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 FACULTY WORKING 
 PAPER NO. 1154 
 
 The Present Value Model of Stock Prices: 
 Regression Tests and Monte Carlo Results 
 
 Louis O. Scott 
 
 College of Commerce and Business Administration 
 Bureau of Economic and Business Research 
 University of Illinois, Urbana-Champaign 
 
BEBR 
 
 FACULTY WORKING PAPER NO. 1154 
 College of Commerce and Business Administration 
 University of Illinois at Urbana- Champaign 
 June, 1985 
 
 The Present Value Model of Stock Prices: 
 Regression Tests and Monte Carlo Results 
 
 Louis 0. Scott, Assistant Professor 
 Department of Finance 
 
 A shorter version of this paper will appear in the Review of Economics 
 and Statistics , 1985. 
 
 Helpful comments were provided by Ron Michener, Tom Glaessner, C. F. 
 Lee, Yoon Dokko, Bob Korajczyk, and Stephen LeRoy. Comments by Fisher 
 Black and Robert Merton on an earlier version of the research motivated 
 some of the extensions. 
 
ABSTRACT 
 
 The variance bounds tests of the present value model of stock prices 
 are re-examined in this paper. A direct test of the model based on 
 ordinary least squares estimation of a simple regression equation is 
 proposed as an alternative and it is shown that this regression 
 approach has several advantages over the variance bounds tests. This 
 test is easily adapted to the important case in which the percentage 
 changes in real dividends and real stock prices are stationary pro- 
 cesses. The t*ests are applied to quarterly data for the Standard & 
 Poor's Index of 500 Common Stocks and tJie results are much more 
 
 > 
 
 conclusive than those obtained by previous tests. 
 
THE PRESENT VALUE MODEL OF STOCK PRICES: 
 REGRESSION TESTS AND MONTE CARLO RESULTS 
 
 Shiller (1981a and 1981b) and LeRoy and Porter (1981) have tested 
 the present value model of stock prices by examining the implicit re- 
 strictions of this model on the variation of stock prices. Their 
 results suggest that actual stock prices vary too much to be con- 
 sistent with this model. If we examine their results closely, we find 
 that Shiller does not construct formal statistical tests of the model 
 and that most of the tests in LeRoy and Porter are not statistically 
 significant because the standard errors of the variance estimates are 
 quite large. In addition, some critics have argued that the relevant 
 time series are not covariance stationary, even after the removal of a 
 time trend, and that the variance estimates are unreliable. This 
 argument has been made by Kleidon (1982, 198A) and by Marsh and Merton 
 (1983, 1984). An alternative would be to model the percentage changes 
 in dividends, earnings, and stock prices as covariance stationary time 
 series. Shiller (1981a) considers this alternative specification, but 
 notes that it does not lead to tractable variance bounds for stock 
 prices. In this paper, a direct regression test of the present value 
 model is developed as an alternative to the variance bounds test, and 
 the test is extended to handle the case in which the percentage changes 
 in dividends, earnings, and stock prices are covariance stationary. 
 
 The present value model of stock prices has the following form: 
 
 oo 
 
-2- 
 
 where P is the asset price at the end of period t and D is the divi- 
 dend paid during period t. E is the conditional expectations operator, 
 conditional on information available in period t. E without a subscript 
 will represent the unconditional expectations operator. P and D are 
 expressed in real terms. This simple expectations model follows from 
 the expected real rate of return being constant and is generally asso- 
 ciated with risk neutrality in asset pricing models, but some empiri- 
 cal researchers argue that expected real rates of return are approxi- 
 mately constant even in the presence of risk aversion. If we let 
 
 00 
 
 * i 
 P = T P~ D , . , we observe that the model implies 
 
 c j-i t+1 
 (i) »* - p t + n t , 
 
 where n is the forecast error. The assumption of rational expectations 
 (or market efficiency) requires that P and n be uncorrelated, so that 
 we get 
 
 Var(P *) = Var(P ) + Var(n ) , 
 
 * 
 and we conclude that Var(P ) < Var(P ). Thus, we have an upper bound 
 
 on the variation of stock prices. 
 
 Two problems are encountered when one attempts a test of this bound 
 on the variation of stock prices. First, the time series must be co- 
 variance stationary; that is, the unconditional means, variances, and 
 covariances of the series must be finite and must not depend on time. 
 Shiller removes a long-term trend from his series on dividends and 
 prices and applies the variance restrictions to the detrended series. 
 LeRoy and Porter argue that there are no apparent trends in their 
 
-3- 
 
 adjusted series for earnings and prices and that further adjustments 
 are not necessary. The second problem deals with the estimation of the 
 variance of P (or a detrended P..), a series which is not observable. 
 Shiller calculates this series recursively by assuming a value at the 
 end of the sample period and then he computes a sample variance for 
 the constructured series, but he does not develop a statistical test 
 of the variance restrictions. LeRoy and Porter formulate and estimate 
 bivariate time series models to compute the variances. This method 
 requires the researcher to formulate a time series model and then test 
 
 the variance bounds conditional on the formulated model. Singleton 
 
 * 
 (1980) has shown that the variance of a series like P can be esti- 
 mated in the frequency domain if we have a prior estimate for the 
 discount factor. The hypothesis tests developed for these variance 
 estimators will depend on the large sample distribution theory for 
 variance estimators. It is well-known that these large sample distri- 
 butions depend on the effects of fourth cumulants of the innovations 
 of the process generating the observed time series, and for this 
 reason the distributions typically used for hypothesis testing are not 
 
 robust if we drop the assumption that the innovations are multivariate 
 
 2 
 normal. It is extremely difficult to estimate the standard errors of 
 
 the variance estimates if we want to relax the assumption of normally 
 distributed innovations (or the assumption that the fourth camulants 
 are all zero). The large sample distributions for many of the stan- 
 dard econometric estimators (such as ordinary least squares, gener- 
 alized least squares, or instrumental variables) do not require the 
 innovations or error terms to be normally distributed. For this 
 
-4- 
 
 reason, the regression tests developed in the next section impose a 
 less restrictive set of assumptions. 
 
