Has the Stock Market Become More Efficient in the Long-Run? Evidence from U.S Corporations

Using the cointegration model to deal with nonstationary time series, we estimate the long-run relationship between the average stock price and the average dividend. The results from U


Introduction
Discount rate appears in the present value model: where PV is present value, i c is the dividend payment in time i , and r is discount rate.The intrinsic value of a firm's stock is defined as the sum of the present values of all cash payments, i.e., the future cash payments discounted at the appropriate risk-adjusted rate of return, r .Here, r is associated with the risk of a stock.The higher risk a firm is taking, the higher will be the appropriate discount rate used in the present value model.Generally, investors would take on higher risk if there was more asymmetric information between investors and a firm.
Markets with more asymmetric information would be working less efficiently, as private information may not be fully incorporated into the current stock prices.In this sense, the average discount rate of all stocks in a market might reflect the overall efficiency of market.The discount rate also represents the return rate that investors demand from the firm based on their judgment of ~ 39 ~ the risk of stock; it is why this rate is also called capital cost.In this sense, the average discount rate of all stocks might represent the opportunity cost of funds, influencing interest rate and many other important variables in economy.Thus, estimating the discount rate and understanding its long-run change is meaningful.If we suppose that a firm will pay a constant dividend d permanently in future, the present value model could be written as r d PV = .To wit, if the market price of a stock reflects its true value and if the firm's dividend is expected to be stable, then r , the discount rate, can be solved or discovered.In light of this model, we can say that a stock price should have a longrun relationship with its current dividend, even though both the stock price and the dividend would fluctuate over time.
Using the long run time series data of the average stock price (Stock Price) and the average dividend (Dividend) across all firms in a county, we can estimate the average discount rate for a county by running the regression model: in which, the reciprocal of β should represent the discount rate, i.e., r=1/β .Now the econometric problem we face is that these time series might be non-stationary processes.For instance, they could be random walks, trends, changing variances or covariances.If we apply OLS regression to the models above, the estimators are not normally distributed, leaving us inconsistent estimation of parameters.Results extracted from non-stationary time series may be spurious in that they may lead to a relationship between two variables where the relationship does not exist.
The usual approach to this problem is to difference the series.If a test fails to reject the hypothesis of non-stationarity of our variables, we can difference the series in question before using them in our regressions.While this is acceptable, differencing may result in a loss of information about the long-run relationship between two variables.Cointegration analysis, developed by Engle and Granger (1987), solves this problem.Sometimes two variables are non-stationary but a linear combination of those variables is stationary.If this is the case, we say that these two variables are cointegrated.We can establish a regression model of these two variables and estimate this model using OLS.Engle and Granger prove that OLS provides consistent estimates of parameters, if cointegration of two variables exists.Furthermore, the residuals of this regression then can be used to test whether these two variables are indeed cointegrated.
In this paper, we apply the cointegration analysis to estimate the discount rate in the United States.We first estimate over a full sample period from 1871 to 2012, and then estimate over two sub-sample periods, 1871-1945 and 1946-2012, respectively.Our primary objective is to compare the discount rates between these two time periods.The long-run change in the discount rate can tell us how the U.S. stock market has been changing in terms of market efficiency and capital cost.The conventional study on the market efficiency hypothesis focuses on whether stock markets are of weak, semi-strong, or strong form efficiency.The empirical evidence has been mixed (Fama, 1970(Fama, & 1998;;Malkiel, 2003).This study contributes to the market efficiency literature by looking at whether a stock market becomes more efficient in a long-run in terms of a measure of market information asymmetry-the average discount rate for all firms in the market.Our results are easily stated; the discount rate is lower in the latter sub-sample period than in the former sub-sample period, indicating that the U.S. stock market has become more efficient and the capital cost has become lower.

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The organization of the paper is as follows.Section 2 describes data used in our study.Section 3 describes the results of the empirical analysis.And Section 4 concludes.

