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ECON61001: Econometric Methods Instructions: • You must answer all five questions in Section A and two out of the four ques- tions in Section B. If you answer more questions than are required and do not indicate which answers should be ignored, we will mark the requisite number of answers in the order in which they appear in your answer submission: answers beyond that number will not be considered. • Your answers could be typed or hand-written (and scanned to a single pdf file that can be submitted) or a combination of a typed answer with included images of algebra or figures. • Where relevant, questions include word limits. These are limits, not targets. Ex- cellent answers can be shorter than the word limit. If you go beyond the word limit the additional text will be ignored. Where a question includes a word limit you HAVE to include a word count for your answer (excluding formulae). • Candidates are advised that the examiners attach considerable importance to the clarity with which answers are expressed. • You must correctly enter your registration number and the course code on your answer.
1. Suppose a researcher is interested in the following linear regression model yi = x ′ iβ0 + ui, i = 1, 2, . . . N, where xi = (1, x2,i, x3,i, x4,i) ′ and β0 = (β0,1, β0,2, β0,3, β0,4) ′ . Given the context, the researcher is able to assume that {ui, x ′ i} N i=1 form a sequence of independently and identically distributed random vectors with E[xix ′ i] = Q, a finite, positive definite matrix of constants, E[ui|xi] = 0 and V ar[ui|xi] = σ 2 0 . Therefore, she estimates the model via Ordinary Least Squares (OLS), obtaining the following estimated equation yˆi = 1.0213 (0.1416) − 0.0020 (0.0005) x2,i − 0.0208 (0.0080) x3,i + 0.0095 (0.0088) x4,i, where the number in parenthesis is the conventional OLS standard error for the coefficient in question. The OLS estimator of σ20 in this model is σˆ 2 N = 0.0081. Given these results, the researcher concludes that β0,4 = 0 and so decides to estimate the model via OLS with x4,i excluded, obtaining the following estimated equation yˆi = 1.1485 − 0.0020x2,i − 0.0211x3,i. (1) Given that the sample size is N = 108 in both estimations, calculate the OLS estimator of the error variance σ20 from the estimation in (1). Be sure to care- fully explain your calculations. Hint: consider the F-statistic for testing β0,4 = 0. [8 marks] 2. Suppose it is desired to predict zt using zˆt = w ′ tγ where wt is a vector of ob- servable variables and γ is a vector of constants that needs to be specified. The choice of γ associated with the linear projection of zt on wt is γ0 where E[(zt − w ′ tγ0)wt] = 0. (a) What optimality property does zˆot = w ′ tγ0 possess? (Word limit: 50) [1mark] (b) Now consider the regression model zt = w ′ tγ0 + vt. Is wt contempora- neously exogenous or strictly exogenous for estimation of γ0 in this model? Justify your answer. (Word limit: 150) [4 marks] (c) Suppose the model in part (b) is dynamically complete. What optimal- ity property does zˆot possess? Briefly justify your answer. (Word limit: 150) [3 marks] Continued over 1 ECON61001 SECTION A continued 3.(a) Let A be n × n nonsingular symmetric matrix. Show that if A is positive definite then A−1 is positive definite. Hint: AA−1 = In. [4 marks] 3.(b) Consider the classical linear regression model y = Xβ0 + u, (2) whereX is the T×k observable data matrix that is fixed in repeated samples with rank(X) = k, and u is a T × 1 vector with E[u] = 0 and V ar[u] = σ20IT where σ 2 0 is an unknown positive finite constant. Let βˆT be the OLS estimator of β0 based on (2) and βˆR,T be the Restricted Least Squares (RLS) estimator of β0 based on (2) subject to the restrictions Rβ0 = r where R is a nr × k matrix of specified constants with rank(R) = nr and r is a specified nr × 1 vector of constants. Assuming the restrictions are correct, prove that βˆR,T is at least as efficient as βˆT . Hint: you may quote the formula for the variance-covariance matrix of the OLS and RLS estimators without proof; you may also take advantage of the stated result in part (a). [4 marks] 4. Consider the linear regression model y = Xβ0 + u, where y