程序代写代做代考 Alastair Hall ECON61001: Semester 1 2020-21 Econometric Methods

Alastair Hall ECON61001: Semester 1 2020-21 Econometric Methods
Problem Set for Tutorial 8
In this question, you investigate the condition for instrument relevance in a simple regression model where β0 is just-identified.
1. Consider the regression model
yi = β0,1 +x2,iβ0,2 +ui = x′iβ0 +ui
where xi = (1, x2,i)′, β0 = (β0,1, β0,2)′. Define zi = (1, z2,i)′, ui = yi − x′iβ, and assume that E[ziui(β0)] = 0. Suppose a researcher estimates β0 using the population moment condition
E[ziui(β0)] = 0. (1)
(a) Assuming (1) holds, show that E[ziui(β)] ̸= 0 for all β ̸= β0 if and only if rank{E[zix′i]} =
2.
(b) Show that rank{E[zix′i]} = 2 if and only if Corr(z2,i, x2,i) ̸= 0 where Corr(a, b) denotes the population correlation between rv’s a and b. Hint: Since E[zix′i] is square, it is full rank if and only if it is nonsingular.
(c) Interpret the condition for instrument relevance in this case.
In this question, you derive the formula for the IV estimator in the over-identified case. You may find it useful to refer back to the matrix differentiation results in Tutorial 1 Question 3 and refer back to how we derived the OLS estimator in lectures.
2. Consider the linear regression model
y = Xβ0 + u (2)
w h e r e y i s T × 1 w i t h t t h e l e m e n t y t , X i s T × k w i t h t t h r o w x ′t , u i s T × 1 w i t h t t h e l e m e n t ut, β0 is a k×1 vector of unknown parameters. Let Z be a T ×q matrix with tth row zt′, and define u(β) = y − Xβ. Assume rank{X′Z} = k and rank{Z′Z} = q. Consider the Instrumental Variables (IV) estimator of β0, βˆIV , defined by
where
βˆIV = argminβQIV (β), (3) QIV (β) = u(β)′Z(Z′Z)−1Z′u(β)
1

(a) By considering the first order conditions for the minimization in (3), show that ˆ 􏰀 ′ ′ −1 ′ 􏰁−1 ′ ′ −1 ′
βIV = XZ(ZZ) ZX XZ(ZZ) Zy. (b) Show that if q = k then: βˆIV = (Z′X )−1 Z′y.
In this question you explore methods for inference in models estimated via IV in cross-sectional data.
3. Consider the linear regression model
yi = x′iβ0 + ui,
where: (i) { (ui,x′i,zi′), i = 1,2,…N} forms an independent and identically distributed sequence; (ii) E[zizi′] = Qzz, finite, p.d.; (iii)) E[zix′i] = Qzx, with rank{Qzx} = k; (iv) E[ui|zi] = 0; (v) V ar[ui|zi] = h(zi), positive, finite constant. Let βˆIV be the IV estimator of β0 basedonE[ziui]=0. SupposeitisdesiredtotestH0 :Rβ0 =rversusHA :Rβ0 ̸=r where R is a nr ×k matrix of specified constants and r is a nr ×1 vector of specified constants. Suggest a suitable decision rule for the test based on βˆIV , being sure to carefully specify how your test statistics is calculated from the data.
This question consider the regression model in Tutorial 7 Question 3. In that question, you showed that if the regressors include the lagged dependent variable and the errors follow a MA(1) process then OLS is an inconsistent estimator of the regression parameters. Here you consider an IV approach to estimation of this model.
4. Consider now the regression model:
yt = β0,1 + β0,2yt−1 + ut, (4)
where |β0,2| < 1 and ut = εt + φεt−1, (5) where φ ̸= 0, |φ| < 1, φ ̸= −β0,2, and εt is white noise. Let βˆT be the OLS estimator of β0 = (β0,1, β0,2)′ based on (4). Suppose that this model is estimated via IV using instrument vector zt = (1, yt−2)′. Show that zt satisfies the orthogonality and relevance conditions. Hint: if yt is generated by (4)-(5) then: (i) yt has the representation yt = β0,1/(1 − β0,2) + 􏰔∞i=0 β0i ,2ut−i; (ii) yt is generated by a stationary ARMA(1,1) process and its first order autocorrelation is: Corr(yt,yt−1) = (φ+β0,2)(1+φβ0,2). 1 + 2φβ0,2 + φ2 2 In this question you consider a simple model in which the instrument relevance condition is not satisfied and the implications of this failure for the large sample behaviour of the IV estimator. 5. Consider the regression model yi = xiβ0 + ui, where xi is a scalar and consider estimation of (scalar) β0 by IV based on the moment condition E[ziui(β0)] = 0 where zi is a scalar and ui(β) = yi − xiβ. Let βˆIV be the resulting IV estimator and suppose that E[zixi] = 0. (a) Verify that the instrument relevance condition is not satisfied in this model. (b) What is E[ziui(β)] for β ̸= β0? (You may assume that E[ziui] = 0.) (c) Would you expect βˆIV to be a consistent estimator for β0 in this case? 3