Are analysts? loss functions asymmetric?

Recent research by Gu and Wu (2003) and Basu and Markov (2004) suggests that the well-known optimism bias in analysts? earnings forecasts is attributable to analysts minimizing symmetric, linear loss functions when the distribution of forecast errors is skewed. An alternative explanation for forecast bias is that analysts have asymmetric loss functions. We test this alternative explanation. Theory predicts that if loss functions are asymmetric then forecast error bias depends on forecast error variance, but not necessarily on forecast error skewness. Our results confirm that the ex ante forecast error variance is a significant determinant of forecast error and that, after controlling for variance, the sign of the coefficient on forecast error skewness is opposite to that found in prior research. Our results are consistent with financial analysts having asymmetric loss functions. Further analysis reveals that forecast bias varies systematically across style portfolios formed on book-to-price and market capitalization. These firm characteristics capture systematic variation in forecast error variance and skewness. Within style portfolios, forecast error variance continues to play a dominant role in explaining forecast error.

Are analysts' loss functions asymmetric?

Recent research by Gu and Wu (2003) and Basu and Markov (2004) suggests that the well-known optimism bias in analysts? earnings forecasts is attributable to analysts minimizing symmetric, linear loss functions when the distribution of forecast errors is skewed. An alternative explanation for forecast bias is that analysts have asymmetric loss functions. We test this alternative explanation. Theory predicts that if loss functions are asymmetric then forecast error bias depends on forecast error variance, but not necessarily on forecast error skewness. Our results confirm that the ex ante forecast error variance is a significant determinant of forecast error and that, after controlling for variance, the sign of the coefficient on forecast error skewness is opposite to that found in prior research. Our results are consistent with financial analysts having asymmetric loss functions. Further analysis reveals that forecast bias varies systematically across style portfolios formed on book-to-price and market capitalization. These firm characteristics capture systematic variation in forecast error variance and skewness. Within style portfolios, forecast error variance continues to play a dominant role in explaining forecast error.

On surrogate loss functions and ff-divergences

The goal of binary classification is to estimate a discriminant function $\gamma$ from observations of covariate vectors and corresponding binary labels. We consider an elaboration of this problem in which the covariates are not available directly but are transformed by a dimensionality-reducing quantizer $Q$. We present conditions on loss functions such that empirical risk minimization yields Bayes consistency when both the discriminant function and the quantizer are estimated. These conditions are stated in terms of a general correspondence between loss functions and a class of functionals known as Ali-Silvey or $f$-divergence functionals. Whereas this correspondence was established by Blackwell [Proc. 2nd Berkeley Symp. Probab. Statist. 1 (1951) 93--102. Univ. California Press, Berkeley] for the 0--1 loss, we extend the correspondence to the broader class of surrogate loss functions that play a key role in the general theory of Bayes consistency for binary classification. Our result makes it possible to pick out the (strict) subset of surrogate loss functions that yield Bayes consistency for joint estimation of the discriminant function and the quantizer.Comment: Published in at http://dx.doi.org/10.1214/08-AOS595 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org