 LeRoy and Porter developed several tests for the implied variance 
 bounds. Even though their point estimates indicate rejection of the 
 present value model, the confidence intervals are so large that the 
 model is rejected in only a few of the cases examined. In a chapter of 
 my dissertation (1982, Ch. 5), I calculated frequency domain variance 
 estimators and found the point estimates for the variances of the price 
 series to be many times greater than the estimates for the upper bounds, 
 but the standard errors associated with these estimates are so large 
 that the tests of the model are barely significant at the 5% level. 
 It appears that these variances are not being estimated with much pre- 
 cision. More recently, Flavin (1983) has examined the finite sample 
 properties of the variance estimators and has found that the resulting 
 tests are biased against the null hypothesis. One can hardly argue 
 that these tests of the present value model are conclusive. The alter- 
 native regression tests are developed in Section I, and the empirical' 
 results are presented in Section II. In Section III, I present the 
 results of a Monte Carlo study which examines the finite sample proper- 
 ties of the proposed test statistics. 
 
 I. The Alternative Regression Test 
 
 In a note, Geweke (1980) has shown that regression tests of expec- 
 tations models are more powerful than variance bounds tests. This 
 testing procedure has been the one most frequently applied in tests of 
 other expectations models (e.g., tests of forward rates as unbiased 
 
-5- 
 
 predictors of spot rates in foreign exchange markets). If the series 
 P were observable, then the relationship in equation (1), implied by 
 
 the present value model, could be easily tested by a least squares 
 
 * * 
 
 regression of P on a constant and P . To calculate P , we need a 
 
 prior estimate of the discount factor P and the entire realization of 
 
 the dividend series (D,., t = 1,2,...,°°). Consider the following esti- 
 
 mate of P from a sample t = 1,...,T,T+1: 
 
 P* - PD T+1 + BP T+1 
 
 ~* * * 
 
 P,,, . = BP„, + pD m 
 T-l T T 
 
 P t , pp^ + pD 
 
 P x - «P 2 + PD 2 , 
 
 where B is determined from a prior estimate. Under the null hypothesis 
 
 of the present value model, we still have the result that E (P,_) - P^ 
 
 * 
 for t = 1,...,T. In fact, this construction of P is similar to 
 
 ' t 
 
 Shiller's except that we use the last value of the stock price instead 
 of a sample average. We again have a regression relation: 
 
 (2) P = a + bP + e , 
 
 where a and b should equal zero and one and e is a forecast error 
 
 H t 
 
 which should be uncorrelated with P according to the null hypothesis. 
 The parameters a and b can be estimated consistently by ordinary least 
 
-6- 
 
 squares (OLS) , but the disadvantages of this approach are that we re- 
 quire a prior estimate of P and e is a serially correlated error 
 
 3 
 term. Hansen and Hodrick (1980) have shown that OLS produces reliable, 
 
 but inefficient estimates and test statistics for expectations models 
 
 of this form. 
 
 Another approach to testing the relationship in equation (1) is to 
 
 restate it in terms of observable dividends and stock prices. First 
 
 we rewrite the equation under the alternative hypothesis using the lag 
 
 operator (L : Lx = x _): 
 
 PL" 1 
 
 D = a + bP^ + n • 
 
 i-pl" 1 
 
 Now we multiply both sides of the equation by the filter — = — to get 
 
 PL 
 
 (3) D t - a-bP t + }p t ^ +V 
 
 where u = (— L - l)n . We now have an equation that is a function of 
 t p t 
 
 observable time series and an error term, and we do not require a prior 
 estimate of P, it becomes another parameter to estimate with a and b. 
 The application of this filter is similar to the usual generalized 
 
 least squares (GLS) correction for first-order autocorrelation in the 
 
 4 
 error term, but this filter is unstable. As a result, u is a 
 
 * t 
 
 * 
 serially uncorrelated error term. In the regression .of P on P , P 
 
 is predetermined but not strictly exogenous, and it is well-known 
 that the standard GLS correction for serial correlation does not pro- 
 duce consistent parameter estimates if the right-hand side variables 
 are not strictly exogenous. In equation (3), P and u are corre- 
 lated, and we must use an instrumental variables (IV) estimator. 
 
-7- 
 
 Under the null hypothesis of the present value model, a = 0, b = 1, 
 
 and E (u .) = for i > 0. Under the alternative hypothesis, we 
 t t+ 1 — 
 
 relax the restrictions on a and b and preserve the restriction on the 
 error term. Hence, u should be uncorrelated with any variable dated 
 t-1 or earlier, and a natural set of instruments includes lagged 
 dividends, lagged stock prices, and lagged values of variables which 
 are important for predicting dividends. Lintner (1956) and Fama and 
 Babiak (1968) have shown that earnings play an important role in the 
 behavior of dividends over time; thus lagged earnings should be 
 included as instrumental variables. The advantage of this second ap- 
 proach to the regression test is that we do not require a prior esti- 
 
 * 
 mate of the discount factor and we do not need to estimate the P 
 
 series. 
 
 If we want to estimate the parameters of equations (2) and (3), 
 
 the relevant time series must have finite second moments, and this 
 
 condition is normally satisfied by requiring that the time series be 
 
 stationary, or at least covariance stationary. This condition is 
 
 necessary for estimating the variances in the variance bounds tests 
 
 and it is also important for the OLS and IV estimators. One approach 
 
 is to follow Shiller and remove a long-term trend from the data as 
 
 follows: p = P (1+g) and d = D (1+g) where p and d are the 
 
 respective detrended series. The discount factor for the model using 
 
 detrended data becomes y = P(l+g), and the following equations can be 
 
 derived for the detrended series: 
 
-8- 
 
 ~* 
 
 (4) p = a + bp t + e t 
 
 (5) d t - a - b P( . +^P t _ 1 + u t> 
 
 ~* 
 where p is computed by using y> detrended dividends, and the terminal 
 
 value of detrended stock prices. For equation (5), we must use an IV 
 
 estimator and we still have the condition that E _. (u ) =0. 
 
