A Critique of the Use of t-ratios in Model Selection

It is a common practice to use t-ratios to select models in empirical asset pricing. In this paper, we show that such a practice could lead to the acceptance of very poor models. As an illustration, we examine a simple version of the widely used cross-sectional regression methodology and find analytically that variables with the highest t-ratios may not be highly correlated with expected returns. Contrary to common belief, a high tratio may in fact be evidence of low explanatory power. The results in this study cast doubt on the economic significance of variables selected only on the basis of high t-ratios and suggest that we should include other diagnostics in addition to t-ratios in model selection. A central issue in finance is the determination of expected returns across different assets. Many theoretical models are proposed to address this issue. These models include the Capital Asset Pricing Model (CAPM) of Sharpe (1964) and Lintner (1965), Intertemporal Capital Asset Pricing Model of Merton (1973), Arbitrage Pricing Theory of Ross (1976), and Consumption Capital Asset Pricing Model of Breeden (1979). At the same time, there are many empirically motivated models which propose some firm-specific variables as explanations of the cross-sectional differences of expected returns. Some notable examples in this category are Banz (1981), Basu (1983), Bhandari (1988), Chan, Hamao, and Lakonishok (1991), and Fama and French (1992). A common feature in all these models is that the expected returns are linear in some firm-specific variables (they can be betas corresponding to some common factors or they can be some accounting ratios). In the face of so many competing models, one of the important tasks of empirical researchers is to find out which model does the best job in describing the cross-sectional differences of expected returns. In practice, this question is often addressed using regression methodologies such as cross-sectional regressions (CSR), seemingly unrelated regressions (SUR), or generalized method of moments (GMM). A generally accepted practice is to test whether a variable is “priced” by itself (i.e., its slope coefficient has a “significant” t-ratio) and whether it is “priced” in competition with other variables. For example, in the recent debate of the validity of the CAPM, the focal point is whether the CAPM betas and other competing variables are statistically significantly priced. The currently known properties of the t-ratios in various methodologies were obtained under the assumption that the model is correctly specified. Suppose a theory suggests that a certain set of variables can completely explain the cross-sectional differences of expected returns, then any other variable will not provide additional explanatory power and therefore it will not be priced. Such a test is a form of specification test of the proposed theory and it is only under this scenario that the use of t-ratios can