Copyright 1996 Lawrence C. Marsh 0 PowerPoint Slides for Undergraduate Econometrics by
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partial derivatives: f’(X
t ,b (ο)
) and y is replaced by y ∗ (ο) (i.e. “y” = y* (ο)
and “X” = f’(X,b (ο)
) ) 14.12
Copyright 1996 Lawrence C. Marsh Recall that: y * (o) ≡
y −
f(X,b (o) ) + f’(X,b (ο)
) b (ο)
Now define: y ∗∗ (ο) ≡
−
∗ (ο) = y ∗∗ (ο) + f’(X,b (ο)
) b (ο)
b =
[ f’(X,b (ο)
) T f’(X,b (ο)
) ] -1 f’(X,b (ο)
) T y ∗ (ο) ^ Now substitute in for y ∗
in Gauss-Newton solution: to get: b = b (o) +
f’(X,b (ο)
) T f’(X,b (ο)
) ] -1 f’(X,b (ο)
) T y ∗∗ (ο) ^ 14.13
Copyright 1996 Lawrence C. Marsh b = b (o) + [ f’(X,b (ο)
) T f’(X,b (ο)
) ] -1 f’(X,b (ο)
) T y ∗∗ (ο) ^ b (1) = b (ο)
+
f’(X,b (ο)
) T f’(X,b (ο)
) ] -1 f’(X,b (ο)
) T y ∗∗ (ο) Now call this b value b (1)
as follows: ^ More generally, in going from interation m to iteration (m+1) we obtain the general expression: b (m+1) = b (m) +
f’(X,b (m )
T f’(X,b (m )
] -1 f’(X,b (m )
T y ∗∗
14.14
)
T f’(X,b (m )
] -1 f’(X,b (m) ) T y * (m) b (m+1) = b (m) +
f’(X,b (m) ) T f’(X,b (m) ) ] -1 f’(X,b (m) ) T y ∗∗
Thus, the Gauss-Newton (nonlinear OLS) solution can be expressed in two alternative, but equivalent, forms: 1. replacement form : 2. updating form: 14.15
Copyright 1996 Lawrence C. Marsh For example, consider Durbin’s Method of estimating the autocorrelation coefficient under a first-order autoregression regime: y t = b 1 + b 2
t 2 + . . . + b K X t K + ε
t
ε
ρ ε t - 1 + u t where u t satisfies the conditions E u t = 0 , E u 2 t = s u 2 , E u t u s = 0 for s ≠
Therefore, u t is nonautocorrelated and homoskedastic. Durbin’s Method is to set aside a copy of the equation, lag it once, multiply by ρ
from the original equation, then move the ρ
t-1 term to the right side and estimate ρ
s by OLS. 14.16
Copyright 1996 Lawrence C. Marsh Durbin’s Method is to set aside a copy of the equation, lag it once, multiply by ρ
from the original equation, then move the ρ
t-1 term to the right side and estimate ρ
y t = b 1 + b 2
ε
t
where ε
= ρ ε t - 1 + u t ρ
t-1 = ρ
1 + ρ
2
t -1, 2 + ρ
3 X t -1, 3 + ρ ε t -1 Lag once and multiply by ρ:
ρ
(1
ρ )
+ b 2 (
t 2 - ρ
t-1, 2 )
+ b 3 (X t 3 − ρ
X t-1, 3 )+ ρ
t-1 + u t 14.17
62 Copyright 1996 Lawrence C. Marsh y t = b 1 (1
ρ )
+ b 2 (
t 2 - ρ
t-1, 2 )
+ b 3 (X t 3 - ρ
t-1, 3 ) + ρ
t-1 + u t Now Durbin separates out the terms as follows: y t = b 1 (1
ρ )
+ b 2 X t 2 - b 2 ρ
t-1 2 + b 3 X t 3 - b 3 ρ
t-1 3 + ρ
t-1 + u t The structural (restricted,behavorial) equation is: The corresponding reduced form (unrestricted) equation is: y t = α
α
X t ,
2 + α
X t-1 ,
+ α
X t ,
3 + α
X t-1 ,
+ α
y t-1 + u t α
= b 1 (1
ρ ) α
2 = b 2 α
= - b 2 ρ
α 4 = b 3
α 5 = - b 3 ρ α 6 = ρ 14.18 Copyright 1996 Lawrence C. Marsh Given OLS estimates: α
α
α
α
α
α
ρ :
ρ = − α 3 α 2 ^ ^ ^ ρ =
− α 5 α 4 ^ ^ ^ ρ = α
6
^ ^ These three separate estimates of ρ
It is difficult to know which one to use as “the” legitimate estimate of ρ.
