Narrow Results

The classical capital asset pricing model postulates a linear relationship between stock returns and stock risks. However, a number of subsequent empirical studies have revealed some anomalies in this relationship, especially for firms with small size and high book-to-market values. A possible explanation for the anomalies is the existence of threshold effects in the proxies of stock risks. However, conventional threshold models only allow for one threshold variable, which limits their applicability in this context. In this paper, we address this issue by applying the econometric technique developed by Bai et al. (2012). We estimate the joint threshold effects of firm size and book-to-market equity ratio on the stock returns using a sample of 5,271 US firms. The test results yield clear evidence for the existence of threshold effects in both firm features. We find that abnormal returns exist when the firm size falls below 52.04 million USD and the book-to-market ratio exceeds 0.4085.