# D 10 points what would be the advantages and

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d.) (10 points) What would be the advantages and disadvantages of adding a constant term ( x t 3 = 1 for all t ) to the regression? 2.) (20 points total) Let { Z 1 , ..., Z m } and { W 1 , .... , W n } be sets of independent standard Normal variables, that is, each variable is N (0 , 1) and is independent of all the others. a.) (10 points) What name would you give for the distribution of the following variable? ( Z 2 1 + Z 2 2 + · · · + Z 2 m ) /m ( W 2 1 + W 2 2 + · · · + W 2 n ) /n b.) (10 points) What name would you give for the limiting distribution of the above variable as n → ∞ ?

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–2– Econ 220B, Winter 2011 James Hamilton 3.) (45 points total) Let { y t , x t } be strictly stationary and ergodic and consider y t = x 0 t β + ε t . Let y be a ( T × 1) vector of observed values for y t and X a ( T × k ) matrix of observations on x t . Suppose there is a true value β 0 such that for ε = y X β 0 it is the case that E ( ε | X ) = 0 E ( εε 0 | X ) = σ 2 V ( X ) V ( X ) ( T × T ) = x 11 0 · · · 0 0 x 21 · · · 0 . . . . . . · · · . . . 0 0 · · · x T 1 where x t 1 denotes the fi rst element of x t . You can also assume that X has rank k. In answering the questions below you do not have to derive the results: just give the formula for the correct answer. a.) (15 points) Calculate the value of β for which T 1 P T t =1 ( y t x 0 t β ) 2 is minimized.
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• Winter '08
• Hamilton,J

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