Alastair Hall ECON61001: Semester 1 2020-21 Econometric Methods
Practice Questions
1. A researcher is interested in testing the null hypothesis H0 : β0,2 = β0,3 versus the alternative H1 : β0,2 ̸= β0,3 in the model
y i = x ′i β 0 + u i , i = 1 , 2 , . . . , N
where xi is 3 × 1 and β0 = (β0,1, β0,2, β0,3)′. To do so, the researcher estimates the model via OLS and plans to test the hypothesis using the decision rule: reject H0 if SN > τN−k(1−α/2) where τN −k (1 − α/2) is the 100(1 − α/2)th percentile of the Student’s t distribution,
βˆ N , 2 − βˆ N , 3 SN= dN
and βˆN = (βˆN,1, βˆN,2, βˆN,3)′ is the OLS estimator of β0. Let y be the N × 1 vector with ith element yi, X be the N × 3 matrix with ith row x′i, and assume that the regression model satisfies Assumptions SR1-SR6 given on pp.54-5 of the Lecture Notes (with N replacing T ). Let σˆN2 be the OLS estimator of σ02 given in Assumption SR6.
(a) In order for the test above to have the probability of a Type I error implied by the
significance level, dN must equal f (y, X, N ), a particular function of {y, X, N }: f(y,X,N).
(b) Given that βˆN = (1.136, 0.913, 0.952)′, σˆN2 = 0.539, N = 30 and
0.058 −0.015 0.007 (X ′X )−1 = −0.015 0.010 −0.001
state
0.007 −0.001 0.009
perform the test using a 5% significance level. Be sure to explain your calculations
clearly.
2. Consider the linear regression model
y = Xβ0 + u (1)
where y is 25 × 1 vector of observations on the dependent variable, X is 25 × 5 data matrix of observations on the explanatory variables, β0 is 5×1 vector of unknown regression coefficients and u is the 25 × 1 vector containing the error term. Suppose that X = (ι25, X2) where ι25 is a 25×1 vector of ones and X2 is a 25×4 matrix, and β0 = (β0,1,β0′,2)′ where β0,1 is a scalar and β0,2 is a 4 × 1 vector. Let F be the F-statistic for testing H0 : β0,2 = 04 versus H0 : β0,2 ̸= 04 where 04 is the 4 × 1 null vector. If the p-value for this test is 0.05 then what is the adjusted R2, R ̄2, for the estimated version of (1)? Be sure to justify your answer carefully.
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3. A researcher estimates the unknown parameter vector γ0 in the model x = Zγ0 + e
where x is a N × 1 vector with ith element xi, Z is N × q matrix of observable explanatory variables with ith row zi′ based on the population moment condition,
E[yi(xi − zi′γ0)] = 0, (2)
where yi is k × 1 vector. Assume that {zi, yi, ei}Ni=1 form a sequence of independently and identically distributed random variables with: (i) E[ziyi′] = Mzy, a matrix of constants with full row rank; (ii) E[yiyi′] = My,y, a nonsingular matrix of constants; (iii) E[ei|yi] = 0; (iv) V ar[ei|yi] = a(yi).
(a) Derive the condition for γ0 to be identified by the population moment condition in (2).
(b) Derive the formula for the IV estimator for γ0 based on (2).
(c) Show that the IV estimator of γ0 based on (2) is consistent under the conditions above.
(d) The researcher wishes to test the hypothesis H0 : β(γ0) = 0 versus H1 : β(γ0) ̸= 0 where β( · ) is a c × 1 vector of continuous differentiable functions. Propose a test based on the IV estimator of γ0 in part (b), being sure to: explain clearly how to calculate the test statistic from the data; to provide the decision rule; to state any additional conditions (beyond those above) that must be satisfied in order for your decision rule to be valid.
4. Suppose {Vi}Ni=1 is a sequence of i.i.d. normal random variables with mean μ0 and variance σ02.
(a) Derive the Wald, Likelihood Ratio (LR) and Lagrange Multiplier (LM) statistics for test- ing H0 : μ0 = 0 versus H1 : μ0 ̸= 0. Hint: you may quote the form of the log likelihood function, score equations and maximum likelihood estimators without derivation.
(b) Given a sample of size N = 100, {vi}Ni=1, for which Ni=1 vi = 12.31 and Ni=1 vi2 = 135.28, what is the outcome of the Wald, LR and LM tests in part (a). Be sure to justify your conclusions carefully.
5. Consider the case where yi ∈ {0, 1, 2} and the value of yi∗ is latent variable generated via yi∗ = x′iβ0 + ui
where xi is vector of observable explanatory variables, β0 is an unknown parameter vector, and ui ∼ N (0, 1). Suppose that the outcome of yi is determined as follows:
yi =0,ifyi∗<0,
= 1, if 0 ≤ yi∗ < 2, = 2,if2≥yi∗.
Assume that {xi, ui}Ni=1 is an independently and identically distributed sequence, and that yi is observable.