 In the alternative specif icatioa, the dividend process is not 
 
 mean-reverting. Here we assume that the percentage changes in 
 
 dividends and stock prices as well as the price-dividend ratio 
 
 f— — , — r-, —J are stationary. Shiller (1981a) notes that the terminal 
 
 condition for the present value model is not necessarily satisfied if 
 
 we assume that only — and — — are stationary. If the terminal 
 
 condition is not satisfied, we have the undesirable result that there 
 
 is no solution for the stock price. Shiller imposes the terminal 
 
 c. 
 
 condition by requiring the price-dividend ratio to be stationary. 
 Kleidon, Marsh, and Merton have presented evidence that this kind of 
 model for dividends is not rejected by the data in tests with Shiller's 
 trend model as the alternative hypothesis. For the remainder of the 
 paper, I focus on this specification of dividends and stock prices, 
 but I include the tests on detrended data so that the results can be 
 compared with Shiller's variance bounds tests. The relationship in 
 equation (1) is now modified as follows: 
 
 P P n 
 
 (6) £ . a* + b* £ + £, 
 
 t t t 
 
 and again a* = and b* = 1 for the present value model. To develop 
 the regression test, we replace P in (6) with its estimate P . To 
 
-9- 
 
 derive the corresponding filtered equation, we apply the same filter 
 as before to both sides of the equation after rewriting it as follows: 
 
 P* = PL , D = a*D + b*P + r\ 
 1-PL 
 
 D t - a*(|- D t _ x - D t ) + b*<± P^ - P t ) + (i Vl - n t ) 
 
 -n + - n 
 D t - "t-1 " bP t + ? P t-1 + Va* "^ 
 
 a* b* 
 
 where a = — 7-= rr arid b = - — . Then the equation can be rewritten 
 
 P(l+a*) 1+a* 
 
 as a function of stationary time series plus an error term. 
 
 bA-i. 
 
 u t-i D t-i p Vi t 
 
 " n t + p* n t-l 
 
 where u = -n rrr;: • Under the null hypothesis for the present valut 
 
 t (l+a*)D _ JV v 
 
 model, a = and b = 1. The error u again has the property that it is 
 
 uncorrelated with variables dated t-1 and earlier, and it is not 
 
 P 
 
 serially correlated. Note that Efu (— )) * 0, and again we must 
 
 t t-1 
 use an IV estimator. 
 
 In addition to separate tests on the coefficients a and b, we can 
 
 use the OLS estimators and the IV estimators to construct joint tests 
 
 of the restrictions for the present value model (joint tests that 
 
 a = and b = 1). These regression tests have several advantages over 
 
 the variance bounds test. The variance bounds test is based on an 
 
 inequality restriction which may contain a substantial amount of 
 
 slack, whereas the regression test constitutes a direct test of the 
 
 model. In the regression tests, we avoid the need to construct 
 
-10- 
 
 hypothesis tests for variances which usually require additional 
 restrictions on the distributional properties of the innovations. For 
 the tests based on IV estimators, we eliminate the need to use a prior 
 estimate for the discount factor. The regression tests described here 
 are more in the spirit of the variance bounds tests than the previous 
 regression tests of market efficiency in the finance literature. The 
 regression tests of market efficiency have focused on regressions of 
 rates of return on past information to see if there is any predictable 
 variation in returns and the results indicate that there is very 
 little. The regression tests derived in this section, like the var- 
 iance bounds tests, are tests of the relation in equation (1), where 
 the underlying issue is the use of stock prices as unbiased predictors 
 of future dividends holding discount rates constant, not the predict- 
 ability of the return series. 
 
 In constructing the tests based on OLS estimates of equations (4) 
 and (6) and IV estimates of equations (5) and (7), we encounter several 
 subtle econometric problems. The error terms in (4) and (6) will be 
 serially correlated, at least under the null hypothesis, and the 
 standard variance matrix for the OLS estimates will be inappropriate. 
 To account for this problem, we apply Hansen's (1979) method for com- 
 puting asymptotic variances: 
 
 Var(f) = T(X'X)" 1 S(X*X)" 1 
 b 
 
 where X is the (Tx2) matrix containing a constant and the right hand 
 
 side variable and S is the spectral density matrix of either 
 
 P t 
 (e ,e p ) or (e ,e t— ) evaluated at the zero frequency. The spectral 
 t ' t p t t ' t D J 
 
-11- 
 
 density matrices are estimated by using the Parzen procedure and a 
 flat window as described in Nerlove, Grether and Carvalho (1978, pp. 
 67-68). Second order autoregressions are used to pre-whiten the 
 series. This approach to estimating the variance matrix incorporates 
 the fact that the stock price variable is not strictly exogenous in 
 the equation and it allows for conditional heteroskedasticity in the 
 error term. 
 
