be tested by a standard Hausman-type statistic.

In terms of the real stock-inflation relation, the PMG estimator offers the best available compromise in the search for consistency and efficiency. This estimator is particularly useful when the long run is given by conditions expected to be homogeneous across countries, while the short-run adjustment depends on country characteristics such as monetary and fiscal adjustment mechanisms, capital market imperfections, and relative price and wage flexibility (e.g., Loayza and Ranciere 2006). Therefore, we use the PMG method to estimate a long-run relationship that is common across countries while allowing for unrestricted country heterogeneity in the adjustment dynamics.

C. Measures of Expected Inflation and Inflation Uncertainty

To estimate the Fisher regression, we need a measure of expected inflation [[pi].sup.e.sub.it]. First, we follow Gultekin (1983) to use contemporaneous inflation rates as proxies for expected inflation. The realized values are used under the assumption that expectations are rational. Second, we rely on a pure autoregressive integrated moving average (ARIMA) model to generate expected inflation and unexpected inflation. Inflation forecasts from ARIMA regressions are used as indicators of expected inflation, while the forecast errors are used as the measure of unexpected inflation [[pi].sup.u.sub.it]. The model is expressed as:

(6) [[pi].sub.t] = [[alpha].sub.0] + [p.summation over (i=1)] [[alpha].sub.i][[pi].sub.t-i] + [q.summation over (j=1)] [[phi].sub.j][[epsilon].sub.t-j],

where [[epsilon].sub.t-j] is a white noise process. Third, to allow for conditional hgteroskedasticity, we assume that [[epsilon].sub.t-j]|[[OMEGA].sub.t-1] = [h.sup.1/2.sub.t][[eta].sub.t] and [[eta].sub.t] ~ NID (0, 1). Particularly, we consider three alternative specifications of the conditional variance [h.sub.t] for each country. The first one is the GARCH(1,1) process set out by Bollerslev (1986), which can be specified as:

(7) [h.sub.t] = [a.sub.0] + [a.sub.1][h.sub.t-1] + [b.sub.1][[epsilon].sup.2.sub.t-1].

By taking account of the asymmetric effects of negative and positive shocks, the second specification for the conditional variance is the exponential GARCH (EGARCH) process proposed by Nelson (1991). The specification can be written as:

(8) In ([h.sub.t]) = [a.sub.0] + [a.sub.1]ln ([h.sub.t-1]) + [b.sub.1][absolute value of [[epsilon].sub.t-1]/[square root of [h.sub.t-1]]] + [c.sub.1][[epsilon].sub.t-1]/[square root of [h.sub.t-1]]].

And the third and last model for the conditional variance is an extension of the basic GARCH model. Engle and Lee (1999) represent the GARCH(1,1) model as characterized by reversion to a constant mean [bar.[mu]], that is,

(9) [h.sub.t] = [bar.[mu]] + [a.sub.1]([h.sub.t-1] - [bar.[mu]]) + [b.sub.1]([[epsilon].sup.2.sub.t-1] - [bar.[mu]]).

In contrast, their component GARCH (CGARCH) process allowing reversion to a time-varying mean [m.sub.t] is modeled as:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By estimating Equation (6) along with Equations (7), (8), and (10), respectively, we can obtain expected and unexpected components of inflation. Moreover, we follow conventional applications such as Asteriou and Price (2005) and Byrne and Davis (2005a, 2005b) to proxy inflation uncertainty by the logarithm of the fitted (conditional) volatility values from Equations (7), (8), and (10), respectively. The corresponding inflation uncertainty measures are denoted as [h.sup.g.sub.it], [h.sup.eg.sub.it], and [h.sup.cg.sub.it], respectively. In addition to expected inflation, they are used to assess the effect of inflation uncertainty on real stock returns.

III. DATA AND EMPIRICAL RESULTS

The generalized Fisher hypothesis is examined using quarterly data on a panel of 16 industrialized OECD countries. These countries include Australia, Austria, Belgium, Canada, Finland, France, Germany, Ireland, Italy, Japan, The Netherlands, New Zealand, Norway, Spain, the United Kingdom, and the United States. The sample period runs from 1957:Q1 to 2000:Q1, with a few exceptions. The original observations are collected by Rapach (2002) from the International Monetary Fund's International Financial Statistics. (6) The inflation rate is computed by taking first differences of the natural logarithm of the consumer price index. And the real stock return is computed as the differences of the natural logarithm of the nominal share price index deflated by the consumer price index. Table 1 reports basic statistics for inflation and real stock return variables for each country. As shown, Germany has the lowest average inflation, while Spain has the highest average inflation. Also, Belgium has the lowest average rate of stock return, whereas Finland has the highest average rate of stock return. Moreover, the correlation coefficients between inflation rates and real stock returns are negative for all countries.

