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Unformatted text preview: ECON 103, Lecture 8: Inference with OLS Maria Casanova April 23rd (version 2) Maria Casanova Lecture 8 Requirements for this lecture: Chapter 5 of Stock and Watson Maria Casanova Lecture 8 1. Distribution of OLS estimators Let’s consider the following linear regression model: Y i = β + β 1 X 1 i + ... + β k X ki + ε i , ε i  X ∼ N (0 , σ 2 ) This model maintains the least square assumptions (assumptions 1. to 4. in lecture 7) plus two additional ones: Ass5: The variance of each ε i is constant given X , that is, ε i is homoskedastic. Ass6: Given X , ε 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 ε is normally distributed conditional on X , so is Y (as it is a linear function of ε ): Y  X ∼ N ( E ( Y  X ) , Var ( Y  X )) The mean of Y is equal to: E ( Y  X ) = E ( β + β 1 X 1 + ... + β k X k + ε  X ) = = E ( β + β 1 X 1 + ... + β k X k  X ) + E ( ε  X ) = β + β 1 X 1 + ... + β k X k Maria Casanova Lecture 8 1. Distribution of OLS estimators The variance of Y is equal to: Var ( Y  X ) = Var ( β + β 1 X 1 + ... + β k X k + ε  X ) = = Var ( β + β 1 X 1 + ... + β k X k  X ) + Var ( ε  X ) = σ 2 Then: Y ∼ N ( β + β 1 X 1 + ... + β k X k , σ 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, ˆ β ∼ N ( β , σ 2 ˆ β ) , σ 2 ˆ β = ∑ X 2 i n ∑ ( X i ¯ X ) 2 σ 2 ˆ β 1 ∼ N ( β 1 , σ 2 ˆ β 1 ) , σ 2 ˆ β 1 = 1 ∑ ( X i ¯ X ) 2 σ 2 More generally: ˆ β j β j σ ˆ β j ∼ N (0 , 1) Maria Casanova Lecture 8 1. Distribution of OLS estimators The (conditional) variance of ˆ β j depends on the unknown parameter σ 2 . In practice, we substitute it with its unbiased estimator: ˆ σ 2 = 1 n k n X i =1 ˆ ε 2 As a consequence of this substitution, the distribution of the standardized ˆ β j is no longer standard normal but a t with n k degrees of freedom: ˆ β j β j s . e . ( ˆ β j ) ∼ t n k The t distribution converges to a normal...
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This note was uploaded on 02/04/2010 for the course ECON 103 taught by Professor Sandrablack during the Spring '07 term at UCLA.
 Spring '07
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