Academic journal article Quarterly Journal of Business and Economics

On the Robustness of Goodness-of-Fit Criteria for Factor Identification: Simulation and Some Korean Evidence

Academic journal article Quarterly Journal of Business and Economics

On the Robustness of Goodness-of-Fit Criteria for Factor Identification: Simulation and Some Korean Evidence

Article excerpt


A recent issue in empirical tests of the arbitrage pricing theory developed by Ross (1976) is factor identification. Roll and Ross (1980), Dhrymes, Friend, and Gultekin (1984), Cho and Taylor (1987), and Gultekin and Gultekin (1987) use the standard likelihood ratio (LR) statistic to estimate the number of factors under the exact factor structure, while Trzcinka (1986) and Brown (1989) use the eigenvalue test under the approximate factor structure. Brown (1989) points out that the eigenvalue test of factor identification has serious limitations; in a simple economy where there are only k equally important factors, eigenvalue analysis may lead one to infer falsely that only one factor is responsible for securities returns. This paper focuses on factor identification by the LR statistic in a maximum likelihood (ML) estimation.

Several financial researchers have cautioned against use of the formal LR statistic in financial applications because several violations of the assumptions of the standard factor model typically occur with actual return data. Brown and Weinstein (1985), for example, indicate that the residuals from daily stock returns are negatively autocorrelated and highly nonnormal. Perry (1982) shows that if daily stock returns are used, the serial independence assumption may not be valid. Thus, serial correlation and nonnormality frequently occur in applications of factor analysis that use actual stock returns.

A second violation may occur in empirical tests of APT. The residual portion of the covariance matrix is not a diagonal matrix and thus can be cross correlated; the residual factor may represent industry or firm size factors. Connor and Korajczyk (1988) use cross-correlated returns for simulation of an asset return series that conforms to an approximate factor model.

Another unrealistic assumption in exploratory factor analysis is that common factors are uncorrelated. In many financial applications, it is more natural to assume that economic factors are correlated. Conway and Reinganum (1988) simulate the correlated two factor structure with prior information in order to assess the robustness of factor identification.

To identify the correct number of factors under the above violations of the assumptions of the standard factor model, several goodness-of-fit criteria can be considered as alternatives to the strict test. This paper examines cross validation (CV), Akaike information criterion (AIC), and Schwartz's Bayesian criterion (SBC).

CV is a general statistical model identification method that determines whether an estimated model reflects stable features of the underlying process. It can be used to identify the stable number of factors by fitting successive models with additional factors and noting when the prediction errors begin to stabilize or increase. Large samples are needed for the CV index because the data are divided into two halves, i.e., a fitted sample and a validation sample.

Conway and Reinganum (1988) use the CV technique to identify the stable number of factors for securities returns. They find that there is one dominant factor and one minor factor that explain securities returns, regardless of the group of securities analyzed. These results contrast with other empirical evidence produced by the standard LR test (e.g., Roll and Ross (1980) find between three and five factors; Dhrymes, Friend, and Gultekin (1984) find more than five factors; and Cho and Taylor (1987) find between six and seven factors, etc.).

Cudeck and Brown (1983), however, argue that AIC and SBC may be used in place of CV for small samples. AIC and SBC include a penalty function that depends on the number of parameters fitted. SBC is an asymptotic version of a Bayes procedure that assigns positive probability to lower dimensional subspaces of the parameter space, while AIC is used to select a model that minimizes the loss function through information theory. …

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