What drives stock prices? Identifying the determinants of stock price movements.

1. Introduction

Before 1981, much of the finance literature viewed the present value of dividends to be the principal determinant of the level of stock prices. However, LeRoy and Porter (1981) and Shiller (1981) found that, under the assumption of a constant discount factor, stock prices were too volatile to be consistent with movements in future dividends. This conclusion, known as the excess volatility hypothesis, argues that stock prices exhibit too much volatility to be justified by fundamental variables. Several papers (Flavin 1983; Kleidon 1986; Marsh and Merton 1986; Mankiw, Romer, and Shapiro 1991) challenged the statistical validity of the variance bounds tests of LeRoy and Porter and Shiller, on the grounds that stock prices and dividends were nonstationary processes; however, much of the subsequent literature found that stock price movements could not be explained solely by dividend variability, as suggested by the present value model with constant discounting (Campbell and Shiller 1987; West 1988a). (1)

By relaxing the assumption of constant discounting, Campbell and Shiller (1988, 1989) and Campbell (1991) attempted to break up stock price movements (returns) into the contributions of changes in expectations about future dividends and future returns. They employed a log-linear approximation of stock returns and derived a linear relationship between the log price-dividend ratio and expectations of future dividends and stock returns. They further assumed that the data- generating process of dividend growth and the log price-dividend ratio could be adequately characterized by a low-order vector autoregression (VAR). By using the VAR to forecast future dividend growth and future stock returns, they were able to decompose the variability of current stock returns into the variability of future dividend growth and future stock returns. They attributed most of the movements in stock prices to revisions in expectations about future stock returns rather than to future dividend growth. Campbell and Ammer (1993) extended the log-linear approximation and the VAR approach to an examination of bond returns as well as stock returns. They found that expectations of future excess returns contributed more to the volatility of stock returns than did movements in expected future dividends. (2)

In this paper, we argue that there is a fundamental problem in identifying the sources of stock price movements. The problem lies in the fact that stock prices (or, more specifically, price-dividend ratios) are very persistent but real dividend growth and excess returns are not. Figure 1 plots the log price-dividend ratio from 1953:II to 2001:IV. This figure clearly indicates that the log price-dividend ratio shows substantial persistence. Standard Dickey-Fuller tests fail to reject the null hypothesis of a unit root in the log price-dividend ratio. (3) In the standard market fundamentals stock price valuation model, movements in the expected values of real dividend growth, real interest rates, and excess returns (the latter two making up required return) explain movements in the price-dividend ratio. Real interest rates, which have a substantial low-frequency component, do not move over time in a manner that would explain the low-frequency movements in the price-dividend ratio. Thus, a market fundamentals explanation of persistent stock price movements also requires movements in excess returns or real dividend growth to persist. However, if we look at movements in real dividend growth and excess returns over time (Figure 2, Panels A and B), we find that these are very volatile, containing little discernable low-frequency movement. Therefore, the log price-dividend ratio exhibits substantial persistence but its market fundamental components do not.

[FIGURES 1-2 OMITTED]

Because of the apparent lack of low-frequency movements in either excess returns or real dividend growth, it is not possible to identify which of these is more important in producing long swings in the price-dividend ratio. More formally, we show that the data cannot distinguish a model in which there are small permanent changes in dividend growth from one in which there are small permanent changes in excess returns. In a five-variable system that includes log price-dividend ratio, real dividend growth, short- and long-term interest rates, and inflation, we find that we cannot reject two alternative vector error correction models (VECMs) that each contain two cointegrating vectors: one corresponding to stationary real dividend growth and stationary term premium and the other corresponding to stationary excess returns and stationary term premium.

The inability of the data to distinguish between these alternative models has enormous consequences for VAR stock price decompositions. We show that the relative importance of dividends and excess returns for explaining stock price volatility is very sensitive to the specification of the long-run properties of the estimated VAR. For the model in which excess returns are assumed to be stationary but real dividend growth is assumed to be nonstationary, real dividend growth (not excess returns) is the key contributor to stock price movements. The relative contributions reverse when we reverse the assumptions about stationarity. Thus, in contrast to much of the previous literature, we argue that the data cannot distinguish between a decomposition in which expectations about future real dividend growth are substantially more important than expectations about future excess returns and a decomposition in which the reverse is true.

The remainder of this paper is organized as follows. In section 2, we review the log-linear, VAR approach to stock price decomposition pioneered by Campbell and Shiller and used in many subsequent studies. In section 3, we show how alternative assumptions about low-frequency movements in stock market fundamentals can be described in terms of restrictions on cointegrating vectors in a VECM. In section 4, we test alternative specifications of the VECM/VAR used to describe the time-series properties of the data and, therefore, to calculate expectations of future stock market fundamentals. These tests include the Johansen (1991) test for cointegration and tests in which the cointegrating vector is prespecified (Horvath and Watson 1995). Section 5 presents stock price decompositions for alternative models and demonstrates how sensitive these are to the specification of the VECM. In section 6, we discuss whether low-frequency movements in real dividend growth or excess returns are plausible, at least on statistical grounds, and discuss why our results differ from much of the previous literature. In section 7, we examine whether our findings could be the consequence of poor power and size properties of our statistical approach. In section 8, we discuss whether our results reflect the existence of a rational bubble in stock prices. We find, however, that the data do not appear to support the existence of a rational bubble. Section 9 provides a summary and conclusion.

2. Stock Price Decompositions

The stock price (returns) decompositions of Campbell and Shiller (1988, 1989), Campbell (1991), and Campbell and Ammer (1993) start with a log-linear approximation of the accounting identity: 1 [equivalent to] [([R.sub.t+1]).sup.-1]([P.sub.t+1]/[D.sub.t+1] + 1)([D.sub.t+1]/[D.sub.t])/([P.sub.t]/[D.sub.t]) where [R.sub.t+1] is gross stock returns, [P.sub.t]/[D.sub.t] is price-dividend ratio, and [D.sub.t+1]/[D.sub.t] is one plus real dividend growth. Log linearizing and breaking the rate of return on stocks into the real return on short-term bonds (the ex-post real interest rate), [r.sub.t], and the excess return of equity over short-term bonds, et, yields:

[p.sub.t] = [E.sub.t][[rho][p.sub.t+1] + [d.sub.t+1] - ([r.sub.t+1] + [e.sub.t+1]) + k], (1)

where [p.sub.t] is the log price-dividend ratio, [d.sub.t] is real dividend growth and [rho] = exp([bar.p])/(1 + exp([bar.p])), and k = log(1 + exp([bar.p])) - [rho][bar.p] where [bar.p] is the average log price-dividend ratio over the sample. Recursively substituting, we obtain:

[p.sub.t] = [[infinity].summation over (j=0)][[rho].sup.j]([E.sub.t][d.sub.t+1+j] - [E.sub.t][r.sub.t+1+j] - [E.sub.t][e.sub.t+1+j]) + [k/[1 - [rho]]]. (2)

Thus, stock prices are a function of expectations of future real dividend growth, future real interest rates, and future excess returns. Similarly, surprises in excess returns can be written as:

[e.sub.t] - [E.sub.t-1][e.sub.t] = [[infinity].summation over (j=1)][[rho].sup.j][([E.sub.t] - [E.sub.t-1]) [d.sub.t+j] - ([E.sub.t] - [E.sub.t-1])[r.sub.t+j] - ([E.sub.t] - [E.sub.t-1])[e.sub.t+j]]. (3)

Surprises in excess returns are a function of revisions in expectations about future real dividend growth, future real interest rates, and future excess returns. One can construct similar decompositions of bond yields and returns (see Campbell and Ammer 1993).

