Econometrics Short Questions Do two out of three) 5 points each ) Let y = Xβ + u and Z be a set of instruments for X When we estimate β with OLS we project y onto the space spanned by X along a path orthogonal to X Write an analogous statement about the instrumental variables estimator of β, and then draw a picture to help explain your sentence 2) Consider a model where an individual receives wage offers from distribution F w; θ) and accepts the first offer greater than ξ You are given a data set of N iid observations, {w i } N i=, and the goal is to estimate θ and ξ using MLE a) What is the MLE of ξ? [Hint: focus on the fact that each person accepts the first offer greater than ξ] b) What goes wrong in the standard proof for consistency of the MLE of ξ? 3) Consider the estimated model for log wages: Variable ˆβ ŝˆβ Constant 234 035 Age 057 0008 Age 2-0003 000 Female -02 0073 Black -053 0043 Educ 0084 0008 Female*Educ -0009 0005 Black*Educ -0008 0003 For each of the questions below either provide the answer it is not necessary to do any arithmetic) or report what information that is missing is needed and provide a formula: a) What is the estimated rate of return to education for women and its standard error? b) What is the estimate of the age where wage stops growing and its standard error? Long Questions Do one out of two) 30 points each
)Let q d t = α 0 + α p t + α 2 y t + u d t qt s = β 0 + β p t + β 2 w t + u s t qt d = qt s a) Describe in detail how to simulate the asymptotic distribution of the 2SLS estimator of the structural parameters in the model b) Describe how you think your simulated parameter estimates would behave if α 2 = 0 2) Let y i = b X i, δ) + u i for i =, 2,, N Assume that there are m elements in the parameter vector δlet u = u, u 2,, u N ) and assume that u N 0, Ω) You want to test H 0 : H δ) = 0 against H A : H δ) 0 where there are k < m nonlinear restrictions implied by H ) Suggest a test statistic and derive its asymptotic distribution under H 0 2
Econometrics: Answer 3 out of 4 questions Each question is equally weighted Let y = Xβ + u, u 0, Ω) Show that the OLS estimator of β is consistent, and derive its asymptotic distribution 2 Consider the model, y i = β 0 + β y 2i + β 2 x i + u i, y 2i = α 0 + α y i + α 2 x 2i + u 2i, ) )) σ σ 0, 2 u 2i σ 2 σ 22 ui Let β be the OLS estimator of β = β 0, β, β 2 ) Let t OLS be a t-statistic with a 5% size to test H 0 : β = 3 vs H A : β 3 using β and ignoring the fact that y 2i is endogenous Show how to compute Pr [Reject H 0 H 0 is true] using the flawed t-statistic 3 Consider the model, g y i, X i, θ) = u i, u i iidf ), i =, 2,, n Sketch a proof that the MLE of θ is consistent, and derive its asymptotic distribution 4 Let u t = ut u 2t ), u t = Au t + ε t, ε t iidn 0, σ 2 I ) Derive the marginal distribution of u t Be specific about any assumptions you need to make about A and/or σ 2
Econometrics: Answer 3 out of 4 questions Each question is equally weighted Total 40 points Let Define y T = Xβ + u, u 0, σ 2 I ) û = y X β where β is the GLS estimator of β a) Derive plim T û û b) Define ι T =,,, ) Find the asymptotic distribution of T ι û 2 Consider the model, y i = β 0 + β y 2i + β 2 x i + u i, y 2i = α 0 + α y i + α 2 x 2i + u 2i, ) )) σ σ 0, 2 u 2i σ 2 σ 22 ui Show how to estimate β using indirect least squares ILS) Provide intuition for why the ILS estimator of β is unique 3 Consider the model, g y i, X i θ) = u i, u i iidf ), E u i X i ) = 0 i =, 2,, n Describe how to estimate g ) and θ semiparametrically 4 Let u t = ρu t + e t, e t = a 0 ε t + a ε t ε t iidn 0, σ 2 I ) Derive the distribution of u t Provide intuition for why you can t identify all of the parameters of the model
