Present value model between prices and dividends with constant and time-varying expected returns: enterprise-level Brazilian stock market evidence from non-stationary panels

The Present Value Model (PVM) – in which current security prices depend upon the present value of future discounted dividends, where the discount rate is equivalent to the required rate of return – is one of the long-standing principles of Finance Theory. The objective of this work is to analyze the validity of the PVM between prices and dividends at the firm level from panel techniques applied to non-stationary and potentially cointegrated processes for the Brazilian stock market. Considering the Present Value Model with Constant and Time-Varying Expected Returns, the evidence that real (log) prices and real (log) dividends are non-stationary I(1) and (log) price-dividend ratio is I(0) cannot be rejected. Regarding FMOLS and DOLS estimators for panel cointegration models, stock prices are found to be overvalued under either constant or time-varying expected returns assumption.


INTRODUCTION
esting expectations and rationality in financial markets, the Present Value Model (PVM) states that current security prices equals the summed discounted value of future dividends, where the discount rate is equivalent to the required rate of return. Scholars have displayed considerable interest in the underlying model due to macroeconomic events (historical collapses in stock prices) in which prices possibly deviated from their fundamental values (low dividends payouts and record high stock prices suggested stock price overvaluation); a theoretical and statistical debate on the possibility to forecast security prices (random walk, martingale properties); estimation and inference in the presence of nonstationarity (stochastic trend), particularly in the panel data framework.
Previous empirical analysis of the PVM and of the long-run relationship between prices and dividends is predominantly based upon two cointegration approaches. First, as in Campbell and Shiller (1987), real prices and real dividends should cointegrate, i.e. exhibit a stable long-run relationship. In this case, the cointegrating parameter provides an estimate of the inverse discount rate. Second, as in Campbell and Shiller (1988a,b), allowing for a time-varying discount rate, the difference between log dividends and log prices must exhibit (0) stationarity. Although cointegration tests do not reveal the existence of bubbles directly, the presence of cointegration can be explained by stock price deviation in vis-a-vis its fundamentals, which enables the indirect inference towards the existence of bubbles.
Assuming rational expectations (RE), the underlying model for stock prices has been tested since the decade of 1980 for U.S. data and many studies indicate that stock prices were more volatile than the PVM theoretically implied. Shiller (1981) developed a seminal work on the model assessment employing variance bounds tests, from which a theoretical and quantitative discussion has emerged, notably through the works of Grossman and Shiller (1981), LeRoy e Porter (1981), Marsh andMerton (1986), Shiller (1989), Scott (1990), Mankiw, Romer e Shapiro (1985, 1991, Gilles e LeRoy (1992), LeRoy and Parke (1992) as well as important statistical criticism in relation to small sample bias and finite sample considerations from Flavin (1983) and Kleidon (1986).
As stock prices assumed higher values due to the rapid development of the stock market throughout the decade of 1990, scholars and market analysts questioned the PVM BBR, Braz. Bus. Rev. (Engl. ed., Online) Vitória, v. 9, n. 4, Art. 3, p. 51-86, oct. -dec. 2012 www.bbronline.com.br validity and the effects of interest rates on the stock price-dividend relation. General consensus is that fundamentals have basically remained unchanged throughout the period considering the large rise in stock prices at the end of last century and their subsequent fall, which implies that the market became significantly overvalued and fundamentals subsequently reasserted themselves. Among the most prominent proponents of this view is Campbell and Shiller (2001). Nevertheless, an alternative view stated that stock prices reflected investors' permanently revised expectations of higher future earnings and dividends due to productivity gains originating from technological change.
The examination of individual firms is unusual, since most time-series studies adopt the S&P 500 index as the analysis benchmark. Thus, Shiller (1987, 1988), Lee (1995), Sung and Urrutia (1995), Timmermann (1995), and Crowder and Wohar (1998) estimate the present value relation on aggregate level over a significant length of time, in accordance to the concept stated by Stoja and Tucker (2004) that the power of unit root and cointegration tests is based on the length of time period rather than the number of observations. Meanwhile, it is recognized that the application of firm-level data allows for observation of patterns and relationships that may not be evidenced through stock market index analysis, since an index application may smooth noise or volatility from individual firms. Cohen, Polk and Vuolteenaho (2001), Vuolteenaho (2002) and Jung and Shiller (2005) suggest a greater likelihood for the PVM to be validated at the firm level rather than at the aggregate level (stock market index). As Jung and Shiller (2005) analyze, although information about cash flows and future prospects of individual companies are well understood by investors, the same degree of clarity may not apply to the market with respect to changes in the pattern of aggregate dividends or earnings flow in a country's stock market. Recent works examining the validity of the PVM at the aggregate stock market index level are those of Campbell and Shiller (1987), Campbell and Shiller (1988a,b), Fama and French (1988), Cecchetti, Lam and Mark (1990), Timmermann (1995), Kim, Morley and Nelson (2001), Dupuis and Tessier (2003), Manzan (2004), Su, Chang and Chen (2007). Some Brazilian stock market empirical findings applying similar methodology as Campbell e Shiller (1987, 1988a were obtained by Anchite and Issler (2001) and Morales (2006). Considering these latter authors, whereas the PVM with timevarying returns is not statistically rejected, evidence points either towards a rejection or failure to reject the model, though with weaker statistical significance.