A. The Effects of Expected Inflation

As a benchmark, we first test the generalized Fisher hypothesis using realized inflation as a proxy for expected inflation via the PMG methodology. The procedure is as follows. First, the estimates of the long-run slope parameters are obtained jointly across countries through a (concentrated) maximum likelihood procedure. Second, the estimation of short-run coefficients along with the speed of adjustment, country-specific intercepts and error variances are done on a country-by-country basis also through maximum likelihood and using the estimates of the long-run slope coefficients previously obtained.

Table 2 displays the results on specification tests and the estimation of long- and short-run parameters linking real stock returns and inflation. (7) We emphasize the results from using the PMG estimator, considering its gains in consistency and efficiency over other panel error-correction estimators. For comparison purposes, We also present the results obtained with the MG estimator.

To check the existence of a long-run relationship (dynamic stability), the coefficient on the error-correction term should be negative and within the unit circle. As can be seen in Table 2, the pooled error-correction coefficient estimates are significantly negative and fall within the dynamically stable range for both PMG and MG estimators, giving evidence of mean reversion to a nonspurious long-run relationship and therefore stationary residuals. In addition, the Hausman test of long-run homogeneity restriction is not rejected, indicating that the PMG estimator is more suitable for the analysis, relative to the MG estimator. Accordingly, the following analysis focuses on the PMG approach.

Regarding the estimated parameters, we find that the long-run estimate of inflation is highly significant and negative, meaning that real stock returns are strongly and negatively linked to realized inflation in the long run. However, the short-run coefficients on inflation tell a different story. As explained, short-run coefficients are not restricted to be the same across countries, so that we do not have a single pooled estimate for each coefficient. Nevertheless, we can still analyze the average short-run effect by considering the mean of the corresponding coefficients across countries. As shown in Table 2, the short-run average relationship between real stock returns and inflation appears to be positive but insignificant. This finding is consistent with the generalized Fisher hypothesis of zero correlation between real stock returns and inflation. Stocks, as claims against productive capital, can fully protect shareholders from changes in (expected) inflation.

To check if these results are sensitive to model specification (time trend and/or seasonal effects), Table 3 reports these robustness tests. The estimation outcome is qualitatively similar to that in Table 2. The signs and statistical significance of both long- and short-run coefficients remain unchanged. Moreover, the pooled error-correction coefficients continue to be significantly negative and within the unit circle. Furthermore, the long- and short-run effects of inflation on real stock returns are not only qualitatively but also quantitatively very similar to those in Table 2.

As another robustness check, we experiment with alternative measures of expected inflation. As mentioned earlier, we extract expected components of inflation from a simple ARIMA model and the ARIMA model with GARCH, EGARCH, and CGARCH processes, respectively. The corresponding results are presented in Table 4. As shown, the pooled error-correction estimates continue to signal long-run cointegration between real stock returns and unexpected inflation, irrespective of alternative unexpected inflation measures. Moreover, the long-run coefficient estimates remain significantly negative, while the short-run coefficient estimates remain insignificant, irrespective of different measures of expected inflation.

Finally, to make sure our results are not pertaining to a specific sample, we segregate the sample into two groups, G7 and non-G7 country categories. The results are reported in Table 5. Specifically, Panel A of Table 5 reports the results for G7 countries, while Panel B of Table 5 reports the results for non-G7 countries. Once again, they are qualitatively similar to those with the full sample, confirming negative long-run correlation but zero correlation between real stock returns and expected inflation in the short run.

B. The Effects of Unexpected Inflation

This section examines the generalized Fisher hypothesis using unexpected inflation. As mentioned, the unexpected inflation is measured as the difference between the actual and the expected inflation extracted above. Table 6 summarizes the estimation results. As can be seen, the estimates for the pooled error-correction coefficient are significantly negative and lie inside the unit circle, indicating the existence of long-run relationship between real stock returns and unexpected inflation. Moreover, the long-run coefficient estimates are negative and highly significant, whereas the short-run coefficient estimates are positive and strongly significant, irrespective of alternative measures of unexpected inflation. The findings suggest that while unexpected inflation has a positive impact on real stock returns in the short run, it tends to affect real stock returns in a negative way in the long run. Accordingly, the evidence of coexistence of a positive short-run effect of unexpected inflation with a negative long-run effect of unexpected inflation implies that stocks are a poor hedge against unexpected inflation.

