Advanced Econometrics - Chapter 8: Heteroskedasticity

Advanced Econometrics Chapter 8: Heteroskedasticity Nam T. Hoang University of New England - Australia 1 University of Economics - HCMC - Vietnam Chapter 8 HETEROSKEDASTICITY Problem of non-constant error variances: 22)( εσσε ≠= iiVar → violated assumption E(εε') = σ2I E(εε') = 1×Σn , diagonal matrix with non-constant elements on diagonal ( 2 iσ ). I. PROPERTIES OF OLS IN PRESENCE OF HETEROSKEDASTICITY: 1. OLSβˆ is still unbiased (still consistent if X is stochastic). 2. OLSβˆ is not best (efficient), because GLS estimators are best OLSβˆ has variance which are large than GLS βˆ 's variances. 3. The standard errors of sj 'βˆ are biased because they are based on incorrect formula. Wrong (OLS) formula: VarCov( OLSβˆ ) = 12 )'( −XXεσ Correct (OLS) formula: VarCov( OLSβˆ ) = 11 )'(')'( −− Σ XXXXXX with )'(εεE=Σ Note: VarCov( OLSβˆ ) = ])'ˆ)(ˆ( ββββ −−E = ])'('')'[( 11 −− Σ XXXXXXE εε = 11 )'(')'( −− Σ XXXXXX If we know the form of the heteroskedasticity, that is: Advanced Econometrics Chapter 8: Heteroskedasticity Nam T. Hoang University of New England - Australia 2 University of Economics - HCMC - Vietnam             =Σ 2 2 2 2 1 00 00 00 nσ σ σ     known. → we can apply "Weighted Least Squares":       +      ++      +      =      i i i ik k i i ii i XXY σ ε σ β σ β σ β σ 221 1 Generally, however, form of heteroskedasticity is unknown. Usually, sources of heteroskedasticity are differences/ in some scale factor (independent variables). EX: iii IC εββ ++= 21 Ci: ith individual expenditures of clothes Ii: ith individual income. II. TESTING FOR HETEROSKEDASTICITY: Because OLS estimator of β is still consistent if there is heteroskedasticity, we can use the OLS residuals to construct (at least asymptotically valid) test for this problem. 1. Goldfeld - Quandt test: iiii ZXY εβββ +++= 321 Advanced Econometrics Chapter 8: Heteroskedasticity Nam T. Hoang University of New England - Australia 3 University of Economics - HCMC - Vietnam )(: 2 iiA XfH =σ f'(.) > ε Xi suspected scale factor: 220 : εσσ =iH (Constant) 2: iAH σ is not a constant. order the observations by size of Xi Divide sample into 3 parts: (a). (n1 observations) the 1st 40% of the observations. (b). (n2 observations) the last 40% of the observations. Apply OLS to (n1) → get ESS1. Apply OLS to (n2) → get ESS2. Form: ),(1 2 1 1 2 2 21 1 2 ~ )( )( knkn kn kn FESS ESS kn ESS kn ESS F −− − − = − − = if n1 = n2. Reject H0 for large F. Example: Education model. 50 observations, 6 parameters (k = 6). 2: iAH σ = f(per capita income). ESS1 = .3678 (1st 20 states) ESS2 = .8013 (last 20 states) → 18.2 3678. 8013.14 14 ==F 5% critical value = 2.46 → do not reject H0. Advanced Econometrics Chapter 8: Heteroskedasticity Nam T. Hoang University of New England - Australia 4 University of Economics - HCMC - Vietnam 2. Park-Glejser test: 22: i XH iA λσ = Apply OLS to original model, save ie′ s estimate: iuXe ii ++= 2 21 2 γγ Reject H0 if γ2 is significant. 3. White test: 2: iAH σ is not a constant. iiii ZXY εβββ +++= 321 iiiii uZXZXZXe iii ++++++= 6 2 5 2 4321 2 γγγγγγ → Regress squared residual on all variables, their squares and their cross products. Test: H0: all slopes = 0 ( 065432 ===== γγγγγ ) → no heteroskedasticity. Using F-test of overall significance )( )( kn ESS k ESSESS F U UR − − = or use LM test (Lagrange Multiplier): nR2 → 2 ][ pχ p: number of slope coefficients in this ei2 regression (p=5 in this case) III. TREATMENT FOR HETEROSKEDASTICITY: 1. No idea of source problem: Use White procedure for computing consistent estimator of )ˆ( OLSVarCov β . This gives us correct standard errors, so we can test hypotheses using t-test. But we stuck with OLSβˆ which are inefficient ( OLSβˆ are efficient). Advanced Econometrics Chapter 8: Heteroskedasticity Nam T. Hoang University of New England - Australia 5 University of Economics - HCMC - Vietnam 2. Heteroskedasticity is due to a particular variable Xi 22 i Xi λσ = the perform WLS:       +      +      +      =      i i i i i i ii i XX XX XX Y ε βββ 321 X 1       i i X ε has constant variance → homoskedasticity If 22 i Xi λσ = → λ λ ε ε ==      =      2 2 2 )( 1 i i X X Var XX Var i ii i If 22: i XH iA λσ = then         i i X ε has constant variance. Cross term: 2 4 2 3 2 21 2 ˆˆˆˆˆ: iiiii eZXZXH iA =+++= λλλλσ If ii ZXi 4 2 λˆσ = → divided by ii ZX If 0,,,, 4321 ≠λλλλ → divided by iiii ZXZX 4 2 3 2 21 ˆˆˆˆ λλλλ +++