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Generalized Least Squares (GLS) Theory, Heteroscedasticity & Autocorrelation Ba Chu E-mail: ba chu@carleton.ca Web: http://www.carleton.ca/∼bchu (Note that this is a lecture note. Please refer to the textbooks suggested in the course outline for details. Examples will be given and explained in the class) 1 Objectives Until now we have assumed that var(u) = σ 2 I but it can happen that the errors have non-constant variance, i.e., var(u1 ) = var(u2 ) = · · · = var(uT ) or are correlated, i.e., E(ut us ) = 0 for t = s. This assumption about the errors is true for many economic variables. Suppose instead that V ar(u) = σ 2 Ω, where the matrix Ω contains terms for heteroscedasticity and autocorrelation. First, we study the GLS estimator and the feasible GLS estimator. Second, we apply these techniques to do statistical inference on the regression model with heteroscedastic errors and test for heteroscedasticity. Third, we learn to estimate regression models with autocor- related disturbances and test for serial correlation. Fourth, we also learn to use the ML technique to estimate the regression models with autocorrelated disturbances. 2 GLS Consider the model: y = Xβ + u, (2.1) 1 where E[u] = 0 and E[u u] = σ 2 Ω with Ω is any matrix, not necessarily diagonal. 2.1 Problem with the OLS If we estimate (2.1) with the OLS, we will obtain βOLS = (X X)−1 X y = β + (X X)−1 X u. It is easy to verify that βOLS is still unbiased. However, since V ar(βOLS = σ 2 (X X)−1 X ΩX(X X)−1 = σ 2 (X X)−1 , inferences based on the estimator σ 2 (X X)−1 , that we have seen, is misleading. Thus, t and F tests are invalid. Moreover, βOLS is not always consistent. This is the main reason for us to learn the GLS. 2.2 Derivation of the GLS Suppose Ω has the eigenvalues λ1 , . . . , λT , by Cholesky’s decomposition, we obtain: Ω = SΛS , where Λ is a diagonal matrix with the diagonal elements (λ1 , . . . , λT ), and S is an orthogonal matrix. Thus, Ω−1 = S −1 Λ−1 S −1 = S −1 Λ−1/2 Λ −1/2 S −1 = PP , √ √ where P = S −1 Λ−1/2 and Λ−1/2 is a diagonal matrix with the diagonal elements ( λ1 , . . . , λT ). It is straight-forward to prove that P ΩP = IT . Now, multiplying both sides of (2.1) by P yields P y = P Xβ + P u. Setting y 0 = P y, X 0 = P X and u0 = P u, we end up with the classical linear regression model: y 0 = X 0 β + u0 , 2 where E[u0 ] = 0 and E[u0 u0 ] = E[P uu P ] = σ 2 P ΩP = σ 2 IT . The OLS estimate of β is given by βGLS = (X 0 X 0 )−1 X 0 y 0 = (X P P X)−1 X P P y = (X Ω−1 X)−1 X Ω−1 y. The last equation is deﬁned as the GLS estimator denoted by βGLS . Note that the subscript GLS is sometimes omitted for brevity. The unbiased estimate of σ 2 is given by 1 1 σ 2 GLS = (y 0 − X 0 β) (y 0 − X 0 β) = (y − X β) Ω−1 (y − X β). T −K T −K 2.3 Properties of the GLS estimator • The GLS estimator is unbiased. • V ar(βGLS ) = σ 2 (X ΩX)−1 . • The GLS estimator is BLUE. • If u ∼ N (0, σ 2 Ω) then β ∼ N (β, σ 2 (X Ω−1 X)−1 ). The F-test formula is given by ( R β − r) [R(X Ω−1 X)−1 R ]−1 (Rβ − r) q×K F = . q σ 2 GLS p • If limT −→∞ X 0 X 0 /T = Q0 > 0, then β =⇒ β as T −→ ∞. √ d −1 • T (β − β) =⇒ N (0, σ 2 Q0 ). 