The results also hold for different subsamples as shown in Table 7, where countries are split into G7 and non-G7 country groups. In either country group, the pooled error-correction coefficient estimates fall within the dynamically stable range, meaning that real stock returns and unexpected inflation are cointegrated. Moreover, the sign and statistical significance of all long- and short-run coefficients remain unchanged, and the magnitude of long- and short-run coefficients is qualitatively similar to those with full sample. The data again confirm that stocks are a poor hedge against unexpected inflation, irrespective of time frequency, unexpected inflation indicators, and sample size considered.

C. The Effects of Inflation Uncertainty

This section evaluates the impacts of inflation uncertainty on real stock returns. As suggested by Friedman (1977), an increase in inflation variability adversely affects economic activity because it reduces the role of prices in guiding market activity and increases the cost of assimilating information. This negative real effect of inflation volatility is supported by Fischer (1981) and Holland (1988), among others. On the other hand, Berument, Ceylan, and Olgun (2007) point out that modeling inflation volatility is necessary in that risk-averse agents tend to consider both anticipated macroeconomic variables and associated risk of the variables in their decision-making process. In that sense, a real effect of inflation uncertainty is expected. For example, based on a precautionary motive and the assumption of risk-averse agents, Dotsey and Sarte (2000) suggest that more inflation uncertainty raises saving and hence investment and economic growth. The impact of inflation volatility is thus an empirical matter.

To explore whether stocks are a good hedge against variable inflation, we need an indicator of inflation uncertainty. Since there are different ways to model inflation uncertainty, here, as mentioned, the inflation uncertainty is measured by conditional variance obtained from the GARCH, EGARCH, and CGARCH processes, respectively. Table 8 reports the estimation results when both expected inflation and inflation uncertainty measures are included as explanatory variables. Clearly, the pooled error-correction coefficient estimates continue to be negative and significant, suggesting a long-run equilibrium relationship among real stock returns, expected inflation, and inflation uncertainty. As expected, anticipated inflation measures maintain significantly negative long-run impacts along with insignificant short-run impacts on real stock returns in all regressions. Notably, inflation uncertainty appears to have a negative long-run effect but an insignificant short-run impact on real stock returns, irrespective of alternative inflation uncertainty indicators. The evidence indicates that both the level and the uncertainty of inflation contribute to lower real stock returns in the long run but exert little short-run influence on real stock returns. Moreover, the observed negative long-run effect of inflation uncertainty agrees with the proposition of malfunctioning price mechanism due to more volatile inflation.

IV. CONCLUSIONS

In the Fisherian world, inflation is a purely monetary phenomenon in the sense that it has no real effect on economy activities. However, because of imperfect capital markets, inertia, and/or contractual obligation, changes in inflation may have different influence on real activities in the short run and long run. By recognizing this possibility, this article estimates an encompassing empirical model of the long-and short-run effects of inflation on real stock returns using a sample of cross-country and time series observations. We implement the analysis using the PMG methodology of Pesaran, Shin, and Smith (1999), which allows for short-run heterogeneity across countries but long-run homogeneity among sample countries.

Our empirical results are summarized as follows. First, in terms of anticipated component of inflation, we find that the Fisher effect appears to hold at short horizons, where expected inflation has insignificant short-run impacts on real stock returns but not at longer horizons and where anticipated inflation is found to have negative long-run impacts on real stock returns. Second, regarding unanticipated inflation, we find that a negative long-run effect coexists with a positive short-run effect. The evidence thus invalidates the Fisher hypothesis and suggests that stocks are a poor hedge against unexpected inflation. Third, as for inflation uncertainty similar to the expected inflation case, we find that while inflation uncertainty has a negative long-run effect on real stock returns, it has an insignificant short-run effect on real stock returns. Finally, these results are found to be robust to alternative measures of expected and unexpected inflation and inflation uncertainty.

