GMM and present value tests of the C-CAPM: evidence from the Danish, German, Swedish and UK stock...

Pergamon Journal of International Money and Finance, Vol. 15, No. 4, pp. 497-521, 1996

Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved

S0261-5606(96)00018-6 0261-5606/96 $15.00 + 0.00

GMM and present value tests of the C-CAPM: evidence from the Danish,

German, Swedish and UK stock markets

JESPER L U N D

Department of Information Science, The Aarhus School of Business, DK-8210 Aarhus V, Denmark

AND

TOM ENGSTED*

Department of Information Science, The Aarhus School of Business, DK-8210 Aarhus V, Denmark

In this paper we test the consumption oriented capital asset pricing model with constant relative risk aversion using long time series data from four European stock markets. Two different methodologies are applied: Han- sen's GMM method and the VAR approach proposed by Campbell and Shiller. Overall the statistical tests are unable to reject the C-CAPM, and the dividend-price ratio generally predicts future dividend growth in the direction implied by the model. However, the estimates of the relative risk aversion parameter are mostly implausible and imprecise, and the dividend-price ratio tends to predict future consumption growth in the wrong direction. Hence, discount rates are time-varying in a way that is inconsis- tent with the C-CAPM/CRRA specification. (JEL G12). Copyright © 1996 Elsevier Science Ltd

Since Fama's (1970) classic article the efficiency of the stock market has been the subject of a great number of empirical studies. Most of these have found that stock returns are predictable, especially over long horizons. See Fama (1991) for a survey and discussion of these results. However, a test of stock market efficiency is always conditioned on a particular equilibrium model, so these tests can only be interpreted as evidence against the efficiency of the stock market if equilibrium expected returns are constant. In general, this

* This paper was presented at the Annual Meetings of the European Finance Association, Brussels, 1994. We thank Adrian Pagan, Morten Ravn and two anonymous referees for useful comments, and Marco Becht, Andrea Beltratti, Per Frennberg, Lars Svensson and Ian Tonks for supplying part of the data. Financial support from the Danish Social Science Research Council is gratefully acknowledged.

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GMM and present value tests of the C-CAPM: J Lund and T Engsted

property only holds under risk-neutrality (Leroy, 1989), so it is natural to consider other equilibrium models. An obvious choice is the consumption oriented model of Lucas (1978)--the Consumption-Capital Asset Pricing Model (C-CAPM)--since it provides a consistent intertemporal framework for study- ing variations in asset returns.

Since Hansen and Singleton's (1982, 1983) seminal papers most of the empirical studies using the C-CAPM have been on US data. Among the exceptions are Hamori (1992a, 1993) and Braun et al . (1993). However, in these papers only the GMM principle is used, and the time periods analyzed are much shorter than our sample periods. As a consequence of the focus on US data, very little is known about the robustness of the results generated, since the data sets used in the different studies have not been independent. The present paper sheds light on the robustness issue by estimating and testing the C-CAPM using long time series from the Danish, German, Swedish and UK stock markets. With a constant relative risk aversion parameterization of the utility function of the representative agent we use both Hansen's (1982) GMM estimator and Campbell and Shiller's (1988a) VAR approach. The former is based on the orthogonality condition given by the first order condition of the intertemporal optimization problem (the Euler equation), whereas the latter approach is based on the linearized present value model which can be derived from the Euler equation.

The empirical results show that stock prices to some extent reflect rationally expected future dividends, but also that (except for pre-World War I Germany) stock prices are excessively volatile compared to dividends, when discount rates are assumed to be constant. When discount rates are allowed to vary in terms of consumption growth, the excess volatility disappears in some cases, but the estimates of the risk aversion parameter are often of the wrong sign, which makes it difficult to interpret the results as evidence in favor of the C-CAPM.

The rest of the paper is organized as follows. The next section presents the C-CAPM of Lucas (1978), and Section II briefly describes Hansen's (1982) GMM method and presents the GMM estimates for the Danish, German, Swedish and UK data. Section III describes Campbell and Shiller's (1988a) VAR approach and presents the VAR results for the same data. Section IV concludes the paper.

I. The C-CAPM

Following Lucas (1978), we assume the existence of a representative investor who chooses to consume and invest in a single asset (a stock index) so that at each time t he maximizes his expected life-time utility

T M a x E t t + , ,

subject to the budget constraint

c , + R, = (Pt

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where:

W, is the wealth invested in the stock index, C t is the real consumption, P, is the ex-dividend real stock price, D t is the real dividend received between time t - 1 and t.

The first-order condition in this maximization problem is the stochastic Euler equation (Lucas, 1978):

(1) E t fl U---7-~t ~ R t + l - 1 = 0

Lucas (1978) considers a pure exchange economy with one perishable consumption good. This means that we can ignore consumption decisions because by definition the representative investor must consume his entire income. However, with the utility function used below, (1) also obtains in the more general production economy of Breeden (1979) and Cox et al. (1985) where consumption and investment decisions are made jointly.

In order to get testable implications we must specify a utility function for the representative investor. As in most other studies we use the constant relative risk aversion (CRRA) utility function:

Cff +1 - 1 U ( C t ) a + l ' a <_ O.

Note that this specification encompasses risk-neutrality (o~ = 0).

II. GMM estimation and testing of the C-CAPM

II .A. The G M M principle

The GMM principle I uses the first-order condition (1> to estimate the unknown parameters b = (/3, a ) and to test the underlying equilibrium model. In the following x t denotes a 2 × 1 vector consisting of the asset return R t and consumption growth rate C , / C t_ 1 while b 0 denotes the true (but unknown) value of the parameter vector. The basic idea behind the GMM is that the 'excess return'

h(X,+l, b0) =/3Rt+l(Ct+l/Ct )~ - 1

should be unpredictable given information at time t. Therefore, if z t is a k x 1 vector containing variables known at time t we have that

(2) E ( h ( xt+ 1, b o ) z , ) = O.