In order to evaluate the above expressions, Campbell and Shiller (1988, 1989), Campbell (1991), and Campbell and Ammer (1993) propose estimating a VAR to calculate expectations of future real dividend growth, real interest rates, and excess returns. We extend their framework to allow for cointegration among the variables and consider a vector error correction model (VECM). Let the vector of time series given by [y.sub.t] = ([p.sub.t], [d.sub.t], [i.sub.t], [l.sub.t], [[pi].sub.t])', where [p.sub.t] is log price-dividend ratio, [d.sub.t] is real dividend growth, [i.sub.t] is the yield on short-term bonds, [l.sub.t] is the yield on long-term bonds, and [[pi].sub.t] is the inflation rate. Consider the following vector error correction model:

[DELTA][y.sub.t] = [mu] + [alpha][beta]'[y.sub.t-1] + [[m-1].summation over (i=1)][C.sub.i][DELTA][y.sub.t-i] + [v.sub.t], (4)

where [y.sub.t] is the 5 x 1 vector of possibly I(1) variables as defined, [beta] is a 5 x r matrix whose r columns represent the cointegrating vectors among the variables in [y.sub.t], [alpha] is a 5 x r matrix whose five rows represent the error correction coefficients, and [C.sub.i] is a 5 x 5 matrix of parameters. If the matrix [alpha][beta]' is of rank 5 (or r = 5), then the VECM system is a standard-levels VAR. The intercept term plays no role in our stock market decompositions but does have an effect on inference about the rank of [alpha][beta]'.

We can take the above VECM (or VAR) and write it as a first-order linear system (suppressing the intercept):

[Y.sub.t] = [AY.sub.t-1] + [V.sub.t], (5)

with [Y.sub.t] = ([y'.sub.t], [y'.sub.t-1], ..., [y'.sub.t-m+1])',

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (6)

and

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (7)

Equation 5 is the companion form of the VAR. This can be used to calculate expectations of variables in the system according to the formula [E.sub.t][Y.sub.t+k] = [A.sup.k][Y.sub.t].

Given the variables in our system, we can evaluate expectations of [d.sub.t] and [r.sub.t] in Equations 2 and 3:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (8)

where [H.sub.d] = (0, 1, 0 ..., 0), [H.sub.i] = (0, 0, 1, 0 ..., 0), and [H.sub.[pi]] = (0, 0, 0, 0, 1, 0 ..., 0) are 1 x 5m vectors that select real dividend growth, short-term interest rate, and inflation, respectively. The formula [H.sub.d][A.sup.j+1][Y.sub.t] is the expectation of real dividend growth at t + j + 1 while the formula ([H.sub.i][A.sup.j] - [H.sub.[pi]][A.sup.j+1])[Y.sub.t] is the expectation of real return on short-term bonds at t + j + 1.

Using Equation 8, we can decompose the log price-dividend ratio into the contributions of expectations of future real dividend growth, real interest rates, and excess returns. The contribution of expectations of future real dividend growth is expressed by [H.sub.d][(I - [rho]A).sup.-1][AY.sub.t] while the contribution of expectations of future real interest rates is -([H.sub.i] - [H.sub.[pi]],A)[(I - [rho]A).sup.-1][Y.sub.t]. The contribution of expected returns can be treated as a residual after subtracting the contributions of future real dividend growth and future real interest rates from the actual log price-dividend ratio. For actual excess returns,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (9)

Once again the contribution of expected future values of [e.sub.t] can be calculated as a residual. Note that in order to evaluate Equations 8 and 9, the roots of the matrix [rho]A must be less than 1. (4)

3. Specification of the VAR

As can be seen from Equations 8 and 9, the companion matrix from the VAR, A, is crucial in the stock price decomposition. Therefore, care must be taken in the specification of the underlying VAR. This includes a careful assessment of the number of stochastic trends in the system. In addition to specifying the number of stochastic trends in the system, we can evaluate alternative economic interpretations of the cointegrating vectors. These correspond to alternative assumptions about the presence of permanent changes in real dividend growth, excess returns, real interest rate, and other measures.

For example, consider the aforementioned system, which includes the log price-dividend ratio ([p.sub.t]), real dividend growth ([d.sub.t]), short-term interest rate ([i.sub.t]), long-term interest rate ([l.sub.t]), and inflation ([[pi].sub.t]). Define the ex-post real interest rate as [r.sub.t] = [i.sub.t-1] - [[pi].sub.t]. Using the log approximation employed by Campbell and Shiller, we can write excess stock returns over short-term bonds as:

[e.sub.t] = [rho][p.sub.t] - [p.sub.t-1] + [d.sub.t] - (i.sub.t-1] - [[pi].sub.t]) + k. (10)

Further, rewriting excess returns yields:

[e.sub.t] = ([rho] - 1)[p.sub.t] + [d.sub.t] - [i.sub.t] + [[pi].sub.t] + k [DELTA][p.sub.t] + [DELTA][i.sub.t]. (11)

If we make the assumption that [DELTA][p.sub.t] and [DELTA][i.sub.t] are stationary, then stationary excess returns implies that ([rho] - 1)[p.sub.t] + [d.sub.t] - [i.sub.t] + [[pi].sub.t] should also be stationary. Thus we can examine a linear combination of variables to evaluate whether excess returns are stationary even though the excess returns variable is not directly included in the system. Similarly, if the term premium is stationary (and [DELTA][i.sub.t] is stationary) then the interest rate spread, [l.sub.t] - [i.sub.t], will be stationary. Thus, given our five-variable system of ([p.sub.t], [d.sub.t], [i.sub.t], [l.sub.t], [[pi].sub.t]), a model in which excess returns are stationary implies the cointegrating vector ([rho] - 1, 1, -1, 0, 1), while a stationary term premium implies the cointegrating vector (0, 0, -1, 1, 0). Alternatively, a model in which real dividend growth is stationary implies the trivial cointegrating vector (0, l, 0, 0, 0). We can also evaluate a model in which the real interest rate is stationary by examining the cointegrating vector (0, 0, 1, 0,-1). (5)

Given that stock prices are determined solely by expectations of future real dividend growth, real interest rates, and excess returns (i.e., no bubbles), then the particular structure of the cointegrating vectors provides some insight as to the economic interpretation of stochastic trends in the system. Consider our five- variable system that includes log price-dividend ratio, real dividend growth, short-and long-term nominal interest rates, and inflation. For example, if there are two cointegrating vectors, this implies that there are three stochastic trends in our system. If the cointegrating vectors correspond to stationary excess stock returns and term premium, then the three stochastic trends would correspond to stochastic trends in real dividend growth, real interest rate, and inflation, if, on the other hand, real dividend growth and the term premium are stationary, then stochastic trends are present in excess returns, real interest rate, and inflation. (6)

4. Empirical Results

The data employed in this paper are quarterly and cover the period from 1953:II to 2001:IV. The price-dividend ratio is the S&P 500 composite stock price index for the last month of each quarter divided by nominal dividend flow for the S&P 500 composite index over the previous year.7 In our empirical analysis, we will consider the ten-year Treasury bond rate and the three-month Treasury bill yield as our interest rate series. Inflation is calculated as the growth in the Consumer Price Index over the quarter. Real dividend growth is nominal dividend growth less CPI inflation.