It is noted that this article focuses on advanced countries. It will be interesting and vital to test the generalized Fisher hypothesis for both developed and developing countries to gain the whole picture of the real stock returns-inflation relation. Moreover, since inflation exerts different long- and short-run impacts on real stock returns, it would be important for both theoretical models and empirical studies to explore the exact mechanisms through which inflation, both expected and unexpected, and inflation uncertainty affect real stock returns at different time lengths. Finally, since the financial infrastructure has changed dramatically for the sample countries, it is important for future study to incorporate this element into the PMG estimation to see whether structural changes play a role in the relationship between real stock returns and inflation. (8)

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(5.) Provided that there is a unique vector defining the long-run relationship among variables involved, and the lag orders p and q are suitably chosen, MG and PMG estimates of an ARDL regression yield consistent estimates of that vector, regardless of whether the variables involved are I(1) or I(0).

(6.) All the data used in this article can be downloaded from the Web site of Rapach at http://pages.slu.edu/ faculty/rapachde/.

(7.) Loayza and Ranciere (2006) suggest that when the main interest is on the long-run parameters, the lag order of the ARDL can be selected using some consistent information criteria on a country-by-country basis; however, when there is also interest in analyzing and comparing the short-run parameters, it is recommended to impose a common lag structure across countries. Thus, in this article, we use the latter procedure and set p = q = 1, for simplicity. For robustness check, we have also tried different orders for p and q selected by Akaike information criterion, Schwarz Bayesian criterion, and Hannan and Quinn, respectively. We found qualitatively and quantitatively similar results.

(8.) We thank one anonymous referee for this point.

ABBREVIATIONS

ARDL: Autoregressive Distributed Lag

ARIMA: Autoregressive Integrated Moving Average

CGARCH: Component Generalized Autoregressive Conditional Heteroskedasticity

EGARCH: Exponential Generalized Autoregressive Conditional Heteroskedasticity

GARCH: Generalized Autoregressive Conditional Heteroskedasticity

MG: Mean Group

OECD: Organization for Economic Cooperation and Development

PMG: Pooled Mean Group

doi: 10.1111/j.1465-7295.2009.00193.x

(1.) Originally, Fisher (1930) maintains that nominal interest rates fully reflect available information about the future values of inflation rates, and thus, nominal interest rates should move one-for-one with inflation, both ex ante and ex post, under the assumption that real interest rates are independent of movements in inflation. Fama and Schwert (1977) argue that this Fisher hypothesis, based on a complete dichotomy between the real and the monetary sectors of the economy, can be generalized to stocks and other forms of assets.

(2.) Several theoretical explanations have been put forth for the negative relationship of stock returns with inflation. For example, the Mundell-Tobin hypothesis is based on the Pigou real wealth effect in explaining a negative relationship between asset returns and inflation and in supporting monetary nonneutrality. However, the money illusion hypothesis states that stock market investors suffer from money illusion and incorrectly discount real cash flows with nominal discount rates, thereby causing the market's subjective expectation of the future equity premium to deviate systematically from the rational expectations (e.g., Asness 2000; Cohen, Polk, and Vuolteenaho 2005; Modigliani and Cohn 1979; Sharpe 2002). In addition, the proxy hypothesis suggests that the observed negative relationship between stock returns and inflation is proxying for a positive relation between stock returns and real variables and a negative relation between inflation and real activity (e.g., Fama 1981, 1983; Geske and Roll 1983).

(3.) As long as there exists long-run relationship, whether variables are integrated is not required for the PMG methodology.

(4.) As defined in Bodie (1976, 460), a security is an inflation hedge if and only if its real return is independent of the rate of inflation.

Shu-Chin Lin *

* I am grateful to M. Hashem Pesaran and David E. Rapach for making publicly available their computer code and data used in this article, respectively. I also thank Vincenzo Quadrini (the editor) and River Huang for helpful suggestions and comments and Albert Suen for excellent research assistance. Any remaining errors are my own responsibility.

