Questions and Answers on Heteroskedasticity, Autocorrelation and Generalized Least Squares

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1 Questions and Answers on Heteroskedasticity, Autocorrelation and Generalized Least Squares L Magee Fall, Consider a regression model y = Xβ +ɛ, where it is assumed that E(ɛ X) = 0 and E(ɛɛ X) = Σ The OLS estimator of β is b = (X X) 1 X y (a) First, suppose that you allow for heteroskedasticity in ɛ, but assume there is no autocorrelation (i) What do we know about the numerical values of Σ? (ii) Describe how to compute the heteroskedasticity-robust variance-covariance matrix estimator (HCCME) of Var(b) (b) Next, suppose that you allow for heteroskedasticity and autocorrelation in ɛ (i) What do we know about the numerical values of Σ? (ii) Describe how to compute the Newey-West HAC variance-covariance matrix estimator of Var(b) 2 The White heteroskedasticity-consistent covariance estimator estimates the matrix (X X) 1 X (σ 2 Ω)X(X X) 1 by replacing the (i, i) th element of σ 2 Ω (this element is E(ɛ 2 i x i) with e 2 i, where e i is from the OLS residual vector e = y Xb, and it replaces the (i, j) th element of σ 2 Ω with zero The Newey-West autocorrelation-consistent covariance estimator attempts to deal with autocorrelation as well as heteroskedasticity By analogy with the White estimator, one might expect that the Newey-West estimator would replace the (i, j) th element of σ 2 Ω (which is E(ɛ i ɛ j x i )) with e i e j But instead, the Newey-West estimator does something more complicated Why doesn t the simpler method work? 3 Suppose ɛ t follows a stationary AR(1) process: ɛ t = ρɛ t 1 + u t, t = 1, 2, 3 where u t is white noise Let ɛ = [ɛ 1 ɛ 2 ɛ 3 ] (a) Defining E(ɛɛ ) = σɛ 2 Ω, where σɛ 2 = E(ɛ 2 i ), express the 3 3 matrix Ω as a function of ρ (b) For this Ω, write a 3 3 matrix P for which P ɛ is not autocorrelated 1

2 (c) Calculate every element of the following 3 3 matrices as functions of ρ only, using standard matrix multiplication: (i) P Ω (ii) (P Ω)P (iii) P P (iv) (P P )Ω (d) Suppose that you have obtained an estimate of ρ for this model as ˆρ = 040 Calculate the OLS, the Prais-Winsten, and the Cochrane-Orcutt estimators of β in the model y = xβ + ɛ for the data: i x y Prais-Winsten is FGLS Cochrane-Orcutt is like FGLS, but it omits the first observation of the transformed data y and X from the calculation 4 Consider a regression model: y = Xβ + ɛ where E(ɛ X) = 0 and E(ɛɛ X) = Σ (a) Describe how to compute the heteroskedasticity-robust variance-covariance matrix estimator (HCCME) of Var(b) (b) Describe how to compute the Newey-West HAC variance-covariance matrix estimator of Var(b), which is valid when ɛ has autocorrelation and heteroskedasticity 5 One way to write the GLS estimator of β in the model y = Xβ + ɛ, E(ɛ X) = 0, E(ɛɛ ) = Σ is ˆβ = (X X ) 1 X y, where X = P X and y = P y for a certain n n matrix P (a) Write a mathematical relation involving P and Σ that must hold for ˆβ to be the GLS estimator (b) Suppose ɛ t follows a stationary AR(1) process: ɛ t = ρɛ t 1 + u t, t = 1, 2,, n where u t is white noise (i) Describe the Σ and P matrices in this case 2

3 Answers (ii) Describe how the vector y is related to the original dependent variable vector y in this case 1 (a) (i) off-diagonal elements equal zero diagonal elements are 0, not necessarily equal (ii) Compute OLS residual vector e = y Xb e e Construct S = diag(e i ) = e 2 n Then the HCCME is (X X) 1 X SX(X X) 1 (b) (i) Σ is symmetric and positive semidefinite diagonal elements are 0, not necessarily equal (ii) Like in the answer to q8(a)(ii) except now the (i, j) th element of S is S ij = w ij e i e j where w ij = { 1 i j L if i j < L 0 if i j L L is an increasing function of n A common choice is L = n 1/4 2 Replacing the (i, j) th element of σ 2 Ω with e i e j, is the same as replacing the matrix σ 2 Ω with the matrix ee, where e is the vector of OLS residuals Then the covariance matrix estimator would be (X X) 1 X (ee )X(X X) 1 = (X X) 1 (X e)(e X)(X X) 1 = (X X) 1 (X e)(x e) (X X) 1 But since X e = 0, this covariance matrix estimator would always consist of a matrix of zeroes 3 (a) Ω = 1 ρ ρ 2 ρ 1 ρ ρ 2 ρ 1 (b) 3

4 (c) (i) P = P Ω = 1 ρ ρ ρ 2 ρ 1 ρ 2 ρ 2 1 ρ ρ 2 ρ ρ 3 (ii) 1 ρ 2 ρ 1 ρ 2 ρ 2 1 ρ 2 1 ρ 2 ρ 0 (P Ω)P = 0 1 ρ 2 ρ ρ ρ (iii) P P = = 1 ρ ρ = (1 ρ 2 )I 1 ρ 2 ρ 0 1 ρ ρ ρ 1 0 (iv) = (P P )Ω = 1 ρ 0 ρ 1 + ρ 2 ρ 1 ρ 0 ρ 1 + ρ 2 ρ 1 ρ ρ 2 ρ 1 ρ ρ 2 ρ 1 = 1 ρ ρ 2 0 = (1 ρ 2 )I (d) OLS = i=1 x iy i i=1 x2 i = = 220 FGLS (Prais-Winsten) = i=1 x iy i i=1 x2 i =

5 where i x i y i and Cochrane-Orcutt = i=2 x iy i i=2 x2 i = = 50 4 (a) Compute the OLS residual vector e = y Xb e e Construct S = diag(e i ) = e 2 n Then the HCCME is (X X) 1 X SX(X X) 1 (b) Let the (i, j) th element of S be S ij = w ij e i e j where { 1 i j w ij = L if i j < L 0 if i j L L is an increasing function of n A common choice is L = n 1/4 5 (a) P ΣP = I, or P P = Σ 1 (b) (i) 1 ρ ρ 2 ρ n 1 ρ 1 ρ ρ n 2 Σ = σɛ 2 ρ n 2 ρ 1 ρ ρ n 1 ρ 2 ρ 1 or describe as: the (i, j) th element of Σ is σ 2 ɛ ρ i j 5

6 1 ρ P = 1 ρ σ u 0 0 y 1 1 ρ 2 y 1 y (ii) Letting y = 2 then y y = 2 ρy 1 y n y n ρy n 1 6