Ex-post rational price approximations and the empirical reliability of the present-value relation

Excess volatility and regression tests have resulted in apparent rejections of the present-value relation when ex-post price approximations are employed. These approximations are based upon a sample terminal condition for prices, are not ergodic time-series, and do not result in statistics with readily calculable standard errors. Kleidon (1986a) has demonstrated that ex-post price approximations can subtly affect the reliability of certain volatility tests. We use a bootstrapped cointegration model to demonstrate some of these same effects in Mankiw, Romer and Shapiro's (1985) volatility statistics. The volatility statistics rarely have positive expected value in finite samples and still do not reject the present-value relation. Approximations based upon a `rolling' terminal condition result in volatility statistics which have calculable large-sample errors, but even these standard errors greatly overstate the accuracy of volatility statisics in small samples. Regression tests of the present value relation are also affected by the price approximations.