Lin: Professor, Department of Economics, Tamkang University, 151 Ying-Chun Road, Tamsui 25137, Taipei County, Taiwan, Republic of China. Phone +886-2-26215656 ext 3359, Fax +886-2-26209654, E-mail: econscl@mail.tku.edu.tw

TABLE 1

Basic Statistics and Sample Coverage

 corr
Country Name r [pi] (r, [pi])

Australia 0.5445 1.3271 -0.3035
Austria 0.3968 0.9277 -0.0516
Belgium -0.2152 1.0359 -0.2377
Canada 0.5339 1.0832 -0.2070
Finland 1.7829 1.4586 -0.4056
French 0.6694 1.3688 -0.2257
Germany 0.9859 0.8136 -0.2497
Ireland 1.1471 1.6229 -0.1212
Italy 0.2872 1.7502 -0.1028
Japan 0.9841 1.0227 -0.2725
The Netherlands 1.0647 0.9773 -0.1766
New Zealand 0.0579 1.7439 -0.2684
Norway 0.5207 1.3270 -0.2264
Spain 0.2744 2.0856 -0.1661
United Kingdom 0.8567 1.5860 -0.1137
United States 1.0598 1.0533 -0.3895

Country Name Period Observations

Australia 1957:02 to 2000:01 172
Austria 1957:02 to 1998:04 167
Belgium 1957:02 to 1996:02 157
Canada 1957:02 to 2000:01 172
Finland 1957:02 to 2000:01 172
French 1957:02 to 2000:01 172
Germany 1970:02 to 2000:01 120
Ireland 1957:02 to 2000:01 172
Italy 1957:02 to 2000:01 172
Japan 1957:02 to 2000:01 172
The Netherlands 1957:02 to 2000:01 172
New Zealand 1961:02 to 2000:01 156
Norway 1957:02 to 2000:01 172
Spain 1961:02 to 2000:01 156
United Kingdom 1958:02 to 1999:01 164
United States 1957:02 to 2000:01 172

Notes: The reported values are the means of the real stock return
(r) and inflation ([pi]). In addition, the correlation coefficients
between r and [pi] for all countries over alternative sample
periods and corresponding (quarterly) observations are also
reported.

TABLE 2

The Effect of Inflation on Real Stock Return

 PMG MG
Long-run coefficient
 [[pi].sub.it] -1.8405 *** (0.1843) -1.8799 *** (0.2056)
Error-correction
 coefficient
 [phi] -0.7564 *** (0.0231) -0.7613 *** (0.0229)
Short-run coefficients
 [DELTA][[pi].sub.it] 0.0037 (0.2419) 0.1345 (0.2554)
 c 2.3810 *** (0.1638) 2.4454 *** (0.3389)

 Hausman Test
Long-run coefficient
 [[pi].sub.it] 0.1878 (0.6647)
Error-correction
 coefficient
 [phi]
Short-run coefficients
 [DELTA][[pi].sub.it]
 c

Notes: The values in the parenthesis are the standard errors
of corresponding coefficients. For Hausman test of long run
parameter homogeneity, the value in the bracket is the
corresponding p value.

*** indicates significance at 1% level.

TABLE 3

Robustness Check

 T Qs

Long-run coefficient
 [[pi].sub.it] -2.1497 *** (0.2257) -2.0727 *** (0.1987)
Error-correction
 coefficient
 [phi] -0.7546 *** (0.0195) -0.7361 *** (0.0193)
Short-run coefficients
 [DELTA][[pi].sub.it] 0.3596 (0.2738) -0.1329 (0.2328)
 T -0.0569 (0.4174)
 Q1 0.6800 (0.7766)
 Q2 0.2583 (0.9859)
 Q3 -0.2060 (0.7801)
 c 2.6969 *** (0.2633) 2.3554 *** (0.6372)

 T plus Qs

Long-run coefficient
 [[pi].sub.it] -2.2462 *** (0.2257)
Error-correction
 coefficient
 [phi] -0.7397 *** (0.0200)
Short-run coefficients
 [DELTA][[pi].sub.it] -0.0639 (0.2347)
 T -0.0576 (0.4394)
 Q1 0.6862 (0.7768)
 Q2 0.2620 (0.9898)
 Q3 -0.2010 (0.7802)
 c 2.5531 *** (0.6746)

Notes: T denotes trend and Qs denote seasonal dummies.
The values in the parenthesis are the standard errors
of corresponding coefficients.

