ECONOMETRICS

ADVANCED ECONOMETRICS Midterm 2 (Take Home)

Due: Dec.25, 2014

Answer all questions. You should not discuss solutions with your peers but me. Good luck! Prof. Dr. H. Taştan ,

First Name:................................................... Last Name:................................................

No:...................................................

1 (20) In class we have shown that when the number of instrumental variables is larger than the number of endogenous variables the generalized IV estimator (or 2SLS) can be written as

β̂IV = ( X>PzX

)−1 X>Pzy

where Pz = Z(Z>Z)−1Z>. In this formulation X is n× k and Z is n× l, l > k.

(a) Show that β̂IV can be obtained as a solution to the following minimization problem

min β Q(β) = (y −Xβ)>Pz (y −Xβ)

(b) Show that when k = l the generalized IV estimator reduces to the simple IV estimator:

β̂IV = ( Z>X

)−1 Z>y

2 (20) Consider the following simple consumption model as a function of permanent income

ci = β1 + β2y ∗ i + ui, ui ∼ iid (0, σ2

u)

where ci is the logarithm of consumption by household i, and y∗i is the permanent income of household i which is not observed. Instead we observe current income, yi

yi = y∗i + vi, vi ∼ iid (0, σ2 v)

where vi is assumed to be uncorrelated with y∗i and ηi. We run the following regression

ci = β1 + β2yi + ηi

(a) Show that yi is negatively correlated with ηi. You can assume β2 > 0.

(b) Evaluate the plim of the OLS estimator β̂2:

β̂2 =

∑n i=1(yi − ȳ)ci∑n i=1(yi − ȳ)2

In particular, show that this plim is less than the true β2.

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3 (30) Use card.dta to answer the following questions. Also read Card (1993), “Using Geographic Variation in College Proximity to Estimate the Return to Schooling”, NBER Working Paper.

(a) Run the OLS regression of log(wage) on educ, exper, exper2, black, smsa, south, smsa66, reg662 to reg669. Comment on the coefficient estimate of educ.

(b) Estimate the same model by 2SLS using nearc4 as an instrument for educ. Compare the OLS and IV coefficient estimates on educ. (Note that we partly did this in class). Carry out the Hausman test.

(c) Use both nearc2 and nearc4 as instruments for educ. Run the reduced form model for educ. Compare 2SLS estimates to the results obtained in the previous section. Carry out the OID test.

(d) Discuss the plausibility of Card (1993)’s econometric methodology and empirical findings. Do you agree with his conclusions?

4 (30) A continuous time model for short term interest rates may be written as a stochastic differential equation

dr = (α + βr)dt+ σrγε √ dt

where r is the short term interest rate, ε is standard normal random variable, dt is a short time interval and α, β, γ, σ are parameters. Discrete time approximation is given as

rt+1 − rt = α + βrt + εt+1

with E(εt+1) = 0, E(ε2t+1) = σ2r2γt

In this question we will consider the restricted version of the model in which γ = 1/2 (for details, see Chan et al 1992). Assume that γ is known so that the parameter vector is given by θ = (α, β, σ)>. The vector of population moment conditions is given as E(f(θ, rt)) = 0 where

f(θ, rt) =

 εt+1

εt+1rt ε2t+1 − σ2rt

(ε2t+1 − σ2rt)rt

 with εt+1 = rt+1 − rt − α − βrt. STATA data file intrate.dta contains 294 weekly observations on 1-month average interest rates on bank deposits. You need to use tsset date to use the data set. Also note that, in the data set, r denotes rt+1 and rlag denotes rt.

(a) Using this data, and the population moment conditions described above, estimate the parameter vector using GMM. You may set initial values as θ0 = (0, 0, 1)>. Choose initial weighting matrix as identity, and also use HAC Newey-West procedure for the long run covariance matrix with bandwidth set to 5.

(b) Compute the Hansen J-test. Interpret the result.

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intrate.dta