ABSTRACT
We propose a weighted Fama-MacBeth (FMB) two-step panel regression procedure and compare the properties of the usual unweighted versus our proposed weighted FMB procedures through a Monte Carlo simulation study. We find evidence that when the cross-sectional regression explanatory power changes over time as well as the standard errors of the coefficient estimates, the proposed weighted FMB procedure produces more efficient coefficient estimators and more powerful tests compared to the usual unweighted FMB procedure across various model specifications in terms of the sampling distribution, sample size, and time-series distribution.
Disclosure statement
No potential conflict of interest was reported by the authors.
Notes
1 Pertersen (Citation2009) presents a list of empirical studies that have used the FMB procedure.
2 It should be noted that the FMB procedure considered in this paper refers to the above two-step procedure to estimate and its standard error in the panel regression of Equation (1). We are not referring to the so-called FMB two-pass regression procedure that is used to get factor risk premia estimates for a linear multifactor asset pricing model (Shanken Citation1992; Shanken and Zhou Citation2007; Bai and Zhou Citation2015).
3 For example, the STATA built-in module ‘xtfmb.ado’ implements the above two-step panel regression procedure. Another STATA module ‘xtfmbj.ado’ made available by Judson Caskey enables us to use an adjusted standard error estimate in the second step that considers heteroskedasticities and autocorrelations in the time series estimates using the Newey and West (Citation1987) procedure. However, in both modules, , the equal-weight average of , is used as an estimator of , which differs from our study.
4 The theorem is a well-known result in statistics and econometrics (see, e.g., Amemiya (Citation1994, p. 130)).
5 represents the t distribution with as the degree of freedom. If X~t(df), then .
6 For example, in Table 7 of Petersen (Citation2009), which shows a firm-level corporate finance study, the average over time is 13%. In addition, in empirical finance studies, the average of about 15% is considered to be fairly large. For example, in of Petersen (Citation2009), which shows a stock-level asset pricing study, the average over time is only 0.08%. In many country-level international finance studies; however, the average over time could be larger with about 30% to 50% (e.g. Morck, Yeung, and Yu Citation2000; Jin and Myers Citation2006; Eun, Wang, and Xiao Citation2015). Thus, in our later analysis, we also considered simulation setups in which was varied over [10%, 50%] or [30%, 70%] evenly over the entire time period considered.
7 For Models 1A and 1B, the regression for time period is . In our simulation setup, modeling a particular form of time-varying regression is equivalent to modeling a corresponding particular form of time-varying error variance .
8 For the multivariate t distribution, refer to Muirhead (Citation1982, p. 48) and Anderson (Citation2003, p. 289).
9 We also considered or in Models 2A and 2B. The results obtained therefrom remained qualitatively the same as those reported here.
10 The mean squared error (MSE) of an estimator is defined as the sum of the bias-squared and variance of the estimator. Mathematically, if is an estimator of the true parameter value , then the MSE of is defined as (see, e.g., Greene (Citation2012, p. 1097) or Shao (Citation2010, p. 123)).
11 In statistics and econometrics, the usual criterion to rank estimators is the MSE (see, e.g., Definition 7.2.1 in Amemiya (Citation1994, p. 123) or Definitions C.3 and C.4 in Greene (Citation2012, pp. 1096–1097)). Specifically, let and be two estimators of . Then, is said to be more efficient than if for all and at least one .
12 In empirical finance studies that use the FMB two-step panel regression procedure, the average over time of 30% or 50% is considered to be very large.