 Our theory tells us nothing about the variances of the forecast 
 errors or the error terms of the equations; hence there is nothing in 
 the theory that requires conditional variances to be constant. The 
 OLS estimators for (4) and (6) are still consistent and the variance 
 matrix for the estimators accounts for the possibility of conditional 
 heteroskedasticity. If the conditional variance of the error term in 
 either (5) or (7) is constant, then the appropriate estimator is the 
 nonlinear two-stage least squares estimator of Amemiya (1974). Note 
 that if the conditional variance of the error term is constant in 
 either (5) or (7), then it cannot be constant in the other equation. 
 To account for conditional heteroskedasticity in (5) and (7), we apply 
 Hansen's (1982) generalized method of moments (GMM) estimator. Let u 
 be a vector of length T containing the residuals of either (5) or (7); 
 T is the sample size. Let Z be the (Txk) matrix containing the instru- 
 mental variables and z be the t'th row of Z. For equation (5), z. = 
 (l,d ,,p , ,c _), where c , is the detrended earnings measure lagged 
 one period. For equation (7), the instruments are the growth rates in 
 dividends and earnings, the price-dividend ratio, and the price- 
 earnings ratio, all lagged one period so that 
 
-12- 
 
 ' M Vl °t-l P fl P fl, 
 £ t " ' D t _ 2 « C t _ 2 ' D^' C^' 
 
 where C is the earnings measure. Let 6_' = (a,b,y) for (5) or 
 Pi' = (a,b,fl) for (7) and the GMM-IV estimator is computed by 
 
 min i = (I u'Z)w" 1 (I Z'u), 
 
 1 
 
 where W is a weighting matrix which can be a function of the sample 
 
 data. If the error term is conditionally heteroskedastic, then the 
 
 optimal weighting matrix is E(u z z_). In most cases, one must 
 
 t — t — t 
 
 estimate W . The following two-step estimator which replaces W with 
 a consistent estimate W is asymptotically equivalent to the optimal 
 estimator. First, estimate f) using a weighting matrix which produces 
 initial consistent estimates; then use the consistent estimates to 
 form u and estimate W from the sample moments: 
 
 T 
 
 1-2 i 
 
 w~ = — y u z z . 
 
 T T t— t— t 
 
 Finally, re-estimate ^ using W . For the initial weighting matrix, I 
 use W = — Z'Z which produces the nonlinear two-stage least squares 
 estimator. The covariance matrix for is computed as follows: 
 
 Su A _i 3u _i 
 VarCi) = TK^fVZW^Z'C^)] \. 
 
 Using both the OLS estimates and the GMM-IV estimates, we can test 
 
 the restriction of the present value model (a = 0, b = 1) by computing 
 
 2 
 the following y statistic: 
 
 2 m f a *' -l,a n 
 X (2) 'b-l J l b-l J ' 
 
-13- 
 
 where V is the variance-covariance matrix for the estimates a and b. 
 The test statistic has a large sample distribution that is Chi-squared 
 with two degrees of freedom. An additional test for the model can be 
 computed by using the criterion function for the GMM-IV estimators of 
 (5) and (7). We restrict the parameters a and b to their respective 
 values under the null hypothesis and estimate the discount factor 
 
 (B or y)« We can then test the model by applying Hansen's specif ica- 
 
 2 
 tion test in Lemma 4.2: the statistic T min I should have a x dis- 
 tribution with degrees of freedom equal to the number of orthogonality 
 conditions (the number of instrumental variables) minus the number of 
 estimated parameters, in this case one. If the value of this test 
 statistic is too large, then the model is rejected. This test is more 
 in the spirit of the familiar regression tests of market efficiency to 
 which I alluded earlier. If there is significant correlation between 
 the error term u and the lagged variables used as instruments, then 
 the test will indicate rejection of the present value model. 
 
 II. Empirical Results 
 
 The present value model of stock prices is tested by using quarterly 
 data for the Standard and Poor's Index of 500 Common Stocks for the 
 period 1947 to the second quarter of 1983. In its publication Trade and 
 Security Statistics , Standard and Poor's compiles quarterly data on its 
 price index of 500 common stocks as well as indices on earnings and 
 dividends for the companies included in the index. The implicit price 
 deflator for personal consumption expenditures is used to deflate the 
 three series. To detrend the data for estimation of equations (4) and 
 
-14- 
 
 (5), I have used a method similar to that used by Shiller. The 
 following regressions are used to estimate the common growth trend for 
 real dividends and real stock prices: 
 
 in D t = a x + fcnd+g^-t, 
 
 'in P t = a 2 + )in(l+g 2 )*t. 
 
 The two estimates, 2,n(l+g.. ) and £n(l+g„), are averaged to form the OLS 
 estimate of the common growth rate (1+g), and this estimate is used to 
 detrend both series. A separate trend regression is estimated for 
 real earnings. For equations (6) and (7), it is not necessary to 
 detrend the data because the effects of long-term trends or growth 
 rates are removed when we form ratios. 
 
 The initial research effort focused on the GMM-IV estimation of the 
 filtered regression equations (5) and (7). The results are quite ex- 
 treme: the estimates for b range from -.033 to .017, the t statistics 
 
 for the tests that b = 1 range from -411.20 to -75.74, and the 
 
 2 
 X(?\ statistics for the joint tests range from 6173 to 212000. These 
 
 extreme results indicate rejection of the present value model at the 
 
 zero marginal significance level. In a subsequent Monte Carlo study, 
 
 which is described in Section III, I found that there is an extreme 
 
 finite sample bias in the GMM-IV estimates. For 100 simulations of a 
 
 model in which stock prices satisfy the present value model, the GMM-IV 
 
 2 
 estimates of b range from -.068 to .1075 and the Xo\ statistics for 
 
 the joint test have extremely high values. The Monte Carlo results 
 
 indicate that tests based on the GMM-IV estimation of a and b in the 
 
-15- 
 
 filtered regression equations have an extreme bias against the present 
 value model. The detailed results on the GMM-IV estimation of (5) and 
 (7) are omitted for this reason. The Monte Carlo study does not reveal 
 extreme biases for the other estimators and test statistics, and the 
 results are reported in Tables I and II. 
 