Tests for Number of Cointegrating Vectors

We test for cointegration in order to help specify the vector error correction model. Table 1 presents the Johansen result during various sample periods for the system that includes log price-dividends, real dividend growth, short- and long-term interest rates, and inflation. (8) Table 2 presents Johansen test results when we replace real dividend growth in the system with log-linear approximation for excess returns (Eqn. 10). (9) In Tables 1 and 2, column 1 shows the ending date of the sample period, column 2 reports the number of cointegrating vectors that cannot be rejected according to the Johansen (1991) lambda-max test, while column 3 of Table 1 reports the results of the Johansen (1991) trace test. The lambda-max test and trace test do not always agree on the order of cointegration; the lambda-max test generally finds one or two cointegrating vectors while the trace test generally finds two to three cointegrating vectors. However, as the sample period lengthens, evidence for two cointegrating vectors increases. (10)

We next test which of the alternative cointegrating vectors discussed above appear to be consistent with the data. Given that there appear to be two cointegrating vectors, we consider pairs of cointegrating vectors. Column 4 of Tables 1 and 2 reports the chi-squared statistic and p-value for the joint restriction that excess stock returns and term premium are stationary. Column 5 reports the chi-squared statistic and p-value for the joint restriction that real dividend growth and the term premium are stationary. For many of the sample periods, we fail to reject both sets of cointegrating relationships. (11) The likelihood functions for these two models are quite close, which suggests that the data does not strongly favor one model over the other.

Tests Taking Cointegrating Vectors as Known

As previously suggested, we can write several alternative characterizations of stochastic trends among market fundamentals in terms of specific restrictions on known cointegrating vectors. Thus, in addition to the Johansen analysis, we can directly test and evaluate competing assumptions about long-run market fundamentals. This is done by testing restrictions on the VECM, as shown in Horvath and Watson (1995). For this purpose, let [y.sub.t] = ([p.sub.t], [d.sub.t], [i.sub.t], [l.sub.t], [[pi].sub.t])', where [p.sub.t] is log price-dividend ratio, [d.sub.t] is real dividend growth, [i.sub.t] is the short-term nominal interest rate, [l.sub.t] is the long-term nominal interest rate, and [[pi].sub.t] is the inflation rate. Consider the error correction model (suppressing the intercepts): (12)

[DELTA][y.sub.t] = C(L)[DELTA][y.sub.t-1] + [alpha][beta]'[y.sub.t] + [u.sub.t].

Suppose we wanted to test the null hypothesis of stationary excess returns and stationary term premium but nonstationary real dividend growth, real interest rate, and inflation against the alternative hypothesis of nonstationary inflation, stationary real rate, excess returns, real dividend growth, and term premium, similar to the VAR examined by Campbell and Ammer (1993). The alternative hypothesis implies a vector error correction model with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Note that the first cointegrating vector in [beta]' is just a representation for stationary excess returns. For the model with nonstationary real dividend growth, real interest rates, and inflation and stationary term premium and stationary excess returns, we can express the VECM as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

This model is a special case of Campbell and Ammer VECM where the third and fourth columns of error correction terms are zero.

To test the null hypothesis of stationary excess returns and term premium, nonstationary real dividend growth, real interest rate, and inflation against the alternative hypotheses, we simply test [[alpha].sup.CA.sub.j3] = [[alpha].sup.CA.sub.j4] = 0, j = 1, ..., 5. (13) In essence, we are testing the joint hypothesis of nonstationary real dividend growth and nonstationary real interest rate in this vector system. Similarly, if we wished to test the null hypothesis of stationary real dividend growth and term premium and nonstationary excess returns, real interest rate, and inflation against the alternative hypothesis of the Campbell and Ammer specification, we test [[alpha].sup.CA.sub.j1] = [[alpha].sup.CA.sub.j4] = 0, j = 1,..., 5. Other null and alternative hypotheses can likewise be examined by writing the alternative as an error correction model and the null as restrictions on the error correction terms ([alpha]'s). Because of the relatively large number of parameters in the model for our sample size, rather than using the asymptotic tabulated critical values in Horvath and Watson, we present p-values based on an empirical bootstrap of the null model in order to conduct inference. (14)

Table 3 presents tests of competing hypotheses about the stationarity or nonstationarity of various market fundamentals. The tests presented in Table 3 are consistent with those of the Johansen analysis. These results of these tests show that there is evidence for three stochastic trends (and two cointegrating vectors) in our system. We fail to reject at the 0.05 level the null hypothesis of nonstationary real dividend growth, real interest rate, and inflation and stationary excess returns and term premium versus both the Campbell and Ammer model and a model with stationary real dividend growth, excess returns, and term premium (see tests 1 and 2). When we consider a null hypothesis in which excess returns, real interest rate, and inflation were nonstationary but real dividend growth and term premium were stationary, we could not reject this hypothesis against either alternative at the 0.05 level (see tests 4 and 5). On the other hand, we can reject the hypothesis of four nonstationary variables (real dividend growth, excess returns, real interest rates, and inflation) in favor of a model in which excess returns were stationary (test 3). We can likewise reject the hypothesis of nonstationary real dividend growth, excess returns, real interest rate, and inflation in favor of a model in which real dividend growth is stationary (test 6). (15)

Tables 1-3 suggest that when considering the entire system as described by our five-variable VECM/VAR, there is substantial evidence to conclude that either real dividend growth or excess returns is nonstationary, but apparently not both. This is in contrast to the previous literature, which assumed real dividend growth and excess returns to be stationary and is at odds with standard univariate unit-root tests. (16) Unfortunately, the data is not conclusive about whether it is real dividend growth or excess returns that is nonstationary. Assuming normally distributed shocks, the likelihood functions for the two models are very similar, with a slight advantage in favor of the nonstationary real dividend growth model. However, as we show in the next section, the decompositions derived from the VECM/VAR hinge on which variable is assumed to be nonstationary. (17)

5. Stock Price Decompositions

We now decompose historical stock price movements into contributions due to expectations of future real dividend growth, real interest rates, and excess returns using the companion form of the estimated VECM model and Equation 8. Recall that for the system with real dividend growth, the contributions of real dividend growth and real interest rates are calculated directly, while the contribution of excess returns is a residual.