Durbin used the last one. α
= b 1 (1
ρ ) α
2 = b 2 α
= - b 2 ρ
α 4 = b 3
α 5 = - b 3 ρ α 6 = ρ 14.19 Copyright 1996 Lawrence C. Marsh The problem with Durbin’s Method is that it ignores the inherent nonlinear restrictions implied by this structural model. To get a single (i.e. unique) estimate for ρ
the implied nonlinear restrictions must be incorporated directly into the estimation process. Consequently, the above structural equation should be estimated using a nonlinear method such as the Gauss-Newton algorithm for nonlinear least squares. y t = b 1 (1
ρ )
+ b 2 X t 2 - b 2 ρ
t -1, 2 + b 3 X t 3 - b 3 ρ
t -1, 3 + ρ
t-1 + u t 14.20
Copyright 1996 Lawrence C. Marsh y t = b 1 (1
ρ )
+ b 2 X t 2 - b 2 ρ
t-1, 2 + b 3 X t 3 - b 3 ρ
t-1, 3 + ρ
t-1 + u t f’(X
t ,b)
= [
] ∂
t ∂ρ = (1 − ρ
) = ( X t,
2 − ρ
X t-1
, 2 ) = ( X t, 3 − ρ X t-1
, 3 ) ∂ y t ∂ρ = ( - b 1
∂ y t ∂ b 1 ∂ y t ∂ b 2 ∂ y t ∂ b 3 ∂ y t ∂ b 1 ∂ y t ∂ b 2 ∂ y t ∂ b 3 14.21
Copyright 1996 Lawrence C. Marsh where y t ∗
≡
y t - f(X t ,b (
)
)
(m ) β (m+1) = [ f’(X,b (
)
(
)
)
T y ∗
)
(1
ρ )
+ b 2 X t 2 - b 2 ρ
t-1 2 + b 3 X t 3 - b 3 ρ
t-1 3 + ρ
t-1
(m) =
b 1(m)
ρ (m) b 2(m)
b 3(m)
Iterate until convergence. f’(X
t ,b ( m ) ) = [
] ∂
t ∂ρ (m) ∂ y t ∂ b 1(m) ∂ y t ∂ b 2(m) ∂ y t ∂ b 3 (m) 14.22
Copyright 1996 Lawrence C. Marsh Distributed Lag Models Chapter 15 Copyright © 1997 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United States Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein. 15.1
63 Copyright 1996 Lawrence C. Marsh The Distributed Lag Effect Economic action at time t Effect at time t Effect at time t+1 Effect at time t+2 15.2 Copyright 1996 Lawrence C. Marsh Unstructured Lags y t
α +
β 0 x t +
β 1 x t-1 +
β 2 x t-2 + . . . + β n
t-n + e
t “n” unstructured lags no systematic structure imposed on the β ’s the β ’s are unrestricted 15.3 Copyright 1996 Lawrence C. Marsh Problems with Unstructured Lags 1. n observations are lost with n-lag setup. 2. high degree of multicollinearity among x t-j ’s.
3. many degrees of freedom used for large n. 4. could get greater precision using structure. 15.4
proposed by Irving Fisher (1937) the lag weights decline linearly Imposing the relationship: β # = (n - # + 1) γ β 0 = (n+1) γ β
= n γ β 2 = (n-1) γ β
= (n-2) γ . . β n-2 = 3 γ β n-1 = 2 γ β
= γ only need to estimate one coefficient, γ , instead of n+1 coefficients, β 0 , ... , β n . 15.5 Download 0.54 Mb. Do'stlaringiz bilan baham: 
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