Corresponding to the vector of population orthogonality conditions we can define a sample counterpart:

T 1 E h(x,+ , b)z,. g ( b ) = ~ 1

t= l

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To apply the GMM principle, h(x,+ 1, b)zt must be stationary for all values of b. The orthogonality condition (2> is formulated in terms of asset returns and consumption growth rates which are generally regarded as stationary. Hence the stationary condition is fulfilled as long as we use stationary variables as instruments (such as lagged values of xt+ 1 and the dividend-price ratio).

The GMM estimator is the value of b which makes g(b) as close to zero as possible, or more precisely the value which minimizes the quadratic form

J(b) =g(b)'Wg(b).

For all choices of the weighting matrix W the GMM estimator is consistent, but the most GMM efficient estimator is obtained if W is equal to the inverse of the asymptotic covariance matrix of T1/2g(bo) (Hansen, 1982). A consistent estimator of this covariance matrix is given by

1 r q T

<3> S ~- -T E htztzt2 ' + -T1 E E htht-j(ztzt-j' + zt-jz ' t); t=l j=l t=j+l

where h t = h(Xt+l , b* ) and b* is a consistent estimator of b 0. With non-over- lapping returns the theoretical value of the lag length q is zero, but as noted by inter alia Ferson and Constantinides (1991) time-aggregation may cause the disturbance term h(x,+ 1, bo) to follow an MA(1) process. In that case z t should only include variables known at time t - 1. However, if q > 0 <3) may not be positive definite in finite samples. The usual solution to this problem is to weight the autocovariances using the Newey and West (1987) weighting scheme, but in that case the lag length should be larger than the value implied by the MA process in order for (3) to be a consistent estimator of the optimal weighting matrix (Ogaki, 1993, p. 468). In our case (3) always turned out to be positive definite, so we did not employ the Newey-West weighting scheme.

Since we need a consistent estimator of b 0 to compute the optimal weighting matrix W* = S-1, a two step procedure is often applied. In the first step W = I (the identity matrix), and the resulting estimator of b o is used to compute W*. In the second step the efficient GMM estimator b is obtained from W*. However, the Monte Carlo evidence in Ferson and Foerster (1994) and Kocherlakota (1990) shows that an iterated version of the GMM gives better small sample properties. At stage k + 1 the weighting matrix is calculated from the k-stage estimates, and J(b) is minimized. This procedure is repeated until convergence. Note that at each stage we minimize a non-linear function which itself requires an iterative procedure.

The asymptotic covariance matrix of the GMM estimator/~ is given by

^ = 1 [Og(b)'s_ 1 0 g ( b ) ] -1 = 1 T [ Ob 3b' "T (D'S-1D)-I" Cov(b)

Furthermore, if the model is correctly specified T times the minimized value of J(b) is asymptotically xE-distributed with k - 2 degrees of freedom (Hansen, 1982, pp. 1249-50). This test of the overidentifying restrictions is known as the J-test. Although J(b) is a general specification test of the model, there are

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certain forms of misspecification which cannot be detected with the test (Newey, 1985). 2 One of these is structural instability, i.e. non-constancy over time of the parameter vector (Ghysels and Hall, 1990a). Therefore, we also test for structural instability using the test developed by Ghysels and Hall (1990b) and (independently) by Hoffman and Pagan (1989). 3 The idea behind this test is to estimate the parameters in the first sub-sample and check whether the orthogonality condition holds in the second sub-sample, i.e. whether g2(b~) only differs from 0 by sampling error. Specifically, the test statistic is given by

- ]-, G = T 2 g 2 ( b l ) [ S 2 + c D 2 ( D ' I S l l D 1 ) 1D' 2 g2(191) ; c = T 2 / T 1

where the subscript refers to the sub-sample from which the matrix or vector in question is calculated, and c is the ratio of the two sample sizes. Asymptoti- cally, the distribution of G is X 2 with k degrees of freedom.

The greatest problem in using the GMM principle is to choose the vector of instruments (and hence the orthogonality conditions). It can be shown (Han- sen, 1982) that including more orthogonality conditions cannot increase the covariance matrix of the estimator, but it is important to note that this is an asymptotic result. Tauchen (1986) has investigated the small sample properties of the GMM estimator with a different number of instruments. The overall conclusion is that the best performance of the GMM estimator is obtained with a small number of instruments.

l l .B . E m p i r i c a l results

Below we present the empirical results for Germany (1885-1913 and 1952-90), Denmark (1922-90), Sweden (1918-90) and the UK (1919-87). The data are collected from various sources, and a full description is given in the Appendix. 4 For each data set we use four different sets of instruments, referred to as I i - I4 . In I1 and I2 the instruments are a constant as well as lagged values of returns and consumption growth rates (1 and 2 lags, respectively). Ferson and Constantinides (1991) have argued that it is preferable to use other variables as instruments since measurement errors may lead to biased parameter estimates and spurious rejection of the overidentifying restrictions. Therefore, in the third set (I3) we use a constant as well as two-lagged values of the dividend-price ratio as instruments. Finally, in I4 we use a constant and the return and consumption growth rate lagged two periods. Furthermore, in I4 the weighting matrix W* is calculated with q = 1.

The overall impression from Table 1 is that the risk aversion parameter a is estimated with great uncertainty, and only 1 out of 20 estimates is significantly different from zero (i.e. risk neutrality) at the 5 percent level (Denmark with instrument set I4). Furthermore, the estimates of ~ vary considerably over different definitions of z t, and while this can be explained by the high standard errors, it could also be an indication of mis-specification of the utility function. The wrong sign of ~ in 9 out of 20 cases clearly contributes to the latter interpretation.

