Now we relax this assumption and allow that the error variance depends on the independent variables, i.e., heteroskedasticity

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1 ECON 48 / WH Hong Heteroskedastcty. Consequences of Heteroskedastcty for OLS Assumpton MLR. 5: Homoskedastcty var ( u x ) = σ Now we relax ths assumpton and allow that the error varance depends on the ndependent varables,.e., heteroskedastcty Consequences of heteroskedastcty [IMPORTANT] OLS estmators are stll unbased and consstent under heteroskedastcty. Recall that we prove unbasedness under MLR. through MLR.4, but not MLR.5. Also, nterpretaton of R-squared s not changed n the presence of heteroskedastcty. σ u R where the uncondtonal error varance, σ Y heteroskedastcty, (whch refers to the condtonal error varance) Heteroskedastcty nvaldates varance formulas for OLS estmators σ u, s unaffected by The usual F-tests and t-test are not vald under heteroskedastcty because the varance formula for OLS estmator s wrong. Under heteroskedastcty, OLS s no longer the best lnear unbased estmator (BLUE); there mght be more effcent lnear estmator.

2 ECON 48 / WH Hong Heteroskedastcty. Heteroskedastcty-Robust Inference after OLS Estmaton Formula for OLS standard errors and related statstcs have been developed that are robust to heteroskedastcty of unknown form To derve those formula, consder the followngs: Smple regresson model case Estmaton model: y = β0 + βx + u We assumes that the frst four Gauss-Markov assumptons hold, but not the last one. The general form of contan heteroskedastcty: ( u x ) = σ var The subscrpt ndcates that the varance of the error depends upon the partcualar value of x. Then, followng the same argument, we can show that var( ˆ ) n ( ) x x σ = β = SSTx. Therefore, the samplng varance can be estmated by: var ( ˆ ) n ( ) x x uˆ = β = SSTx

3 ECON 48 / WH Hong Heteroskedastcty Multple regresson model case Estmaton model: = β0 + β βk k + y x x u The general form of contan heteroskedastcty: ( u x ) = σ Formula for heteroskedastcty-robust OLS varance s: var ( ˆ j ) n = β = ru ˆ ˆ SSR j j var where rˆj denotes the th resdual from regressng x j on other ndependent varables, and SSRj s the sum of squared resduals from ths regresson. The squared root of ths formula provdes heteroskedastcty-robust OLS standard error, and s also called Whte standard errors. Dscusson on heteroskedastcty-robust OLS standard error All formula are only vald n large samples. Usng the formula, the usual t-test s vald asymptotcally. The usual F-statstc does not work under heteroskedastcty, but heteroskedastcty robust versons are avalable n most software. 3

4 ECON 48 / WH Hong Heteroskedastcty (Example) Hourly wage equaton Estmated equaton ( ) log wage = educ exper exper (0.05) (0.0075) (0.005) (0.000) [0.07] [0.0078] [0.0050] [0.000] The usual OLS standard errors are presented n the parentheses and the heteroskedastcty-roust standard errors are presents n the squared brackets. The heteroskedastcty-roust standard errors may be larger or smaller than ther nonrobust counterparts. The dfferences are often small n practce. The usual t-test sad that all the ndependent varables are sgnfcant. H β β : 0 0 exper = = exper F = 7.95 vs. F = 7.99 => the null s hghly rejected robust F-statstcs are also often not too dfferent. However, f there s strong heteroskedascty, dfferences may be larger. To be on safe sde, t would be better to always compute robust standard errors. 4

5 ECON 48 / WH Hong Heteroskedastcty 3. Testng for Heteroskedastcty It may stll be nterestng whether there s heteroskedastcty because then OLS may not be the most effcent lnear estmator anymore () Breusch-Pagan test for heteroskedastcty Ratonale H ( u x ) = σ 0 :var ; Homoskedastcty var( u ) = E( u ) E( u ) = E( u ) x x x x under MLR.4: E ( u x ) = 0 => ( x ) ( ) E u = E u = σ Ths mples that the mean of u must not vary wth x,..., x k. 5