*** indicates significance at 1% level.

TABLE 4

Alternative Expected Inflation on Real Stock Return

 ARIMA

 ARIMA (a) GARCH (b)

Long-run coefficient
 [[pi].sup.e.sub.it] -1.9063 *** (0.2555) -1.8290 *** (0.2611)
Error-correction
 coefficient
 [phi] -0.7443 *** (0.0184) -0.7437 *** (0.0191)
Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] 0.7154 (0.8088) -0.2265 (0.7557)
 c 2.4293 *** (0.1403) 2.2899 *** (0.1266)

 ARIMA

 EGARCH (b) CGARCH (b)

Long-run coefficient
 [[pi].sup.e.sub.it] -1.8156 *** (0.2551) -1.9146 *** (0.2606)
Error-correction
 coefficient
 [phi] -0.7468 *** (0.0195) -0.7472 *** (0.0194)
Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] -0.1156 (0.7716) -0.3025 (0.6777)
 c 2.3078 *** (0.1274) 2.3699 *** (0.1348)

Notes: The values in the parenthesis are the standard errors of
corresponding coefficients.

(a) The fitted value of inflation ([[pi].sup.e.sub.it]) from a pure
ARIMA model as in Equation (6) for each country.

(b) The fitted value of inflation ([[pi].sup.e.sub.it]) from an
ARIMA model with alternative conditional heteroskedasticity
specifications as in Equations (7), (8), and (l U), respectively.

*** indicates significance at 1% level.

TABLE 5

Subsample Results--Expected Inflation on Real Stock Return

 ARIMA

 ARIMA GARCH

Panel A: G7 countries
 Long-run coefficient
 [[pi].sup.e.sub.it] -1.6812 *** (0.3764) -1.5573 *** (0.3970)
 Error-correction
 coefficient
 [phi] -0.7429 *** (0.0240) -0.7422 *** (0.0239)
 Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] -0.3598 (0.7431) -0.6735 (0.9021)
 C 2.1542 *** (0.0684) 1.9613 *** (0.0494)

Panel B: non-G7 countries
 Long-run coefficient
 [[pi].sup.e.sub.it] -2.1012 *** (0.3470) -2.0429 *** (0.3459)
 Error-correction
 coefficient
 [phi] -0.7460 *** (0.0280) -0.7456 *** (0.0294)
 Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] 1.5536 (1.2955) 0.1239 (1.1859)
 c 2.6918 *** (0.2692) 2.5852 *** (0.2456)

 ARIMA

 EGARCH CGARCH

Panel A: G7 countries
 Long-run coefficient
 [[pi].sup.e.sub.it] -1.4028 *** (0.3688) -1.5484 *** (0.3810)
 Error-correction
 coefficient
 [phi] -0.7418 *** (0.0239) -0.7428 *** (0.0238)
 Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] -0.5332 (0.8409) -0.6632 (0.9189)
 C 1.8857 *** (0.0552) 1.9617 *** (0.0562)

Panel B: non-G7 countries
 Long-run coefficient
 [[pi].sup.e.sub.it] -2.1970 *** (0.3506) -2.2399 *** (0.3555)
 Error-correction
 coefficient
 [phi] -0.7527 *** (0.0303) -0.7521 *** (0.0301)
 Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] 0.2308 (1.2491) -0.0005 (1.0135)
 c 2.7442 *** (0.2643) 2.7781 *** (0.2729

Notes: The G7 countries are Canada, France, Germany, Italy, Japan,
United Kingdom, and United States. The non G7 countries include
Australia, Austria, Belgium, Finland, Ireland, The Netherlands, New
Zealand, Norway, and Spain. The values in the parenthesis are the
standard errors of corresponding coefficients.

*** indicates significance at 1% level.

TABLE 6

The Effects of Unexpected Inflation on Real Stock Return

 ARIMA

 ARIMA GARCH

Long-run coefficient
 [[pi].sup.u.sub.it] -2.5082 *** (0.4046) -2.6229 *** (0.3823)
Error-correction
 coefficient
 [phi] -0.7257 *** (0.0179) -0.7286 *** (0.0177)
Short-run coefficients
 [DELTA][[pi].sup.u.
 sub.it] 0.6815 *** (0.2220) 0.7061 *** (0.2188)
 c 0.5540 *** (0.0924) 0.6447 *** (0.1023)

 ARIMA

 EGARCH CGARCH

Long-run coefficient
 [[pi].sup.u.sub.it] -2.5752 *** (0.3896) -2.6533 *** (0.3973)
Error-correction
 coefficient
 [phi] -0.7258 *** (0.0183) -0.7280 *** (0.0177)
Short-run coefficients
 [DELTA][[pi].sup.u.
 sub.it] 0.7083 *** (0.2242) 0.7482 *** (0.2114)
 c 0.6145 *** (0.0900) 0.6600 *** (0.1022)

Notes: The values in the parenthesis are the standard errors
of corresponding coefficients.