3 2.4 Feasible GLS Since the matrix Ω is unknown, we need to use an estimator, say, Ω. We have βF GLS = (X Ω−1 X)−1 X Ω−1 y. If the following conditions: p lim (X Ω−1 X/T ) = p lim (X Ω−1 X/T ) T −→∞ T −→∞ √ √ p lim (X Ω−1 u/ T ) = p lim (X Ω−1 u/ T ) T −→∞ T −→∞ hold, then βGLS and βF GLS are asymptotically equivalent. 3 Heteroscedasticity Let’s consider the model (2.1) in which the matrix Ω is a diagonal matrix with the diagonal elements 2 2 (ω1 , . . . , ωT ). This is a form of heteroscedasticity deﬁned in terms of inconstant variances. We can estimate this model by using either the FGLS or the modiﬁed OLS technique proposed by White (1980). The FGLS procedure is straight-forward. We shall describe White’s OLS procedure in details. 2 2 2 White (1980) proposes a consistent estimator Ω with the diagonal elements (ω1 , . . . , ωt , . . . , ωT ) given by 2 HC-0 ωt = u2 , where ut is the OLS residual. t 2 T HC-1 ωt = u2 . T −K t 2 HC-2 ωt = b u2 t , where ht is t-th diagonal element of the matrix X(X X)−1 X . 1−ht 2 HC-3 ωt = u2 t b . (1−ht )2 4 The asymptotic variance-covariance matrix of βOLS is given by Avar(βOLS ) = lim (X X/T )p lim(σ 2 OLS X ΩX/T ) lim (X X/T ). T −→∞ T −→∞ Based on this heteroscedasticity-consistent estimator, the general linear hypotheses may be tested by the Wald statistics: d W = (RβOLS − r) [RAvar(βOLS )R ]−1 (Rβ − r) =⇒ χ2 (q). The White procedure has large-sample validity. It may not work well in ﬁnite samples. We can identify whether or not there exists heteroscedasticity in the noise by using White’s test. The null hypothesis is homoscedasticity. To do White’s test, we proceed by regressing the OLS residuals on a constant, original regressors, the squares of the regressors, and the cross products of the regressors and obtain the R2 value. The test can be constructed by T R2 ∼ χ2 (q), where q is the number of variables in the auxiliary regression less one. We reject the null if T R2 is suﬃciently large. 4 Serial Correlation Note that serial correlation and autocorrelation mean the same thing. Consider the model: yt = Xt β + ut , ∀ t = 1, . . . , T, (4.1) where E[ut ] = 0, E[u2 = σ 2 and E[ut us ] = 0 for t = s. t For example, we can model serial correlation by an AR(1) model, i.e., ut = ρut−1 + t , (4.2) 5 where t ∼ IID(0, σ 2 ) and |ρ| < 1. We can compute σ2 E[u2 ] = t , 1 − ρ2 ρ|t−s| σ 2 E[ut us ] = . 1 − ρ2 More generally, we have 2 T −1 1 ρ ρ ... ρ T −2 σ2 ρ 1 ρ ... ρ 2 E[uu ] = . = σu Ω, (4.3) 1−ρ 2 . . . . . . . . . . ... . ρT −1 ρT −2 ρT −3 ... 1 2 where σu and Ω have the obvious meanings. 4.1 Estimate (4.1)-(4.2) by FGLS The FGLS can be done in the following steps: 1. Regress yt on Xt and obtain ut = yt − Xt βOLS . P uu P bub T b t=2 t t−1 2. Regress ut on ut−1 and obtain ρOLS = T 2 . t=2 t−1 3. Construct the FGLS by βF GLS = (X Ω−1 X)−1 X Ω−1 y, where Ω is obtained from (4.3) by substituting ρ with ρOLS . 1 T p p If p limT −→∞ T 1 Xt ut = 0, then βOLS =⇒ β and ρOLS =⇒ ρ. Therefore, βF GLS is consistent and asymptotically normal. 