A bias correction to the Akaike information criterion, AIC, is derived for regression and autoregressive time series models. The correction is of particular use when the sample size is small, or when the number of fitted parameters is a moderate to large fraction of the sample size. The corrected method, called AIC C , is asymptotically efficient if the true model is infinite dimensional. Furthermore, when the true model is of finite dimension, AIC C is found to provide better model order choices than any other asymptotically efficient method. Applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed.

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. Biometrika Trust is collaborating with JSTOR to digitize, preserve and extend access to Biometrika. SUMMARY A bias correction to the Akaike information criterion, AIC, is derived for regression and autoregressive time series models. The correction is of particular use when the sample size is small, or when the number of fitted parameters is a moderate to large fraction of the sample size. The corrected method, called AICC, is asymptotically efficient if the true model is infinite dimensional. Furthermore, when the true model is of finite dimension, AICC is found to provide better model order choices than any other asymptotically efficient method. Applications to nonstationary autoregressive and mixed autoregressive moving average time series models are also discussed.

We analyze tests for long-run abnormal returns and document that two approaches yield well-specified test statistics in random samples. The first uses a traditional event study framework and buy-and-hold abnormal returns calculated using carefully constructed reference portfolios. Inference is based on either a skewnessadjusted t-statistic or the empirically generated distribution of long-run abnormal returns. The second approach is based on calculation of mean monthly abnormal returns using calendar-time portfolios and a time-series t-statistic. Though both approaches perform well in random samples, misspecification in nonrandom samples is pervasive. Thus, analysis of long-run abnormal returns is treacherous. COMMONLY USED METHODS TO TEST for long-run abnormal stock returns yield misspecified test statistics, as documented by Lyon~1997a! and Warner~1997!. 1 Simulations reveal that empirical rejection levels routinely exceed theoretical rejection levels in these tests. In combination, these papers highlight three causes for this misspecification. First, the new listing or survivor bias arises because in event studies of long-run abnormal returns, sampled firms are tracked for a long post-event period, but firms that constitute the index~or reference portfolio! typically include firms that begin trading subsequent to the event month. Second, the rebalancing bias arises because the compound returns of a reference portfolio, such as an equally weighted market index, are typically calculated assuming periodic generally monthly! rebalancing, whereas the returns of sample firms are compounded without rebalancing. Third, the skewness bias arises because the distribution of long-run abnormal stock returns is positively skewed, 165 which also contributes to the misspecification of test statistics. Generally, the new listing bias creates a positive bias in test statistics, and the rebalancing and skewness biases create a negative bias.In this research, we evaluate two general approaches for tests of long-run abnormal stock returns that control for these three sources of bias. The first approach is based on a traditional event study framework and buy-and-hold abnormal returns. In this approach we first carefully construct reference portfolios that are free of the new listing and rebalancing biases. Consequently, these reference portfolios yield a population mean abnormal return measure that is identically zero and, therefore, reduce the misspecification of test statistics. Then we control for the skewness bias in tests of long-run abnormal returns by applying standard statistical methods recommended for settings when the underlying distribution is positively skewed. Two statistical methods virtually eliminate the skewness bias in random samples:~1! a bootstrapped version of a skewness-adjusted t-statistic, and~2! empirical p values calculated from the simulated distribution of mean long-run abnormal returns estimated from pseudoportfolios. The first method is developed and analyzed based on a rich history of research in st...