Ch.8 Nonspherical Disturbance This chapter will assume that the full ideal conditions hold except that the covari- ance matrix of the disturbance, i.e. E(EE)=02Q2, where Q is not the identity matrix. In particular, Q may be nondiagonal and / or have unequal diagonal ele- ments Two cases we shall consider in details are heteroscedasticity and auto- correlation. Disturbance are heteroscedastic when they have different variance Heteroscedasticity usually arise in cross-section data where the scale of the de- pendent variable and the explanatory power of the model tend to vary across observations. The disturbance are still assumed to be uncorrelated across obser vation, so an would be g2 0 2g2 Autocorrelation is usually found in time-series data. Economic time-series often display a"memory"in that variation around the regression function is not independent from one period to the next. Time series data are usually ho- moscedasticity, so aQ2 would be 1p1 PT-1 Pr-2 In recent studies, panel data sets, constituting of cross sections observed at several points in time, have exhibited both characteristics. The next three chap- ter examines in details specific types of generalized regression models Our earlier results for the classical mode: will have to be modified. We first consider the consequence of the more general model for the least squares estima- tors
Ch. 8 Nonspherical Disturbance This chapter will assume that the full ideal conditions hold except that the covariance matrix of the disturbance , i.e. E(εε0 ) = σ 2Ω, where Ω is not the identity matrix. In particular, Ω may be nondiagonal and/or have unequal diagonal elements. Two cases we shall consider in details are heteroscedasticity and autocorrelation. Disturbance are heteroscedastic when they have different variance. Heteroscedasticity usually arise in cross-section data where the scale of the dependent variable and the explanatory power of the model tend to vary across observations. The disturbance are still assumed to be uncorrelated across observation, so σ 2Ω would be σ 2Ω = σ 2 1 0 . . . 0 0 σ 2 2 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . σ 2 T . Autocorrelation is usually found in time-series data. Economic time-series often display a ”memory” in that variation around the regression function is not independent from one period to the next. Time series data are usually homoscedasticity, so σ 2Ω would be σ 2Ω = σ 2 1 ρ1 . . . ρT −1 ρ1 1 . . . ρT −2 . . . . . . . . . . . . . . . . . . ρT −1 ρT −2 . . . 1 . In recent studies, panel data sets, constituting of cross sections observed at several points in time, have exhibited both characteristics. The next three chapter examines in details specific types of generalized regression models. Our earlier results for the classical mode; will have to be modified. We first consider the consequence of the more general model for the least squares estimators. 1
1 Properties of the Least squares Estimators Theorem The OLS estimator B is unbiased. Furthermore, if limT-oo(X'Q2X/T)is finite, B Proof: E(B)=B+E(X'X)-IX'E=B, which proves unbiasedness Also plim B=B+ lim plim- T→∞0 X' has zero mean and covariance matrix If limT-oo(X'QLX/) is finite, then xnX=0. Hence x= has zero mean and its covariance matrix vanishes asymptotically, which implies plim AT=0, and therefore, plim 6=6 Theorem The covariance matrix of B is o(X'X)XnX(XX) Proof: E(B-B)(-B)=E(XX)1x∈∈X(XX)- o(XX-XQX(XX fote that the covariance matrix of B is no longer equal to 2(X'X)-.It may be either"larger"or"smaller", in the sense that(X'X)IX'QX(XX-1 (XX)-can be either positive semidefinite, negative semidefinite, or neither Theorem e'e/(T-k) is(in general)a biased and inconsistent estimator of
1 Properties of the Least Squares Estimators Theorem: The OLS estimator βˆ is unbiased. Furthermore, if limT→∞(X0ΩX/T) is finite, βˆ is consistent. Proof: E(βˆ) = β + E(X0X) −1X0ε = β, which proves unbiasedness. Also plim βˆ = β + lim T→∞ X0X T −1 plim X0ε T . But X0ε T has zero mean and covariance matrix σ 2X0ΩX T 2 . If limT→∞(X0ΩX/T) is finite, then σ 2 T X0ΩX T = 0. Hence X0ε T has zero mean and its covariance matrix vanishes asymptotically, which implies plim X0ε T = 0, and therefore, plim βˆ = β. Theorem: The covariance matrix of βˆ is σ 2 (X0X) −1X0ΩX(X0X) −1 . Proof: E(βˆ − β)(βˆ − β) 0 = E(X0X) −1X0 εε0X(X0X) −1 = σ 2 (X0X) −1X0ΩX(X0X) −1 . Note that the covariance matrix of βˆ is no longer equal to σ 2 (X0X) −1 . It may be either ”larger” or ”smaller”, in the sense that (X0X) −1X0ΩX(X0X) −1 − (X0X) −1 can be either positive semidefinite, negative semidefinite, or neither. Theorem: s 2 = e 0e/(T − k) is (in general) a biased and inconsistent estimator of σ 2 . 2
e(ee)= e(Em trace E(Me trace Mn ≠a2(T-k) Also since E(s2)fo2, it is hard to see that it is a consistent estimator of o2 from convergence in mean square error 2 Efficient Estimators To begin, it is useful to consider cases in which Q2 is a known, symmetric, positive definite matrix. This assumption will occasionally be true, but in most models, Q will contains unknown parameters that must also be estimated ssume that a2 0 therefore. we have a” known”9 2.1 Generalized Least Square(GLS) Estimators Since @2 is a positive symmetric matrix, it can be factored into CA-IC=CA-1/2A-1/2C/=Pp where the column of C are the eigenvectors of Q2 and the eigenvalues of Q2 are d in the diagonal matrix a and p= ca