Concerning the few empirical works testing the validity of the PVM at the firm level, Nasseh and Strauss (2004), Goddard, McMillan and Wilson (2008) employ U.S. and U.K. data, respectively, in order to assess the underlying model under time-varying returns, and point out that panel methods are particularly useful when the available time period is relatively short, providing a gain in power precision and avoiding structural shifts in the data that occur over longer time periods. These authors found that the examination of the rational valuation formula at the firm level appears to be somewhat more supportive of the present value model than previous studies based on aggregated stock price and dividend index data.
Thus, the following research question is posed in this paper: In Brazil, is there a stable long-run relationship between the present value of an asset (real prices) and its respective discounted earnings (real dividends), at the microeconomic level (firm level), in order to validate the Present Value Model and, therefore, expectations and rationality of economic agents in the financial market, through first generation unit root and cointegration tests as well as dynamic panel techniques?
The remainder of the paper is structured as follows. In Section 2, the Present Value Model is briefly discussed. Section 3 provides technical details of the panel unit roots and cointegration tests adopted. Section 4 reports and interprets the results of these tests.
Section 5 summarizes and concludes.

THE PRESENT VALUE MODEL
The PVM relates the present value of expected dividends and the stock price under the implied condition of RE as follows:  Campbell and Shiller (1987) demonstrated that, under the transversality condition, there will be only one possible price in order to exclude the presence of bubbles and, therefore, the possibility of many solutions to the price equation above. Assuming the validity of the model under this assumption, Campbell and Shiller (1987), showed that prices and real dividends are cointegrated and the cointegration vector equals to (1, ), as the following equation below: The methodology employed by Campbell and Shiller (1988), in order to circumvent the nonstationarity of the price and dividends series and hence admit the possibility of time-varying discount rates, suggests the logarithmic transformation of the variables [ = ln( ) ; = ln( ) ; = ln(1 + )] as follows: where ( = − ln(%) + (1 − %)ln (% − 1) and % = 1/[exp( − )] . Again, under the transversality condition, the above equation can be rewritten as: If the variation in the dividend and discount rate is stationary, the spread (ℎ = ln( ⁄ ) = − ) will be stationary and therefore the logarithms of prices and dividends will be cointegrated with the vector equal to (1, −1). Therefore, it would be sufficient to prove that the spread is (0) in order to validate the present value model. The spread and variation in dividends can be modeled with an Autoregressive Vector, with the restriction that returns are unpredictable (ℎ , /ℎ , ∆ ) = 0. As a result, the spread should Granger cause the dividends.
Basically, three types of criticism can be inferred regarding the procedure above.
First, as Froot and Obstfeld (1991) note, it would be to oppose empirical evidence contrary to the results above; second, according to Evans (1991), it would be to examine the assumptions of the model and verify, for instance, the existence of bubbles; and the third, as Gil-Alana (1999) and Caporale and Cerrato (2004) observe, would be the adequacy of the tests in relation to facts such as mean reversion.
Long processes of mean reversion and persistent shocks imply fraction order of cointegration, making traditional test results inconclusive, although it remains a long-term equilibrium relationship between prices and dividends. It is worth stating that the present value model is not incompatible with the existence of bubbles and mean reversion. Hence, the econometric approach should be modified to consider these facts.

NON-STATIONARY PANEL ECONOMETRIC PROCEDURES
Until recently, panel data investigation did not have available the crucial stationarity (ADF and Phillips-Perron) and cointegration (Engle-Granger and Johansen) tests, which has been motivated by the growing involvement of macroeconomic applications in the panel data tradition, whose focus has shifted towards examining the asymptotics of macro panels with large T (length of the time series) and N (number of cross-sections). The adoption of similar tests as available in the time series framework on panel data is yet in progress.
