Copyright 1996 Lawrence C. Marsh 0 PowerPoint Slides for Undergraduate Econometrics by
Partitioned Heteroskedasticity. ( discrete categories/
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Partitioned
Heteroskedasticity. ( discrete
categories/groups) 10.11
Copyright 1996 Lawrence C. Marsh Proportional Heteroskedasticity y t
β 1 + β 2 x t + e
t where
var(e t ) = σ t
2 E(e
t ) = 0
cov(e t , e s ) = 0 t
≠
s σ t 2
= σ
2 x t The variance is assumed to be proportional to the value of x t 10.12 Copyright 1996 Lawrence C. Marsh σ t 2
= σ
2 x t y t = β 1 + β 2 x t
+ e t std.dev. proportional to x t
variance: standard deviation: σ t
= σ
x t
y t 1 x t e t = β 1 + β 2 + x t x t x t x t To correct for heteroskedasticity divide the model by
x t
var(e t ) =
σ t
2 10.13
47 Copyright 1996 Lawrence C. Marsh y t 1 x t e t
β 1 + β 2 + x
x t x t x t y t = β 1 x t1 +
β 2 x t2 + e
t * * * * var(e t ) = var( ) = var(e t ) = σ
2 x t * e t
x t 1 x t 1 x t var(e t ) =
σ
2 * e t is heteroskedastic , but
e t is homoskedastic . * 10.14 Copyright 1996 Lawrence C. Marsh 1. Decide which variable is proportional to the heteroskedasticity (x t in previous example). 2. Divide all terms in the original model by the square root of that variable (divide by x t
3. Run least squares on the transformed model which has new y t , x
t1
and x t2 variables but
. Generalized Least Squares These steps describe weighted least squares: * * * 10.15
Copyright 1996 Lawrence C. Marsh Partitioned Heteroskedasticity y t
β 1 + β 2 x t + e
t var(e
t ) =
σ 1
2 var(e
t ) =
σ 2
2 error variance of “field” corn : error variance of “sweet” corn : y t = bushels per acre of corn x t
gallons of water per acre (rain or other) t = 1, . . .
,100 t = 1, . . .
,80 t = 81, . . .
,100 10.16
Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 x t
+ e t var(e t ) =
σ 1
2 “field” corn : y
= β 1 + β 2 x t
+ e t var(e t ) =
σ 2
2 “sweet” corn : y
1 x t e t = β 1 + β 2 + σ 1 σ 1
σ 1 σ 1
t
1 x t e t = β 1 + β 2 + σ 2 σ 2
σ 2 σ 2
t = 1, . . .
,80 t = 81, . . .
,100 10.17
Copyright 1996 Lawrence C. Marsh Apply Generalized Least Squares Run least squares separately on data for each group. σ 1 2 provides estimator of σ 1
2 using
the 80 observations on “field” corn. ^ σ 2
2 provides estimator of σ 2 2 using the 20 observations on “sweet” corn. ^ 10.18 Copyright 1996 Lawrence C. Marsh 1. Residual Plots provide information on the exact nature of heteroskedasticity (partitioned or proportional) to aid in correcting for it. 2. Goldfeld-Quandt Test checks for presence of heteroskedasticity. Detecting Heteroskedasticity Determine existence and nature of heteroskedasticity : 10.19
48 Copyright 1996 Lawrence C. Marsh Residual Plots e t
x t
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plot residuals against one variable at a time after sorting the data by that variable to try to find a heteroskedastic pattern in the data. 10.20
Goldfeld-Quandt Test The Goldfeld-Quandt test can be used to detect heteroskedasticity in either the proportional case or for comparing two groups in the discrete case. For proportional heteroskedasticity, it is first necessary to determine which variable, such as x t , is proportional to the error variance. Then sort the data from the largest to smallest values of that variable. 10.21
Copyright 1996 Lawrence C. Marsh H o : σ 1 2 = σ 2 2
H 1 : σ 1
2 >
σ 2
2
[T 1
1 , T
2 -K 2 ] σ 1 2 σ 2
2 ^ ^ In the proportional case, drop the middle r observations where r ≈ T/6, then run separate least squares regressions on the first T 1 observations and the last T 2 observations. Small values of GQ support H o while large values support H 1 .