6 4.2 Estimate (4.1)-(4.2) by the ML An application of the conditional probability rules, we have pdf (y1 , y2 ) = pdf (y2 |y1 )pdf (y1 ), pdf (y1 , y2 , y3 ) = pdf (y3 |y1 , y2 )pdf (y1 , y2 ) = pdf (y3 |y1 , y2 )pdf (y2 |y1 )pdf (y1 ), ......... T pdf (y1 , . . . , yT ) = pdf (y1 ) f (yt |yt−1 , . . . , y1 ). t=2 Since yt = Xt β + ut = Xt β + ρut−1 + t = Xt β + ρ(yt−1 − Xt−1 β) + t and t ∼ N (0, σ 2 ), we have yt |yt−1 ∼ N (ρyt−1 + Xt β − ρXt−1 β, σ 2 ). σ 2 Moreover, since y1 ∼ N (X1 β, 1−ρ2 ), we can derive the log-likelihood function: log L(β, ρ, σ 2 ) = log pdf (y1 , . . . , yT ) T = log pdf (y1 ) pdf (yt |yt−1 ). t=2 We then maximize this log likelihood function to obtain the MLEs. The asymptotic variance of the MLEs can be obtained from the CR lower bound. 7 4.3 Estimate (4.1) by the modiﬁed OLS In general, the model (4.1) with general form of heteroscedasticity and autocorrelation can be consistently estimated by the OLS, i.e., βOLS = (X X)−1 X y. βOLS is AN, i.e., √ d T (βOLS − β) =⇒ N (0, σ 2 Q−1 M Q−1 ), where Q = limT −→∞ X X/T and M = p limT −→∞ X ΩX/T under the assumption E[uu ] = σ 2 Ω. Q can be estimated by Q = X X/T ; σ 2 M can be consistently estimated by the Newey-West heteroscedasticity and autocorrelation consistent covariance (HAC) matrix estimator: m HAC = Γ0 + w(j, m)(Γj + Γj ), j=1 where T 1 Γj = Xt ut ut−j Xt−j , ∀ j = 1, . . . , m, T t=j+1 where w(j, m) is a lag window, m is a lag truncation parameter. If w(j, m) = 1 − j/(m + 1), then w(j, m) is called the Barlett window. Hence, the asymptotic covariance matrix of βOLS can be estimated by Q−1 HAC Q−1 . 4.4 Test for serial correlation • The Durbin-Watson test T T DW = (ut − ut−1 )2 / u2 , t 2 1 where ut is the OLS residual, is used to test for the ﬁrst-order serial correlation (i.e., AR(1)). – For H0 : ρ = 0 vs. H1 : ρ > 0, reject H0 if DW ≤ DW , do not reject if DW > DWu , and inconclusive if DW < DW < DWu with DW is the lower critical value and DWu is upper critical value. – For H0 : ρ = 0 vs. H1 : ρ < 0, reject H0 if DW ≥ 4 − DW , accept if DW < 4 − DWu . 8 • The Breusch-Godfrey LM test is used to test for high-order serial correlation (i.e., the AR(p) model as given by p ut = φi ut−i + t ). i=1 p The test is carried out by doing the OLS regression: ut = Xt γ + i=1 bi ut−i + error. Next, d we do either the LM test T R2 =⇒ χ2 (q) under the null hypothesis of no serial correlation or the F-test for b1 = b2 = · · · = bp = 0. 5 Autoregressive Conditional Heteroscedasticity (ARCH) Engle (1982) suggests that heteroscedasticity may occur in time-series context. In speculative markets such as exchange rates and stock market returns, one often observes the phenomenon called volatility clustering – large changes tend to be followed by large changes, of either sign, and small changes tend to be followed by small changes. Observations of this phenomenon in ﬁnancial time series have led to the use of ARCH models. Engle (1982) formulates the notion that the recent past might give information about the conditional disturbance variance. He postulates the relation: yt = Xt β + σt t , where t ∼ IID(0, 1), 2 σt = α0 + α1 u2 + · · · + αp u2 , where ut = yt − Xt β. t−1 t−p Testing for ARCH can be done in the following steps: 1. Fit y to X by OLS and obtain the residuals {et }. 2. Compute the OLS regression, e2 = α0 + α1 e2 + · · · + αp e2 + error. t t−1 t−p 3. Test the joint signiﬁcance of α0 , α1 , . . . , αp . 