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TABLE 1. GMM estimates

Inst [3 s([3) a s (a ) J-test p-value GH-test p-value

Germany1885-1913 11 0.8227 0.0954 10.3616 7.5981 0.1392 0.7091 3.0923 0.3776 12 0.9149 0.0351 2.5282 2.6976 5.0240 0.1700 3.6540 0.6002 13 1.1059 0.1562 - 16.6278 19.0881 0.6361 0.4251 0.4792 0.9234 14 0.9392 0.0564 -0.3700 4.0248 2.9109 0.0880 2.1324 0.5454

Germany1951-1990 I1 0.8967 0.0611 0.8097 1.7292 3.8876 0.0486 5.5327 0.1367 12 0.8819 0.0689 1.9724 2.1093 10.2452 0.0166 5.5353 0.3541 13 0.9726 0.0530 - 1.0685 1.5986 6.9903 0.0082 4.8034 0.1868 14 0.7413 0.1159 5.5554 4.4993 1.1616 0.2811 1.9827 0.5760

Denmark 1922-1990 I1 0.8594 0.0831 4.6767 3.8169 0.1067 0.7439 2.4121 0.4914 I2 0.9891 0.0256 - 1.3905 0.8675 7.9112 0.0479 10.5301 0.0615 13 0.9104 0.0422 2.0634 1.7627 0.0863 0.7689 1.3873 0.7085 14 0.9773 0.0278 - 1.7122 0.7132 1.7099 0.1910 3.2332 0.3570

Sweden 1918-1990 I1 0.9250 0.0333 0.1614 1 . 3 3 0 1 1.2819 0.2575 2.6735 0.4447 12 0.9324 0.0256 -0.8949 0.9764 2.4891 0.4773 3.9753 0.5530 13 0.9271 0.0306 -0.4454 1.6267 0.4795 0.4886 0.6959 0.8742 14 0.9379 0.0332 - 1.2260 1.5942 2.1871 0.1392 1.6473 0.6487

United Kingdom1919-1987 11 0.4658 0.2745 -26.1333 9.0791 0.0004 0.9840 5.1984 0.1578 12 0.9758 0.0621 -5.0249 6.0886 2.6175 0.4544 5.6771 0.3389 13 0.6745 0.1078 10.8144 4.1773 0.0100 0.9203 1.0823 0.7813 14 0.9707 0.0564 -7.1942 10.0596 1.3860 0.2391 1.8034 0.6142

Notes: s([3) and s(a) denote the standard errors of the estimates of [3 and a. For 11 and 12 the instruments are a constant and lagged values of returns and consumption growth rates (1 and 2 lags, respectively). In 13 the instruments consist of a constant and two lagged values of the dividend price ratio. In 14 the instruments are a constant and the return and consumption growth rate lagged two periods. The weighting matrix is calculated assuming an MA(0) error term for 11-13 and an MA(1) error term for 14. J-test is Hansen's (1982) orthogonality test of the over-identifying restrictions. This test is distributed as chi-square with degrees of freedom equal to the number of instruments minus two. GH-test is the Ghysels-Hall test for mid-sample structural instability (except for Denmark with instrument set I1, where the split point is 2 /3 of the sample due to convergence problems when estimating the model in the first sub-sample). The degrees of freedom for GH-test is the number of instruments.

The inability to obtain precise estimates of a is a common characteristic feature of most studies of consumption-based asset pricing models. Another problem is that the estimate of a may be substantially biased because of problems with time-aggregation (Grossman et al., 1987). Specifically, measured

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consumption is a flow variable while stock returns are computed using prices at different time points. The likely effect of this is that the consumption series appears too smooth to explain variations in stock returns. The fundamental problem is that our model specification effectively assumes that consumption decisions only take place once a year. Of course, this assumption is dictated by the availability of consumption data, but it is possible to overcome the problem (at least in principle) by working with the continuous-time version of the C-CAPM as in Grossman et al. (1987).

The discount factor can be estimated somewhat more precisely, but many of the estimates are either unrealistically low or high (we take values in the range 0.95-0.98 as realistic). Note however, that the estimates of a and /3 are negatively correlated, so unrealistic estimates of a imply unrealistic estimates of/3. Overall, the best results are obtained with the Swedish data, even though the estimate of the discount factor/3 is somewhat low.

Given these mostly implausible results we should perhaps expect that the test of the overidentifying restrictions generally rejects the model, but as can be seen from Table 1 this is not the case. In fact, we can only reject the model at the 5 percent level for post-World War II Germany (1952-90) using instrument sets I I - I3 and Denmark with instruments set I2, and most of the p-values are well above the 5 percent level. These rejections may even be spurious because of time aggregation since the test statistics for post-World War II Germany and Denmark are much smaller (and insignificant) if we use I4 as the instrument set. However, this result could also be due to the loss of power that is known to occur when using variables lagged two periods as instruments. Finally, we test for structural instability with a mid-sample shift. As the last two columns in Table 1 show there are no signs of structural instability since all test statistics are insignificant at the 5 percent level. Only for Denmark with instrument set I2 is the p-value close to the 5 percent level.

III. Present value tests of the C-CAPM

III.A. The V A R me thod

Campbell and Shiller (1988a) have developed an alternative test of the C- CAPM. Basically, they extend the test from Campbell and Shiller (1987) so that it can handle time-varying discount factors. The advantage of this approach is that the formal statistical tests can be supplemented by graphical results showing the 'economic' significance of a possible deviation from the equilibrium model.

We begin by defining hit as the logarithm of the utility-adjusted return, i.e.

t (4) h i t = l o g U,(Ct) Rt+ 1 =log P~ 1 + a l o g C, }

= log(exp(6 , - ~+1) + exp(6,)) + Ad~+ 1 + aAc~+ 1,

where 6t = log(Dt/Pt) , dt = log(Dr) and c t = log(C,). Next, we linearize (4)

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around the point 8 t = St+ 1 = 8:

hi t "~ ~lt = t~t - P~t+l --I--Adt+ 1 + olmct+ 1 + k .