Proof: E(e 0 e) = E(ε 0Mε) = trace E(Mεε0 ) = σ 2 trace MΩ 6= σ 2 (T − k). Also since E(s 2 ) 6= σ 2 , it is hard to see that it is a consistent estimator of σ 2 from convergence in mean square error. 2 Efficient Estimators To begin, it is useful to consider cases in which Ω is a known, symmetric, positive definite matrix. This assumption will occasionally be true, but in most models, Ω will contains unknown parameters that must also be estimated. Example: Assume that σ 2 t = σ 2x2t , then σ 2Ω = σ 2x21 0 . . . 0 0 σ 2x22 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . σ 2x2T = σ 2 x21 0 . . . 0 0 x22 . . . 0 . . . . . . . . . . . . . . . . . . 0 0 . . . x2T , therefore, we have a ”known” Ω. 2.1 Generalized Least Square (GLS) Estimators Since Ω is a positive symmetric matrix, it can be factored into Ω −1 = CΛ−1C 0 = CΛ−1/2Λ −1/2C 0 = P 0P, , where the column of C are the eigenvectors of Ω and the eigenvalues of Ω are arrayed in the diagonal matrix Λ and P0 = CΛ−1/2 . 3
Theorem Suppose that the regression model Y=XB+E satisfy the ideal conditions except that @2 is not the identity matrix. Suppose that X9-1x lin is finite and nonsingular. Then the transformed equation PY=PXB+PE satisfies the full ideal condition Proof Since p is nonsingular and nonstochastic. PXis nonstochastic and of full rank if X is.(Condition 2 and 5). Also, for the consistency of OLS estimators -o T= lim X'n-1X lim (PX(PX T is finite and nonsingular by assumption. Therefore the transformed regressors ma- trix satisfies the required conditions, and we need consider only the transformed disturbance pe Clearly, E(PE)=0(Condition 3). Also E(PE)(PE) a2(A1/2C)(CAC)CA-12) o I(Condition 4) Finally, the normality(Condition 6) of Pe follows immediately from the nor- y of e Theorem The blue of B is just B=(X!2-1x)-1xg-1Y Proof: Since the transformed equation satisfies the full ideal conditions, the blue of 6
Theorem: Suppose that the regression model Y = Xβ+ε satisfy the ideal conditions except that Ω is not the identity matrix. Suppose that lim T→∞ X0Ω−1X T is finite and nonsingular. Then the transformed equation PY = PXβ + Pε satisfies the full ideal condition. Proof: Since P is nonsingular and nonstochastic, PXis nonstochastic and of full rank if X is. (Condition 2 and 5). Also, for the consistency of OLS estimators lim T→∞ (PX) 0 (PX) T = lim T→∞ X0Ω−1X T is finite and nonsingular by assumption. Therefore the transformed regressors matrix satisfies the required conditions, and we need consider only the transformed disturbance Pε. Clearly, E(Pε) = 0 (Condition 3). Also E(Pε)(Pε) 0 = σ 2PΩP0 = σ 2 (Λ −1/2C 0 )(CΛC0 )(CΛ−1/2 ) = σ 2Λ −1/2ΛΛ−1/2 = σ 2 I (Condition 4). Finally, the normality (Condition 6) of Pε follows immediately from the normality of ε. Theorem: The BLUE of β is just β˜ = (X0Ω −1X) −1X0Ω −1Y. Proof: Since the transformed equation satisfies the full ideal conditions, the BLUE of β 4
Is Just B=[(PX)(PX)-(PX)(PY) (X)X'Q2-Y Indeed, since B is the OLs estimator of B in the transformed equation, and since the transformed equation satisfies the ideal conditions, B has all the usual de- sirable properties-it is unbiased, BLUE, efficient, consistent, and asymptotically B is the Ols of the transformed equation, but it is a generalized least square (GLS) estimator of the original regression model which take the Ols as a sub- cases when Q=I Theorem The variance-covariance of the GLS estimator B is o(X'Q2-IX) TOO Viewing B as the Ols estimator in the transformed equation, it is clearly has covariance matrix o2(PX)(PX)]-1=a2(X92-1x)-1 Theorem An unbiased, consistent, efficient, and asymptotically efficient estimator of o wheree=Y-XB Proof: Since the transformed equation satisfies the ideal conditions, the desired estimator K(PY-PXB)(PY-PXB=m 7=k(Y-x/9(Y-xB)
is just β˜ = [(PX) 0 (PX)]−1 (PX) 0 (PY) = (X0Ω −1X) −1X0Ω −1Y. Indeed, since β˜ is the OLS estimator of β in the transformed equation, and since the transformed equation satisfies the ideal conditions, β˜ has all the usual desirable properties–it is unbiased, BLUE, efficient, consistent, and asymptotically efficient. β˜ is the OLS of the transformed equation, but it is a generalized least square (GLS) estimator of the original regression model which take the OLS as a subcases when Ω = I. Theorem: The variance -covariance of the GLS estimator β˜ is σ 2 (X0Ω−1X) −1 . Proof: Viewing β˜ as the OLS estimator in the transformed equation, it is clearly has covariance matrix σ 2 [(PX) 0 (PX)]−1 = σ 2 (X0Ω −1X) −1 . Theorem: An unbiased, consistent, efficient, and asymptotically efficient estimator of σ 2 is s˜ 2 = ˜e 0Ω−1˜e T − k , where ˜e = Y − Xβ˜. Proof: Since the transformed equation satisfies the ideal conditions, the desired estimator of σ 2 is 1 T − k (PY − PXβ˜) 0 (PY − PXβ˜) = 1 T − k (Y − Xβ˜) 0Ω −1 (Y − Xβ˜). 5