Stock Price Volatility: Tests Based on the Geometric Random Walk

Abstract

The simplest variance-bounds inequality says that the price of stock should be less volatile than the present value of actual future dividends, assuming that the former can be taken to equal the conditional expectation of the latter. This is so because under the assumption just stated the price of stock equals the present value of dividends less a forecast error which, under rational expectations, is orthogonal to price. Therefore the variance of the present value of future dividends-the ex post rational stock price -should exceed the variance of the actual price by an amount equal to the variance of the forecast error. Despite the intuitive appeal of the variance inequality just stated, empirical implementation is far from straightforward. An important econometric problem is that if the unconditional population variances change over time then sample variances, not being consistent estimates of the population variances that are subject to the inequality, are uninterpretable from the viewpoint of the variance-bounds theorems. Sample variances are of interest only if some data transformation can be found which ensures that the relevant variables are stationary and which at the same time preserves the variance-bounds theorems. The original papers that formulated the variance-bounds relation, Robert J. Shiller (1979, 1981) and LeRoy and Richard D. Porter (1981), employed different corrections to attain stationarity. Shiller (1981) fitted a log-linear trend to stock prices and took the residuals from this trend to be the series restricted by the variance-bounds inequalities. LeRoy and Porter, on the other hand, assumed that the trend in prices was attributable to inflation and retained earnings and implemented a procedure to reverse the effects of these factors on stock prices. Subsequent criticism indicated that both procedures were flawed. Allan W. Kleidon (1986a) showed that if the underlying price series has a unit root, as does a random walk, then the supposedly trend-adjusted series as calculated by Shiller (1979, 1981) will still be nonstationary. Further, if the variance-bounds tests are implemented despite the nonstationarity of the underlying series, they will be strongly biased toward rejection. Kleidon's analysis obviously raised questions about the finding of excess volatility that Shiller reported. Kleidon (1983) also pointed out an error in LeRoy and Porter's implementation of the correction for earnings retention.1 Here we correct for trend in stock prices in an entirely different way: by dividing dividends into price. In other words, we conduct variance-bounds tests on the ratio of

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