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1、The Stata Journal (2015) 15, Number 1, pp. 121134 Fixed-eff ect panel threshold model using Stata Qunyong Wang Institute of Statistics and Econometrics Nankai University Tianjin, China QunyongW Abstract. Threshold models are widely used in macroeconomics and fi nancial analysis for their simple and
2、obvious economic implications. With these models, however, estimation and inference is complicated by the existence of nuisance parameters. To combat this issue, Hansen (1999, Journal of Econometrics 93: 345 368) proposed the fi xed-eff ect panel threshold model. In this article, I introduce a new c
3、ommand (xthreg) for implementing this model. I also use Monte Carlo simulations to show that, although the size distortion of the threshold-eff ect test is small, the coverage rate of the confi dence interval estimator is unsatisfactory. I include an example on fi nancial constraints (originally fro
4、m Hansen 1999, Journal of Econometrics 93: 345368) to further demonstrate the use of xthreg. Keywords: st0373, xthreg, panel threshold, fi xed eff ect 1Introduction Heterogeneity is a common problem of panel data. That is to say, each individual in a study is diff erent, and structural relationships
5、 may vary across individuals. The classical fi xed eff ect or random eff ect refl ects only the heterogeneity in intercepts. Hsiao (2003) considers many varying slope models for this problem. Among these models, Hansen s (1999) panel threshold model has a simple specifi cation but obvious implicatio
6、ns for economic policy. Though threshold models are familiar in time-series analysis, their use with panel data has been limited. The threshold model describes the jumping character or structural break in the re- lationship between variables. This model type is popular in nonlinear time series, one
7、example being the threshold autoregressive (TAR) model (Tong 1983). This model can capture many economic phenomena. For example, using fi ve-year interval averages of standard measures of fi nancial development, infl ation, and growth for 84 countries from 1960 to 1995, Rousseau and Wachtel (2009) s
8、howed that there is an infl ation thresh- old for the fi nance and growth relationship that lies between 1325%. When infl ation exceeds the threshold, fi nance ceases to increase economic growth. Infl ation s eff ect on economic growth depends on the infl ation level. High levels of infl ation are h
9、armful to economic growth, while low levels of infl ation are benefi cial to economic growth. As an- other example, the technical spillover of foreign direct investment (FDI) has been widely studied. Girma (2005) found that the productivity benefi t fromFDIincreases with ab- sorptive capacity until
10、some threshold level, at which point it becomes less pronounced. There is also a minimum absorptive capacity threshold level below which productivity spillovers fromFDIare negligible or even negative. c ? 2015 StataCorp LPst0373 122 Fixed-eff ect panel threshold model using Stata This article is arr
11、anged as follows. In section 2, I review some basic theories about fi xed-eff ect panel threshold models. I then describe the new xthreg command in sec- tion 3. In section 4, I perform Monte Carlo simulations to study test-power distortion and the coverage rate of confi dence interval estimators in
12、fi nite samples. I illustrate use of the command with an example from Hansen (1999) in section 5. In section 6, I conclude the article. 2 Fixed-eff ect panel threshold models 2.1Single-threshold model Consider the following single-threshold model: yit= + Xit(qit )1+ Xit(qit )2+ ui+ eit(1) The variab
13、le qitis the threshold variable, and is the threshold parameter that divides the equation into two regimes with coeffi cients 1and 2. The parameter uiis the individual eff ect, while eitis the disturbance. We can also write (1) as yit= + Xit(qit,) + ui+ eit where Xit(qit,) = ? XitI(qit ) XitI(qit )
14、Given , the ordinary least-squares estimator of is ? = X()?X()1X()?y where yand Xare within-group deviations. The residual sum of squares (RSS) is equal to ? e? e. To estimate , one can search over a subset of the threshold variable qit. Instead of searching over the whole sample, we restrict the ra
15、nge within the interval (,), which are quantiles of qit. s estimator is the value that minimizes theRSS, that is, ? = argmin S1() If is known, the model is no diff erent from the ordinary linear model.But if is unknown, there is a nuisance parameter problem, which makes the estimator s distribution
16、nonstandard. Hansen (1999) proved that ? is a consistent estimator for , and he argued that the best way to test = 0 is to form the confi dence interval using the “no-rejection region” method with a likelihood-ratio (LR) statistic, as follows: LR1() =LR1() LR1(? ) ? 2 Pr Pr(x F1), namely, the proportion of F F1in bootstrap number B. 2.2Multiple-thresholds model If there are multiple thresholds (that is, multiple regimes), we fi t the model sequentially. We use a double-thr
