Testing for Slope Heterogeneity Bias in Panel Data Models

ABSTRACT

Standard econometric methods can overlook individual heterogeneity in empirical work, generating inconsistent parameter estimates in panel data models. We propose the use of methods that allow researchers to easily identify, quantify, and address estimation issues arising from individual slope heterogeneity. We first characterize the bias in the standard fixed effects estimator when the true econometric model allows for heterogeneous slope coefficients. We then introduce a new test to check whether the fixed effects estimation is subject to heterogeneity bias. The procedure tests the population moment conditions required for fixed effects to consistently estimate the relevant parameters in the model. We establish the limiting distribution of the test and show that it is very simple to implement in practice. Examining firm investment models to showcase our approach, we show that heterogeneity bias-robust methods identify cash flow as a more important driver of investment than previously reported. Our study demonstrates analytically, via simulations, and empirically the importance of carefully accounting for individual specific slope heterogeneity in drawing conclusions about economic behavior.

KEY WORDS:

• Bias
• Corporate investment
• Fixed effects
• Individual heterogeneity
• Slope heterogeneity
• Testing.

ACKNOWLEDGMENTS

The authors are grateful to the editor, associate editor, and two anonymous referees for their comments that improved the exposition of the article. In addition, we thank Cecilia Bustamante, Zongwu Cai, George Gao, Erasmo Giambona, Stefan Hoderlein, Hyunseob Kim, Oliver Linton, Mitchell Petersen, Whitney Newey, Suyong Song, Albert Wang, Zhijie Xiao, and the participants at seminars at Boston College, Chapman University, Claremont McKenna College, University of Illinois at Urbana-Champaign, University of Iowa, University of Kansas, University Wisconsin-Milwaukee, 25th Midwest Econometrics Group, and NY Camp Econometrics VIII for their constructive comments and suggestions.

Notes

1 The model in Equation (1(1) $$y_{it}={\alpha }_i+x_{it}^{\top }{\beta }_i+u_{it},i=1,\dots ,N,t=1,\dots ,T,$$(1) ) can also be interpreted as a random coefficient model. Although we focus on the interpretation of heterogeneous slopes, we refer the reader to Hsiao and Pesaran (2008) and Wooldridge (2010) for detailed discussions of random coefficient models.

2 Pesaran and Smith (1995) showed that in a dynamic panel data model, heterogeneity causes bias in any case.

3 This estimator was fully developed by Pesaran (2006) and belongs to the class of minimum distance estimators (see Newey and McFadden 1994).

4 The test does not depend on other relationships that might exist between β _i and X_i . The moment condition depends on the correlation between β _i and X ^⊤ _i M _ι X_i . We thank an anonymous referee for asking us to check this possibility.

5 We conducted other experiments where the errors were serially correlated. The results were similar and are available upon request.

6 Since the data used in many empirical applications face limitations on the times series dimension, T, our main presentation focuses on variations along this dimension. However, in unreported tables we also experiment with variations in the number of individuals, N (e.g., N = 100 and N = 1000). These alternative experiments lead to similar inferences and are readily available from the authors.

7 A number of papers estimate investment–cash flow sensitivities for sample partitions based on proxies for financial constraints (e.g., firm size or existence of bond ratings). These estimations are also subject to the firm heterogeneity biases that we highlight in our article.

8 The smoother we employ is known as the Friedman Super Smoother (see Friedman 1984).