The major differences between time-series and panel unit root and cointegration tests can be summarized as follows: observation of patterns and relationships in the data that may not be detectable at the stock market aggregate level due to data smoothing caused by aggregation; consideration of different degrees of heterogeneity among individuals; in panel data analysis, the validity of rejecting a unit root may be subject of discussion; the power of panel unit root tests increases as N increases, with increased robustness in relation to the standard low-power DF and ADF tests applied to small samples; additional cross-sectional components incorporated in panel data models provide better properties of panel unit root tests; panel cointegration tests have increased power especially for small T, commonly encountered when data is limited to the post war period.
Testing for unit roots in panels is not a common practice as it is testing for unit roots in time series studies. The statistical methods applied in this paper relate to the works by Levin and Lin (1993), Levin, Lin and Chu (2002), Im, Pesaran and Shin (2003), Breitung (2000), Fisher-ADF and Fisher-PP proposed by Maddala and Wu (1999) and Choi (2001) and Hadri (2000). Recent panel cointegration tests applied are those developed by Kao (1999), Pedroni (2000Pedroni ( , 2004 and Maddala and Wu (1999  The purpose of testing for cointegration is primarily related with the investigation of spurious regression, which occurs only in the presence of nonstationarity. Following the same logic as the panel unit root tests, panel cointegration tests can be motivated by the search for more powerful tests than those obtained by applying individual time-series cointegration tests. The latter models have low power especially for short M and short span of the data which is often limited to post-war annual data. In the Johansen-Fisher panel test, Maddala andWu (1999) uses Fisher's (1932) combined test to propose an alternative approach to test cointegration in panel data by the In panel cointegrated regression models, the asymptotic properties of the estimators of the regression coefficients and the associated statistical tests differ from those of the time series cointegration regression models. A long run relationship commonly observed in macroeconomic and financial data is often predicted by economic theory. It is thus significant to estimate regression coefficients and test whether restrictions established are empirically satisfied such as a one-for-one cointegrating equilibrium between prices and dividends, which also implies that the price-dividend ratio is stationary.

RESULTS
In order to assess the present value model at the firm level for the Brazilian stock market, datasets on prices and dividends have been used at an annual frequency for the period of January 1987 to December 2008. The initial period is based on the availability of data platform, considering that the power of unit root and cointegration tests focuses both on cross sections (N) and, more remarkably, on the extension of the time period considered (T), as evidenced by Shiller and Perron (1985) and Hakkio and Rush (1991 The following IBOVESPA companies presented in Table 1 have been analyzed for the evaluation of the present value model with constant and time-varying expected returns from unit root and cointegration techniques for non-stationary panels. Stock quotes have been updated to changes due to financial events such as M&A.  Tables 2 and 3. Considering all tests for the presence of a unit root in the real price series of the companies composing the panel, they reveal sensitivity to the presence of individual effects and individual linear trends as well as to the lag order, as expected and verified in Goddard et al. (2008). Reverting the null hypothesis in order to test for stationarity in all companies using the Hadri test along with the Heterocedastic Consistent Z-stat, in both individual models with intercept and intercept and trend, the null hypothesis of no unit root is rejected at the 1% level, not confirming that / 9 ~ (0), noting that real prices as stationary processes present no theoretical support. Hence, we cannot reject the hypothesis that the real price series of the companies surveyed have a unit root for the entire panel or for most companies analyzed, considering the different null and alternative hypotheses tested.  (2003) paper. In Hadri test, high correlation leads to severe size distortion, leading to over-rejection of the null. Source: Elaborated by the authors.
Regarding the panel unit root tests applied to the dependent variable of the PVM, they also reveal sensitivity to the presence of individual effects and individual linear trends and the lag order. Using the Hadri test along with the Heterocedastic Consistent Z-stat, in both individual models with intercept and intercept and trend, the null hypothesis of no unit root is rejected at the 1% level, not confirming that / 9 ~ (0), noting that real dividends as stationary processes present no theoretical support. Thus, we cannot reject the hypothesis that the real dividends series of the companies surveyed are integrated of order one for the entire panel or for most sample companies.  (2003) paper. In Hadri test, high correlation leads to severe size distortion, leading to over-rejection of the null. Source: Elaborated by the authors.