Figures 3-5 display the actual (demeaned) log price-dividend ratio and the implied contribution of expectations of future real dividend growth, future excess returns, and future real interest rates. Figure 3 displays the contributions for the model that assumes nonstationary dividend growth, real interest rate, and inflation and stationary excess returns and term premium. Panel A displays the contribution of expectations of future real dividend growth and the contribution of expectations of future excess returns, while Panel B displays the contribution of expectations of future real interest rates. What is striking about Figure 3, relative to what might be expected given the previous literature, is the large contribution of real dividend growth; the contribution of excess returns is substantially smaller than either real dividend growth or real interest rates. In addition, there is a large negative correlation between the contribution of dividend growth and the contribution of real interest rates. (18) Also, there are some interesting interpretations of historical stock price movements. The decomposition suggests that the decline in stock prices in the 1970s was due primarily to pessimism about future dividends (although the decline was mitigated to some extent by the decline in real interest rates that occurred during this period), while the run up in stock prices in the late 1990s was driven primarily by optimism about future dividends. This particular decomposition could be consistent with the explanation of Greenwood and Jovanovic (1999) and Hobijn and Jovanovic (2001), who argue that the so-called "information technology revolution" hurt existing "old technology" firms as long ago as the 1970s, two decades before the new technology firms began to prosper.

[FIGURES 3-5 OMITTED]

However, if we alter the specification of the VECM and assume that excess returns are nonstationary while real dividend growth is stationary, the implied contribution of real dividend growth and excess returns changes dramatically. Figure 4 displays contributions for the model that assumes nonstationary excess returns and stationary real dividend growth. A comparison of Figures 3 and 4 demonstrates just how sensitive stock price decompositions are to specification of the underlying VECM. In the model with nonstationary excess returns but stationary real dividend growth, expectations of future excess returns rather than real dividend growth are important. In fact, it is almost as if the contributions of excess returns and real dividend growth shown in Figure 3 are switched in Figure 4. For the model with nonstationary excess returns and stationary real dividend growth, the contribution of expectations of future real interest rates (Figure 4b) is very similar to that for the model with nonstationary real dividend growth and stationary excess returns.

When we use a Campbell-Ammer-type specification for the VECM, stock price movements are nearly entirely driven by changes in expectations of excess returns (see Figure 5). Neither real dividend growth nor real interest rates has a substantial effect on log price-dividend movements in this specification. Recall, however, that we could not reject the specifications underlying Figures 3 and 4 in favor of the Campbell-Ammer specification.

In sum, the nature of stock price decompositions is very sensitive to the specification of the VECM used to model the data and to calculate future expectations of market fundamentals. Depending on which set of restrictions is imposed on the cointegrating vectors, the relative contributions of real dividend and excess returns can change dramatically. The implication is that previous findings that excess returns and not dividends explain most of the stock price variability are not robust to statistically plausible alternative specifications of the data- generating process.

6. Interpretation

As was previously stated, when the cointegrating vectors are restricted to imply stationary excess returns and term premium, these same restrictions suggest stochastic trends are present in real interest rates, real dividend growth, and inflation. Similarly, a system in which real dividend growth and term premium are stationary implies stochastic trends in excess returns, real interest rates, and inflation. The possibility of persistent changes in real dividend growth and, to a lesser extent, excess returns is at odds with much of the previous literature. In this section, we try to reconcile our research with these findings.

Because permanent movements in market fundamentals, in particular real dividend growth or excess returns, are potentially very important contributors to historical stock price movements, we use a multivariate version of the Beveridge and Nelson (1981) decomposition to infer a long-run or permanent real dividend growth (or excess returns) series. The Beveridge-Nelson "trend" or long-run value of variable [x.sub.t], is just [x.sup.[tau]] = [lim.sub.k[right arrow][infinity]] E[[x.sub.t+k]|[[OMEGA].sub.t]], where [[OMEGA].sub.t] denotes information set at time t. Using the companion form of the VECM/VAR, the Beveridge-Nelson trend for real dividend growth is just:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (12)

while for excess returns the permanent component is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (13)

where [H.sub.p], [H.sub.d], [H.sub.i], and [H.sub.[pi]] are (1x5 m) vectors that select the current values of [p.sub.t], [d.sub.t], [i.sub.t], and [[pi].sub.t], respectively.

Figure 6 displays the implied long-run real dividend growth series from the model with nonstationary real dividend growth and stationary excess returns (note that for this model the long-run excess returns are just a constant). The implied long-run real dividend growth series is well within historical variation of actual real dividend growth. The variance of innovations in long-run real dividend growth is substantially smaller than the variance of innovations in actual real dividend growth (3.97 x [10.sup.-2] vs. 1.52), which suggests that the quarterly changes in long-run real dividend growth are relatively small. (19) The implied long-run real dividend growth also seems plausible given historical movements in real dividend growth. In the 1970s, actual real dividend growth was substantially lower than in the previous decade. This was in part reflected (albeit with a lag) in a decline in the implied long-run real dividend growth series. Beginning in 1982, implied long-run real dividend growth increased and was followed later in the decade by an increase in actual real dividend growth. In the mid 1990s, both the implied long-run real dividend growth and actual real dividend growth increased. In 2000-2001, we did see divergence between actual real dividend growth and implied long-run real dividend growth, with actual real dividend growth falling substantially while implied long-run real dividend growth remained relatively high.

[FIGURE 6 OMITTED]

Figure 7 displays the implied long-run excess returns from the model with nonstationary excess returns and stationary real dividend growth (for this model, long-run real dividend growth is a constant). Again, long-run excess returns are well within the historical variation of actual excess returns and are substantially less volatile (the variance of innovations in long-run excess returns are 4.12 x [10.sup.-2], while the variance of innovations in actual excess returns is 50.80). In fact, actual excess returns are so volatile that their movements dwarf those of the implied long-run excess returns series. A relatively small decline in long-run (future) excess returns is consistent with high, but temporary, actual (current) excess returns. The stock price decomposition based on the VECM with nonstationary excess returns suggests that the high excess stock returns of the late 1990s resulted from a relatively small decline in future excess returns (a decline in the equity premium).

[FIGURE 7 OMITTED]

As we suggested earlier, neither real dividend growth nor excess returns shows much persistence. When examined in a univariate context, standard tests reject the unit-root hypothesis for both real dividend growth and excess returns. Standard augmented (with five lags) Dickey-Fuller t-statistics are -3.58 for real dividend growth and -6.26 for excess returns; both reject the unit-root null at conventional significance levels. The fact that innovations in the implied permanent components of real dividend growth and excess returns are several times smaller than innovations in the actual series may explain why standard univariate unit-root tests strongly reject at conventional levels. These small variances for innovations in long-run components suggest that large sample periods are likely to be needed to detect unit-roots in this data. This conjecture is, in fact, borne out when we apply standard Dickey-Fuller tests to simulated real dividend growth data based on the estimated VECM with nonstationary real dividend growth (but stationary excess returns). A Dickey-Fuller t-statistic of -3.58 has a bootstrap p-value of 0.265 (the 0.05 critical value for the augmented Dickey-Fuller t-statistics is -4.38). This suggests that we would actually fail to reject the unit-root null if the appropriate finite sample critical values were used. Similarly, for the VECM in which excess returns were nonstationary and real dividend growth was stationary, a Dickey-Fuller t-statistic of -6.26 for excess returns has a bootstrap p-value of 0.192 (the 0.05 critical value for the augmented Dickey-Fuller t-statistic on excess returns is -6.89). Again, we would fail to reject the unit-root null if the appropriate finite sample critical values were used.

