9. AUTOCORRELATION. [1] Definition of Autocorrelation (AUTO) 1) Model: y t = x t β + ε t. We say that AUTO exists if cov(ε t,ε s ) 0, t s.

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1 9. AUTOCORRELATION [1] Definition of Autocorrelation (AUTO) 1) Model: y t = x t β + ε t. We say that AUTO exists if cov(ε t,ε s ) 0, t s. ) Assumptions: All of SIC except SIC.3 (the random sample assumption). Or, all of WIC except WIC. [E(ε t ε 1,...,ε t-1,x 1,...,x t ) = 0]. Replace WIC. by E(ε t x 1,...,x T ) = 0 (strictly exogenous regressors). Comments: For OLS, the assumption of weakly exogenous regressors is enough. However, for GLS, we need the assumption of strictly exogenous regressors (that is, E( ε t xt, xt 1,,..., x1 ) = 0 for all t). AUTO-1

2 3) Example: AR(1): ε t = ρε t-1 + v t, v t iid N(0,σ v ). cov(ε t,ε t-1 ) = E(ε t ε t-1 ) = E[(ρε t-1 +v t )ε t-1 ] = ρe(ε t-1 ) 0. MA(1): ε t = v t + ψv t-1, v t iid N(0,σ v ). cov(ε t,ε t-1 ) = E(ε t ε t-1 ) = E[(v t +ψv t-1 )(v t-1 +ψv t- )] = ψe(v t-1 ) = ψσ v E1 ε t = ε t-1 + v t, σ v = 1. AUTO-

3 E ε t = 0.5ε t-1 + ν t. AUTO-3

4 E3 ε t = ν t AUTO-4

5 E4 ε t = -0.5ε t-1 + ν t AUTO-5

6 E5 ε t = -ε t + ν t AUTO-6

7 E6 ε t = v t + v t-1 AUTO-7

8 E7 ε t = ν t + 0.5v t-1 AUTO-8

9 E8 ε t = ν t -0.5ν t-1. AUTO-9

10 E9 ε t = ν t - ν t-1. AUTO-10

11 4) Typical in time-series data: i) Presence of unobservable variables. ii) Seasonally adjusted data. Empirical Example: Expectations-Augmented Phillips curve: Data: (Table F5.1 from Greene) From 1950I to 000II. For data descriptions, see p INFL: Annualized inflation rate (%). UNEMP: Unemployment rate (%). Expectation-Augmented Phillips curve: INFL INFL UNEMP t t1 = β1+ β t + εt, where β 1 = -β u * and u * is the natural unemployment rate. AUTO-11

12 Dependent Variable: INFL-INFL(-1) Method: Least Squares Date: 04//03 Time: 17:03 Sample(adjusted): 1950: 000:4 Included observations: 03 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C UNEMP R-squared Mean dependent var Adjusted R-squared S.D. dependent var.8910 S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Residual Actual Fitted AUTO-1

13 [] Stationarity Assumption on ε t (1) Stationarity and Ergodicity Let {s t } be a process of interest. Notation: E(s t ) μ t ; var(s t ) γ 0t ; cov(s t,s t-j ) γ jt [Autocovariance] Definition of Covariance (weak) Stationarity: {s t } is called covariance-stationary (CS) iff μ t = μ (finite) for all t and γ jt = γ j (finite) for all t. [e.g., cov(s 5,s 3 ) = cov(s 4,s ) = γ ] Rough Definition of Ergodicity: γ jt 0 as j. Ergodicity is needed for CLT and LLN. () Stationarity Assumption on ε t in y t = x t β + ε t : E(ε t ) = 0, for all t. var(ε t ) = σ ε, for all t. cov(ε t,ε t-j ) = γ tj = γ j, for all t. γ j 0 as j. AUTO-13

