overdetermined system problems if cov xt xt 1 is

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Unformatted text preview: utocorrelation Expectations Model Partial Adjustment Model Adaptive Adaptive Expectations Model Substituting, we get: Yt = β1 + β2 λXt + β2 (1 − λ)Xte + ut But we still have the unobserved Xte . Substituting repeatedly we get: Yt = β1 + β2 λXt + β2 (1 − λ)Xt −1 + · · · + β2 (1 − λ)s−1 Xt −s+1 + β2 (1 − λ)s Xte s+1 + ut − Taking s large enough, we can ignore the further terms in Xte s+1 . Why? − 15 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Expectations Model Partial Adjustment Model Adaptive Adaptive Expectations Model Yt = β1 + β2 λXt + β2 (1 − λ)Xt −1 + · · · + β2 (1 − λ)s−1 Xt −s+1 + β2 (1 − λ)s Xte s+1 + ut − Can we estimate this with OLS? OLS is consistent but in small samples we have more equations than unknowns (why?) ⇒ overdetermined system problems if cov (Xt , Xt −1 ) is high So we use a nonlinear algorithm. 16 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Expectations Model Partial Adjustment Model Adaptive Adaptive Expectations Model Yt = β1 + β2 λXt + β2 (1 − λ)Xt −1 + · · · + β2 (1 − λ)s−1 Xt −s+1 + β2 (1 − λ)s Xte s+1 + ut − Can we estimate this with OLS? OLS is consistent but in small samples we have more equations than unknowns (why?) ⇒ overdetermined system problems if cov (Xt , Xt −1 ) is high So we use a nonlinear algorithm. 17 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Expectations Model Partial Adjustment Model Adaptive Adaptive Expectations Model Yt = β1 + β2 λXt + β2 (1 − λ)Xt −1 + · · · + β2 (1 − λ)s−1 Xt −s+1 + β2 (1 − λ)s Xte s+1 + ut − Can we estimate this with OLS? OLS is consistent but in small samples we have more equations than unknowns (why?) ⇒ overdetermined system problems if cov (Xt , Xt −1 ) is high So we use a nonlinear algorithm. 18 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Expectations Model Partial Adjustment Model Adaptive Adaptive Expectations Model Yt = β1 + β2 λXt + β2 (1 − λ)Xt −1 + · · · + β2 (1 − λ)s−1 Xt −s+1 + β2 (1 − λ)s Xte s+1 + ut − Can we estimate this with OLS? OLS is consistent but in small samples we have more equations than unknowns (why?) ⇒ overdetermined system problems if cov (Xt , Xt −1 ) is high So we use a nonlinear algorithm. 19 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Expectations Model Partial Adjustment Model Adaptive Adaptive Expectations Model Yt = β1 + β2 λXt + β2 (1 − λ)Xt −1 + · · · + β2 (1 − λ)s−1 Xt −s+1 + β2 (1 − λ)s Xte s+1 + ut − Can we estimate this with OLS? OLS is consistent but in small samples we have more equations than unknowns (why?) ⇒ overdetermined system problems if cov (Xt , Xt −1 ) is high So we use a nonlinear algorithm. 20 / 62 Introduction Time Series and OLS Two Dynamic Models Autocorrelation Expectations Model Partial Adjustment Model Adaptive Adaptive Expectations Model Yt = β1 + β2 λXt + β2 (1 − λ)Xte + ut Long run effect of X → β2 Short run effect of X → λβ2 Why? Yt −1 = β1 +...
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This document was uploaded on 03/12/2014 for the course ECON 202 at University of London University of London International Programmes (Distance Learning).

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