6 Exercises 1. Problems 6.1, 6.2, 6.3 and 6.7 (pp. 202-203, JD97). 9 2. Consider the model yt = βyt−1 + ut , |β| < 1, ut = t +θ t−1 , |θ| < 1, where t = 1, . . . , T and t ∼ IID(0, σ 2 ). (a) Show that the OLS estimator of β is inconsistent when θ = 0. (b) Explain how you would test H0 : θ = 0 vs. H1 : θ = 0 using the Lagrangian multiplier test principle. Is such test likely to be superior to the Durbin-Watson test?. 3. Consider the model yt = βxt + ut , ut = ρut−1 + t , |ρ| < 1, where xt is non-stochastic and t ∼ IID − N (0, σ 2 ). (a) Given a sample {xt , yt }T , construct the log-likelihood function. t=1 (b) Estimate the parameters β, ρ and σ 2 by the ML method. (c) Derive the asymptotic variances of these MLEs. (d) Given xT +1 , what do you think is the best predictor of yT +1 ?. 4. Consider the linear regression model y = Xβ +u, where y is a T ×1 vector of observations on the dependent variable, X is a T × K matrix of observations on K non-stochastic explanatory variables with rank(X) = K < T , β is a K × 1 vector of unknown coeﬃcients, and u is a T × 1 vector of unobserved disturbances such that E[u] = 0 and E[uu ] = σ 2 Ω, with Ω positive deﬁnite and known. (a) Derive the GLS estimator βGLS of β. 10 d (b) Assuming that T −1/2 X Ω−1 u =⇒ N (0, σ 2 limT −→∞ X Ω−1 X/T ), derive the limiting dis- tribution of βGLS as T −→ ∞. State clearly any further assumptions you make and any statistical results you use. 5. In the generalized regression model, suppose that Ω is known. (a) What is the covariance matrix of the OLS and GLS estimators of β? (b) What is the covariance matrix of the OLS residual vector uOLS = y − X βOLS ?. (c) What is the covariance matrix of the GLS residual uGLS = y − X βGLS ?. (d) What is the covariance matrix of the OLS and GLS residual vectors? 6. Suppose that y has the pdf pdf (yt |Xt ) = 1/( Xt β ) exp{−y/((Xt β) )}, y > 0. Then 1×K K×1 2 E[yt |Xt ] = β Xt and V ar[yt |Xt ] = (β Xt ) . For this model, prove that the GLS and MLE are the same, even though this distribution involves the same parameters in the conditional mean function and the disturbance variance. 7. Suppose that the regression model is yt = µ + t , where has a zero mean, constant variance, and equal correlation ρ across observations. Then Cov( t , s ) = σ 2 ρ if t = s. Prove that the OLS estimator of µ is inconsistent. 8. Suppose that the regression model is yt = µ + t , where E[ t |xt ] = 0, Cov( t s |xt , xs ) = 0 for i = j, but V ar[ t |xt ] = σ 2 x2 , xt > 0. t (a) Given a sample of observations on yt and xt , what is the most eﬃcient estimator of µ?, What is its variance?. (b) What is the OLS estimator of µ?, and what is its variance?. (c) Prove that the estimator in part (a) is at least as eﬃcient as the estimator in part (b). 11 References Engle, R. F. (1982). Autoregressive conditional heteroscedasticity with estimates of variance of united kingdom inﬂation, Econometrica 50: 987–1008. White, H. (1980). A heteroskedasticity-consistent covariance matrix estimator and a direct test for heteroscedasticity, Econometrica 48: 817–838. 12

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posted: | 6/16/2010 |

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