Here, p = 1/(1 + exp(6)); while k is an inessential constant from the lineariza- tion. We also introduce a variable ~t defined by

(55 i - 1 i - 1 1 - p i k

= P'~t+i + E PJ(Adt+j+l + ° tAct+j+l) + 1 - p " ~, = E PJ~l,t+j ~ t - i j=O j=O

We assume that E t ( ~ l t+j) is equal to a constant c for all j > 0. From (1) we should expect Et(~ l , t+ ' j ) to be close to - log( /3) , so the assumption is reason- able, at least as a good approximation. Given this assumption we can take conditional expectations on both sides of (5) and let i --) ~. After some simple manipulations we have

o o

(6) 8t - ~-, p J E t ( A d t + j + l + ° t A c t + j + l ) -'1- ¢ - - k = 1 - p '

j=O

since lim piEr(St+ i) = 0 as i---> ~ (otherwise 8 t is non-stationary with an explosive root indicating that stock prices are driven by a rational bubble).

Equation (6) shows that if the model is true, the dividend-price ratio will be the optimal predictor of future dividend growth and (unless agents are risk- neutral) consumption growth. This has the testable implication that 6 t should negatively Granger-cause A d t and positively Granger-cause A c t. A more demanding test of the C-CAPM is obtained by testing the cross-equation restrictions implied by the underlying theory on a VAR model. We define a limited information set H t containing past and present values of x t = (S t , A d t , Ac t ) ' , and assume that expectations conditional on H t are linear projections on the information set. This corresponds to the VAR(p) specification:

(7) Zt+ 1 = a z t + Ut+ 1

where z t = ( x t - E ( x ) . . . . . X t _ p + l - E ( x ) ) is a 3 p × l vector and A is the 3p × 3p companion matrix of the VAR(p) system. See Campbell and Shiller (1987, 1988a) for details. The first-order approximation implies that we cannot test any restrictions on the constant term in the VAR, so we have adjusted out the unconditional means in (7). 5

The conditional expectation of zt+ j given H t is equal to A J z , . Since H t is a subset of the market information set, and since 8 t is contained in H t we can use the law of iterated expectations to transform (6) into

o o o o

(8) e l ' z t = - ~ , p J e 2 ' A j+ l z t - a ~_, pJe3 'A j+ 1Zt, j=0 j=0

where el, e 2 and e3 are vectors that pick out St, A d t and Ac t from the VAR system. Because all variables are now demeaned, there is no constant term in (8). Since (8) must hold for all levels of z t we have the following restrictions

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on the companion matrix A:

(9) e l ' I = - e 2 ' A ( I - p A ) -1 - a e 3 ' A ( I - p A ) - 1,

or alternatively

(10) e l ' ( I - p A ) = - e 2 ' A - a e 3 ' A .

For a given value of a (we discuss below how to estimate a), the set of restrictions (10) is linear in the parameters, so we prefer (10) to (9).

To gain some insight into these restrictions we look at the special case p = 1 where we have three restrictions:

p a l l -- 1 = a2~ + ota31

pal2 = a22 q- ota32

pal3 = a23 -4- ota33.

The restrictions from the case with constant expected returns in Campbell and Shiller (1988b, p. 670) will obtain if the representative investor is risk-neutral (i.e. a = 0) or if consumption changes are unpredictable (i.e. a31 = a 3 2 = a 3 3 =

0). Thus, the properties of the restrictions correspond closely to the theoretical results of Lucas (1978) and Rubinstein (1976).

In addition to the formal statistical tests we compare the actual and theoretical value of 6 t = d t --Pt" T h e theoretical value 6* (apart from a constant) is obtained from (8) and (9), i.e.

(11} 6* = - ( e 2 ' + a e 3 ' ) A ( I - p A ) - X z t =- b ' z t .

It is denoted 'theoretical' because it gives the value that the dividend-price ratio should take if the theory (the C-CAPM) is true. Note that we are using the unconstrained estimates of the VAR parameters in (11). The actual value is given by 6 t = e l ' z , , so if the model is correctly specified the vector b should be close to el which has 1 on its first element and zeros elsewhere. The intuition behind this analysis is that the degree of comovement of the two variables measures the extent to which agents have rationally incorporated the predictable part of future dividend and consumption growth in setting present stock prices.

From (11} it is seen that the relative importance of time-varying discount factors, in terms of consumption growth, can be inferred by examining how much 6", computed with a = 0 imposed, deviates from 6" computed with the estimated value of ot imposed.

The advantage of this more informal method is that in contrast to the formal statistical test of the cross-equation restrictions in (10}, the graphical compar- ison of 6 t with 6" is robust to statistically significant--but minor, and e c o n o m i c a l l y unimportant--deviations from the theory; see Campbell and Shiller (1987) for elaboration on this point.

We conclude this section by discussing how the risk aversion parameter can be estimated. Campbell and Shiller (1988a) use a two-step procedure. The first step is OLS (or ML under normality) estimation of the VAR model. The estimate of a is obtained in the second step by minimizing the Wald test

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statistic for the restrictions (9) or (10) with respect to a. This reduces the degrees of freedom from 3p to 3p - 1. As noted by Campbell and Shiller (1988a) this procedure closely resembles Hansen's (1982) GMM estimator.

Our approach is to estimate all parameters simultaneously since this is more efficient. Under (10) the 9p parameters in the matrix A are expressed as non-linear functions of 9p - (3p - 1) = 6p + 1 parameters (including a), and these parameters can be estimated by FIML. This allows us to test the restrictions by an LR test thus avoiding the well-known ambiguity of the Wald test when the restrictions are non-linear (see Gregory and Veall, 1985). Note that a is unidentified unless the restrictions in (10) are imposed.

