1.4 Model selection (bonus) Consider the linear regression model with only an intercept Y₁ = ß+ et. Assume that et ~ N(0

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1 4 Model Selection Bonus Consider The Linear Regression Model With Only An Intercept Y Ss Et Assume That Et N 0 1 (121.72 KiB) Viewed 54 times

1.4 Model selection (bonus) Consider the linear regression model with only an intercept Y₁ = ß+ et. Assume that et ~ N(0,02) with density 1 e 202 V2πσ2 • Write the likelihood function for et written as a function of Yt and Xț(= 1). • Write the full likelihood function for all T observations, assuming each observation is independent of the other (call this function L()). • Write the full log likelihood function (call this function In L()). • Maximize the full log likelihood function wrt to B and o2. You will find two formulas (estimators). One for 3 and one for ². • Interpret the estimator (what will it estimate?) • Substitute your solution for o², ô², into ln L and show that it can be written as T In L = - In(27) 2 T T 2 2 -In(SSR/T) = C- T 2 In(SSR/T) • The AIC is equal the −2 ln L()+2K where K is the total number of parameters estimated (2 here). Ignoring the constant term C show that the AIC found above is the same as the one from your notes. The AIC contains two parts: 1) −2 ln L measures the amount of information contained in our model/data (this measures the benefits of a model) and 2) 2K which counts the number of parameters in a model (this measures the costs of a model; adding parameters increases model complexity).