8.1 The Nature of Heteroskedastcty 8. Usng the Least Squares Estmator 8.3 The Generalzed Least Squares Estmator 8.4 Detectng Heteroskedastcty
E( y) = β+β 1 x e = y E( y ) = y β β x 1 y = β+β x + e 1
Fgure 8.1 Heteroskedastc Errors
Ee e e e ( ) = 0 var( ) =σ cov(, j) = 0 var( y ) = var( e) = h( x ) yˆ = 83.4 + 10.1x eˆ = y 83.4 10.1x
Fgure 8. Least Squares Estmated Expendture Functon and Observed Data Ponts
The exstence of heteroskedastcty mples: The least squares estmator s stll a lnear and unbased estmator, but t s no longer best. There s another estmator wth a smaller varance. The standard errors usually computed for the least squares estmator are ncorrect. Confdence ntervals and hypothess tests that use these standard errors may be msleadng.
y =β 1+β x + e var( e) =σ var( b ) = N = 1 σ ( x x) y =β 1+β x + e var( e) =σ
var( b ) N ( ) N x x σ = 1 = wσ = N = 1 ( x x) = 1 var( b ) N ( ) ˆ N x x e = 1 ˆ = we = N = 1 ( x x) = 1
yˆ = 83.4 + 10.1x (7.46) (1.81) (Whte se) (43.41) (.09) (ncorrect se) Whte: b ± t se( b ) = 10.1±.04 1.81 = [6.55, 13.87] c Incorrect: b ± t se( b ) = 10.1±.04.09 = [5.97, 14.45] c
y =β +β x + e 1 ( ) = 0 var( ) =σ cov(, j) = 0 Ee e e e
( e ) = σ =σ var x y 1 x e =β 1 +β + x x x x y 1 x e y = x1 = x = = x e = x x x x
y = β x +β x + e 1 1 var( e ) = var e 1 1 = var( e) = σ x =σ x x x
To obtan the best lnear unbased estmator for a model wth heteroskedastcty of the type specfed n equaton (8.11): 1. Calculate the transformed varables gven n (8.13).. Use least squares to estmate the transformed model gven n (8.14).
The generalzed least squares estmator s as a weghted least squares estmator. Mnmzng the sum of squared transformed errors that s gven by: N N N 1/ e = = ( x e) = 1 = 1 x = 1 When s small, the data contan more nformaton about the regresson functon and the observatons are weghted heavly. When e x x s large, the data contan less nformaton and the observatons are weghted lghtly.
yˆ = 78.68 + 10.45x (se) (3.79) (1.39) β 垐 ± se( β ) = 10.451 ±.04 1.386 = [7.65,13.6] t c
var( e) = σ =σ x γ ln( σ ) = ln( σ ) +γln( x) ( x ) σ = exp ln( σ ) +γln( ) = exp( α +α z ) 1
σ = exp( α +α z + L+α z ) 1 s S ln( ) σ =α +α 1 z y = E( y ) + e =β +β x + e 1
ˆ ln( e ) = ln( σ ) + v =α 1+α z + v ˆ ln( σ ) =.9378 +.39 z σ = α 垐 +α ˆ exp( 1 1z) y 1 x e =β 1 +β + σ σ σ σ
e 1 1 var = var( ) 1 e = σ = σ σ σ y 1 x y = x 1 = x = σ垐? σ σ y = β x +β x + e 1 1
y = β+β x + L+β x + e 1 k K L var( e) =σ = exp( α 1+α z + +αszs)
The steps for obtanng a feasble generalzed least squares estmator β, β, K, βk for 1 are: 1. Estmate (8.5) by least squares and compute the squares of the least squares resduals ˆ. e α1, α, K, αs. Estmate by applyng least squares to the equaton ˆ ln e = α+α 1 z + L+α S z S + v
3. Compute varance estmates 1 S S. 4. Compute the transformed observatons defned by (8.3), ncludng f K >. 5. Apply least squares to (8.4), or to an extended verson of (8.4) K > f. x,, 3 K xk yˆ = 76.05 + 10.63x (se) (9.71) (.97) σ = α 垐 +α z + L+α? ˆ exp( ) z
WAGE = 9.914 + 1.34 EDUC +.133EXPER + 1.54METRO (se) (1.08) (.070) (.015) (.431) WAGE =β 1+β EDUC +β 3EXPER + e = 1,, K, N M M M M M M WAGE =β 1+β EDUC +β 3EXPER + e = 1,, K, N R R R R R R b M 1 = 9.914 + 1.54 = 8.39
var( e ) = σ var( e ) =σ M M R R 垐 M 31.84 R 15.43 σ = σ = b = 9.05 b = 1.8 b =.1346 M1 M M3 b = 6.166 b =.956 b =.160 R1 R R3
WAGE M 1 EDUC M EXPER M e M =β M 1 +β +β 3 + σm σm σm σm σm = 1,, K, NM WAGE R 1 EDUC R EXPER R e R =β R1 +β +β 3 + σr σr σr σr σr = 1,, K, NR
Feasble generalzed least squares: ˆ M ˆ R 1. Obtan estmated σ and σ by applyng least squares separately to the metropoltan and rural observatons. σ ˆ M when METRO = 1. σ ˆ = σ ˆ R when METRO = 0 3. Apply least squares to the transformed model WAGE 1 EDUC EXPER METRO e =β R1 +β +β 3 +δ + σ垐垐垐 σ σ σ σ σ
WAGE = 9.398 + 1.196 EDUC +.13EXPER + 1.539METRO (se) (1.0) (.069) (.015) (.346)
Remark: To mplement the generalzed least squares estmators descrbed n ths Secton for three alternatve heteroskedastc specfcatons, an assumpton about the form of the heteroskedastcty s requred. Usng least squares wth Whte standard errors avods the need to make an assumpton about the form of heteroskedastcty, but does not realze the potental effcency gans from generalzed least squares.