TABLE 2. Summary statistics f rom V A R models: D e n m a r k

D e n m a r k 1922-1990

lags = 1 lags = 2 lags = 3

1. A I C 2. BIC

&equa t ion 3. adj. R 2

4. AC--~ $

A d-equa t ion 5. adj R 2

6. Ac ~ Ad 7. 8 ~ A d 8. sum of 8 coefficients

Ac-equat ion 9. adj. R E

10. Ac unpred ic tab le 11. ~ - ~ A c 12. sum of ~ coefficients

13. M L est imate of a

14. L R test of restr ict ions

a = M L est imate 15. cor re la t ion(8 , 8 " ) 16. t r ( 8 * ) / o ' ( 6 )

a = 0.0 17. cor re la t ion(8 , 8 " ) 18. o ' ( ~ * ) / o ' ( ( ~ )

- 5.290 - 5.236 - 5.064 - 13.507 - 13.153 - 12.674

0.65 0.65 0.64 0.031 0.016 0.004

0.15 0.14 0.14 0.106 0.132 0.091 0.116 0.355 0.323

- 0.092 - 0.086 - 0.111

0.06 0.12 0.10 0.090 0.000 0.000 0.019 0.169 0.064

- 0.050 - 0.046 - 0.062

2.783 0.994 0.584 (1.907) (1.074) (0.987) 0.702 0.434 0.498

0.996 0.992 0.981 1.058 0.674 0.703

0.998 0.997 0.988 0.524 0.516 0.595

Notes: A I C and BIC are the Akaike and Schwarch criterion, respectively. The numbers in rows 4, 6, 7, 10 and 11 are margina l significance levels for the Granger-causal i ty tests. The n u m b e r in parenthes is in row 13 is the s tandard er ror of the es t imate of c~. The n u m b e r in row 14 is the marginal significance level for the chi-squared test of the cross-equat ion restrictions.

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III.B Empirical results

Tables 2 to 5 contain the results of the present value tests applied to the Danish, Swedish, UK and German data. In order to investigate the robustness of the results with respect to lag-length we analyze VAR models with lag-lengths 1, 2 and 3, except for Germany where only lag-lengths 1 and 2 are analyzed due to the much smaller samples available for this country. Reporting results for different lag-lengths is important because if 6 t is much more persistent than Ad t and Ac t (which it is in reality), then low-order VAR models tend to produce spuriously high correlations of 6 t and 6", see Campbell and Shiller (1987, pp.1084-85), and Campbell and Shiller (1988a, p.216). In the tables we report the Akaike (AIC) and the Schwarz (BIC) measures. For pre-World War I Germany and Denmark both of these criteria select a one-lag model. For Sweden they select a two-lag model, while for post-World War II Germany and

TABLE 3. Summary statistics from VAR models: Sweden

Sweden 1919-1990


1. A I C 2. B I C

&equation 3. adj. R 2

4. A c - - . 6

Ad-equation 5. adj R 2

6. Ac ~ A d

7. 6 ~ A d

8. sum of 6 coefficients

ac-equation 9. adj. R 2

10. Ac unpredictable 11. 6 ~ Ac

12. sum of 6 coefficients

13. M L estimate of a

14. L R test of restrictions

c~ = M L estimate 15. co r r e l a t i on ( 6 , 6 * ) 16. o - ( 6 " ) / o - ( 6 )

a = 0.0 17. c o r r e l a t i on ( 6 , 6 " ) 18. o ' ( 6 * ) / o ' ( 6 )

- 6.133 - 6.621 - 6.583 - 14.358 - 14.552 - 14.215

0.65 0.67 0.68 0.220 0.000 0.002

0.22 0.29 0.34 0.827 0.226 0.019 0.000 0.000 0.000 0.150 - 0 . 1 2 9 - 0 . 1 5 7

0.13 0.29 0.28 0.001 0.000 0.000 0.723 0.022 0.111

- 0.007 0.020 0.006

- 0.476 - 1.010 - 0.726 (1.331) (1.033) (1.021)

0.394 0.586 0.089

0.997 0.994 0.928 0.718 0.648 1.253

0.995 0.982 0.929 0.793 0.585 1.204

Notes: See notes to Table 2.

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the UK AIC and BIC select a two-lag and a one-lag model, respectively. Figures 1 to 10 show time series plots of 6, and 6" from the VAR(2) models only because of the above mentioned problems with the VAR(1) model. (Note also that variables plotted in the figures are demeaned.)

First we look at the Danish results (Table 2, Figures 1 and 2). As seen, the dividend-price ratio predicts dividend growth in the direction implied by the C-CAPM, although not significantly. In the one-lag model the dividend-price ratio significantly predicts consumption growth, but in the direction opposite to that implied by the C-CAPM. Consumption growth has significant predictive power both for the dividend-price ratio and dividend growth, so the augmenta- tion of the standard two-equation VAR model with an equation for consumption has proven to be important. When testing the cross-equation parameter restrictions implied by the C-CAPM, the likelihood ratio test statistics are

TABLE 4. Summary statistics f rom V A R models: U n i t e d Kingdom

U n i t e d Kingdom 1919-1987


1. A I C 2. BIC

&equa t ion 3. adj. R 2

4. Ac-~ 8

Ad-equa t ion 5. adj R E

6. Ac ~ Ad 7. 6 ~ A d 8. sum of 8 coefficients

Ac-equa t ion 9. adj. R 2

10. Ac unpred ic tab le 11. 8 - ~ A c 12. sum of 8 coefficients

13. M L es t imate of a

14. L R test of restr ict ions

a = M L es t imate 15. cor re la t ion(& 6 " ) 16. o'(8*)/0-(8)

a = 0.0 17. cor re la t ion(6 , 6 " ) 18. ¢r(8")/o '(6)

