Exam. Econometrics - Exam 1
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1 Econometrcs - Exam 1 Exam Problem 1: (15 ponts) Suppose that the classcal regresson model apples but that the true value of the constant s zero. In order to answer the followng questons assume just one ndependent varable. 1. Gve the formulae for the two least squares slope estmators (the one wth and the one wthout the constant).. Calculate ther varances. 3. Compare the varance of the least squares slope estmator computed wthout a constant term wth that of the estmator computed wth an unnecessary constant term. Soluton: 1. y = β 1 +β x+ε β = (x x)(y ȳ) (x x) y = β x+ε β = x y. Var(β ) = Var( β ) = σ (x x) σ
2 Econometrcs - Exam 3. The rato of these two varances s Var( β ) Var(β ) = = σ σ (x x) (x x) (x x) = (x x x+ x) = x n x x+n x = x n x = n x = 1 n x 1 It follows that fttng the constant term when t s unnecessary nflates the varance of the least squares estmator f the mean of the regressor s not zero.
3 Econometrcs - Exam 3 Problem : (15 ponts) Suppose that y has the pdf f(y x) = 1 x),y > 0. Then E[y x] = β x e y/(β β x and Var[y x] = (β x). For ths model, prove that the GLS and MLE estmators are the same, even though ths dstrbuton nvolves the same parameters n the condtonal mean functon and the dsturbance varance. Soluton: Frst the GLS estmator: ˆβ GLS = (X Ω 1 X) 1 X Ω 1 y = Next the MLE estmator: L = ( 1 β x e y /(β x ) ) 1 ( ) x x x y (β x ) (β x ) lnl = ln(β x ) y /(β x ) lnl β y x = x (β = x ) β x y x (β = x ) ( ˆβ MLE = x β x + x x β (β x ) x x (β x ) y /(β x ) x = 0 Now wrte ) 1 y x (β x ) x β x = x x β (β x )
4 Econometrcs - Exam 4 Problem 3: (15 ponts) The followng model s estmated usng a balanced panel of fve frms over 0 years: I t = β 1 F t +β C t +ε t, where the regressors are market value (F) and captal (C) and the dependent varable s nvestment (I). Suppose that the true error structure of the model s ε t = α +η t, where α s uncorrelated wth the regressors. 1. If the model s estmated as a fxed effects model, what wll be the statstcal propertes, n terms of effcency and consstency, of the estmates?. The estmates for pooled OLS, fxed effects (usng dummes) and random effects models are gven n the table below. Use the statstcs shown to decde whether the data support a fxed effects or random effects specfcaton. Carefully explan your reasonng. Dependent Varable s Investment Estmaton Constant Market Value Captal (a) OLS (-.36) (9.36) (7.019) (b) Fxed Effects (6.669) (14.348) (c) Random Effects (-0.775) (6.859) (14.350) (t-ratos are shown n brackets) Breush-Pagan LM test for random effects (1 df): Hausman test of fxed vs random effects ( df): 1.7 Soluton: 1. If the ndvdual effects are strctly uncorrelated wth the regressors then a random effects model s the approprate model. However, f a fxed effect model s estmated the estmates wll be consstent but not effcent.. Breush-Pagan LM test: Test statstc s 453.8, the crtcal value from the ch-squared table s 3.84, so the null hypothess that random effects are not needed can be rejected. Hausman Test: Test statstc s 1.7, the crtcal value from the chsquared tables5.99, so thenull hypothess oftherandomeffects model cannot be rejected.
5 Econometrcs - Exam 5 Problem 4: (15 ponts) Consder the stochastc processes gven below. For each process determne what the effects of frst dfferencng the process,.e. computng y t y t 1, on autocorrelaton are, e.g. reducton of the autocorrelaton. 1. y t = y t 1 +ε t, where ε t s normally dstrbuted whte nose.. y t = β 0 +β 1 t+ε t, where ε t s normally dstrbuted whte nose. 3. y t = β x t + ε t, where ε t = ρε t 1 + u t and u t s normally dstrbuted whte nose. [Hnt: Compare the autocorrelaton of ε t and the autocorrelaton of (ε t ε t 1 ).] Soluton: 1. y t = y t y t 1 = ε t, whte nose, no more autocorrelaton. y t = y t y t 1 = β 1 + ε t ε t 1. Ths s an MA(1) process wth autocorrelaton θ = 1 = 1+θ y t = y t y t 1 = β (x t x t 1 )+v t, where v t = ε t ε t 1. Var(ε t ) = σ u 1 ρ Var(v t ) = Var(ε t ε t 1 ) = Var(ρε t 1 ε t +u t ) = Var[(ρ 1)ε t 1 +u t ] = (ρ 1) σ u 1 ρ +σ u = σ u 1+ρ Cov[v t,v t 1 ] = Cov[ε t ε t 1,ε t 1 ε t ] Cov[v t,v t 1 ] Var[v t ] = E[ε t ε t 1 ε t 1 ε tε t +ε t 1 ε t ] = ρ σ u 1 ρ σ u σ 1 ρ ρ u 1 ρ +ρ σ u 1 ρ = σ u (ρ 1 ρ ) 1 ρ σu = (ρ 1) (ρ 1)(ρ+1) = σ u (ρ 1) ρ+1 = ρ 1
6 Econometrcs - Exam 6 Compare the two autocorrelatons: ρ > ρ 1 Assume ρ > 0 and ρ < 1 ρ > ρ 1 ρ > 1 3 If the orgnal autocorrelaton s greater than 1/3 (For economc data, ths s lkely to be farly common.) the dfferenced process has a smaller autocorrelaton.
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