N.E. Savin

Real and Spurious Long-Memory Properties of Stock-Market Data

We test for the presence of long memory in daily stock returns and their squares using a robust semiparametric procedure of Lobato and Robinson. Spurious results can be produced by nonstationarity and aggregation. We address these problems by analyzing subperiods of returns and using individual stocks. The test results show no evidence of long memory in the returns. By contrast, there is strong evidence in the squared returns.

The power problems of unit root test in time series with autoregressive errors

Abstract Monte Carlo methods are used to study the size and power of serial-correlation-corrected versions of the Dickey-Fuller (1979,1981) unit root tests appropriate when the time series has unknown mean. The modifications do not cause serious size distortions or power deterioration in the white noise case. While studies in the literature have investigated the operating characteristics of these tests in the presence of moving average errors, of particular concern in this paper is the performance of these procedures in the presence of autoregressive errors. The Philips and Perron (1988) and Choi and Philips (1991) procedures are found to suffer from serious size distortions and have very low power when errors are autoregressively correlated. We conclude that even in the most favorable cases, these tests perform poorly against trend-stationary alternatives which are plausible for annual, quarterly, and monthly macroeconomic time series. The augmented Dickey-Fuller procedure, on the other hand, is reasonably well-behaved.

Three analyses of the firm size premium

Abstract The size premium for smaller companies is one of the best-known academic market anomalies. The relevant issue for investors is whether size premium for small-cap stocks is still positive, and, if so, whether its magnitude is substantial. In our analysis, we use annual compounded returns, monthly cross-sectional regressions, and linear spline regressions to investigate the relation between expected returns and firm size during 1980–1996. All three methodologies report no consistent relationship between size and realized returns. Hence, our results show that the widespread use of size in asset pricing is unwarranted.

Empirically relevant critical values for hypothesis tests: A bootstrap approach

Tests of statistical hypotheses can be based on either of two critical values: the Type I critical value or the size-corrected critical value. The former usually depends on unknown population parameters and cannot be evaluated exactly in applications, but it can often be estimated very accurately by using the bootstrap. The latter does not depend on unknown population parameters but is likely to yield a test with low power. The critical values used in most Monte Carlo studies of the powers of tests are neither Type I nor size-corrected. They are irrelevant to empirical research.

TESTING FOR ZERO AUTOCORRELATION IN THE PRESENCE OF STATISTICAL DEPENDENCE

The problem addressed in this paper is to test the null hypothesis that a time series process is uncorrelated up to lag K in the presence of statistical dependence. We propose an extension of the Box–Pierce Q-test that is asymptotically distributed as chi-square when the null is true for a very general class of dependent processes that includes non-martingale difference sequences. The test is based on a consistent estimator of the asymptotic covariance matrix of the sample autocorrelations under the null. The finite sample performance of this extension is investigated in a Monte Carlo study.

Testing for Autocorrelation Using a Modified Box-Pierce Q Test

This article investigates the finite-sample performance of a modified Box-Pierce Q statistic (Q*) for testing that financial time series are uncorrelated without assuming statistical independence. The finite-sample rejection probabilities of the Q* test under the null and its power are examined in experiments using time series generated by an MA (1) process where the errors are generated by a GARCH (1, 1) model and by a long memory stochastic volatility model. The tests are applied to daily currency returns.

The Level and Power of the Bootstrap t Test in the AR(1) Model With Trend

This article considers a first-order autoregressive (AR) model that may include an intercept and trend in which the innovations are independently and identically distributed. The innovation distribution is assumed unknown. The AR parameter is tested using the conventional t statistic. The article presents Monte Carlo estimates of the rejection probability of the test with bootstrap-based critical values. The results show that the test with the bootstrap-based critical value has essentially the right rejection probability for sample sizes comparable to or smaller than those that occur in practice and essentially the same power as the test with level-corrected critical values.

Multiple optima and asymptotic approximations in the partial adjustment model

This paper examines statistical problems which arise in empirical applications of the partial adjustment model with autoregressive errors when the model is nearly nonidentified. The results of Monte Carlo experiments show that the NLS estimation criterion function is multipeaked with high probability when the model is nearly nonidentified. In the cases examined the finite-sample distributions of the NLS estimators and the Wald test statistics are poorly approximated by their asymptotic distributions. The asymptotic approximation works better for the likelihood ratio (LR) test statistics, but still can be unsatisfactory. When the Wald and LR tests are based on bootstrap critical values the size distortions are effectively eliminated.

The exact moments of the least-squares estimator for the autoregressive model. Corrections and extensions

Abstract For a first-order autoregressive model with unknown intercept and normal errors the mean and variance of the finite-sample distribution of the least-squares estimator of the autoregressive parameter are numerically computed using formulae for evaluating the first and second moments of a ratio of two quadratic forms in normally distributed variables. Previous results in the literature are corrected and results for the case where the value of the autoregressive parameter is near unity in absolute value are presented.

Mirror-Image and Invariant Distributions in ARMA Models

The finite sample distributions of estimators and test statistics in ARMA time series models are generally unknown. For typical sample sizes, the approximations provided by asymptotic distributions are often unsatisfactory. Hence simulation or numerical integration methods are used to investigate the distributions. In practice only a limited part of the parameter space is examined using these methods. Thus any results which allow us to infer properties from one portion of the parameter space to another or to establish symmetry are most welcome.