 The results for the OLS estimation of equations (4) and (6) are 
 presented in Table I. Both equations are estimated for a longer sample 
 period 1947-83 and a more recent sample period 1960-83. The purpose 
 of the second sample period is to show the results for a more recent 
 
 Q 
 
 period. Because the results for the two sample periods are similar,* 
 the discussion will concentrate onthe longer sample period. The prior 
 
 estimates for the discount factor are computed from estimates of the 
 
 p + d 
 — t t 
 average real return. Let r be the sample mean of , and the 
 
 P t-1 
 
 estimate of y is the reciprocal of r 1 . Let r~ be the sample mean of 
 
 P t + D t - 
 
 — , and the estimate of 8 is the reciprocal of r . The estimated 
 
 t-1 
 
 trend coefficients used for detrending the data in (4) are shown in the 
 
 middle of the table. For the longer sample period, the trend coeffi- 
 cients for 2,n D and £n P are .004398 and .005628, respectively, so 
 that the resulting estimate of (1+g) is 1.005026. The same trend coef- 
 ficients are used for the data for the more recent sample period because 
 the estimated trend coefficient over the shorter period is negative for 
 real stock prices. This result alone indicates that the data fit the 
 model poorly or that stock prices deviated substantially from their long 
 term trend. The estimates for a and b in the longer sample period are 
 35.47 and -.02478, and the respective t statistics for a = and b = 1 are 
 
-16- 
 
 26.33 and -42.82. The separate t tests on a and b indicate rejection 
 
 2 
 at extremely low marginal significance levels. The Xfj\ statistic 
 
 for the joint test is 5014 which also has an extremely low marginal 
 
 o 
 
 significance level. In addition, the sample standard deviation of p 
 
 * 
 
 is sixteen times greater than a(p ), the estimate of the theoretical 
 
 upper bound. In Shiller's study, a(p ) exceeds a(p._) by a factor of 
 five times for the S&P series and by a factor of thirteen for the Dow 
 Jones Industrial Average. The results of the OLS tests on detrended 
 data imply the same dramatic rejection of the present value model. 
 
 The results for equation (6) are contained in the bottom panel of 
 Table I, and again we have dramatic rejection of the model. The OLS 
 estimates of a and b are 73.34 and -.02092, and the respective t 
 statistics for a = and b = 1 are 6.53 and -10.02. The separate t 
 
 tests are not quite as dramatic, but the results indicate rejection 
 
 2 
 at extremely low significance levels. The ^ . , statistic for the 
 
 joint test has an extreme value of 366, and the sample standard devia- 
 tion of the price-dividend ratio exceeds the estimate of its upper 
 bound by a factor of 1.67. Even though the violation of the upper 
 bound is not dramatic, the tests based on the OLS estimates indicate 
 overwhelming evidence against the model. 
 
 Despite the problems with the GMM-IV estimates of a and b in the 
 
 filtered regression equation, we can restrict a and b to zero and one 
 
 2 
 and compute a x test based on Hansen's specification error test. The 
 
 results of these tests are contained in Table II. The top panel con- 
 tains the results for equation (5) and the bottom panel contains the 
 
 2 
 results for equation (7). The x tests are all significant at the 
 
-17- 
 
 1% level except for the test associated with equation (7) for the 
 shorter sample period, which is significant at the 2.5% level. Al- 
 though these tests are significant at standard significance levels, 
 the results are not as dramatic as those obtained with variance bounds 
 tests or the OLS regression tests. As I pointed out earlier, this 
 test is similar to the regression tests of market efficiency, and the 
 results are similar. This test can also be criticized by observing -' 
 that earnings in period t-1 are being used to predict dividends and 
 prices in period t, but earnings are normally reported with some lag. 
 
 III. The Monte Carlo Simulations 
 
 The regression tests reported in Section II indicate dramatic re- 
 jection of the present value model. Kleidon and Flavin have presented 
 evidence that the variance bounds tests can be biased against the 
 present value model, and for this reason a Monte Carlo study of the 
 estimators and test statistics developed in Section I has been per- 
 formed. The purpose of the simulation study *is to examine the per- 
 formance of the test statistics when stock prices do in fact satisfy 
 the present value model. I have already noted that the Monte Carlo 
 study of the GMM-IV estimator indicates a severe bias against the 
 model. 
 
 The simulated model for dividends and earnings has the following 
 form: 
 
 in D t - in D t _ 1 = a 1 - a 2 in D + a 2 in X fc + e lt 
 
 in X t = a 3 + in X + e^, 
 
-18- 
 
 where e, and p_ are normally distributed and independent. The 
 It 2t 
 
 coefficient on earnings is restricted so that we get a steady state 
 
 dividend payout ratio which does not depend on the level of earnings; 
 
 a l 
 in this model the steady-state dividend payout ratio is exp( ). The 
 
 a 2 
 
 form of the dividend equation is motivated by Lintner's well-known 
 
 study of dividend policy. In the model, earnings follow a multiplica- 
 tive random walk with drift, so that D /D , and X /X , are 
 stationary time series. I then compute stock prices using the present 
 value model: 
 
 P t ^u\(D t+j ) 
 
 a^Cl-a, 
 t "t -a. 
 
 00 a m « ^ i a o v1 a ((1-a _ 
 T. P^D (1_a 2 ) X 1 " (1 " a 2) expf -J I 
 
 = 7. P^'D z X *' exp 
 
 i=i t t "2 
 
 .1 (1-a ) 2j -l 
 
 + a a,( T. (j+l-k)(l-a ) K l ) + ± a 
 
 n v w vri ~ /Vi ~2' ' 2 "1 (-a 2 )(2-a 2 ) 
 
 . (l-(l-a 9 ) 2i ) (l-(l-a V 1 ) 
 
 + ir;[(l-a 9 )^ ^— + 2(l-a ? ) p— -f + j]} 
 
 2 (l-(l-a 2 ) 2 ) ( " a 2 ) 
 
 2 2 
 where p is the discount factor and a, and a 9 are the respective vari- 
 ances of f, and F_ . As i gets large, the terms in the summation are 
 It 2t 
 
 approaching: 
 
 P^V ,-^) = X. expf(-i) +L 2 (— 7 L- - T ) 
 t t+j t a_ 2 1 a.(2-aj 
 
 2 
 (l-a_) 1-a . 2 . 
 