- 6.346 - 6.405 - 6.369 - 14.564 - 14.322 - 13.980

0.29 0.24 0.22 0.307 0.022 0.027

0.27 0.41 0.46 0.947 0.979 0.639 0.000 0.000 0.000

- 0.131 - 0.019 - 0.026

0.08 0.04 0.04 0.032 0.172 0.050 0.014 0.111 0.013

- 0.052 - 0.041 - 0.050

7.121 17.825 8.965 (4.155) (15.926) (5.128) 0.198 0.255 0.425

0.946 0.382 0.461 1.014 1.918 1.160

0.954 0.001 0.061 0.367 0.350 0.579


508


TABLE 5. Summary statistics from V A R models: Germany

Germany 1885-1913 Germany 1951-1990

lags = 1 lags = 2 lags = 1 lags = 2

1. A I C 2. BIC

6-equation 3. adj. R 2 4. A c ~ 6

A d-equation 5. adj R 2 6. Ac ~ Ad 7. ~--, Ad 8. sum of ~ coefficients

Ac-equation 9. adj. R 2

10. Ac unpredictable 11. 6--* Ac 12. sum of 6 coefficients

13. ML estimate of a

- 11.005 - 10.584 - 7.121 - 7.292 - 19.083 - 18.220 - 15.242 - 15.014

0.57 0.57 0.34 0.47 0.658 0.825 0.689 0.997

0.86 0.87 0.56 0.50 0.095 0.072 0.304 0.675 0.000 0.000 0.000 0.000

- 0.716 - 1.007 - 0.387 - 0.459

- 0.06 - 0.09 0.64 0.55 0.862 O. 163 0.000 0.000 0.521 0.056 0.000 0.000

- 0.015 0.076 - 0.033 - 0.031

- 34.626 - 16.664 - 0.076 - 2.044 (89.747) (20.691) (1.632) (2.140)

14. LR test of restrictions 0.825 0.681 0.970 0.002

a = ML estimate 15. correlation(6, ~*) 0.887 0.800 0.999 16. c r ( 8 * ) / ~ ( ~ ) 1.191 1.320 1.004

a=O.O 17. correlation(6, ~*) 0.952 0.919 0.999 18. c r ( 8 * ) / c r ( $ ) 1.127 1.047 1.014

0.889 0.654

0.857 0.897


highly insignificant. The ML estimates of the risk aversion parameter a, however, turn out to be of the wrong sign, but with large standard errors, so that the hypothesis of risk-neutrality cannot be rejected. Finally, we compare the actual dividend-price ratio with an unrestricted VAR forecast of the present value of future dividend and consumption growth. With a set equal to zero this produces a correlation close to unity. The standard deviation of 8* is, however, only half the standard deviation of 8t (see also Figures 1 and 2). This implies that under the assumption of risk neutrality the dividend-price ratio m o v e s c l o s e l y w i t h r a t i o n a l l y e x p e c t e d f u t u r e g r o w t h in d i v i d e n d s a n d c o n -

s u m p t i o n , b u t w i t h s o m e d e g r e e o f o v e r r e a c t i o n in t h e s t o c k p r i c e . I t is

i n t e r e s t i n g ( a n d c u r i o u s ) t h a t i f w e i m p o s e t h e e s t i m a t e o f a , so t h a t w e

a s s u m e t h a t a g e n t s a r e r isk- lovers , t h e o v e r r e a c t i o n is t o t a l l y e l i m i n a t e d f o r t h e

o n e - l a g m o d e l , a n d s o m e w h a t r e d u c e d f o r t h e h i g h e r - o r d e r V A R m o d e l s . T h i s

509

GMM and present oalue tests of the C-CAPM: J Lund and T Engsted

8 - -

H ~

%%

0 0 - -

I #~,

i V

- 8 ~

t -1 2 - -

182H 1930 1938 19H2 18H0 185H 1960 1966 1972 1978 198H 1990 I OELYR

DELTR _ I

FIGURE 1. Ac tua l and theore t ica l d iv idend-p r i ce ratio. D e n m a r k 1924-1990 ( u = 0.994).

8 -

H m

0 0--

I r<!

- 8-- ~ I -1 2 - -

192H 1930 1938 18't2 19H9 185H 1960 1966 1872 1978 198~i 1980

I DELTR I DELTR _ - - - I

FIGURE 2. A c t u a l and theore t i ca l d i v i d e n d - p r i c e ra t io . D e n m a r k ] 9 2 4 - 1 9 9 0 ( a -- 0.0).

510

5--


3 m

1 - - v

1 - I

- 3 - -

- 7 - -

- 5 - -

1521 1527 1933 1939 19~5 1951 1557 1563 1565 1575 1981 1967

I DELTR 0ELTR= - - - i

FIGURE 3. Ac tua l and theoret ical d iv idend-pr ice ratio. Sweden 1921-1990 (~z = - 1.010).

indicates that consumption has an important role to play, but because of the wrong sign of a one should be careful in interpreting this as evidence in support of the C-CAPM. Further, as previously discussed, care should be exercised in interpreting the degree of comovement of 6 t and 6" based on the one-lag VAR model, because it tends to produce a spuriously high degree of comovement.

Next, we look at the Swedish results (Table 3, Figures 3 and 4). These are more favorable to the C-CAPM than the Danish results reported in Table 2. The dividend-price ratio significantly predicts both dividend growth and consumption growth in the right direction for the two-lag model, which is selected by both information criteria. In turn, consumption growth significantly predicts the dividend-price ratio. Furthermore, the ML-estimates of a are correctly signed, though statistically insignificant, and the cross-equation parameter restrictions are not rejected at a 5 percent level. The correlation between 6, and 6* is very high for all VAR models, and although 6 t m o v e s too much compared to 6* in the one-lag and two-lag models, 6 t moves too little in the three-lag model, so the evidence regarding overvariability in stock prices is inconclusive. When we impose risk-neutrality, a = 0, the correlations and standard deviation ratios hardly change. These results suggest that although time-variation in discount rates in terms of consumption growth helps explain

511

GMM and present value tests of the C-CAPM." J Lund and T Engsted

5 - -

3--

1 - -

- 1 - -

- 9 - -

- 5 - -

- 7 - -

- 9 - -

V

1921 1927 1933 1939 19~5 1951 1957 1963 1969 1975 1~81 1987 D E L T f l D E L T R . - - - I

FIGURE 4. Actual and theoretical dividend-price ratio. Sweden 1921-1990 (a ffi 0.0).

the behavior of the Swedish dividend-price ratio, the additional explanatory power is only marginal.

