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Unformatted text preview: ECON 103, Lecture 8: Inference with OLS Maria Casanova February 4th (version 0) Maria Casanova Lecture 8 Requirements for this lecture: Chapter 5 of Stock and Watson Maria Casanova Lecture 8 1. Distribution of OLS estimators Lets consider the following linear regression model: Y i = + 1 X 1 i + ... + k X ki + i , i  X N (0 , 2 ) This model maintains the least square assumptions (assumptions 1. to 4. in lecture 7) plus two additional ones: Ass5: The variance of each i is constant given X , that is, i is homoskedastic. Ass6: Given X , i is normally distributed. Maria Casanova Lecture 8 1. Distribution of OLS estimators Under assumptions 1. to 6. we can derive exact distribution for the OLS estimators. In particular, if is normally distributed conditional on X , so is Y (as it is a linear function of ): Y  X N ( E ( Y  X ) , Var ( Y  X )) The mean of Y is equal to: E ( Y  X ) = E ( + 1 X 1 + ... + k X k +  X ) = = E ( + 1 X 1 + ... + k X k  X ) + E (  X ) = + 1 X 1 + ... + k X k Maria Casanova Lecture 8 1. Distribution of OLS estimators The variance of Y is equal to: Var ( Y  X ) = Var ( + 1 X 1 + ... + k X k +  X ) = = Var ( + 1 X 1 + ... + k X k  X ) + Var (  X ) = 2 Then: Y N ( + 1 X 1 + ... + k X k , 2 ) Normality may be a bad assumption, for example for nonnegative variables (e.g. wages, prices) or for variables that take on only a small number of values. Sometimes taking a nonlinear transformation of Y (e.g. taking the natural logarithm) makes normality more plausible. Maria Casanova Lecture 8 1. Distribution of OLS estimators Normality is a convenient assumption because it implies that the OLS estimators are exactly normally distributed (since they are linear functions of Y ). Therefore, N ( , 2 ) , 2 = X 2 i n ( X i X ) 2 2 1 N ( 1 , 2 1 ) , 2 1 = 1 ( X i X ) 2 2 More generally: j j j N (0 , 1) Maria Casanova Lecture 8 1. Distribution of OLS estimators The (conditional) variance of j depends on the unknown parameter 2 . In practice, we substitute it with its unbiased estimator: 2 = 1 n k n X i =1 2 As a consequence of this substitution, the distribution of the standardized j is no longer standard normal but a t with n k degrees of freedom: j j s . e . ( j ) t n k The t distribution converges to a normal...
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This note was uploaded on 03/15/2010 for the course ECON 103 taught by Professor Sandrablack during the Winter '07 term at UCLA.
 Winter '07
 SandraBlack
 Econometrics

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