Recall that innovations in the implied long-run component of real dividend growth and excess returns are so small, relative to innovations in the actual series themselves, that actual real dividend growth and excess returns may have relatively little information about low-frequency movements in these series. It is, in fact, the log price-dividend ratio that contains most of the information about long-run real dividends or excess returns. Because stock prices depend on expectations of future real dividends, real interest rates, and excess returns, persistent innovations in these variables result in large changes in current stock prices. Thus, small permanent changes in market fundamentals can have relatively large effects on the log price--dividend ratio.

To illustrate this idea, suppose [d.sub.t], [r.sub.t], and [e.sub.t] are described by an unobserved components model with a permanent or trend component and a stationary component,

[x.sub.t] = [x.sup.[tau].sub.t] + [x.sup.c.sub.t], x = d, r, and e (14)

with [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (15)

and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

where [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are white-noise error terms. When we evaluate Equation 2 using Equations 14-16, we find

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] (17)

A small, permanent change in a market fundamental ([d.sup.[tau].sub.t], [r.sup.[tau].sub.t], [e.sup.[tau].sub.t]) can cause a large change in log price-dividend ratio; in our data the term 1/(1 - [rho]) has a value of 124.2. On the other hand, a temporary change in a market fundamental may have a substantially smaller effect on stock prices. For example, when theta is equal to 0.8, the effect on [p.sub.t] is just 3.88. Thus, a permanent change in market fundamentals that is barely reflected in their current value may nonetheless have an important effect on the price-dividend ratio.

To further demonstrate that log price-dividend contains most of the information about long- run real dividend and/or long-run excess returns, Figure 8 overlays long-run real dividend growth, [d.sup.[tau].sub.t] implied by the model with nonstationary real dividend growth (but stationary excess returns) with the negative of long-run excess returns, -[e.sup.[tau].sub.t], from the model with nonstationary excess returns (but stationary real dividend growth). We observe that these two series are nearly identical. The figure suggests that most of the information about long-run real dividend growth or excess returns is coming from the log price-dividend series and not directly from real dividend growth or excess returns.

[FIGURE 8 OMITTED]

As we noted above, the VECM with nonstationary real dividend growth and stationary excess returns and the VECM with stationary real dividend growth and nonstationary excess returns have very similar likelihood values; they appear to explain the data equally well. The reason is that stock prices contain almost all of the information about long-run real dividend growth or long-run excess returns. It is not possible to determine whether real dividend growth or excess returns is responsible for the low-frequency movements in stock prices; it is similar to having only one equation (log price/dividend ratio) with which to solve for two unknowns (long-run real dividend growth and excess returns).

7. Power Properties of the Cointegration Tests

It is well known that unit-root and tests with a null of no cointegration can have low power against persistent stationary alternatives. Could our findings of three stochastic trends be the result of poor power properties of the tests we employ? As Horvath and Watson pointed out, an advantage of the multivariate or systems approach to testing for cointegration is that, by adding variables that covary with the variable of interest, one can increase the power of the test. But this is at the cost of adding additional parameters, which tends to lower the power of the test. It is not clear which effect dominates.

In order to determine the size and power properties of the tests we employed, we conduct a small Monte Carlo experiment. First, we use actual data to estimate a maintained model, which is then assumed to be the true data-generating process in the subsequent Monte Carlo experiment. This datagenerating process is used to generate pseudo-data (by resampling actual residuals) to which we apply the test for (no) cointegration. In these tests, we set the null hypothesis to be either the VECM with nonstationary real dividend growth and stationary excess returns or the VECM with stationary real dividend growth and nonstationary excess returns. The alternative hypothesis in each case is the VECM with stationary real dividend growth and stationary excess returns. We generate 500 pseudodata samples, each time using a bootstrap (of the pseudo-data and null model) to make statistical inferences, and count the percentage of times the null model is rejected. If the data-generating process corresponds to the null model, then our experiment examines the size of the test. If the data-generating process corresponds to the alternative model, then our experiment examines the power of the test. For comparison, we also examine a finite sample Dickey-Fuller test whose critical values are also based on a bootstrap of the null model.

From Tables 4 and 5, we observe that the multivariate test has quite reasonable power properties against an alternative of stationary real dividend growth, excess returns, and term premium, certainly compared to a bootstrap Dickey-Fuller test. When the data-generating process implies stationary real dividend growth and excess returns, regardless of whether the null is nonstationary real dividend growth or excess returns, at a 0.05 significance level we correctly reject the null more than 75% of the time. This is substantially more frequently than when we use the bootstrap Dickey-Fuller test. Furthermore, and perhaps not surprisingly, the nominal size of the bootstrap inference appears to be close to the actual size (although the Dickey-Fuller statistics tend to reject slightly less often than the nominal size of the test; see Table 5). Thus, our finding that in our system there appear to be only two cointegrating vectors rather than three is not solely the result of the poor power properties of our tests. (20)

8. Could Our Results Be Due to the Presence of a Rational Bubble?

What we have shown thus far is that there are some statistically plausible market fundamentals representations that explain low- frequency stock price movements. As we argued above, stock prices may have more information about long-run movements in market fundamentals than the market fundamentals themselves. Nonetheless, could our results reflect non-fundamental behavior such as irrational exuberance or a rational bubble? Indeed, the fact that stock prices are very persistent, while real dividend growth and excess returns are apparently stationary, could be interpreted as evidence of a bubble (see Hamilton and Whiteman 1985; Diba and Grossman 1988).

To see how the presence of a rational bubble would affect our analysis, consider the expectational log-linear approximation given by Equation 1. Let the market fundamentals solution, given by Equation 2, be denoted by [f.sub.t]. Then a solution to Equation 1 can be characterized by:

[p.sub.t] = [f.sub.t] + [b.sub.t], (18)

where [b.sub.t] is a stochastic process that satisfies the equation [E.sub.t][b.sub.t+1] = (1/[rho])[b.sub.t]. The [b.sub.t] is similar to a standard stock market bubble except that it holds for the log-linear approximation rather than the standard present-value model. (21) This so-called "bubble" term would imply a nonstationarity log price-dividend ratio even if market fundamentals were stationary.