14 [3] Properties of OLS (1) Autocorrelation: ρ j = γ j /γ 0 = γ j /var(ε t ). () Autocorrelation Matrix under the stationarity assumption: R 1 ρ ρ ρ... ρ 1 3 ρ 1 ρ ρ ρ ρ 1 ρ... ρ T 1 T 1 1 T 3 = ρ3 ρ ρ ρ T 4 : : : : : ρt1 ρt ρt3 ρt ρ. Let σ = var(ε t ) γ 0. Cov(ε) = σ Ω = σ R. Theorem: The OLS estimator ˆ β is unbiased (and consistent). But it is inefficient. Theorem: Cov ˆ = X X X ΩX X X 1 ( β) ( ) σ ( ) 1. AUTO-14

15 Implications: The t and F-tests based on s (X X) -1 are all irrelevant. Need to estimate Δ = X (σ Ω)X. If ˆΔ is available, a consistent estimate of 1ˆ 1 ( XX ) Δ ( XX ). Cov( ˆ β ) is Using this estimate, we can compute t and Wald statistics. (3) Nonparametric Estimation of X (σ Ω)X. Definitions: Let e t = OLS residuals; and g t = e t x t. Define: S 0 = 1 Σ tgg t t ; T 1 T t= j+ t t j S j = Σ 1gg ; T q i ˆΔ = T S0 +Σ i= 1 k ( Sj + Sj ) q + 1, where q is called bandwidth and k( ) is a kernel function. Can also control for heteroskedasticity. AUTO-15

16 Choice of q? If we believe that ρ j = 0 for j > τ [it happens if ε t follows MA(τ)], choose q = τ or more. If don't know? Choose q such that q and q/t 0, as T. Newey-West Method (1987, ECON) Use Bartlett's kernel, k(z) = 1 - z. Choose q = T 1/5 or q = T /9 (same for other methods) so that q/t 1/4 0. Gallant Method (1987, Nonlinear Statistic Models) Use Pazen kernel, k(z) = 1-6z + 6z 3 for 0 z 1/; = (1-z) 3, 1/ z 1 and = 0, otherwise. Andrews (1991, ECON) Quadratic kernel: 3 sin(6 π z /5) kz ( ) = cos(6 π z/5) (6 πz/ 5) 6 πz/5. 1 ˆ T T i Δ= S0 +Σ i= 1k ( Si + Si ) T p q+ 1. AUTO-16

17 [4] Detection of First-Order AUTO [Testing H o : ρ 1 = 0 against H a : ρ 1 0 assuming ρ =... = 0] (1) Notation: 1 T Σ t = j + t t j Sample Autocovariance: r j = 1ee T j. Sample Autocorrelation: a j = r j r (-1 a j 1). 0 Note: Under the CS and Ergodicity assumptions, r j p γ j as T ; a j p ρ j as T. () Durbin-Watson Test (Assuming regressors are strictly exogenous) D-W test statistic: DW = Intuition: Σ ( e e ) T t= t t1 T Σt= 1et. DW = (1-a 1 ) - e + e Σ 1 T T t= 1et (1-a 1 ). If H o is correct, DW. For formal test with DW, see book. AUTO-17

18 (3) LM-type Test (Regressors need not be strictly exogenous) STEP 1: Regress e t on x t and e t-1, and get R. STEP : t-test for Ho: the coefficient on e t-1 = 0. Or T R p χ (1). (4) Note on these tests. Suppose ρ j 0, j =, 3,.... Then, these tests may fail to detect autocorrelation. Accepting Ho: ρ 1 = 0 does not mean that ρ j = 0, j =, 3,.... AUTO-18

19 [5] Detection of Higher-Order AUTO [H o : ρ 1 = ρ =... = ρ m = 0] (1) LM-type test. STEP 1: Regress e t on x t and e t-1,..., e t-m. STEP : LM = TR d χ (m). () Box-Pierce Test. BP = TΣ a d χ (m). m j= 1 j (3) Ljung-Box Test LB = TT ( + ) Σ m j= 1 T a j j d χ (m). AUTO-19