 + ^t l —r + at ( *•)} [Pexpfa + ± ap ] J 
 
 2 2 l-(l-a ) 2 _a 2 3 2 2 
 
-19- 
 
 As an approximation for some large k, we have 
 
 X (8exp{a -K-a } ) 
 
 I B J E t (D ) --£ 12 . 
 
 j=k+l J o l-Bexp{a 3 +^o 2 ) 
 
 2 
 
 * a l 1 2 1 12 {-*-~ a o 2 2 
 
 where X = X exp{ (—) + -*j ( , . ) + -so =- + cr (— -)} 
 
 C t a 2 z 1 a 2 U a 2 ; I l (l-(l-a ) Z ) Z a 2 
 
 I experimented with values of 100, 200, and 300 for k and found that 
 the stock price using k = 100 is extremely close to the value using 
 k = 300. 
 
 To get simulated data that resemble actual dividends and earnings, 
 I estimated the parameter values by fitting the model with seasonal 
 dummies to actual data. (Earnings for the S&P 500 have not been nega- 
 tive during the post World War II period.) The following model was 
 used in the simulations: 
 
 in D - £n D = -0.3938 - .5811 in D + .5811 in X + e 1( . 
 
 ^n X fc - in X = .0042 + e 2t 
 
 Var(e lt ) - .030032, VarU^) = .006979, 
 
 where the coefficients for the dummy variables are omitted and each 
 intercept represents the average of the quarterly intercepts. The 
 model was simulated without the seasonality, and the Q value was set 
 at .9801625 to be consistent with average real returns on a quarterly 
 basis. 
 
 The results of 1000 simulations are contained in Table III and 
 there is some evidence of a finite sample bias in sample sizes of 145. 
 
-20- 
 
 The mean of the 1000 estimates for the slope coefficient b is .9048, 
 compared to the true mean of one. This suggests that there is a small 
 downward bias in this estimator. But the slope coefficients range from 
 .2563 to 2.596 and never have a value as low as those estimated from 
 actual data. The large sample approximation for the t statistics is 
 the standard normal distribution, and the results in the middle of 
 Table III Indicate that neither of the separate t tests has a bias 
 against the present value model. In fact, the simulations indicate 
 
 that the Type I error at various critical values is being overstated. 
 
 2 
 The asymptotic X(?\ distribution, however, appears to be a poor approxi- 
 mation for the joint test on a and b. The critical value for the 5 
 percent significance level is 5.99, but 294 (or 29%) of the simulations 
 produced test statistics exceeding 5.99. In fact, 189 (or 19%) of the 
 simulations produced values that exceeded the critical value at 0.5 per- 
 cent level. The maximum simulated value for the test statistic was 
 96.16. The large sample distribution is a poor approximation in this 
 application, but the joint test statistic computed from actual data in 
 
 the sample period of 145 observations is much greater than the maximum 
 
 2 
 value produced by the simulations. The unusually large Xo\ statistics 
 
 seem to be the result of large negative covariances between the estima- 
 tes of a and b. In summary, the simulations indicate that despite a 
 small finite sample bias in the estimates of a and b, the estimates and 
 
 the separate t statistics are not biased against the present value 
 
 2 
 model. The ioint test based on the y,„ s statistic does have a bias 
 
 *(2) 
 
 against the model. One final observation concerns the sample standard 
 
-21- 
 
 p* P p 
 
 deviations of jr— and -r-— . The sample standard deviation of -jr— exceeds 
 
 P* t t t 
 
 that of jr— in only 16 of the 1000 simulations, and it never exceeds the 
 
 estimated upper bound by a factor of more than 1.16. In the sample of 
 
 p t 
 145 actual observations for equation (6), the standard deviation of -=r- 
 
 P t l 
 
 exceeds that of — by a factor of 1.67. 
 
 In Table IV, I present the results of 1000 simulations of the 
 
 2 
 X//,\ statistic with a and b restricted in the filtered equation. For 
 
 this test, we first estimate p with the GMM estimator allowing for 
 
 conditional heteroskedasticity in the error term. The test statistic 
 
 2 
 under the null hypothesis is distributed asymptotically as a x with 
 
 4 degrees of freedom, and the asymptotic distribution appears to be a 
 
 good approximation for the upper tail of the distribution. 
 
 IV. Conclusion 
 
 The results for the S&P 500 data suggest that the present value 
 model of stock, prices with rational expectations does not adequately 
 describe the behavior of asset prices in the stock market. The alter- 
 native tests described here do not require some of the additional 
 assumptions required to derive tests based on estimates of variances, 
 and the results are much more conclusive than those previously 
 obtained for tests of variance bounds. In addition, this test is 
 modified to handle the case in which the percentage changes in real 
 dividends and real stock prices are stationary and the results are 
 found to be qualitatively the same. 
 
 There are several competing hypotheses in the literature which 
 attempt to explain the results of the variance bounds tests. 
 
-22- 
 
 Blanchard and Watson (1982) interpret Shiller's results as evidence of 
 an arbitrary term in the rational expectations solution for stock 
 prices, specifically a stochastic bubble. Another more appealing 
 explanation is that risk aversion and varying real interest rates are 
 responsible for the observed volatility of stock prices. Kleidon, 
 Marsh, and Merton have argued that combining the present value model 
 with nonstationary processes for dividends and stock prices is capable 
 of explaining stock price variability, but the results presented here 
 for equation (6) reject this explanation. These tests do not allow us 
 to pinpoint the causes of stock price volatility, but we can conclude 
 that some of the variation in stock prices is due to something other 
 than the variation in dividends or the market's expectations of future 
 dividends. Indeed, a casual reading of the financial press suggests 
 that stock prices are quite sensitive to interest rate changes. 
 
