Threshold effects in non-dynamic panels: Estimation, testing, and inference
Introduction
Are regression functions identical across all observations in a sample, or do they fall into discrete classes? This question may be addressed using threshold regression techniques. Threshold regression models specify that individual observations can be divided into classes based on the value of an observed variable. Despite their intuitive appeal, econometric techniques have not been well developed for threshold regression.
This paper introduces econometric techniques appropriate for threshold regression with panel data. Least squares estimation methods are described. An asymptotic distribution theory is derived which is used to construct confidence intervals for the parameters. A bootstrap method to assess the statistical significance of the threshold effect is also described. The methods are similar to those developed in earlier work by the author (Hansen 1996, Hansen 1999).
The methods are used to investigate whether financial constraints affect the investment practices of firms. The classical theory of the firm suggests that financing should have no allocative effects (e.g., the Modigliani-Miller theorem). Investment decisions should only be based on the marginal Q of a specific project, since banks will be willing to extend finance. In the context of imperfect information, external financing may be limited, and debt-constrained firms may need to finance investment out of cash flow. If this is the case, investment will be correlated with cash flow for constrained firms. This observation led Fazzari et al. (1988) to divide a sample of US firms into classes based on their degree of financial constraints and estimate the differing effects of cash flow on investment among these classes. Their analysis suffered from two problems. First, they used an endogenous variable (dividend to income ratio) rather than an exogenous variable to form their sample splits. Second, they used an ad hoc method to select their sample splits. We repeat their analysis on an analogous data set using appropriate econometric techniques and find qualitatively similar results.
Other authors have investigated the implications of non-linear q models of investment. Abel and Eberly (1994) propose a model which implies that the response of investment to q may be non-linear in q. Abel and Eberly (1996) use panel data to estimate a similar model, and find evidence for non-linearities in the investment function. Barnett and Sakellaris (1998) find similar results using a threshold regression approach. Hu and Schiantarelli (1998) use a switching regression framework to study the same problem. Our paper extends and reinforces this growing literature.
The next section introduces the model and notation. Section 3 discusses estimation by fixed effects. Section 4 outlines our asymptotic theory of inference. A distribution theory is developed for the threshold estimate and the slope coefficients. Section 5 reports the empirical application to firms’ investment decisions. Section 6 concludes. Proofs of the asymptotic theory are provided in the appendix. GAUSS programs and data which replicate the empirical work are available from the author's homepage.
Section snippets
Model
The observed data are from a balanced2 panel . The subscript i indexes the individual and the subscript t indexes time. The dependent variable yit is scalar, the threshold variable qit is scalar, and the regressor xit is a k vector. The structural equation of interest iswhere I(·) is the indicator function. An alternative intuitive way of writing (1) is
Least squares estimation
One traditional method to eliminate the individual effect μi is to remove individual-specific means. While straightforward in linear models, the non-linear specification (1) calls for a more careful treatment. Note that taking averages of (1) over the time index t produceswhere , andTaking the difference between , yieldswhereand
Testing for a threshold
It is important to determine whether the threshold effect is statistically significant. The hypothesis of no threshold effect in (1) can be represented by the linear constraintUnder H0 the threshold γ is not identified, so classical tests have non-standard distributions. This is typically called the ‘Davies’ Problem’ (see Davies 1977, Davies 1987) and has been recently investigated by Andrews and Ploberger (1994) and Hansen (1996). The fixed-effects equations (4) fall in the class of
Multiple thresholds
Model (1) has a single threshold. In some applications there may be multiple thresholds. For example, the double threshold model takes the formwhere the thresholds are ordered so that γ1<γ2. We will focus on this double-threshold model since the methods extend in a straightforward manner to higher-order threshold models. We discuss three relevant statistical issues: (1) Estimation; (2) Testing for the presence of a double threshold;
Investment and financing constraints
Classical models of the firm assume the existence of perfect financial markets on which firms can borrow the needed resources for investment projects. Alternative models of financing place restrictions on the extent of external financing. An important empirical question is whether or not there exist firms which behave as though they are subject to such constraints.
A well-cited paper which explored the empirical implications of financing constraints is Fazzari et al. (1988), henceforth FHP.
Conclusion
This paper has developed new empirical methods for panel data. We have defined a threshold regression model with individual-specific effects, and shown that the model is rather straightforward to estimate using a fixed-effects transformation. The asymptotic theory is non-standard, but confidence intervals for the threshold can be constructed by inverting the likelihood ratio statistic, and this construction is a natural by-product of the estimation method.
The methods are applied to the
Acknowledgements
This research was supported by a grant from the National Science Foundation and a Sloan Foundation Research Fellowship. Special thanks go to Fabio Schiantarelli for many helpful discussions and motivation, to Richard Blundell for an insightful discussion, two referees for correcting some of my errors, and to Maria Laura Parisi for research assistance.
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Homepage: http://www.ssc.wisc.edu/bhansen/.