How would the presence of this bubble affect our analysis above? In the presence of a bubble, actual excess returns are given by

[e.sub.t] = [rho][p.sub.t] - [p.sub.t-1] + [d.sub.t] - [i.sub.t-1] + [[pi].sub.t] + k = [rho][f.sub.t] - [f.sub.t-1] + [d.sub.t] - [i.sub.t-1] + [[pi].sub.t] + k + [v.sup.b.sub.t], (19)

Where [v.sup.b.sub.t] is unpredictable random shock to the "bubble" term. Thus, the presence of a bubble will not manifest itself as explosive actual excess returns. However, when we tested for stationarity of excess returns in our system, we tested the stationarity of

([rho] - 1)[p.sub.t] + [d.sub.t] - [i.sub.t] + [[pi].sub.t] = ([rho] - 1)[f.sub.t] + [d.sub.t] - [i.sub.t] + [[pi].sub.t] + ([rho] - 1)[b.sub.t]. (20)

This term does in fact depend on the "bubble." Thus, the presence of a bubble, in addition to implying nonstationarity of [p.sub.t], would imply that our test would fail to reject nonstationary excess returns.

We have two possible explanations to these findings. While our model could not reject the null hypothesis of nonstationary excess returns and stationary real dividend growth, neither could our model reject the null hypothesis of nonstationary real dividend growth and stationary excess returns; the second hypothesis is inconsistent with the presence of a "bubble." Second, the "bubble" in log price-dividend implies greater testable restrictions than only the nonstationarity of [p.sub.t]--it implies that [DELTA][p.sub.t] is nonstationary as well. (22)

We test for the nonstationarity of [DELTA][p.sub.t] in the systems context as we did in section 4. Consider the VECM implied by the market fundamentals model in which real dividend growth and term premium were stationary, but log price-dividend (and excess returns), real interest, and inflation were nonstationary:

[DELTA][y.sub.t] = C(L)[DELTA][y.sub.t] 1 + [alpha][beta]'[y.sub.t-1] + [u.sub.t],

with [y.sub.t] = ([p.sub.t], [d.sub.t], [i.sub.t], [l.sub.t], [[pi].sub.t])' and [beta]' = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. Because the level of the log price--dividend ratio, [p.sub.t], does not appear in any of the cointegrating vectors, we can rewrite this VECM in terms of [DELTA][p.sub.t]:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

with [[??].sub.t] = ([DELTA][p.sub.t], [d.sub.t], [i.sub.t], [l.sub.t], [[pi].sub.t])' and [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. To test the null hypothesis of nonstationarity of [DELTA][p.sub.t], we simply test whether the elements of the first column of [??] are zeros. (23) We can in fact strongly reject this null hypothesis. The Wald statistic is 64.26 with a p-value of 0.000.

Strictly speaking, the VECM testing approach of Horvath and Watson is a two- sided test while test of stationarity is really a one-sided test (Horvath and Watson 1995). This raises the possibility that we could reject the null hypothesis that [[??].sub.j1] = 0 for j = 1, ..., 5 simply because the coefficient on [DELTA][p.sub.t] in the [[DELTA].sup.2][p.sub.t] equation, [[??].sub.11], is positive, as would be the case if a bubble were present. However, it is unlikely that this is the case in our application. The coefficient [[??].sub.11] is estimated to be -1.048 (with a t-statistic of -7.55), so the rejection is not due to [[??].sub.11] being positive. (24) Overall, this evidence does not support the presence of a bubble in the log-linear approach taken in this paper. Of course, these results do not rule out the possibility that other non-fundamental factors, such as irrational exuberance, herd behavior, or fads, play a role in stock price variation, but it is not clear what testable restrictions these types of models would have for our data.

9. Concluding Comments

This paper argues that the data do not speak strongly to the determinants of low-frequency movements in stock prices. Our results for VAR-type decompositions are similar to those found in some of our previous work, in which we used an entirely different methodology (Balke and Wohar 2002). In that work, we employed a state-space model to model the dynamics of the log price-dividend ratio along with long-term and short-term interest rates, real dividend growth, and inflation. We showed that the decompositions of stock price movements are very sensitive to what assumptions are made about the presence of permanent changes in either real dividend growth or excess stock returns. When we allowed real dividend growth to have a permanent component but excess stock returns only to have a transitory component, real dividend growth was found to explain much more of the movement in stock prices than did excess stock returns. When we reversed this assumption, the relative contributions of excess stock returns and real dividend growth are also reversed. The results in the current paper suggest that the sensitivity of stock market decompositions is present in VAR decompositions as well.

The possibility that market expectations about changes in future real dividend growth may be a more important determinant of stock prices than they have typically been ascribed in the literature has been recognized in some other studies. For example, Barsky and DeLong (1993) show that large swings in the stock market could be rationalized if market participants believe that permanent changes in dividend growth are possible. They went on to show that an ARIMA(0,1,1) model for dividend growth with a large negative moving average term can explain both dividend growth and stock prices. In a more recent paper, Timmerman (2001) proposes that structural breaks in the underlying dividend process, about which investors have only imperfect information, can explain stock price movements. In the time periods immediately following a structural break in the dividend process, investors cannot rely on historical data to arrive at a new revised estimate of mean dividend growth and instead gradually update their beliefs as new information arises. Timmerman argues that his model can explain several stock price (ir)regularities, including skewness, excess kurtosis, volatility clustering, and serial correlation in stock returns.

One puzzle that arises from our analysis (both in this paper and that in the state-space approach of Balke and Wohar 2002) is that, from a statistical point of view, persistent changes in real dividend growth and excess returns are equally likely. In order to identify the relative importance of real dividend growth or excess stock returns for stock price variability, one is likely to need additional information beyond that of stock price, real dividend growth (or excess returns), and interest rate data typically used in the stock price decomposition literature. For example, information on relative transactions costs and their effect on investors' asset allocations (Heaton and Lucas 1999), information about the underlying determinants of a time- varying equity premium (Campbell and Cochrane 1999), or indicators of long-run economic growth might be helpful in distinguishing between changes in expectations of future real dividend growth and excess returns. Alternatively, one might attempt to tie real interest rate and risk premium movements not only to the level of assets prices as done in Balke and Wohar (2002) but also to movement in the covariance structure of asset prices. Finally, one might formally incorporate prior information about the relative importance of real dividend growth and excess returns by taking a Bayesian approach to stock price decompositions.

The authors would like to thank David Rapach, John Duca, Tom Fomby, and two anonymous referees for helpful comments on a previous draft of this paper. The views expressed in this paper are those of the authors and do not necessarily reflect the views of the Federal Reserve Bank of Dallas or those of the Federal Reserve System.

Received March 2004; accepted October 2005.

References

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(1) Cochrane (1991, 1992), Epstein and Zin (1991), and Timmerman (1995) have argued that fluctuations in stock prices can be explained by time-varying discount rates and future excess returns. Other studies (e.g., Marsh and Merton [1987], Lee [1996, 1998], and Bulkley and Harris [1997]) found that expectations of future earnings contribute more to fluctuations in stock prices. On the existence of bubbles or fads, see West (1988b) and Flood (1990).

(2) Cochrane (1992), by using an alternative methodology to decompose the variability of stock prices, also found the variability of excess returns to be more important than the variability of dividend growth.

(3) We computed augmented Dickey-Fuller t-statistics for lag lengths from 1 to 12 (with no time trend). The t-statistics range from -1.23 to 0.42, none of which are significant at conventional significance levels.

