Heteroskedasticity. in the Error Component Model

Transcription

1 Heteroskedasticity in the Error Component Model Baltagi Textbook Chapter 5 Mozhgan Raeisian Parvari ( )

2 Content Introduction Cases of Heteroskedasticity Adaptive heteroskedastic estimators (EGLS, GLSAD) Testing for Heteroskedasticity

3 Introduction The standard error component model given by equation y it = X it β + u it i = 1,, N ; t = 1,, T u it = μ i + ν it assumes that the regression disturbances are homoskedastic with the same variance across time and individuals. This may be a restrictive assumption for panels, where the crosssectional units may be of varying size and as a result may exhibit different variation.

4 The Consequences of Heteroscedasticity Assuming homoskedastic disturbances when heteroskedasticity is present will still result in consistent estimates of the regression coefficients, but these estimates will not be efficient. Also, the standard errors of these estimates will be biased and one should compute robust standard errors correcting for the possible presence of heteroskedasticity.

5 Cases of Heteroskedasticity Case 1: The heteroskedasticity is on the individual specific error component (Mazodier and Trognon (1978)) μ i ~ 0, ω i for i = 1,, N In vector form, ν it ~IID(0, σ ν ) μ~ (0, μ ) where μ = diag[ω i ] ν~(0, σ ν I NT )

6 Ω = E uu = Z μ Σ μ Z μ + σ ν I NT Ω = diag ω J T + diag σ ν I T Using the Wansbeek and Kapteyn(198b, 1983) trick, Baltagi and Griffin (1988) derived the transformation as follow: Ω = diag Tω + σ ν JT + diag σ ν E T Ω r = diag[(τ i ) r ] JT + diag (σ ν ) r E T σ ν Ω 1 = diag σ ν τ i JT + (I N E T )

7 Hence, has a typical element where y = σ ν Ω 1 y y it = y it θ i y i θ i = 1 (σ ν τ i ) Baltagi and Griffin (1988) provided feasible GLS estimator including Rao s (1970,197) MINQUE estimator for this model. Phillips (003) argues that there is no guarantee that feasible GLS and true GLS will have the same asymptotic distributions.

8 Case : The heteroskedasticity is on the remainder error term (Wansbeek (1989)) In this case, μ i ~ IID(0, σ μ ) ν it ~(0, ω i ) Ω = E uu = diag σ μ J T + diag ω i I T Ω = diag Tσ μ + ω i JT + diag ω i E T Ω r = diag[(τ i ) r ] JT + diag (σ ν ) r E T

9 Ω 1 = diag 1 τ i JT + diag[1 ω i ] E T y = Ω 1 y y it = (y i / τ i ) + (y it y i )/ ω i y it = 1 ω i (y it θ i y i ) where θ i = 1 (ω i τ i ) Case 3: The heteroskedasticity is both on the individual specific error Component and on the remainder error term (Randolph (1988)) Var (μ i ) = σ i E νν = diag[σ it ] i = 1,, N ; t = 1,, T

10 Adaptive heteroskedastic estimators (EGLS, GLSAD): Roy (00)-EGLS: E μ i X i. = 0 var μ i X i. = ω(x i. ) ω i σ ν = N i=1 T t=1 [ y it y i. (X it X i. ) β] N T 1 k β is the fixed effects or within estimator of β. ω i = N j=1 N j=1 T t=1 T t=1 K i.,j. u jt K i.,j. σ μ where the kernel function is given by K i.,j. = K X i. X j.

11 Monte Carlo results suggest that ignoring the presence of heteroskedasticity on the remainder term has a much more dramatic effect than ignoring the presence of heteroskedasticity on the individual specific error. Hence, the Li and Stengos (1994) estimator should be preferred to the Roy (00) estimator when a researcher does not know the source of the heteroskedasticity.

12 Li and Stengos (1994)-GLSAD: μ i ~ IID(0, σ ν ) E ν it X it = 0 with var ν it X it = γ(x it ) γ it σ it = E u it X it = σ μ + γ it σ μ = u it denote the OLS residual. N j=1 T t s u it u is NT(T 1) γ it = N j=1 N j=1 T s=1 T s=1 K it,js u js K it,js σ μ

13 Testing for Heteroskedasticity Verbon (1980) Lagrange Multiplier test H 0 : Homoskedasticity H 1 : Heteroskedasticity ) μ i ~(0, σ μi ν it ~(0, σ νi ) = σ μ f Z i θ σ μi σ νi = σ ν f Z i θ 1 Lejeune (1996) ML estimation and LM Testing general heteroskedastic one way error component (H 0 = σ μ = 0) ) μ i ~(0, σ μi ν it ~(0, σ νit ) = σ μ μ F i θ σ μi = σ ν ν Z it θ 1 σ νit Holly & Gardiol (000) Score test Homoskedasticity in a one way error component model (H 0 : θ = 0) H 1 : ) μ i ~N(0, σ μi = σ μ μ F i θ σ μi

14 Baltagi, Bresson, Pirotte (005) Joint LM test Homoskedasticity in one way error component (H 0 : θ 1 = θ = 0) Baltagi et al. (005) LM test H 0 : Homoskedasticity of the individual Random Effect assuming homoskedasticity of remainder error Monte Carlo experiments showed that the joint LM test performed well when both error components were heteroskedastic, and performed second best when one of the components was homoskedastic while the other was not. In contrast, the marginal LM tests performed best when heteroskedasticity was present in the right error component.

15 References Baltagi, B. H., (005): Econometric Analysis of Panel Data. John Wiley & Sons, Chichester, England. Castilla, Carolina. (008), Heteroskedasticity in Fixed- Effects One-way Error Components Models. Hsiao, C., Pirotte, A. (006), Heteroskedasticity and Random Coefficient model on Panel Data.

16 Thank you for attention