Econometrics: Answer 3 out of 4 questions Each question is equally weighted Let n y t = β i x ti + u t i=0 u t iid 0, σ 2) Suggest how to estimate β = β 0, β,, β n ) subject to the restriction, and show that it is consistent 2 Consider the model, β + 2β 3 = 4, y i = β 0 + β y 2i + β 2 x i + u i, y 2i = α 0 + α 2 x 2i + u 2i, ) )) σ σ 0, 2 u 2i σ 2 σ 22 ui Show that β = β 0, β, β 2 ) is identified, and derive the asymptotic distribution of the 2SLS estimator of β 3 Let x i iidu 0, θ), i =, 2,, n Derive the MLE, the MOM estimator, and the Bayesian estimator of θ For the Bayesian estimator, use an exponential prior 4 Provide detailed instructions on how to do a Monte Carlo experiment to learn about the small sample properties of probit estimators
Econometrics: Answer 3 out of 4 questions Each question is equally weighted Let y t = X t β + u t, u t = ρ i u t i + ε t, i=,2 ε t 0, σ 2) Provide a consistent estimate of the covariance matrix of u = u, u 2,, u T ), show that it is consistent, and show that the Feasible GLS estimator of β is consistent 2 Consider the model, y i = β 0 + β y 2i + β 2 x i + u i, ui u 2i y 2i = α 0 + α 2 x 2i + u 2i, ) σ σ 0, 2 σ 2 σ 22 )) Let β be the OLS estimator of β = β 0, β, β 2 ) Derive the asymptotic bias of β 3 Let x i iidbernoulli p) p U 0, ) Use Bayes Theorem to form a posterior for p x, x 2,, x n 4 Let y t = x t β + u t, u t = ρu t + e t, e t = exp {αv t } ε t, ε t iidn 0, ), v t = γv t + η t, η t iidn 0, σ 2 η) Construct the likelihood function for {y t, x t } T t=, and show, in detail, how to simulate it
Econometrics: Answer 3 out of 4 questions Each question is equally weighted Consider the model, y i = X i β + z i γ + u i where Ez i u i 0 and γ = 0 Find the asymptotic distribution of the OLS estimator of β, γ) Provide an example of an empirical problem where this would be relevant and explain the implications of your asymptotics results 2 Consider the model, y it = X it β + e i + u it, e i iidn 0, σ 2 e), u it iidn 0, ), y it = y it > 0) Construct the likelihood function for this model 3 Consider the model, y t = X t β + z t γ + u t Assume that z t is not observed, so the econometrician estimates the model, y t = X t b + e t What are the statistical properties of the OLS estimator of b? 4 Consider the model, y t = ρy t 2 + u t, u t iid 0, σ 2) Derive the necessary and suffi cient conditions for y t to be stationary Find V ar y t ) if the stationarity conditions are not satisfied
2 Econometrics Component 60 Points) Instructions: Answer three out of the four following questions We suggest you allocate one hour for the completion of this part of the examination ) Consider the model with Consider the test, y = Xβ + u u 0, σ 2 I ) H 0 : Aβ = c vs H A : Aβ c Suggest a consistent estimate of β and use it to construct a test statistic Derive the asymptotic distribution of the test statistic Hint: it is not enough to specify a test statistic and assert its distribution; derive the distribution 2) Consider the model q d i = α d + β d p i + γ d z d i + u d i q s i = α s + β s p i + γ s z s i + u s i q d i = q s i where qi d is demand for bananas, qs i is supply of bananas, p i is price of bananas, and zi d, ) zs i are two different exogenous variables Under what conditions are all of the structural parameters identified? How might you test the identification assumption? 3) Show under reasonable conditions that the maximum likelihood estimator is consistent and derive its asymptotic distribution 4) Consider the model y i = m x i ) + u i, u i iid 0, σ 2), i =, 2,, n where x i is an exogenous scalar and m ) is an unspecified function Suggest how to estimate m ) using a) kernel estimation, b) polynomial approximations, and c) spline functions in slopes For each one, explain how your estimation procedure changes as n and why that provides a consistent estimate of m )