Data
The data set used in our paper is the stock market data coming from Professor Robert J. Shiller's website.This data set contains monthly average stock price and dividend data of a broad stock index of the Standard and Poor's Portfolio, starting January 1871.The monthly dividend data before 1926 are from Cowles and associates (1939), interpolated from annual data, and the monthly dividend data since 1926 are computed from the S&P, with linear interpolation to monthly figures.Stock price data are monthly averages of daily closing prices.Using The CPI-U (Consumer Price Index-All Urban Consumers) published by the U.S. Bureau of Labor Statistics for post-1913 period and Warren and Pearson's price index for prior-1913 period, Professor Shiller computes the real term of the variables.Campbell and Shiller (1987) use the similar data set to estimate the discount rate in U.S. over a period from 1871 to 1987.
We use the data of these two variables from 1871 to 2012 to estimate the model.The term Stock Price represents the Standard and Poor's composite stock price index, and the term Dividend represents of the average dividend of the firms in the index, both in real term.To be consistent with Campbell and Shiller (1987) in terms of methodology, we convert them into yearly measures.

Empirical Analysis
The Engle-Granger cointegration test involves three steps.First, determine the orders of integration for each of the variables under consideration; that is, difference each series successively until stationary series emerge.Usually, one-order difference would be a stationary series.Second, estimate cointegration regressions with OLS using variables with the same order of integration.Finally, test for stationarity of residuals from cointegration regressions (see Engle & Granger, 1987;Miller, 1991).We develop our analysis in this order.

. Testing for Stationarity of Three Time-Series Variables: Stock Price and Dividend
We follow the methodology of Dickey and Fuller (1981) to test the stationarity of a time-series variable.Table 1 reports the results of the Dickey-Fuller (DF) and augmented Dickey-Fuller (ADF) tests for the stationarity of the two time-series variables-Stock Price and Dividend, respectivelyas well as of their first-differences.For each test, we present four statistics: three t-statistics for zero mean, single mean, trend, and F-statistic.The three t-statistics for zero mean, single mean, and trend are computed based on the following regression models.The p-values, calculated by SAS program based on the Dickey-Fuller test, are reported in parentheses below each t-statistic.We run a Dickey-Fuller unit root test on each variable series by estimating the unrestricted regression: and the restricted regression:  The results shown in Table 1 are straightforward.In terms of levels, the t-statistics and F-statistic do not reject the null hypothesis of non-stationary for the time series Stock Price and Dividend in both DF and ADF tests.After first order differencing, the t-statistics in both DF and ADF test reject the null hypothesis of non-stationarity at the 1 percent level, and the F-statistic does so at the 5 percent level.The results of the stationarity indicate that using the level variables of Stock Price and Dividend in OLS regression might lead to inconsistent estimation of parameters.We have to apply the cointegration analysis to get the consistent estimation.

. Cointegration Analysis
After verifying the stationarity of first differences of Stock Price, and Dividend, we apply OLS to the model in the introduction.If cointegration exists between Stock Price and Dividend, we say that the OLS estimator is normally distributed and our estimations of parameters are consistent.We run the model and use Durbin-Watson and Dickey-Fuller methods to check their cointegration.We first run the model from 1871 to 2012 to check the cointegration over the full-sample period.The results are presented in Panel A, Table 2.

Table 2. Analysis of cointegration regressions
This table shows the results of the test for the cointegration between Stock Price and Dividend.Panel A reports the results for the full-sample period (1871-2012), and Panel B reports the results for a sub-sample period .The term Stock Price represents the Standard and Poor's composite stock price index, and the term Dividend represents of the average dividend of the firms in the index, both in real term.The errors from the cointegration models are recovered to perform the augmented Dickey-Fuller (ADF) non-stationarity tests based on the following regression: where t µ is the error from the cointegration equation, t e is a stationary random error, and the null hypothesis of non-stationarity is rejected when 1 σ is significant negative.The Dickey-Fuller (DF) tests for non-stationarity delete the summation.The results in Panel A show that the ADF test rejects the null hypothesis of non-stationary residual at the 5 percent level.But the DF test and D-W test do not reject the null hypothesis.So, we say that the cointegration exists under the ADF test at the level of 5%, but not under the D-W test and DF test.This result is not consist with the results of Campbell and Shiller (1987), who tested the cointegration between stock prices and dividends using data from 1871 to 1987 and PPI based on 1967.We repeated their test and got very similar results.The repeated test, reported in Panel B, Table 2, indicates that Stock Price and Dividend are cointegrated under all tests.Miller (1991) points out that the high 2 R and low D-W statistics suggest possible spurious regression.
Considering that 2 R is not too high (0.699), D-W statistics is not too low (0.235), we conclude that there is weak evidence supporting the cointegration between Stock Price and Dividend.
To follow the change in discount rate over time, we follow Miller (1991) and split the sample period into two sub-sample periods at 1945 to estimate conintegration model respectively.In this way, we can test for the structural shifts and compare the estimated discount rates from two different time period.Panel A and B in Table 3 report the results of the cointegration regressions for the sub-sample periods 1871-1945 and 1946-2012, respectively.