20 [EMPIRICAL EXAMPLE] Data: ustc96.wf1 (Quarterly data) FEDFRQ = Fed Fund Rate (%) GDP INF = annualized quarterly inflation rate (%) M1Q = M1 MQ = M TBILLQ = treasury bill rate (%) GGDP = 400*ln(GDP/GDP(-1)): Annualized growth rate of GDP (%). GM1 = 400*LOG(M1Q/M1Q(-1)): Annualized growth rate of M1 (%) Research project: How Fed reacts to the economy? Model to estimate: (fedfrq-tbillq) = β 1 + β ggdp t-4 + β 3 inf t-1 + ε t. Why (fedfrq t -tbillq t ) instead of fedfrq? The variable fedfrq appears to be nonstationary. But (fedfrq-tbillq) is stationary. Why not ggdp t-1, ggdp t-,... and inf t-, inf t-3...? They appear to be insignificant. AUTO-0

21 TBILLQ INF FEDFRQ GM GGDP AUTO-1

22 OLS Results: Dependent Variable: DEVFEDTBILL Method: Least Squares Sample(adjusted): 1955: 1996:1 Included observations: 164 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C INF(-1) GGDP(-4) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) AUTO-

23 DW = Implies AUTO. view/residual tests/serial correlation LM test lags to include: 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic.0850 Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Date: 04/16/0 Time: 15:4 Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C INF(-1) GGDP(-4) RESID(-1) RESID(-) RESID(-3) RESID(-4) R-squared Mean dependent var -4.56E-16 Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) AUTO-3

24 OLS with NEWEY-WEST option/heteroskedasticity/newey-west Dependent Variable: DEVFEDTBILL Method: Least Squares Sample(adjusted): 1955: 1996:1 Included observations: 164 after adjusting endpoints Newey-West HAC Standard Errors & Covariance (lag truncation=4) Variable Coefficient Std. Error t-statistic Prob. C INF(-1) GGDP(-4) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) AUTO-4

25 [6] AR(1) (1) Assumptions: ε t = ρε t-1 + v t, v t iid N(0,σ v ), ρ < 1. {ε t } starts at the infinite past (- ) () Derivation of Ω and V. ε t = ρ(ρε t- +v t-1 ) + v t = ρ ε t- + v t + ρv t-1 = ρ (ρε t-3 +v t- ) + v t + ρv t-1 = ρ 3 ε t-3 + v t + ρv t-1 + ρ v t- = ρ j j ε t- + Σ j= 0ρ vt j = Σ j= 0ρ vt j. j γ 0 = var(ε t ) = var ( j = 0ρ vt j) Σ = j j = 0ρ var( vt j) Σ = σ v 1 ρ. γ 1 = E(ε t ε t-1 ) = E[(ρε t-1 +v t )ε t-1 ] = ρe(ε t-1 ) = ρvar(ε t-1 ) = ρσ v 1 ρ. γ = E(ε t ε t- ) = E[(ρ ε t- +ρv t-1 +v t )ε t- ] = ρ E(ε t- ) = ρ σ v 1 ρ. : In general, γ j = ρ j var(ε t ): ρ j = γ j γ 0 = ρ j. AUTO-5

26 3 T 1 1 ρ ρ ρ... ρ T 1... Cov(ε) = σ 1 ρ ρ ρ ρ T 3 Ω = σ v ρ ρ 1 ρ... ρ. 1 ρ : : : : : T1 T T3 T4 ρ ρ ρ ρ... 1 Set ρ ρ ρ.. ρ ρ 1 ρ ρ... T 1 T * T 3 Ω = ρ ρ 1 ρ... ρ 1 ρ : : : : : T1 T T3 T4 ρ ρ ρ ρ... 1 ρ. (Ω * ) -1 = 1 ρ ρ 1+ ρ ρ ρ 1 + ρ ρ : : : : : : ρ ρ ρ 1 V = ρ ρ : : : : : : ρ 1 1 ρ V V = (Ω * ) -1. AUTO-6

