ETW2410 Week 4 (Chp 4)

ETW2410 Week 4 (Chp 4) - ETW2410 Introductory Econometrics...

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ETC2410 1 Introductory Econometrics Lecture Slides Week 4 ETW2410
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ETC2410 2 4. Multiple Regression Analysis y = b 0 + b 1 x 1 + b 2 x 2 + . . . b k x k + u 2. Inference
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ETC2410 3 At the end of this chapter, students will (1) have a comprehensive understanding of the underlying assumptions of the classical linear regression (2) be able to interpret, evaluate and apply inferential methods to multiple linear regression (3) to test a single hypothesis involving more than one parameter and (4) determine whether a group of independent variables can be omitted from a model by testing multiple restrictions
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ETC2410 4 Assumptions of the Classical Linear Model (CLM) So far, we know that given the Gauss- Markov assumptions, OLS is BLUE In order to do classical hypothesis testing, we need to add another assumption (beyond the Gauss-Markov assumptions) Assume that u is independent of x 1 , x 2 ,…, x k and u is normally distributed with zero mean and variance s 2 : u ~ Normal(0, s 2 )
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ETC2410 5 CLM Assumptions (cont) Under CLM, OLS is not only BLUE, but is the minimum variance unbiased estimator We can summarize the population assumptions of CLM as follows y| x ~ Normal( b 0 + b 1 x 1 +…+ b k x k , s 2 ) While for now we just assume normality, clear that sometimes not the case Large samples will let us drop normality
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ETC2410 6 . . x 1 x 2 The homoskedastic normal distribution with a single explanatory variable E( y | x ) = b 0 + b 1 x y f( y|x ) Normal distributions
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ETC2410 7 Normal Sampling Distributions j Under the CLM assumptions, conditional on the sample values of the independent variables ˆ ˆ ~ Normal , , so that ˆ ~ Normal 0,1 ˆ ˆ is distributed normally because it is a linear combina j j j j j j Var sd b b b b b b b tion of the errors (refer to Wooldridge, page 120) (Wooldridge- page 740 Appendix B)
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ETC2410 8 The t Test 1 : freedom of degrees the Note ˆ by estimate to have we because normal) (vs on distributi a is this Note ~ ˆ ˆ s assumption CLM Under the 2 2 1 j k n t t se k n j j s s b b b
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ETC2410 9 The t Test (cont) Knowing the sampling distribution for the standardized estimator allows us to carry out hypothesis tests Start with a null hypothesis For example, H 0 : b j =0 Under the null, x j has no effect on y , controlling for other x ’s
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ETC2410 10 The t Test (cont) 0 ˆ j H , hypothesis null reject the o whether t determine to rule rejection a with along statistic our use then will We ˆ ˆ : ˆ for statistic the" " form to need first e our test w perform To t se t t j j j b b b b
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ETC2410 11 t Test: One-Sided Alternatives Besides our null, H 0 , we need an alternative hypothesis, H 1 , and a significance level H 1 may be one-sided, or two-sided H 1 : b j > 0 and H 1 : b j < 0 are one-sided H 1 : b j 0 is a two-sided alternative If we want to have only a 5% probability of rejecting H 0 if it is really true, then we say our significance level is 5%
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ETC2410 12 One-Sided Alternatives (cont) Having picked a significance level, a , we look up the (1 a ) th percentile in a t distribution with n k 1 df and call this c ,
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  • Three '14
  • Null hypothesis, Statistical hypothesis testing, significance level, critical value

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