Table 3. Analysis of cointegration regressions
This table shows the results of the test for the cointegration between Stock Price and Dividend.Panel A reports the results for the full-sample period (1871-1945, and Panel B reports the results for a sub-sample period .The term Stock Price represents the Standard and Poor's composite stock price index, and the term Dividend represents of the average dividend of the firms in the index, both in real term.The errors from the cointegration models are recovered to perform the augmented Dickey-Fuller (ADF) non-stationarity tests based on the following regression:  (10) where t µ is the error from the cointegration equation, t e is a stationary random error, and the null hypothesis of non-stationarity is rejected when 1 σ is significant negative.The Dickey-Fuller (DF)   tests for non-stationarity delete the summation.~ 44 ~ It is not a surprise that the results in sub-sample period 1871-1945 are consistent with the results in sub-sample period 1871-1967, with all statistics suggesting cointegration.Comparing with the full sample, the D-W and DF do reject non-stationary error much more strongly.For the sub-sample period 1946-2012, the D-W, DF, and ADF tests support the cointegration only at 10% level.Furthermore, we can check the autocorrelation and partial-autocorrelation to corroborate our judgment.The autocorrelation and partial-autocorrelation charts resemble stationary error structures; the autocorrelation for the residual of Model 1 decline to zero quickly, while the partialautocorrelation function has a spike of 0.82959 at lag one and a spike of -0.3579 at lag two only.The evidence supports that Stock Price is cointegrated with Dividend throughout the two subsample periods.
After verifying the conintegration, we can now look at the estimates of the coefficients of Dividend in OLS model.We can safely say that the estimates of the coefficients in the two subsample periods are consistent and can use the estimates to compute the discount rate in the two subsample periods.They are 1/20.42=0.049 and 1/75.88=0.0132 in first and second periods, respectively.The decrease in the discount rate over more than one hundred years indicates that the stock market in the U.S. has a long trend of increasing the market efficiency and lowering the capital costs.

Conclusion
The discount rate contains a great deal of information.An average discount rate of all stocks in a market can tell market efficiency and opportunity cost.A long run discount rate can be estimated by regressing the stock price against the dividend.However, the problem of non-stationarity of time series variables might result in inconsistent estimation of the model.Application of the cointegration analysis enables us to deal with this problem.In this paper, we use the cointegration analysis to estimate the long run relationship between stock price and dividend over two periods, 1871-1945 and 1946-2012.We found that the discount rate lowered from 4.9% in the first period to 1.32% in the second period.This result indicates that the stock market in the U.S. States has become more efficient and capital cost has become lower in the long run.This evidence, along with the well-known decline in the bid-ask price (Corwin & Schultz, 2012) and the greater speed of stock market order execution (Hendershott & Moulton, 2011), bolsters the case of growing market efficiency in the long run.

1 α
estimated regression coefficient for first lag of the series, and let se be the standard error of 1 α .The t-statistic is: restrictions by calculating an F ratio and comparing it to the critical values provided by Dickey and Fuller (1981, p.1063).Augmented Dickey-Fuller tests are carried out by expanding the regressions above to include right-hand side of the equations.The t-statistics and F-statistic are calculated in the same way.

R
is the adjusted coefficient of determination and D-W is the Durbin-Watson statistic.The values in parentheses are p-value.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.

Table 1 .
Test for stationarity This table shows the results of the test for stationarity of 2 time-series variables, Stock Price and Dividend, from 1871 to 2012.The term Stock Price represents the Standard and Poor's composite stock price index, and the term Dividend represents of the average dividend of the firms in the index, both in real term.Panel A shows the results of DF Test, and Panel B shows the results of ADF Test.The values in parentheses are p-value.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.

2
R is the adjusted coefficient of determination and D-W is the Durbin-Watson statistic.The values in parentheses are p-value.***, **, and * denote statistical significance at the 1%, 5%, and 10% levels, respectively.