27 (3) GLS when ρ is known. 1) Prais-Winston GLS: Vy = : T T y y y y y y y ρ ρ ρ ρ ; VX = , :... T T x x x x x x x ρ ρ ρ ρ. OLS on Vy = VXβ + Vε. ) Cochrane-Orcutt GLS y * = ; X : T T y y y y y y ρ ρ ρ * = , :... T T x x x x x x ρ ρ ρ. Asymptotically, P-W GLS = C-O GLS. When T is small, CO GLS is not desirable. AUTO-7

28 (4) FGLS when ρ is unknown 1) Consistent estimator of ρ T Σt= ee t t1 Cochrane-Orcutt: ˆ ρ =. Σ T e t= 1 t T ( Σt= ee t t1)/( T 1) DW Theil: ˆ ρ = or ˆ ρ = 1. T ( Σ e )/( T k) t= 1 t ) Two-Step FGLS of β Do GLS with ˆρ. Asymptotically identical to MLE of β if regressors are strictly exogenous. Two-Step FGLS and GLS (or MLE) can have different distributions if regressors are only weakly exogenous. AUTO-8

29 3) Iterative FGLS of β. STEP 1: Get Two-Step FGLS. STEP : Using T-S FGLS, compute residuals. STEP 3: Using these residuals, estimate ρ again. STEP 4: Using this estimate ρ, compute FGLS. STEP 5: Keep doing this until the estimates of β do not change. Comments: Asymptotically identical to MLE of β and two-step GLS if regressors are strictly exogenous. Iterative FGLS and GLS (or MLE) can have different distributions if regressors are only weakly exogenous. AUTO-9

30 (5) Nonlinear Least Square Approach (Estimate β and ρ jointly) Durbin Equation: y t = ρy t-1 + x t β + x t-1, (-ρβ) + v t. (y t = x t β + ε t ) - (ρy t-1 = x t-1, (ρβ) + ρε t-1 ) y t = ρy t-1 + x t β + x t-1, (-ρβ) + v t. [Durbin Equation] Regress y t on y t-1, x t and x t-1,. The estimated coefficient on y t-1 is a consistent estimate of ρ. Estimate this equation by NLLS Both estimates of β and ρ are asymptotically identical to MLE if the v t are normal. Okay with weakly exogenous x t. (6) MLE (Estimate β, ρ and σ v jointly): See Greene. y ~ N(Xβ, σ Ω): f(y) = ( π) 1 1 σ Ω σ T / y Xβ y Xβ, 1 T / exp ( ) Ω ( ) where σ σ v =. 1 ρ AUTO-30

31 l T (θ) = lnf(y) = - (T/)ln(π) - (T/)ln(σ v ) + (1/)ln(1-ρ ) - [1/(σ v )](Vy-VXβ) (Vy-VXβ) = - (T/)ln(π) - (T/)ln(σ v ) + (1/)ln(1-ρ ) - [(1-ρ )/(σ v )](y 1 -x 1 β) -[1/(σ T v )] Σ ( y ρ y x β + x ( ρβ)) where θ = (β,ρ,σ v ). t= t t 1 t t 1,, If T is large: (1/)ln(1-ρ ) - [(1-ρ )/(σ v )](y 1 -x 1 β) is negligible. Full MLE partial MLE (called MLE conditional on y 1 ) on - (T/)ln(π) - (T/)ln(σ v ) - [1/(σ v )] T Σ ( y ρ y x β + x ( ρβ )). t= t t 1 t t 1, Conditional MLE = NLLS. In EViews, y c x1 x x3 ar(1). AUTO-31

32 [EMPIRICAL EXAMPLE] Data: ustc96.wf1 FEDFRQ = Fed Fund Rate (%) GDP INF = annualized quarterly inflation rate (%) M1Q = M1 MQ = M TBILLQ = treasury bill rate (%) GGDP = 400*ln(GDP/GDP(-1)): Annualized growth rate of GDP (%). GM1 = 400*LOG(M1Q/M1Q(-1)): Annualized growth rate of M1 (%) Model to estimate: (fedfrq-tbillq) = β 1 + β ggdp t-4 + β 3 inf t-1 + ε t. From OLS (p. AUTO_4), ˆρ = 1- DW/ = / = AUTO-3

