1.4 Model selection (bonus) Consider the linear regression model with only an intercept Yt = B + et. Assume that et ~ N(0, 02) with density 202 V2702 . Write the likelihood function for et written as a function of Yt and Xt(= 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 B and one for 62. . Interpret the estimator B (what will it estimate?) . Substitute your solution for o, 62, into In L and show that it can be written as In L = - 2 -In (2TT) - T T 2 2 In(SSR/T) = C- - In(SSR/T) . The AIC is equal the -2In 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) -2In 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)