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(a) Derive the probability distribution function for yi conditional on xi.
(b) WritedownCLLFN(β),theconditionallog-likelihoodfunctionbasedonasample{yi,xi}Ni=1.
6. Suppose that yt is generated via where β0 = 0,
yt = β0yt−1 + ut (3) ut = εt + φεt−1, (4)
φ ̸= 0 and {εt} is sequence of independently and identically distributed random variables with mean zero and variance σ2. Let βˆT be the OLS estimator of β0 based on (3) that is,
ˆ Tt=1 yt−1yt βT = T y2 .
t=1 t−1
(a) Derive the probability limit of βˆT and verify whether or not this estimator is consistent
for β0.
(b) Suppose that (3) is estimated via Instrumental Variables (IV) using yt−2 as instrument for yt−1 that is, estimation is based on the moment condition E[yt−2(yt − β0yt−1)] = 0. Verify whether or not this choice of instrument satisfies the orthogonality and relevance conditions associated with IV estimation.
(c) Let β ̃T be the IV estimator of β0 based on the population moment condition in part (b). ShowthatT1/2(β ̃T −β0)→d N(0,H)whereHisamatrixthatyoumustspecifyaspart
of your answer.
(d) Suppose now that φ = 0. How does T1/2(β ̃T − β0) behave as T → ∞? Justify your answer briefly.
Hint: yt is a covariance stationary process and you may quote the generic form of: (i) the Weak Law of Large Numbers for covariance stationary and weakly dependent processes,
T−1 Tt=1 wt →p μw, but must verify the specific forms of the limits relevant to the quantities
analyzedinyouranswer;(ii)theCentralLimitTheorem,T−1/2Tt=1(wt−μw)→d N(0,Vw) but must verify the specific form of μw and Vw relevant to the quantities in your answer.
7. Consider the model
y = Xβ0 + u (5)
where y is a T × 1 observable random vector, X is a T × k observable matrix that is fixed in repeated samples and u is a T ×1 vector of unobservable errors. It is assumed that the model satisfies Classical Assumptions CA1-CA6 given in the Lecture Notes pp.10-11. In addition, it is assumed that Rβ0 = r where R is a nr × k matrix of constants with rank(R) = nr, and r is a nr × 1 vector of constants. If (5) is estimated by least squares subject to Rβ0 = r then
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the resulting RLS estimator of β0 is denoted βˆR,T , and the RLS estimator of σ02 is denoted σˆ2 . Recall that
R,T
βˆR,T = βˆT + (X′X)−1R′[R(X′X)−1R′]−1(r − RβˆT ) σˆ 2 = e ′ e / ( T − k + n )
(a) Define Q = I − X(X′X)−1X′ + X(X′X)−1R′[R(X′X)−1R′]−1R(X′X)−1X′. Show that Q is an orthogonal projection matrix.
R,T RR r
where βˆT is the OLS estimator of β0 based on (5) and eR = y − XβˆR,T .
(6) ( 7 )
(b) Show that e′ReR = u′Qu.
(c) Show that E[σˆ2 ] = σ2.
R,T 0 (d)Showthat(T−k+nr)σˆ2 /σ02∼χ2 .
R,T T −k+nr
(e) Show that σˆ2 is a more efficient estimator of σ2 than the OLS estimator, σˆ2 .
R,T 0 T Hint:Ifv∼χ2n thenE[v]=nandVar[v]=2n.
8.(a) For conformable matrices, A, B and C show that: tr(ABC) = tr(BCA) = tr(CAB) where tr( · ) denotes the trace operator.
8.(b) Consider f = x′Az where x is a m×1 vector, A is a m×n matrix and z is a n×1 vector. Derive and state the dimensions of: (i)∂f/∂x; (ii)∂f/∂z.
9. Consider the linear regression model
yi = x′iβ0 + ui
where xi = (1,x′2,i)′, {(ui,x′2,i)′, i = 1,2,...N} forms an independent and identically distributed (i.i.d.) sequence, E[xix′i] = Q, a finite, positive definite matrix of constants, E[ui|xi] = 0, V ar[ui|xi] = σ02, a positive, finite constant. Consider the statistic
NN
SN =uix′iMNxiui
i=1 i=1
where MN is a k × k random matrix.
(a) Propose a choice of MN such that SN →d χ2k being sure to justify your answer care- fully. Hint: you may quote the generic form of the Weak Law of Large Numbers, N−1 Ni=1 zi →p μz, but must verify the specific forms of the limits relevant to the quan- tities analyzed in your answer; you may quote the generic forms of the Central Limit Theorem, N −1/2 Ni=1 (zi − μz ) →d N (0, Vz) but must verify the specific forms of μz and Vz limits relevant to the quantities analyzed in your answer.
(b) Suppose now that V ar[ui|xi] = h(xi). Is it still true that SN →d χ2k for your choice of MN in part (a)? Justify your answer briefly but formal derivations are not required.
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