Autocorrelation structure of forecast errors from time-series models: Alternative assessments of the causes of post-earnings announcement drift

Abstract

This paper demonstrates that the evidence supporting the hypothesis that post-earnings announcement drift (PEAD) is caused by investors’ failure to incorporate the implications of current earnings for future earnings is (also) consistent with researchers’ over-differencing an already stationary time-series. Specifically, we show the evidence is driven by a subset of firms where over-differencing of quarterly earnings in estimating earnings surprises is most likely to have occurred. Given the persistence of the PEAD over time, our alternative explanation suggests that the prior research investigating the causes for the PEAD overestimates investors’ naivete.

Introduction

Among the documented departures from market efficiency, the post-earnings-announcement drift (PEAD) is arguably the most puzzling. To quote Brennan (1991):

• Perhaps the most severe challenge to financial theorists posed by recent work on market reactions to earnings announcements is… the finding by Bernard and Thomas, 1989, Bernard and Thomas, 1990 that market reaction to the current quarters’ earnings can be predicted on the basis of information contained in the previous quarter's announcement, and that this phenomenon can be explained by a model in which the market uses a naive model of the stochastic process for earnings.

The evidence suggests that independent of the time period, specific markets examined or firm size, investors under-react to earnings surprises. While the PEAD is well documented in the literature, the reasons for the persistent under-reaction to earnings announcements are not well understood. Bernard and Thomas (1990), (henceforth BT) and others suggest that this phenomenon may be caused by investors’ reliance on naive earnings forecasting models. Although the exact interpretations of this explanation are still being debated (e.g., Ball and Bartov, 1996; Rangan and Sloan, 1998; Soffer and Lys, 1999), the emerging consensus is that the PEAD is a consequence of investors’ failure to fully incorporate the implications of current earnings for future earnings.

In this paper, we investigate the techniques used by researchers to determine the causes of the PEAD. Typically, the evidence relies on the following elements: First, researchers define earnings surprises relative to a `naive’ model and show that these surprises are predictable. For example, BT show that if investors’ expectations are based on the seasonal random walk model, while earnings follow a more complicated process, then the correlation between current earnings surprises and those in the subsequent four quarters will show a (+,+,+,−) pattern. Second, researchers show that similar correlation patterns are observed in abnormal returns (Bernard and Thomas, 1989; Freeman and Tse, 1989). Finally, researchers regress abnormal returns on preceding earnings surprises and show that the regression coefficients also have the (+,+,+,−) pattern. Jointly these results are interpreted as evidence that the PEAD is caused by investors’ failure to incorporate the implications of current earnings for future earnings.

In this paper we show that the evidence supporting the hypothesis on the causes of the PEAD may be partly the result of the empirical procedures used by researchers (i.e., over-differencing stationary time series). It is important to emphasize, however, that our results do not exclude the explanations given in prior research on the causes of the PEAD. Rather, we offer an alternative, and possibly simpler, explanation for the findings in prior research and suggest that discerning the degree of investor naiveté from prior studies may be problematic.

Our analysis is centered around the pattern of the autocorrelations of time-series forecast errors. Foster (1977) documents that the autocorrelations in seasonally differenced quarterly earnings are positive with declining magnitudes at the first three lags, and strongly negative at the fourth. This pattern of autocorrelation plays a central role in the discussion of the causes for the PEAD (see BT; Ball and Bartov, 1996; Rangan and Sloan, 1998; Soffer and Lys, 1999). Following BT, predictions and explanations for the PEAD are derived from the sign and magnitude of these autocorrelations and, in particular, the negative autocorrelation at lag four.

We begin our analysis by deriving the theoretical autocorrelations of earnings surprises for the seasonal random walk model as a function of the underlying time-series process of quarterly earnings. The results indicate that the pattern of autocorrelations may be a statistical artifact induced by over-differencing a stationary process and, therefore, may not be an innate property of quarterly earnings. This, in turn implies that the interpretation given to these correlations in explaining the PEAD needs to be re-examined. Our results also have implications for other studies that have interpreted serial correlation coefficients in differenced series (e.g., Foster, 1977; Dechow, 1994).

We also examine the autocorrelation structure of errors from a second time-series model, namely the first-order auto-regressive model in seasonal differences (the Foster model). This model has been extensively used in accounting studies to model quarterly earnings and was used by BT as an approximation to the `true’ model generating quarterly earnings. We find a similar pattern in the autocorrelation structure of errors from this model as we find in the errors from a seasonal random walk model. This finding makes it difficult to discriminate, from the tests in BT, whether investors rely on the naive seasonal random walk model or the more sophisticated Foster model in forming their expectations.

We then test our theoretical results using data from the same time-period as BT. For each sample firm, we compute the probability of a seasonal unit root at lag four using the procedure outlined in Dickey et al. (1984).¹ This probability is used as a metric to determine whether seasonal differencing is likely to induce the familiar (+,+,+,−) pattern in the serial correlation of the earnings surprises. In a first set of tests, we then show that the (+,+,+,−) correlation pattern occurs only for firms for which the unit root at lag four was rejected: that is, for firms for which the autocorrelation pattern is likely to have been induced by the differencing process. This result holds independent of whether earnings surprises are measured using the seasonal random walk or the Foster model.

Next, we replicate the tests performed by BT to investigate whether the PEAD is caused by investors’ failure to fully incorporate the implications of current earnings for future earnings. Consistent with our alternative hypothesis, we find that the regression coefficients have the (+,+,+,−) pattern only for firms where the serial correlation of earnings surprises may have been induced by the research approach. Thus, we conclude that the BT evidence is consistent with either investors’ reliance on naive earnings expectations models or researchers inducing serial correlation by effectively over-differencing an earnings series. Unfortunately, it is difficult to discriminate between these two explanations.

The persistence of the PEAD phenomenon is rather puzzling. If the cause were investors’ naiveté, as BT propose, one would expect arbitragers to take advantage of this naiveté causing the phenomenon to disappear over time. We believe that this persistence is suggestive that the extent of investor naiveté is not so extreme as some prior investigations of the PEAD seem to indicate. Our alternative explanation complements the work of Ball and Bartov (1996) who show that BT overestimate investors’ naiveté.

The remainder of the paper is organized into four sections. The next section discusses the autocorrelation structure of the errors from a seasonal random walk and the implications of this pattern for accounting studies. Section 3 discusses the autocorrelation structure of errors from the Foster model. Section 4 applies the analyses of 2 Expected value of autocorrelation coefficients in a seasonally differenced time series, 3 Expected value of autocorrelation coefficients of errors from the Foster model to prior PEAD studies. Finally, Section 5 summarizes our findings and presents conclusions.