intro-ects-handout-5

intro-ects-handout-5 - Introductory Econometrics ECON0701...

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Introductory Econometrics ECON0701 (2009) 66 5. Multiple regression analysis: Inference We now introduce hypothesis testing and interval estima- tion We begin by f nding the distributions of the OLS estimators un- der the added assumption that the population error is normally distributed. In developing the inference procedures we will be using the nor- mal distribution and other related distributions; you may want to review the materials in Section 2 5.1. Sampling distributions of the OLS estimators the results in the previous section depend on assumptions MLR1 to MLR5 with these assumptions, we know E ³ b β j ´ in (4.6) and var ³ b β j ´ in (4.9): E ³ b β j ´ = β j , (4.6) var ³ b β j ´ = σ 2 TSS j ³ 1 R 2 j ´ , (4.9) where TSS j = n X i =1 ( x ij x j ) 2 is the total sample variation in x j ,and R 2 j is the R -squared from regressing x j on all other explanatory variables (and a constant).
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Introductory Econometrics ECON0701 (2009) 67 in order to perform statistical inference, we need to know more than just the f rst two moments of b β j ; we need to know the full sampling distribution of the b β j . To make the sampling distributions of the b β j tractable, we now assume that the unobserved error is normally distributed in the population. Assumption MLR6 (Normality): The population error ε is independent of the explanatory vari- ables x 1 ,x 2 , ..., x k and is normally distributed with zero mean and variance σ 2 ε N ³ 0 , σ 2 ´ (5.1) Assumption MLR6 is much stronger than any of our previous assumptions: if we make assumption MLR6, then we are neces- sarily assuming MLR4 and MLR5. For cross-sectional regression applications, assumptions MLR1 to MLR6 are called the classical linear model (CLM) assump- tions. Result: Normal Sampling Distributions (Theorem 4.1 in Wooldridge, 2008) Under the CLM assumptions MLR1 to MLR6, conditional on the sample values of the explanatory variables, b β j N ³ β j ,var ³ b β j ´´ . (5.2) Therefore, b β j β j sd ³ b β j ´ N (0 , 1) . (5.3)
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Introductory Econometrics ECON0701 (2009) 68 Why? (a) the OLS estimator of β j , b β j , can be expressed as a linear combination of y i ’s (b) we make the additional assumption that the random error ε i is normally distributed with mean 0 and variance σ 2 therefore, b β j is normal since linear combinations of normal random vari- ables are also normal. 5.2. An important random variable Result: t Distribution for the Standardized Estimators (Theorem 4.2 in Wooldridge, 2008) using b β j and se ³ b β j ´ for j =0 , 1 , ..., k as well as the normality assumption, the random variable t can be formed, and it has a t -distribution with n ( k +1) degrees of freedom t = b β j β j se ³ b β j ´ t ( n k 1) ,j =0 , 1 , ..., k (5.4) Why? (a)
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intro-ects-handout-5 - Introductory Econometrics ECON0701...

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