A weighted Fama-MacBeth two-step panel regression procedure

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 $R^2$ distribution.

KEYWORDS:

• Fama-MacBeth regression
• simulation
• efficiency
• power

JEL CLASSIFICATION:

• G00
• C23

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

¹ Pertersen (2009) 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 $b$ 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 1992; Shanken and Zhou 2007; Bai and Zhou 2015).

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 ${\hat{b}}_t$ using the Newey and West (1987) procedure. However, in both modules, $\hat{b}$, the equal-weight average of ${\hat{b}}_t$, is used as an estimator of $b$, which differs from our study.

⁴ The theorem is a well-known result in statistics and econometrics (see, e.g., Amemiya (1994, p. 130)).

⁵ $t\left(df\right)$ represents the t distribution with $df$ as the degree of freedom. If X~t(df), then $Var\left(X\right)=df/\left(df-2\right)$.

⁶ For example, in Table 7 of Petersen (2009), which shows a firm-level corporate finance study, the average $R^2$ over time is 13%. In addition, in empirical finance studies, the average $R^2$ of about 15% is considered to be fairly large. For example, in Table 6 of Petersen (2009), which shows a stock-level asset pricing study, the average $R^2$ over time is only 0.08%. In many country-level international finance studies; however, the average $R^2$ over time could be larger with about 30% to 50% (e.g. Morck, Yeung, and Yu 2000; Jin and Myers 2006; Eun, Wang, and Xiao 2015). Thus, in our later analysis, we also considered simulation setups in which $R_t^2$ was varied over [10%, 50%] or [30%, 70%] evenly over the entire time period considered.

⁷ For Models 1A and 1B, the regression $R^2$ for time period $t$ is $R_t^2=b_1^{2}Var\left(x_{1,t}\right)/\left[b_1^{2}Var\left(x_{1,t}\right)+Var\left({ϵ}_t\right)\right]$. In our simulation setup, modeling a particular form of time-varying regression $R_t^2$ is equivalent to modeling a corresponding particular form of time-varying error variance ${\sigma }_t^2$.

⁸ For the multivariate t distribution, refer to Muirhead (1982, p. 48) and Anderson (2003, p. 289).

⁹ We also considered $\rho =0$ or $\rho =-0.5$ 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 $\hat{\theta }$ is an estimator of the true parameter value $\theta$, then the MSE of $\hat{\theta }$ is defined as $E\left[{\left(\hat{\theta }-\theta \right)}^2\right]={\left(E\left[\hat{\theta }\right]-\theta \right)}^2+E\left[{\left(\hat{\theta }-E\left[\hat{\theta }\right]\right)}^2\right]$ (see, e.g., Greene (2012, p. 1097) or Shao (2010, 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 (1994, p. 123) or Definitions C.3 and C.4 in Greene (2012, pp. 1096–1097)). Specifically, let ${\hat{\theta }}_1$ and ${\hat{\theta }}_2$ be two estimators of $\theta$. Then, ${\hat{\theta }}_1$ is said to be more efficient than ${\hat{\theta }}_2$ if $E\left[{\left({\hat{\theta }}_1-\theta \right)}^2\right]\le E\left[{\left({\hat{\theta }}_2-\theta \right)}^2\right]$ for all $\theta$ and $E\left[{\left({\hat{\theta }}_1-\theta \right)}^2\right]<E\left[{\left({\hat{\theta }}_2-\theta \right)}^2\right]$ at least one $\theta$.

¹² In empirical finance studies that use the FMB two-step panel regression procedure, the average $R^2$ over time of 30% or 50% is considered to be very large.