lecture_note_102316

lecture_note_102316 - Econ 5280 Applied Econometrics...

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Chapter 4 Linear Model without Normal Residuals 4.1 Introduction and Preliminaries Recall that the assumptions of the classical linear model. 1. Dataset is a sequence of IID observations: {( Y t , X t ) R k + 1 t = 1 , 2 , ,n } ; 2. For some unknown β * , E ( Y t X t ) = X t β * , where X t = X t 1 X t 2 X tk , and β = β 1 β 2 β k ; 3. n t = 1 ( X t X t ) is invertible; 4. U t X t N ( 0 , σ 2 * ) , where U t = Y t - E ( Y t X t ) . From now, we remove the normality assumption in the fourth. This is too strong in the sense that most data do not reveal distributional properties to researchers. The main goal of this elimination is to acquire the applicability of model analysis. Nevertheless, we cannot expect that data analysis without the final assumption will deliver the same or better results than we could obtain from the CLM case. We instead suppose that the researcher has many observations to complement this missing information. When there are many data observations, we can derive similar results to the CLM case from the OLS estimator although the normality assumption may not hold. 46
Here are some properties we exploit many times from now: 1. Law of Large Numbers (LLN): If a sequence of random variables { X t } is IID and E [ X t ] < , then 1 n n t = 1 X t a.s. E [ X t ] . 2. Central Limit Theorem (CLT): If a sequence of random variables { X t } is IID and E [ X 2 t ] < , then 1 n n t = 1 ( X t - E [ X t ]) A N [ 0 , var ( X t )] . 3. Cauchy-Schwarz Inequality • For any two sequences { a i } and { b i } ( i = 1 , 2 , ,n ) , n i = 1 a i b i n i = 1 a 2 i 1 2 n i = 1 b 2 i 1 2 . • If f ( x ) and g ( x ) are given such that b a f ( x ) 2 dx < and b a g ( x ) 2 dx < , then b a f ( x ) g ( x ) dx b a f ( x ) 2 dx 1 2 b a g ( x ) 2 dx 1 2 . • For any random variable X with PDF φ ( ) , f ( x ) and g ( x ) are such that f ( x ) 2 φ ( x ) dx < and g ( x ) 2 φ ( x ) dx < , then f ( x ) g ( x ) φ ( x ) dx f ( x ) 2 φ ( x ) dx 1 2 g ( x ) 2 φ ( x ) dx 1 2 . That is, E φ [ f ( X ) g ( X ) ] E φ [ f ( X ) 2 ] 1 2 E φ [ g ( X ) 2 ] 1 2 , where subscript φ indicates that the random variable X has PDF φ ( ) . Unless confusions otherwise arises, we will omit the subscript indicating PDF from now. 4.2 Extension 1: Model with Conditional Homoskedasticity 4.2.1 Assumptions 1. Dataset is a sequence of IID observations: {( Y t , X t ) R k + 1 t = 1 , 2 , ,n } ; 47

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2. For some unknown β * , E ( Y t X t ) = X t β * ; 3. E [ X t X t ] is positive definite ; 4. E [ X 2 tj ] < ,j = 1 , 2 ,...,k ; 5. U t X t ( 0 2 * ) with σ 2 * < , where U t = Y t - E [ Y t X t ] . 4.2.2 Properties of OLS Estimator As before, we note that there are two unknown parameters β * and σ 2 * . We estimate this by the OLS estimator and the unbiased estimator defined before: ̂ β n = ( X X ) - 1 X Y ; ̂ σ 2 n = ( n - k ) - 1 ̂ U ̂ U , where ̂ U = Y - X ̂ β n .
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