This paper provides new evidence concerning the probability of informed trading (PIN) and the PIN-return relationship. We take measures to overcome known estimation biases and improve the quality of quarterly PIN estimates. We use the average of a firm’s PIN estimates in four consecutive quarters to smooth out the effect of seasonal variation in trading activities. We find that when high-quality PIN estimates are used, the Fama–MacBeth cross-sectional regressions show stronger evidence for the positive PIN-return relationship than documented in the prior literature. This finding is robust to controls for the January, liquidity, and momentum effects.

We report evidence that boundary solutions can cause a bias in the estimate of the probability of informed trading (PIN). We develop an algorithm to overcome this bias and use it to estimate PIN for nearly 80,000 stock-quarters between 1993 and 2004. We obtain two sets of PIN estimates by using the factorized likelihood functions in both Easley et al., 2010, Lin and Ke, 2011, respectively. We find that the estimate based on the EHO factorization is systematically smaller than the estimate based on the LK factorization, meaning that there is a downward bias associated with the EHO factorization. In addition, we find that boundary solutions appear with a very high frequency when the LK factorization is used. Thus it is necessary to use the LK factorization together with the algorithm in this paper. At last, we document several interesting empirical properties of PIN.

Existing studies have found that ex post stock returns are positively skewed, but such skewness is only persistent for individual stocks, not for portfolios. This implies that the ex post knowledge of skewness may not be useful in ex ante portfolio selection. However, the portfolios in these studies are not optimally formed because preferences for skewness are not taken into consideration when forming these portfolios. It is more meaningful to see if the positive skewness would persist in optimally formed mean–variance–skewness efficient portfolios. Using stocks from both Japanese and US markets and a bootstrap method, we find that the portfolios optimally formed by using a polynomial goal programming method, which considers preference for skewness, greatly enhances skewness persistence over time. Our results are robust across both Japanese and US stocks. However, the skewness persistence is stronger for portfolios formed with monthly data than that with weekly data. These findings have practical implications to investors with skewness preferences.