The results for panel cointegration tests are presented in Tables 4, 5 and 6. We apply the residual Kao (1999) tests and multiple Pedroni (2000Pedroni ( , 2004 tests, both based on Engle-Granger. Regarding the Kao (1999) t -11,7603 16.22261 11.39772 -20,1559   Concerning the Pedroni (2000Pedroni ( , 2004 tests, although they display residual sensitivity to the inclusion of linear trends and the lag order established, the prevalence is evident in relation to the rejection of the null hypothesis of no cointegration between prices and dividends considering the companies examined, hence validating the PVM with constant expected returns. 3678 ---Note: ***, **, * represent test statistics significant to the 1%, 5%, and 10% levels, respectively. S1 represents the statistics, and S2 denotes the weighted statistics. Newey-West bandwidth selection using Bartlett kernel. Source: Elaborated by the authors. Regarding the trace test and maximum eigenvalue of Johansen-Fisher panel data, in the absence of trend in data, the model with intercept (no trend) in CE and VARparticularly suitable for the PVM analysis -rejects the hypothesis of zero cointegrating relationship in both statistical tests based on the trace and maximum eigenvalue at the 1% level; concerning the hypothesis of at most 1 cointegrating vector, it is also rejected in both trace and maximum eigenvalue statistics at the 1% level. This is in line with the hypothesis that real prices and real dividends exhibit a stationary relationship.

PRESENT VALUE MODEL: TIME-VARYING EXPECTED RETURNS
Analogously to the previous section, as presented in Tables 8 and 9, we cannot reject the hypothesis that the log prices series are integrated of order one for the entire panel or for most companies comprising it. [0.0000]*** Note: ***, **, * represent test statistics significant to the 1%, 5%, and 10% levels, respectively. Probabilities for Fisher tests are computed using an asymptotic chi-square distribution. All other tests assume asymptotic normality. LLC, Fisher-PP and Hadri: Newey-West bandwidth selection using Bartlett kernel. Critical t-bar values obtained from original Im, Pesaran e Shin (2003) paper. In Hadri test, high correlation leads to severe size distortion, leading to over-rejection of the null. Source: Elaborated by the authors.
In Table 9, although the diagnosis of (0) stationarity or nonstationarity (1) of ln( / 9 ) shows sensitivity to the inclusion or exclusion of trend as well as to the lag order established, we cannot reject the hypothesis that the log dividends series have a unit root for the entire panel or for most companies analyzed. In PVM with time-varying expected returns, it is expected that the log pricedividend ratio is (0) stationary, as discussed in the presented literature. In relation to the panel unit root tests applied to the log price-dividend ratio ln( / ) in Table 10, we cannot reject the hypothesis that the log price-dividend series is stationary for the entire panel or for most companies surveyed. Hence, we cannot reject the PVM with timevarying expected returns from the panel unit root tests applied. Note: ***, **, * represent test statistics significant to the 1%, 5%, and 10% levels, respectively. Probabilities for Fisher tests are computed using an asymptotic chi-square distribution. All other tests assume asymptotic normality. LLC, Fisher-PP and Hadri: Newey-West bandwidth selection using Bartlett kernel. Critical t-bar values obtained from original Im, Pesaran e Shin (2003) paper. In Hadri test, high correlation leads to severe size distortion, leading to over-rejection of the null. Source: Elaborated by the authors.
Once verified that log real prices and log real dividends are predominantly (1), we apply the panel cointegration tests. The results are presented in Tables 11, 12 and 13. As in the previous section, we employ the residual Kao (1999) and multiple Pedroni (2000Pedroni ( , 2004 tests based on Engle-Granger. Regarding the Kao (1999) tests, under the model with individual intercept, we fail to reject the hypothesis of no cointegration by the automatic lag selection criterion. Analyzing the sensitivity of the results, we reject the hypothesis of no cointegration only for fixed lag of order 1.   Note: ***, **, * represent test statistics significant to the 1%, 5%, and 10% levels, respectively. Newey-West bandwidth selection using Bartlett kernel. Source: Elaborated by the authors.
Examining the Pedroni (2000Pedroni ( , 2004 tests, although they display residual sensitivity to the inclusion of linear trends and the lag order established, the prevalence is evident in relation to the rejection of the null hypothesis of no cointegration between log real prices and log real dividends, considering the companies examined, hence validating the PVM with time-varying expected returns.  ---Note: ***, **, * represent test statistics significant to the 1%, 5%, and 10% levels, respectively. S1 represents the statistics, and S2 denotes the weighted statistics. Newey-West bandwidth selection using Bartlett kernel. Source: Elaborated by the authors. Regarding the Maddala and Wu (1999) cointegration tests that combine the pvalues from the trace test and maximum eigenvalue of Johansen-Fisher, in the model with intercept (no trend) in CE and VAR -particularly suitable for the PVM analysis -we reject the hypothesis of zero cointegrating relationships in both statistics based on the trace test and maximum eigenvalue at the 1% level; in relation to the hypothesis of at most 1 cointegrating vector, it is also rejected in both trace statistics and maximum eigenvalue at the 1% level.