Test Statistic Use F Table 10.22
Copyright 1996 Lawrence C. Marsh σ t 2
=
σ 2 exp{ α 1 z t1 +
α 2 z t2 } More General Model Structure of heteroskedasticity could be more complicated: z t1 and
z t2 are any observable variables upon which we believe the variance could depend. Note: The function exp{ . } ensures that σ t
is positive. 10.23
Copyright 1996 Lawrence C. Marsh σ t 2 =
σ
2 exp{ α 1 z t1 + α 2 z t2 } More General Model ln (σ t 2 )
= ln (σ
2 ) + α 1 z t1 + α 2 z t2 ln (σ t 2 ) = α 0
+ α 1 z t1 + α 2 z t2 where
α 0 = ln
(σ
2 ) H o : α 1
= 0, α 2 = 0
H 1 : α 1
≠ 0,
α 2
≠ 0
and/or Least squares residuals, e t ^ ln ( e t 2 )
= α 0
+ α 1 z t1 + α 2 z t2 + ν t ^
10.24
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. 11.1
49 Copyright 1996 Lawrence C. Marsh The Nature of Autocorrelation For
(accurate estimation/prediction) all systematic information needs to be incor- porated into the regression model. Autocorrelation is a systematic pattern in the errors that can be either attracting (
) or repelling ( negative ) autocorrelation. 11.2
Postive
Auto. No Auto. Negative Auto.
e t
0 e
0 e t 0 t t t . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . crosses line not enough (attracting) crosses line randomly crosses line too much (repelling) 11.3
y t = β 1 + β 2 x t
+ e t Regression Model E(e t ) = 0 var(e t ) = σ
2 zero mean: homoskedasticity: nonautocorrelation: cov(e
t , e
s ) =
0 t
≠
s autocorrelation: cov(e
t , e
s ) ≠ 0
t
≠ s 11.4 Copyright 1996 Lawrence C. Marsh Order of Autocorrelation y t
β 1 + β 2 x t + e
t e t = ρ e t − 1 + ν t e t
= ρ 1 e t − 1
+ ρ 2 e t − 2
+ ν t e t
= ρ 1 e t − 1
+ ρ 2 e t − 2
+ ρ 3 e t − 3 + ν t 1st Order: 2nd Order: 3rd Order: We will assume First Order Autocorrelation: e t = ρ e t − 1 + ν t AR(1) : 11.5
Copyright 1996 Lawrence C. Marsh First Order Autocorrelation y t
β 1 + β 2 x t + e
t e t = ρ e t − 1 + ν t where − 1 < ρ < 1
E( ν t ) = 0 var(
ν t ) = σ ν 2 cov( ν t , ν s ) = 0
t
≠ s These assumptions about ν t imply the following about e t : E(e t ) = 0 var(e t ) = σ
e 2 = cov(e t , e
t − k ) = σ e 2 ρ k
for k
>
0 corr(e t , e t − k ) = ρ k
for k
>
0 σ ν 2 1 − ρ 2 11.6
Copyright 1996 Lawrence C. Marsh Autocorrelation creates some Problems for Least Squares : 1. The least squares estimator is still linear and unbiased but it is
. 2. The formulas normally used to compute the least squares standard errors are no longer correct and confidence intervals and hypothesis tests using them will be
. 11.7 50 Copyright 1996 Lawrence C. Marsh Generalized Least Squares y t = β 1 + β 2 x t
+ e t e t =
ρ e t − 1
+ ν t y t = β 1 + β 2 x t +
ρ e t − 1
+ ν t substitute in for e t
e t − 1 (continued) AR(1) : 11.8
Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 x t
+ e t y t =
β 1 + β 2 x t +
ρ e t − 1
+ ν t e t = y t
− β 1 − β 2 x t e t − 1 = y t − 1 −
β 1 − β 2 x t − 1 y t =
β 1 + β 2 x t +
ρ( y t − 1
−
β 1 − β
2 x t − 1 ) + ν t lag the
errors once
(continued) 11.9
Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 x t + ρ( y t − 1
−
β 1 − β
2 x t − 1 ) + ν t y t = β 1 + β 2 x t + ρ y t − 1
− ρ
β 1 − ρ β
2 x t − 1
+ ν t y t − ρ y t − 1 = β 1 (1 −ρ ) + β 2 (x t −ρ x t − 1 )
+ ν t y t =
β 1 + β 2 x t2 +
ν t * * * y t = y
t −
ρ y t − 1
* β 1 = β 1 (1 −ρ ) * x t2 = (x t −ρ x t − 1 )
* 11.10
Copyright 1996 Lawrence C. Marsh y t = β 1 + β 2 x t2
+ ν t * * * y t = y t −
ρ y t − 1
* β 1 = β 1 (1 −ρ ) * x t2 = x t − ρ x t − 1 * Problems estimating Download 0.54 Mb. Do'stlaringiz bilan baham: 
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