(4) If we include excess returns in the VAR (as in Campbell and Ammer 1993) rather than dividend growth, then the contribution of dividend growth is a residual. It does not substantially affect the results if the model is estimated with excess returns instead of dividend growth.

(5) We can similarly derive restrictions for stationary excess returns, stationary real dividend growth, and stationary term premium for a system that includes excess returns rather than dividend growth ([p.sub.t], [e.sub.t], [i.sub.t], [l.sub.t], [[PI].sub.t]).

(6) Campbell and Ammer (1993) in their analysis treat log(p/d), real interest rate, excess returns, and the interest rate spread as stationary series; only nominal interest rates were treated as nonstationary. For the five-variable system that we examine, the tour cointegrating vectors implied by Campbell and Ammer's (1993) model are: (1, 0, 0, 0, 0), or stationary log price-dividend ratio; (0, 0, l, 0, 1), or stationary real interest rate; (0, 0, 1, 1,0), or stationary term premium; and (0, 1, 0, 0, 0), or stationary real dividend growth. Note that these cointegrating vectors also imply stationary excess returns.

(7) For dividends, we use the end-of-quarter S&P 500 dividend yield multiplied by the end-of-quarter S&P 500 composite index. We convert the price-dividend ratio and dividends to quarterly flows by dividing dividends by four.

(8) Note that we calculate a new value of P for each sample period.

(9) An intercept is included in the VAR/VECM; however, in the Johansen analysis, this is restricted to be in the so-called equilibrium error (the intercept, no drift case). The number of lags in the VAR is set at four. This is the number of lags selected if one sequentially adds lags until the additional lag is not statistically significant. The Akaike Information Criterion chooses two lags for our system, but there is substantial residual serial correlation remaining in some of the equations (we use LM-test for serial correlation of order four). With four lags, only one of the equations displays significant serial correlation. We consider four lags to be a good compromise between parsimony and adequately capturing the dynamics in the data. The results are essentially unchanged if we increase the number of lags to six, but fewer lags tend to suggest fewer than three stochastic trends.

(10) These results are consistent with those of Goyal and Welch (2003), who found that the log price-dividend ratio has become more persistent in recent time periods.

(11) For all sample periods, we reject restrictions on the cointegrating vectors that correspond to (i) the joint hypothesis of stationary excess returns and stationary real dividend growth and (ii) the joint hypothesis of a stationary real interest and a stationary term premium.

(12) As in the aforementioned Johansen analysis, we consider the case in which there is an intercept in the VECM but no drift in the stochastic trends when conducting inference. However, like Horvath and Watson, we estimate the VECM without restrictions.

(13) Because the cointegrating vectors are assumed to be known, there is no problem of having parameters that are not identified under the null of no cointegration.

(14) The empirical bootstrap was based on the estimated null model, which is then used to generate pseudo-data by resampling vectors of empirical residuals. The pseudo-data are generated so that there is no drift in the stochastic trends. The alternative model is then estimated by using the pseudo-data, and a Wald statistic is calculated for the null hypothesis such that the relevant error-correction parameters are zero. A distribution of sample Wald statistics was generated by conducting the aforementioned experiment 10,000 times.

(15) The test results are essentially unchanged if we replace real dividend growth with excess returns in the system.

(16) An exception is Barsky and DeLong (1993).

(17) Strictly speaking, the Campbell-Shiller approximation (in which the value of p is a function of the sample average of the log price-dividend ratio) only holds if the log price-dividend ratio is stationary. However, our results are essentially unchanged when we use the minimum log price-dividend ratio over the sample to calculate p and when we use the maximum log price-dividend ratio over the sample. Thus, our results do not appear to be very sensitive to reasonable values of p.

(18) Recall that an increase in future real dividend growth has a positive effect on current stock prices, whereas an increase in future real interest rates has a negative effect. This suggests that the underlying correlation between future real dividend growth and future real interest rates is positive. Note that a positive correlation between real dividend growth and real interest rate is consistent with a consumption-based asset pricing model with diminishing marginal utility.

(19) The variance of innovations to long-term real dividend growth is [H.sub.D] [A.sub.LR][OMEGA][A'.sub.LR][H'.sub.D], where [A.sub.LR] = [lim.sub.k[right arrow][infinity]] [A.sup.k] and [OMEGA] is variance/ covariance matrix of residuals from the estimated VECM.

(20) At the suggestions of a referee, we also examined a smaller system consisting of ([p.sub.t], [d.sub.t], [i.sub.t] - [[PI].sub.t]). For this smaller system, we find evidence consistent with our larger system. The data appear to support the presence of a stochastic trend in either dividend growth or excess returns (hut not both) and a stochastic trend for the real interest rate.

(21) We continue to work with log-linear approximation because, with time varying returns, the basic present value equation is now a nonlinear difference equation, which makes examination of a rational bubble substantially more difficult.

(22) In fact, a bubble implies no amount of differencing, which will render [p.sub.t] stationary.

(23) More precisely, we test the null hypothesis of nonstationary Apt, nonstationary real interest rate, and inflation but stationary real dividend growth and term premium against the alternative hypothesis of stationary Apt (but nonstationary Pt and excess returns), nonstationary real interest rate and inflation, and stationary real dividend growth and term premium.

(24) When we use the univariate Dickey-Fuller test (a one-sided test of nonstationarity), we also reject strongly the null of nonstationarity in favor of stationary alternative.

Nathan S. Balke * and Mark E. Wohar ([dagger])

* Department of Economics, Southern Methodist University, Dallas, TX 75275, USA, and Research Department, Federal Reserve Bank of Dallas, Dallas, TX 75201, USA; E-mail [e-mail]; corresponding author.

([dagger]) Department of Economics, RH-512K, University of Nebraska at Omaha, Omaha, NE 68182, USA; E-mail [e-mail].

Table 1. Cointegration Results (Lag Length in Levels VAR = 4):
Real Dividend Growth Specification

Date on          Cointegrating     Cointegrating
which Sample         Vectors          Vectors
Period Ended      Determined by    Determined by
(Year:Quarter)   Lambda-Max Test    Trace Test

1987:IV                1 *             1 **
1990:IV                0               3 *
1995:IV                1 *             2 **
1999:IV                2 *             2 *
2000:IV                2 *             2 **
2001:IV                2 **            2 **

                    Chi-Square Test          Chi-Square Test
Date on              of Restriction         of Restriction of
which Sample         of Stationary           Stationary Real
Period Ended       Excess Returns and      Dividend Growth and
(Year:Quarter)   Term Premium (p-Value)   Term Premium (p-Value)

1987:1V             14.1105 (0.028)          17.2348 (0.008)
1990:1V             10.6893 (0.098)          13.9440 (0.030)
1995:1V              8.8732 (0.181)          12.2647 (0.056)
1999:1V              6.9910 (0.322)           9.4382 (0.150)
2000:1V             13.0324 (0.043)          12.0268 (0.061)
2001:1V             10.0794 (0.121)          11.2786 (0.080)

* 90% confidence level.