*** indicates significance at 1% level.

TABLE 7

Subsample Results--Unexpected Inflation on Real Stock Return

 ARIMA

 ARIMA GARCH

Panel A: G7 countries
 Long-run coefficient
 [[pi].sup.u.sub.it] -2.5162 *** (0.6845) -2.7490 *** (0.6124)
 Error-correction
 coefficient
 [phi] -0.7230 *** (0.0258) -0.7270 *** (0.0257)
 Short-run coefficients
 [DELTA][[pi].sup.u.
 sub.it] 0.9255 ** (0.4031) 0.9032 ** (0.4192)
 c 0.6174 *** (0.0905) 0.7485 *** (0.0889)

Panel B: non-G7 countries
 Long-run coefficient
 [[pi].sup.u.sub.it] -2.5041 *** (0.5017) -2.5438 *** (0.4898)
 Error-correction
 coefficient
 [phi] -0.7278 *** (0.0261) -0.7298 *** (0.0257)
 Short-run coefficients
 [DELTA][[pi].sup.u.
 sub.it] 0.4926 ** (0.2438) 0.5598 ** (0.2320)
 c 0.5047 *** (0.1517) 0.5663 *** (0.1689)

 ARIMA

 EGARCH CGARCH

Panel A: G7 countries
 Long-run coefficient
 [[pi].sup.u.sub.it] -2.7003 *** (0.6310) -2.9373 *** (0.6655)
 Error-correction
 coefficient
 [phi] -0.7236 *** (0.0272) -0.7264 *** (0.0263)
 Short-run coefficients
 [DELTA][[pi].sup.u.
 sub.it] 0.9224 ** (0.4331) 0.9680 ** (0.412)
 c 0.6442 *** (0.0897) 0.7429 *** (0.0979)

Panel B: non-G7 countries
 Long-run coefficient
 [[pi].sup.u.sub.it] -2.5004 *** (0.4957) -2.4984 *** (0.4956)
 Error-correction
 coefficient
 [phi] -0.7275 *** (0.0263) -0.7292 *** (0.0254)
 Short-run coefficients
 [DELTA][[pi].sup.u.
 sub.it] 0.5500 ** (0.2322) 0.6015 *** (0.2215)
 c 0.5896 *** (0.1491) 0.5984 *** (0.1680)

Notes. The G7 countries are Canada, France, Germany, Italy, Japan,
United Kingdom, and United States. The non G7 countries include
Australia, Austria, Belgium, Finland, Ireland, The Netherlands, New
Zealand, Norway, and Spain. The values in the parenthesis are the
standard errors of corresponding coefficients.

*** and ** indicate significance at 1% and 5% levels.

TABLE 8

The Effect of Inflation Uncertainty

 ARIMA

 GARCH EGARCH

Long-run coefficients
 [[pi].sup.e.sub.it] -1.6872 *** (0.2934) -1.1020 *** (0.3312)
 [h.sup.g.sub.it] -0.2919 (0.3164)
 -1.1948 *** (0.3510)
 [h.sup.cg.sub.it]
Error-correction
 coefficient
 [phi] -0.7440 *** (0.0192) -0.7508 *** (0.0193)
Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] -0.2033 (0.7480) -0.5196 (0.7107)
 [DELTA][h.sup.g.
 sub.it] -0.3659 (0.6442)
 [DELTA][h.sup.eg.
 sub.it] 0.4025 (1.0679)
 [DELTA][h.sup.cg.
 sub.it]
 c 1.9457 *** (0.1310) 0.7608 *** (0.1807)

 ARIMA

 CGARCH

Long-run coefficients
 [[pi].sup.e.sub.it] -1.6661 *** (0.2872)
 [h.sup.g.sub.it]
 [h.sup.cg.sub.it] -0.6100 *** (0.2931)
Error-correction
 coefficient
 [phi] -0.7482 *** (0.0199)
Short-run coefficients
 [DELTA][[pi].sup.e.
 sub.it] -0.3993 (0.6604)
 [DELTA][h.sup.g.
 sub.it]
 [DELTA][h.sup.eg.
 sub.it]
 [DELTA][h.sup.cg.
 sub.it] 1.7715 (1.1036)
 c 1.7203 *** (0.1611)

Notes: The values in the parenthesis are the standard errors
of corresponding coefficients.

*** indicates significance at 1% level.