33 Cochrane-Orcutt GLS: codevfedtbill = devfedtbill 0.58*devfedtbill(-1); coinf = inf(-1) 0.58*inf(-); coggdp = ggdp(-4) 0.58*ggdp(-5). Dependent Variable: CODEVFEDTBILL Method: Least Squares Sample(adjusted): 1955:3 1996:1 Included observations: 163 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C COINF COGGDP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) AUTO-33

34 Residual Tests: Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C COINF COGGDP RESID(-1) RESID(-) RESID(-3) RESID(-4) R-squared Mean dependent var 1.0E-17 Adjusted R-squared S.D. dependent var Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) AUTO-34

35 NLLS Estimation: Joint estimation of ρ and β Dependent Variable: DEVFEDTBILL Method: Least Squares Sample(adjusted): 1955:3 1996:1 Included observations: 163 after adjusting endpoints Convergence achieved after 10 iterations Variable Coefficient Std. Error t-statistic Prob. C INF(-1) GGDP(-4) AR(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots.80 AUTO-35

36 [7] AR(p) (1) Assumptions: ε t = ρ 1 ε t-1 + ρ ε t ρ p ε t-p + v t, v t iid N(0,σ v ). The solutions of 1 - ρ 1 z ρ p z p = 0 are all outside of (-1,1). Gaurantees CS and ergodicity. {ε t } starts at the infinite past (- ) () Estimation θ s can be consistently estimated by regressing e t on e t-1,..., e t-p. The estimated coefficient on e t-j is a consistent estimate of ρ j. Use Cochrane-Ocutt type FGLS: y = y ˆ ρ y ˆ ρ y... ˆ ρ y ; * t t 1 t1 t p tp x = x ˆ ρ x ˆ ρ x... ˆ ρ x. * t t 1 t 1, t, p tp, AUTO-36

37 Use NLLS: (y t = x t β + ε t ) - (ρ 1 y t-1 = x t-1, (ρ 1 β) + ρ 1 ε t-1 ) (ρ p y t-p = x t-p, (ρ p β) + ρ p ε t-p ) y t - ρ 1 y t ρ p y t-p = x t β + x t-1, (-ρ 1 β) x t-p, (-ρ p β). y t = ρ 1 y t ρ p y t-p + x t β + x t-1, (-ρ 1 β) x t-p, (-ρ p β). Do NLLS on this equation = MLE conditional on y 1,..., y p. Full MLE if T is large. In EViews, y c x1 x x3 ar(1) ar()... ar(p). AUTO-37

38 [8] MOVING AVERAGE: MA(q) (1) Assumption 1) ε t = v t + ψ 1 v t ψ q v t-q, v t iid N(0,σ v ). ) The ψ j are finite: Guarantees CS. () Estimation 1) FGLS is possible for MA(1), but is messy. (See Fomby, et al) ) Use conditional MLE (See Hamilton, Ch. 5.) This is a kind of NLLS. In EViews, y c x1 x x3 ma(1) ma()... ma(q) AUTO-38

39 [EMPIRICAL EXAMPLE] Data: ustc96.wf1 FEDFRQ = Fed Fund Rate (%) GDP INF = annualized quarterly inflation rate (%) M1Q = M1 MQ = M TBILLQ = treasury bill rate (%) GGDP = 400*ln(GDP/GDP(-1)): Annualized growth rate of GDP (%). GM1 = 400*LOG(M1Q/M1Q(-1)): Annualized growth rate of M1 (%) DEVFEDTBILL = fedfrq tbillq. Model to estimate: (fedfrq-tbillq) = ggdp t = β 1 + β ggdp t-4 + β 3 inf t-1 + ε t. AUTO-39