** 95% confidence level.

Table 2. Cointegration Results (Lag Length in Levels
VAR = 4): Excess Return Specification

Date on          Cointegrating     Cointegrating
which Sample         Vectors          Vectors
Period Ended      Determined by    Determined by
(Year:Quarter)   Lambda-Max Test    Trace Test

1987:IV               1 **            3 *
1990:IV               1 *             3 **
1995:IV               1 **            3 *
1999:IV               2 *             2 *
2000:IV               1 **            2 **
2001:IV               2 *             2 **

                                      Chi-Square Test
                  Chi-Square Test    of Restriction of
Date on          of Restriction of    Stationary Real
which Sample     Stationary Excess    Dividend Growth
Period Ended     Returns and Term        and Term
(Year:Quarter)   Premium (p-Value)   Premium (p-Value)

1987:IV           13.5078 (0.036)     13.6942 (0.033)
1990:IV            8.9164 (0.178)      9.9641 (0.126)
1995:IV            9.048 (0.171)      10.6195 (0.101)
1999:IV            6.9388 (0.327)      7.2588 (0.298)
2000:IV           13.3699 (0.038)     10.2782 (0.113)
2001:IV           10.2783 (0.113)      8.8277 (0.184)

* 90% confidence level.

** 95% confidence level.

Table 3. Test of Alternative Specifications of VECM: Real
Dividend Growth in System

                                                            Wald
Test    Null Hypothesis          Alternative Hypothesis   Statistic

1 (a)   Nonstationary:           Nonstationary:             23.13
          Real dividend growth     Inflation
          Real interest rate     Stationary:
          Inflation                Real dividend growth
        Stationary:                Real interest rate
          Excess returns           Excess returns
          Term premium             Term premium
2       Nonstationary:           Nonstationary:             14.57
          Real dividend growth     Real interest rate
          Real interest rate       Inflation
          Inflation              Stationary:
        Stationary:                Real dividend growth
          Excess returns           Excess returns
          Term premium             Term premium
3       Nonstationary:           Nonstationary:             28.96
          Real dividend growth     Real dividend growth
          Real interest rate       Real interest rate
          Inflation                Inflation
          Excess returns         Stationary:
        Stationary:                Excess returns
          Term premium             Term premium
4 (a)   Nonstationary:           Nonstationary:             24.91
          Excess returns           Inflation
          Real interest rate     Stationary:
          Inflation                Excess returns
        Stationary:                Real interest rate
          Real dividend growth     Real dividend growth
          Term premium             Term premium
5       Nonstationary:           Nonstationary:             15.97
          Excess returns           Excess returns
          Real interest rate       Real interest rate
          Inflation                Inflation
        Stationary:              Stationary:
          Real dividend growth     Real dividend growth
          Term premium             Term premium
6       Nonstationary:           Nonstationary:             27.48
          Excess returns           Excess returns
          Real interest rate       Real interest rate
          Inflation                Inflation
          Real dividend growth   Stationary:
        Stationary:                Real dividend growth
          Term premium             Term premium

Test    Null Hypothesis          Alternative Hypothesis    p-Value

1 (a)   Nonstationary:           Nonstationary:            0.265
          Real dividend growth     Inflation
          Real interest rate     Stationary:
          Inflation                Real dividend growth
        Stationary:                Real interest rate
          Excess returns           Excess returns
          Term premium             Term premium
2       Nonstationary:           Nonstationary:            0.107
          Real dividend growth     Real interest rate
          Real interest rate       Inflation
          Inflation              Stationary:
        Stationary:                Real dividend growth
          Excess returns           Excess returns
          Term premium             Term premium
3       Nonstationary:           Nonstationary:            0.001
          Real dividend growth     Real dividend growth
          Real interest rate       Real interest rate
          Inflation                Inflation
          Excess returns         Stationary:
        Stationary:                Excess returns
          Term premium             Term premium
4 (a)   Nonstationary:           Nonstationary:            0.176
          Excess returns           Inflation
          Real interest rate     Stationary:
          Inflation                Excess returns
        Stationary:                Real interest rate
          Real dividend growth     Real dividend growth
          Term premium             Term premium
5       Nonstationary:           Nonstationary:            0.074
          Excess returns           Excess returns
          Real interest rate       Real interest rate
          Inflation                Inflation
        Stationary:              Stationary:
          Real dividend growth     Real dividend growth
          Term premium             Term premium
6       Nonstationary:           Nonstationary:            0.001
          Excess returns           Excess returns
          Real interest rate       Real interest rate
          Inflation                Inflation
          Real dividend growth   Stationary:
        Stationary:                Real dividend growth
          Term premium             Term premium

(a) Alternative hypothesis from Campbell and Ammer (1993).

Table 4. Power and Size of Bootstrap Inference for Cointegration:
Examination of Power

                                                 Null Hypothesis
                                               Correctly Rejected (%)

                                                        Multivariate
                                                        Approach with
                                              Nominal     Bootstrap
Test   Null Hypothesis                        p-Value     Inference

1      Nonstationary: real dividend growth,    0.01         0.454
         real interest rate, inflation         0.05         0.770
       Stationary: excess returns,             0.10         0.896
         term premium

2      Nonstationary: excess returns,          0.01         0.534
         real interest rate, inflation         0.05         0.802
       Stationary: real dividend               0.10         0.910
         growth, term premium

                                               Null Hypothesis
                                              Correctly Rejected
                                                     (%)

                                                  Bootstrap
Test   Null Hypothesis                        Dickey-Fuller Test

1      Nonstationary: real dividend growth,         0.096
         real interest rate, inflation              0.428
       Stationary: excess returns,                  0.668
         term premium

2      Nonstationary: excess returns,               0.022
         real interest rate, inflation              0.116
       Stationary: real dividend                    0.202
         growth, term premium

Data-generating process (alternative hypothesis): stationary real
dividend growth, excess returns, term premium, nonstationary real
interest rate, and inflation.

Table 5. Power and Size of Bootstrap Inference for Cointegration:
Examination of Size

                                                 Null Hypothesis
                                              Incorrectly Rejected (%)

                                                         Multivariate
                                                           Approach
Test             Null Hypothesis              Nominal   with Bootstrap
                                              p-Value     Inference

1      Nonstationary: real dividend growth,    0.01         0.012
         real interest rate, inflation         0.05         0.050
       Stationary: excess returns,             0.10         0.078
         term premium

2      Nonstationary: excess returns,          0.01         0.016
         real interest rate, inflation         0.05         0.070
       Stationary: real dividend               0.10         0.112
         growth, term premium

                                               Null Hypothesis
                                                 Incorrectly
                                                 Rejected (%)

                                                  Bootstrap
Test             Null Hypothesis              Dickey-Fuller Test

1      Nonstationary: real dividend growth,         0.006
         real interest rate, inflation              0.018
       Stationary: excess returns,                  0.074
         term premium

2      Nonstationary: excess returns,               0.002
         real interest rate, inflation              0.018
       Stationary: real dividend                    0.048
         growth, term premium

Data-generating process: null hypothesis. For these tests, the
alternative hypothesis is stationary real dividend growth, excess
returns, and term premium with nonstationary real interest rate
and inflation.