Turning to the UK data, Table 4 and Figures 5 and 6 show that the results are somewhat similar to the Danish results: The dividend-price ratio predicts dividend growth in the right direction and consumption growth in the wrong direction. The cross-equation parameter restrictions are not rejected at a 5 percent level, but the values of ot again have the wrong sign and are very imprecisely estimated. Interestingly, for the UK the degree of comovement of 8 t and 6* is highly sensitive to the lag-length of the VAR models. The degree of comovement is quite high for the one-lag model, but quite low for the higher order models. This resembles Campbell and Shiller's (1988a) results for the USA, and again emphasizes that care should be exercised in interpreting a high degree of comovement based on a one-lag VAR as strong support for the underlying model.

Finally, we look at the German data. Table 5 and Figures 7 to 10 give the results. According to Table 5 the standard present value model with constant discount rates seems to explain the pre-World War I German stock prices quite well. An increase in the dividend-price ratio strongly predicts a subsequent fall in dividend growth, while there are no significant correlations between consumption growth on the one hand and the dividend-price ratio and dividend growth on the other hand. The LR test statistics of the parameter

512

1 5 - -


,o- !

- ^if / II. ~"

0 0--

"~' ' , ~ ~ " v u ' ~ . i t . I t I I;'- i ; tl t i t!

- 5 - - - - ~ t / V V ~j

-1 0 - -

-1 5 - -

1921 1927 1933 1939 19q5 1951 1957 1983 1969 1975 1991 1987 I DELTA

DELTA. - - - [

l ~ o t n ~ 5. Actual and theoret ical d iv idend-pr ice ratio. United Kingdom 1921-1987 (oc = 17.825).

1 0 ~

6--

5--

'4--

2--

O O--

- 2--

- H--

, "' ',j '~ '~;,,/ ! v ~/

1921 1927 1933 193~ 19~5 1951 1957 1~63 1969 1975 1981 1987 DELTA DELTA= - - - ]

FmURE 6. Actual and theoret ical d iv idend-pr ice ratio. Uni ted Kingdom 1921-1987 ( a = 0.0).

513


N

2 I

0 0

- 2 - -

A A

/ / ' ' \ ,/¢'/! .,',q ; ,

P¢,i \ V /', // !\ ' / ' , , / / ',,, /, / / / ~\ I i V', z~-J ' , /

/ x¢"- V I \w i ! ,I

i i

V

- 6 I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I '

1 8 8 8 1 8 9 0 1 8 9 2 1 6 9 H 1 8 9 8 1 8 9 8 1 9 0 0 1 9 0 2 1 9 0 N 1 9 0 6 1 9 0 8 1 9 1 0 1 9 1 2

I OELTR DELTRI - - - I

FIGURE 7. A c t u a l a n d t h e o r e t i c a l d i v i d e n d - p r i c e rat io. G e r m a n y 1 8 8 8 - 1 9 1 3 ( a = - 16.664).

restrictions in the VAR models are highly insignificant, and although the estimates of a are numerically quite high the standard errors are enormous, so risk-neutrality cannot be rejected. With a set equal to 0 the correlations between 8 t and 8* are close to unity in both VAR models, and so is the ratio of their standard deviations. This implies that for this period the German dividend-price ratio moves very closely with rationally expected future growth in dividends, with no significant time-variation in the real discount factor.

For the post-World War II period, German stock prices move in less accordance with the present value model. The dividend-price ratio still significantly predicts dividend growth in the right direction, and consumption growth does not help to predict either the dividend-price ratio or dividend growth; however, the dividend-price ratio now significantly predicts consumption growth in the wrong direction. The ML-estimates of a are correctly signed, but with high standard errors. More important is, however, that the parameter restrictions implied by the C-CAPM are now strongly rejected in the two-lag model. Interestingly, most of the J-tests reported in section II.B also rejected the model for post-World War II Germany. With a = 0 the correlations between 8 t and 8* are still high, and 8* varies almost as much as 8 t, so despite this rejection the model contains an important element of truth in the sense that a substantial part of the predictable movement in future dividend growth is incorporated in the present dividend-price ratio. In contrast to the Danish, Swedish and UK results, the standard deviation ratio drops further

514


!

. - ..

' - 1/ " - . J ;

'I / J // ,\ :s o o l / I t

",\ .'7 ,'.: '/

' - Y " " Iv,'_ - 2 I t.. ~

- 3 I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I ' I '

1888 1880 le92 1894 1896 1898 1800 1902 190H 1806 1908 1910 1912

I D E L T R O E L T R = - - - I

FIGUI~ 8. Actual and theoretical dividend-price ratio. Germany 1888-1913 ( a = 0.0).

below unity when the estimated value of o~ is imposed. This is not surprising since the parameter restrictions are strongly rejected in this case. The results with respect to Germany are quite similar to those of de Long and Becht (1992) who use a volatility test.

The main conclusions that emerge from our results are as follows. First, the finding of Campbell and Shiller (1988a) that the US dividend-price ratio moves with rationally expected future growth in dividends, seems to be very robust, since we obtain the same result using stock market data from four European countries. Only for the UK is this result highly sensitive to the lag-length of the VAR models. Second, the standard present value model with a constant discount rate appears to give a quite accurate description of stock price fluctuations in the pre-World War I period in Germany. However, for Den- mark, Sweden, the UK and post-World War II Germany, discount rates cannot be considered constant, and introducing time-variation in terms of consumption growth has not been very successful; the risk-aversion parameter is often of the wrong sign and very imprecisely estimated. So, although likelihood ratio tests generally are unable to reject the consumption based asset pricing model with constant relative risk-aversion, we are reluctant to interpret the results as evidence in favor of this model.