40 NLLS with the MA(1) error assumption Dependent Variable: DEVFEDTBILL Method: Least Squares Date: 04/16/0 Time: 16:3 Sample(adjusted): 1955: 1996:1 Included observations: 164 after adjusting endpoints Convergence achieved after 1 iterations Backcast: 1955:1 Variable Coefficient Std. Error t-statistic Prob. C INF(-1) GGDP(-4) MA(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted MA Roots -.50 AUTO-40

41 view/residual tests/serial correlation LM test lags to include: 4 Breusch-Godfrey Serial Correlation LM Test: F-statistic Probability Obs*R-squared Probability Test Equation: Dependent Variable: RESID Method: Least Squares Presample missing value lagged residuals set to zero. Variable Coefficient Std. Error t-statistic Prob. C INF(-1) GGDP(-4) MA(1) RESID(-1) RESID(-) RESID(-3) RESID(-4) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) AUTO-41

42 [9] ARMA(p,q) (1) Assumption 1) ε t = ρ 1 ε t ρ p ε t-p + v t + ψ 1 v t ψ q v t-q, v t iid N(0,σ v ). ) The ψ j are finite and all solutions of 1- ρ 1 z - ρ z ρ p z p = 0 are all outside of (-1,1): Guarantees CS and ergodicity. () Estimation Use conditional MLE (See Hamilton, Ch. 5.) This is a kind of NLLS. In EViews, y c x1 x x3 ar(1)... ar(p) ma(1)... ma(q). AUTO-4

43 Dependent Variable: DEVFEDTBILL Method: Least Squares Sample(adjusted): 1955:3 1996:1 Included observations: 163 after adjusting endpoints Convergence achieved after 11 iterations Backcast: 1955: Variable Coefficient Std. Error t-statistic Prob. C INF(-1) GGDP(-4) AR(1) MA(1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) Inverted AR Roots.81 Inverted MA Roots.03 AUTO-43

44 [10] Forecasting (1) AR(1) Model: y = x β + ε ; ε = ρε + ν. t t t t t1 t Eˆ( y Ω ) = xˆ ˆ β + Eˆ( ε Ω ) = xˆ ˆ β + ρε T+ 1 T T+ 1, T+ 1 T T+ 1, = xˆ ˆ β + ˆ ρ( y x ˆ β). T+ 1, T T Eˆ( y Ω ) = xˆ ˆ β + Eˆ( ε Ω ) = xˆ ˆ β + ρε T+ T T+, T+ T T+, T+ 1 ˆ ˆ = xˆ β + ρε = xˆ β+ ˆ ρ( y x ˆ β). T+, T T T T Eˆ ( y ) x ˆ β ρ ( y x ˆ β). n ˆ ˆ T+ n Ω T = T+ n, + T T T () MA(1) Model: Estimating ν T : Set ˆ ν 0 = 0. ˆ ν = ˆ ε ψˆˆ = ( ˆ β). 1 1 vo y1 x1 = β + ε ; ε = ψ + ν. yt xt t t vt 1 t ˆ ν = ˆ ε ψˆ = ˆ β + ψˆ. : vˆ 1 ( y x ) v0 ˆ ν = ( y x ˆ β) ψˆˆ v. T T T T1 AUTO-44

45 Forecasting: Eˆ( y Ω ) = xˆ ˆ β + Eˆ( ε Ω ) = xˆ ˆ β + v + ψv T+ 1 T T+ 1, T+ 1 T T+ 1, T+ 1 = xˆ ˆ β + ψˆ vˆ. T+ 1, T Eˆ( y Ω ) = xˆ ˆ β + Eˆ( ε Ω ) = xˆ ˆ β + v + ψ T+ T T+, T+ T T+, T+ v T + 1 = xˆ ˆ β. T +, : Eˆ ( y ) ˆ ˆ T+ n Ω T = x T+ n, β. T AUTO-45