-23- 
 
 FOOTNOTES 
 
 1. For a discussion of this model and other expectations models, see 
 LeRoy's survey (1982). 
 
 2. For the asymptotic distribution of sample variances for time series, 
 see Hannan (1970, pp. 209-12). In a footnote (footnote 15, p. 567), 
 LeRoy and Porter recognize this subtle assumption in the derivation 
 of their tests. 
 
 3. It is easy to show that the error term should be a first order 
 autoregression that is stable in the forward direction. 
 
 e t = Pe t+1 + ^t+l' w ^ ere n t+l * s a variable which is uncorrelated 
 with variables known at time t or earlier including itself. 
 
 4. The root of (1-PL ) lies inside the unit circle, but the filter 
 is stable going in the forward direction. 
 
 5. This subtle point is frequently ignored in applied econometrics. 
 To apply Hansen's GMM estimator for example, we need to have sta- 
 tionary time series. 
 
 6. In a comment on Shiller's paper, Copeland (1983) has argued that 
 the long term growth rate for dividends may be changing. His argu- 
 ment is confined to the effect of a one-period change, but we must 
 consider how the growth rate changes over time. When we assume 
 that the percentage changes in dividend and stock prices are sta- 
 tionary, we normally think of a constant growth rate for the series. 
 If the random growth rate for dividends is itself a stationary pro- 
 cess, then the series {•=:— , ■=— J will be stationary. If we assume 
 
 that the random growth rate is not stationary, then we again run the 
 risk of not satisfying the terminal condition for the present value 
 model. 
 
 7. The data for dividends are reported as a twelve-month moving total. 
 LeRoy and Porter obtained the data necessary to recompute the quar- 
 terly dividends and their numbers are available in a technical 
 appendix to their original paper. 
 
 8. Also, the S&P index was expanded from 90 stocks to 500 stocks in 
 1957. 
 
-24- 
 
 REFERENCES 
 
 Takeshi Amemiya, "The Nonlinear Two-Stage Least-Square Estimator," 
 Journal of Econometrics 2 (1974): 105-10. 
 
 Olivier J. Blanchard and Mark W«. Watson, "Bubbles, Rational Expecta- 
 tions, and Financial Markets," Paul Wachtel, Ed., Crisis in the 
 Economic and Financial Structure (Lexington: P. C. Heath and 
 Company, 1982). 
 
 Basil L. Copeland, "Po Stock. Prices Move Too Much to be Justified by 
 Subsequent Changes in Pividends? : Comment," American Economic 
 Review (March 1983): 234-35. 
 
 Eugene F. Fama and Harvey Babiak, "Pividend Policy: An Empirical 
 Analysis," Journal of the American Statistical Association 
 (Pecember 1968): 1132-61. 
 
 Marjorie Flavin, "Excess Volatility in the Financial Markets: A 
 Reassessment of the Empirical Evidence," Journal of Political 
 Economy (Pecember 1983): 929-56. 
 
 John Geweke, "A Note on the Testable Implications of Expectations 
 
 Models," University of Wisconsin-Madison, Revised November 1980, 
 SSRI Workshop Series. 
 
 Clive W. J. Granger, "Some Consequences of the Valuation Model When 
 
 Expectations are Taken to be Optimum Forecasts," Journal of Finance 
 (March 1975): 135-45. 
 
 E. J. Hannan, Multiple Time Series (New York: John Wiley & Sons, 1970). 
 
 Lars P. Hansen, "The Asymptotic Pistribution of Least Squares Estimators 
 with Endogenous Regressors and Pependent Residuals," Carnegie-Mellon 
 University, March 1979. 
 
 , "Large Sample Properties of Generalized Method of Moments 
 
 Estimators," Econometrica (July 1982): 1029-54. 
 and Robert J. Hodrick, "Forward Exchange Rates as Optimal 
 
 Predictors of Future Spot Rates: An Econometric Analysis," 
 Journal of Political Economy (October 1980): 829-53. 
 
 Allan W. Kleidon, "Bias in Small Sample Tests of Stock Price Rationality,' 
 Stanford University, May 1982. 
 
 , "Variance Bounds Tests and Stock Price Valuation Models," 
 
 Stanford University, June 1984. 
 
-25- 
 
 Stephen F. LeRoy, "Expectations Models of Asset Prices: A Survey of 
 Theory," Journal of Finance (March 1982): 185-217. 
 
 , "Efficiency and the Variability of Asset Prices," American 
 
 Economic Review (May 1984): 183-87. 
 
 and Richard D. Porter, "The Present-Value Relation: Tests 
 
 Based on Implied Variance Bounds," Econometrica (May 1981): 555-74, 
 
 John Lintner, "Distribution of Incomes of Corporations Among Dividends, 
 Retained Earnings, and Taxes," American Economic Review (May 1956): 
 97-113. 
 
 Robert Lucas, "Asset Prices in an Exchange Economy,"* Econometrica 
 (November 1978): 1429-45. 
 
 Terry A. Marsh and Robert C. Merton, "Aggregate Dividend Behavior and 
 Its Implications for Tests of Stock Market Rationality," 
 Massachusetts Institute of Technology, September 1983. 
 
 , "Earnings Variability and Variance Bounds Tests for the 
 
 Rationality of Stock Market Prices," Massachusetts Institute of 
 Technology, March 1984. 
 
 Mark Nerlove, David M. Grether, and Jose L. Carvalho, Analysis of 
 Economic Time Series: A Synthesis (New York: Academic Press, 
 1979). 
 
 Louis 0. Scott, "Risk Aversion, Rational Expectations, and Constant 
 Expected Real Rates of Return," Unpublished Ph.D. dissertation, 
 Department of Economics, University of Virginia, 1982. 
 
 Robert J. Shiller, "The Use of Volatility Measures in Assessing Market 
 Efficiency," Journal of Finance (May 1981a): 291-304. 
 
 , "Do Stock Prices Move Too Much to be Justified by Subsequent 
 
 Changes in Dividends?" American Economic Review (June 1981b): 421- 
 36. 
 