8.4.1 Resdual Plots Estmate the model usng least squares and plot the least squares resduals. Wth more than one explanatory varable, plot the least squares resduals aganst each explanatory varable, or aganst yˆ, to see f those resduals vary n a systematc way relatve to the specfed varable.
8.4. The Goldfeld-Quandt Test σˆ σ F = F σ σ M M R R ˆ M M R R ( N K, N K ) H : σ =σ aganst H : σ σ 0 M R 0 M R F ˆ M 31.84 ˆ R 15.43 = σ = = σ.09
8.4. The Goldfeld-Quandt Test σ ˆ = 3574.8 1 σ ˆ = 1,91.9 F ˆ 1,91.9 = σ = = σˆ 3574.8 1 3.61
8.4.3 Testng the Varance Functon y = E( y ) + e =β +β x + L+β x + e 1 K K L var( y) =σ = E( e ) = h( α 1+α z + +αszs) h( α +α z + L+α z ) = exp( α +α z + L+α z ) 1 S S 1 S S 1 ( ) h( α+α z ) = exp ln( σ ) +γln( x )
8.4.3 Testng the Varance Functon h( α +α z + L+α z ) =α +α z + L+α z 1 S S 1 S S h( α +α z + L+α z ) = h( α ) 1 S S 1 H : α =α = L =α = 0 0 3 S H : not all the α n H are zero 1 s 0
8.4.3 Testng the Varance Functon var( y) =σ = E( e ) =α 1+α z + L+αSzS e = E( e ) + v =α +α z + L+α z + v 1 S S e = α+α z + L+α z + v ˆ 1 S S χ = N R χ ( S 1)
8.4.3a The Whte Test E( y ) = β+β x +βx 1 3 3 z = x z = x z = x z = x 3 3 4 5 3
8.4.3b Testng the Food Expendture Example SST = 4,610,749, 441 SSE = 3,759,556,169 R SSE = 1 =.1846 SST χ = N R = 40.1846 = 7.38 χ = N R = 40.18888 = 7.555 p-value =.03
Prncples of Econometrcs, 3rd Edton Slde 8-38
Prncples of Econometrcs, 3rd Edton Slde 8-39
y = β+β x + e 1 Ee e e e j ( ) = 0 var( ) =σ cov(, j) = 0 ( ) b = β + we (8A.1) w = x x ( x ) x Prncples of Econometrcs, 3rd Edton Slde 8-40
( ) ( ) ( ) = β + E b E E we ( ) = β + we e =β Prncples of Econometrcs, 3rd Edton Slde 8-41
var ( b ) ( ) = var we ( ) ( ) = w var e + ww cov e, e = w σ j j j (8A.) ( x x) σ = ( x x) Prncples of Econometrcs, 3rd Edton Slde 8-4
var( b ) σ ( x x) (8A.3) Prncples of Econometrcs, 3rd Edton Slde 8-43
e = α+α z + L+α z + v ˆ 1 S S (8B.1) F = ( SST SSE)/( S 1) SSE /( N S) (8B.) ( ) N N 垐 and? = 1 = 1 SST = e e SSE = v Prncples of Econometrcs, 3rd Edton Slde 8-44
SST SSE χ = ( S 1) F = χ SSE /( N S ) ( S 1) (8B.3) var( e ) var( v) = = SSE N S (8B.4) χ = SST SSE var( e ) (8B.5) Prncples of Econometrcs, 3rd Edton Slde 8-45
χ = SST SSE ˆ 4 σ e (8B.6) e 1 var var( ) var( ) = e = e = σ 4 σe σe 4 e 1 var( e ) = ( e e ) = N 垐 N = 1 SST N (8B.7) Prncples of Econometrcs, 3rd Edton Slde 8-46
χ = SST SSE SST / N = N 1 SSE SST (8B.8) = N R Prncples of Econometrcs, 3rd Edton Slde 8-47