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Unformatted text preview: 1 File: C:\WINWORD\ECONMET\Lecture4.DOC UNIVERSITY OF STRATHCLYDE ECONOMETRICS LECTURE NOTES STOCHASTIC REGRESSORS, INSTRUMENTAL VARIABLES AND WEAK EXOGENEITY 1. INTRODUCTION In certain circumstances, the OLS estimator is optimal; it is the best estimator to use in those circumstances. But the OLS estimator is not always optimal. The main purpose of these notes is to explain when another estimator  the instrumental variables (IV) estimator should be used in preference to OLS. We also show how it can be used. It is necessary to begin by establishing some notation, and to present some key results that will underpin our analysis. This comes in the following section (2). Required reading: Thomas, Chapter 8, Stochastic explanatory variables. You may find the following references useful as additional reading: Stewart, Jon: Econometrics , Ch 4 (large sample theory) and Ch 5 (Instrumental Variables) Maddala, G S: Introduction to Econometrics (2nd ed), pp 2326, 366373, 461 464, 506509. Charemza and Deadman (1992): New Directions in Econometric Practice , Ch 7. 2. SOME RESULTS CONCERNING UNBIASEDNESS AND CONSISTENCY The material in this section is a much condensed version of what is contained in Ref1.doc, and consists mainly of results explained at more length there. Suppose we are interested in obtaining "good" estimates of the parameters in the following multiple linear regression model: 2 which in matrix notation is written as The OLS estimator of the parameter vector β is given by Substituting for Y in (2) from (1) we obtain ) 3 ( u X ) X X ( + = ˆ 1 ′ ′ β β Let us now go through the properties of the OLS estimator under a number of alternative scenarios. CASE 1: NONSTOCHASTIC REGRESSORS AND ALL ASSUMPTIONS OF THE NORMAL CLASSICAL LINEAR REGRESSION MODEL SATISFIED Here, the OLS estimator is unbiased and fully efficient. CASE 2: STOCHASTIC REGRESSORS AND X INDEPENDENT OF u Just as in Case (1), the OLS estimator of β is unbiased. It is also true that the OLS estimator is efficient and consistent. CASE 3: STOCHASTIC REGRESSORS AND X and u are “asymptotically uncorrelated” In some modelling contexts, the most that can be assumed is that the equation error and the regressors are uncorrelated in the limit as the sample size goes to infinity, even though they may be correlated in finite size samples. That is, X and u are “asymptotically uncorrelated”. This may be denoted in the following way: plim 1 T X u = 0 j k j t t t T t , , ,..., . = = ∑ = 1 1 or in matrix terms t 1 2 2,t k k,t t Y = + X + X + u t = 1,...,T β β β ... + Y = X + u β (1) β = (X X) X Y1 ′ ′ (2) 3 plim 1 T X u = 0 ′ If this condition is satisfied, and provided the other assumptions of the LRM are satisfied, the OLS estimator will have some desirable asymptotic properties,. In particular, it will be consistent. Furthermore, given that estimators of the error variance (and so of coefficient standard errors) will also have this property, then the basis for valid statistical inference...
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 Fall '07
 RogerPerman
 Econometrics, Regression Analysis, OLS, ols estimator, disturbance term

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