 , "Reply," American Economic Review (March 1983): 236-37. 
 
 Kenneth J. Singleton, "Expectation Models of the Term Structure and 
 Implied Variance Bounds," Journal of Political Economy (December 
 1980): 1159-76. 
 
 D/18 2 
 
TABLE I 
 OLS Models 
 
 Sample Period 
 1947:1 to 1983:11 
 
 14 5 
 
 Sample Period 
 1960:1 to 1983: II 
 
 93 
 
 Detrended Data: p = 
 
 (Prior Estimate) 
 
 a + bp + e 
 
 o 
 e 
 
 2 
 
 X (2) 
 
 .9846 
 
 35.47 
 (1.347) 
 
 -.02478 
 (.02393) 
 
 .8887 
 
 5014.13 
 
 .9917 
 
 43.95 
 
 (3.611) 
 
 .1790 
 
 (.05632) 
 
 9.431 
 
 305.91 
 
 Trend Coefficients 
 
 Dividends 
 Stock Prices 
 Earnings 
 (1+g) 
 
 .004398 
 
 .005628 
 
 .005572 
 
 1.005026 
 
 .004398 
 
 .005628 
 
 .005572 
 
 1.005026 
 
 Ratio Data : ■=— - = a + b — — + e 
 
 (Prior Estimate) 
 
 .9797 
 
 73.34 
 (11.24) 
 
 -.02092 
 (.1019) 
 
 .9868 
 
 92.96 
 (9.979) 
 
 -.04153 
 (.09948) 
 
 e 
 2 
 : (2) 
 
 277.1 
 366.27 
 
 69.52 
 128.84 
 
 Standard errors are in parentheses. The standard errors and X/ ? \ 
 statistics are based on error terms with serial correlation 
 and conditional heteroskedasticity. 
 
TABLE II 
 Tests of Restricted Filtered Equations 
 
 Sample Period Sample Period 
 
 1947:1 to 1983:11 1960:1 to 1983:11 
 
 93 
 -9881 
 
 30.16 * 
 
 1.70 
 12.85 
 
 Detrended 
 
 Data: 
 
 d t 
 
 + 
 
 p t 
 
 ■ YP t . 
 
 -1 + U t 
 
 T 
 
 
 
 
 
 
 
 14 5 
 
 
 
 
 
 
 
 
 .9876 
 
 -2 
 a 
 u 
 
 
 
 
 
 
 
 14.46 
 
 D.W. 
 
 
 
 
 
 
 1.70 
 
 T min I 
 
 ■ > 
 
 2 
 
 
 
 
 
 13.59 
 
 (a=0, b=l) 
 
 Trend coefficients are the same as in Table I. 
 
 D + P P 
 Ratio Data : — = P + u 
 
 T 144 92 
 
 P .9823 .9904 
 
 a 2 60.05 73.61 
 
 u 
 
 D.W. 1.71 1.68 
 
 T min I = y f . 14.04 13.12 
 
 (a=0, b=l)^ ; 
 
 Estimates are based on conditional heteroskedasticity for error terms 
 
TABLE III 
 
 Results of 1000 Simulations of OLS Model 
 Sample Sizes = 145 
 
 6 (Prior Estimate) 
 
 a 
 
 b 
 
 t(a=0) 
 
 t(b=l) 
 
 2 
 X (2) 
 
 Mean 
 
 Range 
 
 .9800 
 
 .9580 
 
 to 
 
 1.0002 
 
 2.787 
 
 -95.65 
 
 to 
 
 61.20 
 
 .9048 
 
 .2563 
 
 to 
 
 2.596 
 
 .06366 
 
 -1.483 
 
 to 
 
 3.047 
 
 .2699 
 
 -4.961 
 
 to 
 
 1.908 
 
 6.513 
 
 .000367 
 
 to 
 
 96.19 
 
 Number 
 
 of 
 
 Times 
 
 Number 
 
 of 
 
 Times 
 
 
 
 |t(a)| > 
 
 X 
 
 (%) 
 
 |t(b=l)| 
 
 > 
 
 x (%) 
 
 PR[ 
 
 It 1 > x] 
 
 4 
 
 
 (.4%) 
 
 24 
 
 
 (2.4%) 
 
 
 .10 
 
 2 
 
 
 (.2%) 
 
 11 
 
 
 (1.1%) 
 
 
 .05 
 
 1 
 
 
 (.1%) 
 
 7 
 
 
 (.7%) 
 
 
 .025 
 
 1 
 
 
 (.1%) 
 
 4 
 
 
 (.4%) 
 
 
 .01 
 
 1.645 
 1.96 
 2.326 
 2.576 
 
 Number of Times 
 
 2 2 
 x. x_( 2 ) Exceeds x (%) £liX( 2 ) > X J 
 
 5.99147 294 (29.4%) .05 
 
 7.37776 247 (24.7%) .025 
 
 9.21034 212 (21.2%) .01 
 
 10.5966 189 (18.9%) .005 
 
TABLE IV 
 
 Results of 1000 Simulations for Restricted Filtered Model 
 
 Sample Sizes = 144 
 
 Mean Values 
 
 R .9808 
 
 2 
 Test Statistic, y,,, 3.956 
 
 9.4877 
 11.1433 
 13.2767 
 14.8602 
 
 Number of Times 
 Test Statistic 
 Exceeds x (%) 
 
 Pr tX( 2 4) > *J 
 
 37 
 
 (3.7%) 
 
 .05 
 
 19 
 
 (1.9%) 
 
 .025 
 
 7 
 
 (.7%) 
 
 .01 
 
 5 
 
 (.5%) 
 
 .005 
 
HECKMAN 
 
 BlNDERV INC. 
 
 JUN95 
 
 Bound -Ti 
 
 .ru..~t 
 
 ...MANCHESTER, 
 INDIANA *