تخمین مبتنی بر برنامه ریزی خطی در رگرسیون خطی ساده

In this paper we introduce a linear programming estimator (LPE) for the slope parameter in a constrained linear regression model with a single regressor. The LPE is interesting because it can be superconsistent in the presence of an endogenous regressor and, hence, preferable to the ordinary least squares estimator (LSE). Two different cases are considered as we investigate the statistical properties of the LPE. In the first case, the regressor is assumed to be fixed in repeated samples. In the second, the regressor is stochastic and potentially endogenous. For both cases the strong consistency and exact finite-sample distribution of the LPE is established. Conditions under which the LPE is consistent in the presence of serially correlated, heteroskedastic errors are also given. Finally, we describe how the LPE can be extended to the case with multiple regressors and conjecture that the extended estimator is consistent under conditions analogous to the ones given herein. Finite-sample properties of the LPE and extended LPE in comparison to the LSE and instrumental variable estimator (IVE) are investigated in a simulation study. One advantage of the LPE is that it does not require an instrument.

The use of certain linear programming estimators in time series analysis is well documented. See, for instance, Davis and McCormick (1989), Feigin and Resnick (1994) and Feigin et al. (1996). LPEs can yield much more precise estimates than traditional methods such as conditional least squares (e.g. Datta et al., 1998 and Nielsen and Shephard, 2003). The limited success of these estimators in applied work can be partially explained by the fact that their point process limit theory complicates the use of their asymptotics for inference (e.g. Datta and McCormick, 1995). In regression analysis, it is well known that the ordinary least squares estimator is inconsistent for the regression parameters when the error term is correlated with the explanatory variables of the model. In this case an instrumental variables estimator or the generalized method of moments may be used instead. In economics, such endogenous explanatory variables could be caused by measurement error, simultaneity or omitted variables. To the authors’ knowledge, however, there has so far been no attempt to investigate the statistical properties of LP-based estimators in a cross-sectional setting. In this paper we show that LPEs can, under certain circumstances, be a preferable alternative to LS and IV estimators for the slope parameter in a simple linear regression model. We look at two types of regressors which are likely to be of practical importance. First, we introduce LPEs to the simple case of a non-stochastic regressor. Second, we consider the general case of a stochastic, and potentially endogenous, regressor. For both cases we establish the strong consistency and exact finite-sample distribution of a LPE for the slope parameter. The LPE can be used in situations where the regressor is strictly positive. For example, in empirical finance, we can consider regressions involving volatility and volume. In labor economics a possible application is the regression between income and schooling, for example. The remainder of the paper is organized as follows. In Section 2, we establish the strong consistency and exact finite-sample distribution of the LPE when (1) the explanatory variable is non-stochastic, and (2) the explanatory variable is stochastic and potentially endogenous. In Section 3, we discuss how our results can be extended to other endogenous specifications and give conditions under which the LPE is consistent in the presence of serially correlated, heteroskedastic errors. We also describe how the LPE can be extended to the case with multiple regressors. Section 4 reports the simulation results of a Monte Carlo study comparing the LPE and extended LPE to the LSE and IVE. Section 5 concludes. Mathematical proofs are collected in the Appendix. An extended Appendix available on request from the authors contains some results mentioned in the text but omitted from the paper to save space.

In this paper we have established the exact finite-sample distribution of a LPE for the slope parameter in a constrained simple linear regression model when (1) the regressor is non-stochastic, and (2) the regressor is stochastic and potentially endogenous. The exact distribution may be used for statistical inference or to bias-correct the LPE. In addition, we have shown that the LPE is strongly consistent under fairly general conditions on the related distributions. In particular, the LPE is robust to various heavy-tailed specifications and its functional form indicates that it can be insensitive to outliers in yiyi or xixi. We have also identified a number of cases where the LPE is superconsistent. In contrast to existing results for the LPE, in a time series setting, our results in a cross-sectional setting are valid also in the case when the slope parameter is negative. We provided conditions under which the LPE is consistent in the presence of serially correlated, heteroskedastic errors and described how the LPE can be extended to the case with multiple regressors. Our simulation results indicated that the LPE and extended LPE can have very reasonable finite-sample properties compared to the LSE and IVE, also in the presence of heteroskedastic or serially correlated errors. Clearly, one advantage of the LPE is that, in contrast to the IVE, it does not require an instrument.