Lecture 7

# Lecture 7 - LECTURE 7: INFERENCE WITH OLS PDFs of OLS...

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LECTURE 7: INFERENCE WITH OLS PDF’s of OLS ESTIMATORS Need to add one more assumption to CLRM: Classical Linear Regression Model (Revisited) Assumptions of the CLRM: 1. For each i, the population regression function of Y i given (X i1 , X i2 , X i3 , …, X ik ) is linear , i.e. Y i = β 0 + β 1 X i1 + β 2 X i2 + β 3 X i3 + … + β k X ik + ε i (PRF) 2. (X i1 , X i2 , X i3 , … X ik ) are nonstochastic variables (i.e., their values are fixed numbers in repeated samples) 3. The expected, or mean, value of the disturbance term ε i is zero: E( ε i ) = 0 4. The variance of each ε i is constant for all i, that is, ε i is homoskedastic : Var( ε i ) = σ 2 5. There is no correlation between two error terms. This is the assumption of no autocorrelation or no serial correlation : Cov( ε i , ε j ) = 0 i j 6. No exact collinearity exists between X 1 and X 2 . 7. In the PRF Y i = β 0 + β 1 X i + ε i ε i is normally distributed ε i ~ N(0, σ 2 )

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Assumption 6 is equivalent to assuming that Y is normally distributed with mean equal to β 0 + β 1 X 1 + β 2 X 2 + … + β k X k and variance equal to σ 2 : Y ~ N( β 0 + β 1 X 1 + β 2 X 2 + … + β k X k , σ 2 ) Normality may be a bad assumption, for example for non-negative variables (e.g. wages, prices) or for variables that take on only a small number of values. Sometimes taking a nonlinear transformation (e.g. taking the natural logarithm) of the dependent variable makes normality plausible. Normality is a convenient assumption because it implies normality of the OLS estimators (since they are linear functions of the normal Y’s). We know that a linear function of a normally distributed variable is itself normally distributed. If our PRF is: Y i = β 0 + β 1 X i + ε i Since our OLS estimators for 0 β ˆ and 1 β ˆ are linear functions of ε i , then they themselves are normally distributed. ) σ , β ( N ~ β ˆ 2 β ˆ 0 0 0 2 2 i 2 i 0 2 β ˆ σ ) X X n X ) β ˆ var( σ 0 = = ) σ , β ( N ~ β ˆ 2 β ˆ 1 1 1 = = 2 i 2 1 2 1 β ˆ ) X X ( σ ) β ˆ var( σ More generally: ) 1 , 0 ( N ~ ) β ˆ ( se β β ˆ j j j
FUN FACTS The (conditional) standard error of j β ˆ depends on the unknown σ 2 . If we use the unbiased estimator = = n 1 i 2 i 2 ε ˆ k n 1 σ ˆ For estimating the standard error of j β ˆ , then the distribution of the standardized j β ˆ is no longer standard normal but will be asymptotically normal: ) 1 , 0 ( N ~ hat ))^ β ˆ ( se ( β β ˆ A j j j Even if assumption 7 does not hold

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## This note was uploaded on 03/20/2009 for the course ECON 103 taught by Professor Sandrablack during the Winter '07 term at UCLA.

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Lecture 7 - LECTURE 7: INFERENCE WITH OLS PDFs of OLS...

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