and u are T × 1 vectors, X is T × k matrix, and β0 is the k × 1 vector of unknown regression coefficients. Assume that X is fixed in repeated sam- ples with rank(X) = k, and u ∼ N ( 0, σ20IT ) where σ 2 0 is an unknown positive constant. Let θˆT denote the maximum likelihood estimator of the unknown pa- rameter vector θ0 = (β ′ 0, σ 2 0) ′. Derive the information matrix for this model. Hint: you may state the form of the log likelihood function and score function for this model without proof. [8 marks] 5. Consider the model yi = x ′ iβ0 + ui, i = 1, 2, . . . , N, where β0 is the k × 1 vector of unknown regression coefficients, {(x ′ i, ui)} N i=1 is a sequence of independently and identically distributed random vectors with E[ui|xi] = 0, V ar[ui|xi] = σ 2 0, an unknown finite positive constant and E[xix ′ i] = Q, a finite positive definite matrix of constants. Let σˆ2N be the OLS estimator of σ20. Show that N 1/2(σˆ2N − σ 2 0) d → N( 0, µ4 − σ 4 0) where µ4 = E[u 4 i ]. Hint: You may assume that: N−1 ∑N i=1 xix ′ i p → Q; (ii) N−1/2 ∑N i=1 vi d → N(0,Ω) where vi = (u 2 i − σ 2 0, x ′ iui) ′, and Ω = V ar[vi] is a finite, positive definite (k + 1) × (k + 1) matrix whose elements you must specify as needed to develop your answer. [8 marks] Continued over 2 ECON61001 SECTION B 6. Consider the regression model yi = x ′ iβ0 + ui, i = 1, 2, . . . , N, where β0 is the k × 1 vector of unknown regression coefficients, {(x ′ i, ui)} N i=1 is a sequence of independently and identically distributed random vectors with E[ui|xi] = 0, V ar[ui|xi] = σ 2 0, an unknown finite positive constant and E[xix ′ i] = Q, a finite positive definite matrix of constants. You may further assume that: (i) N−1 ∑N i=1 xix ′ i p → Q; (ii) N−1/2 ∑N i=1 xiui d → N(0, σ20Q). Let βˆR,N denote the RLS estimator based on the linear restrictionsRβ = r where R is a nr × k matrix of pre-specified constants with rank equal to nr and r is a nr × 1 vector of pre-specified constants, and let λˆN be the vector of Lagrange Multipliers associated with this RLS estimation. Assuming Rβ0 = r, answer the following questions. (a) Show that N1/2(βˆR,N − β0) d → N ( 0, VR ) where VR = σ 2 0 ( Q−1 − Q−1R′(RQ−1R′)−1RQ−1 ) . Hint: you may quote the formulae for βˆR,N and βˆN , the Ordinary Least Squares estimator of β0, without proof. [10 marks] (b) A colleague proposes testing H0 : Rβ0 = r versus H1 : Rβ0 6= r using the decision rule of the form: reject H0 at the (approximate) 100α% significance level if λˆ′NMN λˆN > cnr(1− α) where cnr(1− α) is the 100(1 − α) th percentile of the χ2nr distribution. How- ever, your colleague is unsure what the matrix MN should be in order that this decision rule has the properties implied by the stated significance level. Provide a suitable choice of MN , being sure to justify your choice carefully. Hint: you may quote without proof: (i) the formulae for λˆN and βˆN ; (ii) that both the OLS and RLS estimators of σ20 are consistent under the conditions of the question. [20 marks] Continued over 3 ECON61001 SECTION B continued 7.(a) Let {vt} T t=−3 be a weakly stationary time series process. Consider the following statistic, ρˆ4,T = ∑T t=1 vtvt−4∑T t=1 v 2 t . Let {εt} ∞ t=−∞ denote a sequence of independently and identically distributed (i.i.d.) random variables with mean zero and variance σ2ε . (i) Assume that vt = εt. Show that T 1/2ρˆ4,T d → N(0, 1). [6 marks] (ii) Assume that vt = θ4vt−4 + εt, where |θ4| < 1. What is the probability limit of ρˆ4,T as T → ∞? Be sure to justify your answer carefully. Hint: vt has the following representation, vt = ∑ ∞ i=0 θ i 4εt−4i. [9 marks] 7.(b) A researcher wishes to test the simple efficient-markets hypothesis in the foreign exchange market. Let st = ln(St) and ft,n = ln[Ft,n], where St and Ft,n are the levels of the spot exchange rate at time t and the n−period forward exchange rate at time t. The simple efficient-markets hypothesis is that ft,n = E[st+n | It] where It is the information set at time t which for the purposes of this question can be taken to be It = {st, ft,n, st−1, ft−1,n, st−2, ft−2,n, . . .}. Using daily spot and thirty-day forward exchange rate data for the US dollar UK pound exchange rate, the researcher estimates the model,