Thus, from panel cointegration tests of Kao (1999), Pedroni (2000Pedroni ( , 2004 and Maddala and Wu (1999), we cannot reject the hypothesis of no cointegration between real log prices and real log dividends, considering the sample companies examined, validating, therefore, the present value model between prices and dividends with time-varying expected returns developed seminally in Campbell and Shiller (1988a,b   log real dividends have a unit root and follow, therefore, an U (1) process as provided in the literature. Additionally, we cannot reject that the log price-dividend ratio series is a stationary (0) process, representing the validity of the time-varying returns hypothesis.
Applying the panel cointegration tests, Kao tests do not show predominance that log real prices and log real dividends are cointegrated; Pedroni tests, moreover, clearly indicate that we cannot reject the hypothesis that the underlying series are cointegrated, validating the PVM under time-varying returns; finally, Johansen-Fisher panel tests proposed by Maddala and Wu (1999), particularly in the model with intercept (no trend) in CE and VAR, suitable for the assessment of the PVM, reject the null hypothesis that zero cointegrating relations exists, also rejecting the hypothesis that at most one cointegrating relationship exists. Thus, the panel unit root tests reveal that log real price and log real dividends have a unit root and the log price-dividend ratio is stationary; the panel   Rejection of Null 4 Rejection of Null Pedroni (1997Pedroni ( , 1999Pedroni ( , 2000Pedroni ( , 2004 No

CONCLUSION
The empirical evidence on the long-term relationship between stock prices and dividends remains scarce. As stock prices rose, analysts questioned whether the fundamental value of a share related to innovations in dividends, since low dividend payouts and record-high stock prices suggested an overvaluation. From then on, the validity of the Present Value Model (PVM) has been subject of debate, because the recent collapse of stock prices underlines the importance of traditional measures in the valuation of stocks, since they relate stock prices to the fundamental value of corporations.
While most studies focusing on the relationship between prices and dividends have examined the long-term relationship between a stock price index and an index of dividends of a particular country of interest, the empirical analysis in this paper is based on prices and dividends at the firm level through first generation panel unit root and panel cointegration estimation methods to test the long-term relationship between stock prices and dividends for the Brazilian stock market. The use of firm level data allows the analysis of patterns and relationships that can be obscured at the aggregate stock market level through averaging in the aggregation process. Thus, the power increase and precision obtained by the procedures allow the application of recent data, as well as possible structural changes in the data that occur more frequently over longer periods, and the more accurate assessment regarding the consistency of the present value model under considerable fluctuations in the stock market.

Regarding the results obtained in the Present Value Model with Constant Expected
Returns, from the panel unit root tests, the statistics reveal sensitivity to the presence of individual effects and individual linear trends and to the lag order. The ambivalent results of the tests are expected and also found in Goddard et al. (2008). However, there is an inclination to the failure of rejecting the hypothesis that real prices and real dividends series have a unit root for the entire panel or for most companies surveyed, considering the different null and alternative hypotheses tested. From the panel cointegration tests of Kao (1999), Pedroni (1997Pedroni ( , 1999Pedroni ( , 2000Pedroni ( , 2004 and Maddala and Wu (1999), results fail to reject the hypothesis of no cointegration between real prices and real dividends considering the different sample companies examined, validating, therefore, the Present Value Model  Campbell and Shiller (1987).
Analyzing the Present Value Model with Time-Varying Expected Returns, the apparent ambivalence of the unit root tests is expected and verified, in which the diagnosis of (0) stationarity or nonstationarity (1) depends on whether or not the trend is included, as well as upon the lag order established. However, results cannot reject the hypothesis that real log prices and real log dividends series have a unit root for the entire panel or for most companies comprising it, considering the different null and alternative hypotheses tested.
In accordance to the theory, results do not reject that log price-dividend ratio is a (0) stationary process, indicating the validity of the Present Value Model. Finally, from the cointegration tests for panel data, statistical results cannot reject the hypothesis of cointegration between real prices and real dividends, considering the different sample companies observed, hence validating the Present Value Model between prices and dividends with Time-Varying Expected Returns developed seminally in Campbell and Shiller (1988a,b).
Finally, it is presented that, for panel cointegrated regression models, the asymptotic properties of the estimators of the regression coefficients and the associated statistical tests are different from those of the time series cointegration regression models.
Panel cointegration models direct to the assessment of long-term relationships verified in macroeconomic and financial data. Thus, results from the FMOLS and DOLS estimators applied to cointegrated panels, individual companies show evidence of overvaluation of stock prices for most examined companies, assuming either the hypothesis of constant or time-varying expected returns.