IV. Concluding remarks

Most tests of the consumption based asset pricing model have been carried out

515

GMM and present oalue tests of the C-CAPM: J Lund and T Engsted

7 - -

A ,x / " ' , _ " ,

3--

7--

1S5H 1957 1900 1983 1985 1969 1972 1S75 1978 1981 199q 1997 1990 I DELTA

DELTR. - - - I

PI61.JI~ 9. Actual and theoretical dividend-price ratio. Germany 1954-1990 ( a = -2.044).

using US data. The results have mostly been negative, in the sense that the C-CAPM/CRRA specification has not been able to fit the data very well. 6 In this paper we have investigated the robustness of these results by testing the C-CAPM/CRRA specification using long time series from the Danish, Ger- man, Swedish and UK stock markets. Our results tend to support the findings in the US studies: there is a significant forward-looking element in the dividend-price ratio with respect to future dividend growth, but the dividend-price ratio moves 'too much' to be consistent with the simple present value model with constant discount rates, and introducing time-variation of discount rates in terms of consumption growth does not provide a credible explanation for the over-variability, since most of the estimates of the risk- aversion parameter have the wrong sign and are very imprecisely estimated.

In the literature various extensions of the basic C-CAPM/CRRA specification have been considered, including time-nonseparability in the form of durability of goods and habit persistence, see e.g. Ferson and Constantinides (1991). All these studies are based exclusively on the GMM method and the extensions have met with rather limited success: the fit of the models generally improves, but the parameters continue to be very imprecisely estimated and often of the wrong signs. Although it would be interesting to develop a method of incorporating these extensions into Campbell and Shiller's VAR approach, we suspect that the results would not differ markedly from the results obtained using the GMM methodology. The inability to explain time-varying discount

516

7--


5 - -

3 - -

1--

- 1 m

- 5 - -

- 5 - -

- 7 - -

/ t

• "\ / ,' 'I A !', \ /1 '~ I / l ,~i

l\ /I 1 1VI \ J i ! ill J V',,/

195"1 1557 1560 1963 1966 1565 1972 1975 1976 1961 198'1 1987 1550 DELTA DELTA. - - - [

FIGURI/10. Actual and theoretical dividend-price ratio. Germany 1954-1990 ( a = 0.0).

rates in terms of simple and theoretically consistent consumption based models continues to pose a major challenge to the profession.

Appendix

The data for Sweden, Germany and the UK are documented elsewhere in the literature, so only a brief description is given. A thorough description of the Danish stock market data is given below since it is only available (in Danish) in Lund (1992).

Denmark

The data for Danish stock prices and dividends were computed by one of us (see Lund, 1992). A nominal stock index and accompanying dividend series were constructed from the original price quotation sheets from the Copenhagen Stock Exchange. In order to avoid a possible tax-induced distortion due to the well-known January effect, the stock index at the end of year t is defined as the value in (mid) February of year t + 1. Similarly, dividends for year t are defined as dividends paid out between February of year t and February of year t + 1. However, no Danish companies pay dividends in January, so the dividend series is effectively the dividends paid during year t. Corrections are made for stock splits and new equity issues below the market price using techniques similar to those described in Shiller (1981). The stock index is a value-weighted portfolio consisting of approximately 16 individual stocks (companies), which are generally chosen in order to obtain the maximum coverage of the 'market ' index of the Copenhagen Stock Exchange. The composition of the

517


stock index is changed about every 10 years, and the weights for the individual stocks are only adjusted in connection with changes in the composition of the stock index. In spite of these limitations and the relatively few individual stocks, the stock index constructed in Lund (1992) tracks the official market index of the Copenhagen Stock Exchange quite well (these results are available upon request or see Figure 1 in Lund, 1992).

The consumption and population series are from Hansen (1974) and various publications from the Danish Central Statistical Bureau. The consumption series covers private consumption of durable and non-durable goods. Unfortunately, there is no price deflator for private consumption, so nominal prices, dividends and consumption are deflated using the consumption deflator for total consumption.

Germany

The German stock price and dividend series are from de Long and Becht (1992), which also contains a detailed description of the construction of the series. Both series are computed as annual averages (de Long and Becht only tabulate the real values of prices and dividends; the deflation is obtained using the consumer price index). The population and real consumption series are from Liesner (1989) and OECD Main Economic Indicators.

Sweden

The data for Swedish real stock prices and dividends were supplied to us by Per Frennberg (see also Frennberg and Hansson, 1992a, 1992b). The stock price is computed at the end of the year, while dividend is the dividend income payable throughout that year. The consumption series was supplied to us by Lars Svensson (see also Englund et al., 1992). All series are deflated by the implicit consumption deflator. The population series is from Liesner (1989) and OECD Main Economic Indicators.

UK

The data for UK stock prices and dividends were supplied to us by Ian Tonks (see also Bulkley and Tonks, 1989). The stock price is the de Zoete and Wedd value-weighted annual Equity Price Index computed at the end of the year. Dividend is the dividend income payable on that index throughout the year. These data are similar to the UK data used by Shiller (1989), Shiller and Beltratti (1992) and Beltratti and Shiller (1993). The consumption and population series are from Liesner (1989) and OECD Main Economic Indicators. Nominal stock prices, dividends and consumption are deflated by the implicit consumption deflator.

For all countries consumption is measured per capita.

Notes

1. For recent surveys of the GMM principle, see Hall (1993) and Ogaki (1993). 2. More formally, there are misspecification directions for which the test J(b) has power

equal to size. 3. Hoffman and Pagan (1989), Ghysels and Hall (1990a), as well as Hamori (1992b), have

previously applied this test to the C-CAPM. 4. The data are available upon request from the authors. 5. Note that by letting the VAR model be formulated in terms of changes in dividends and

consumption and in terms of the dividend-price ratio instead of stock prices themselves, the results are robust to possible non-stationarity of Pt, dt and c t.

518

6.


Although Hansen and Singleton (1982) originally obtained results quite favorable to the C-CAPM, they later corrected their results due to an error discovered in their consumption series, see Econometrica, 1984, 52: 267-268.

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