6 ECON 48 / WH Hong Heteroskedastcty How to test Consder the followng model: ˆ δ0 δ... δk k u = + x + + x + error Regress squared resduals on all explanatory varables Hypothess testng: H0 : δ =... = δ k = 0 The hypothess means that all the explanatory varables do not explan the squared resduals, mplyng homoskedastcty. F-statstc: F = uˆ ( R )/( n k ) uˆ R / k ~ F kn, k where s the R-squared from the regresson of û on all explanatory varables R û LM test (alternatve test statstc) LM = n R χ ~ uˆ k Ths test s only asymptotcally vald. 6

7 ECON 48 / WH Hong Heteroskedastcty () Whte test for heteroskedastcty How to test Consder the followng model wth 3 ndependent varable case: û = δ + δ x + δ x + δ x + δ x + δ x + δ x + δ x x + δ x x + δ x x + e rror Regress squared resduals on all explanatory varables, ther squares, and nteractons. Hypothess testng: H0 : δ =... = δ9 = 0 The Whte test detects more general devatons from heteroskedastcty than the Breush- Pagan test LM = n R χ ~ ˆ u 9 Dsadvantage of ths form of the Whte test: (we can also use an F-test for ths hypothess.) Includng all squares and nteractons leads to a large number of estmated parameters (e.g. k = 6 leads to 7 parameters to be estmated) Alternatve form of the Whte test uˆ = δ + δ yˆ + δ yˆ + error 0 Hypothess testng: H : δ = δ = 0, LM = n R χ 0 ~ uˆ 7

8 ECON 48 / WH Hong Heteroskedastcty 4. Weghted Least Squared Estmaton () The Heteroskedastcty s known up to a multplcatve constant (Weghted LS) Assume that ( u ) = σ h( ) x x, h( x ) = h > 0 var The functonal form of the heteroskedastcty,.e., y = β0 + βx βkxk + u y x x k u => = β0 + β βk + h h h h h <=> * * * * y = β x + β x + + β x + u ; Transformed model k k * h( x ), s known. Example: Savngs and ncome sav β0 βnc u = + +, ( u nc ) = σ var nc sav nc = β0 + β + u nc nc nc Note that ths regresson model has no constant. * 8

9 ECON 48 / WH Hong Heteroskedastcty The transformed model s homoskedastc ( ) E u ( x) * u E u σ h x = E x = = = σ h h h Provded that the other Gauss-Markov assumptons hold as well, OLS appled to the transformed model s the best lnear unbased estmator. OLS n the transformed model s called as weghted least squares (WLS) y x x n k mn b0 b... bk = h h h h <=> n = ( ) mn y b bx... b x / h 0 k k Observaton wth a large varance gets a smaller weght n the optmzaton problem Why s WLS more effcent than OLS n the orgnal model? Observaton wth a large varance are less nformatve than observaton wth small varance and therefore should get less weght. WLS s a specal case of generalzed least squares (GLS) 9

10 ECON 48 / WH Hong Heteroskedastcty () Unknown heteroskedastcty functon (feasble GLS) ( u x) = σ ( δ + δ x + + δ x ) = σ h( x ) var exp 0... k k Assumed general form of heteroskedastcty. That s, h( x ) s unknown and therefore should be estmated before runnng the estmaton model. exp-functon s used to ensure postvty. u = σ exp ( δ + δ x δkxk) v where v s multplcatve error 0 ( ) = log( ) => log u α0 δx... δkxk e where Steps for estmaton of FGLS. Regress y on all ndependent varables,. Estmates the model, ( ˆ ) => log α0 δ... δk k ( ˆ α0 δ δ ) h ˆ = exp + ˆ x ˆ x k k α = σ + δ. 0 0 x,..., x k, and obtan the resduals, u = ˆ + ˆx + + ˆ x + error, n order to obtan h.. Apply WLS usng hˆ obtaned n the prevous step,.e., estmate the followng model: uˆ ˆ * * * * * y y = β0x0 + βx βkxk + u where * y =, hˆ x * j xj = for j = 0,,.., k, and hˆ u * = u hˆ 0

11 ECON 48 / WH Hong Heteroskedastcty (Example) Demand for cgarettes Estmaton by OLS ( ) ( ) cgs = log ncome 0.75log cgprc 0.50educ (4.08) (0.78) (5.773) (0.67) 0.77age age.83restaurn (0.60) (0.007) n = 807, R = (.) cgs: cgarettes smoked per day; cgprc : cgarettes prce restaurn : smokng restrcton n restaurants p value = => reject homoskedastcty Breusch Pagan

12 ECON 48 / WH Hong Heteroskedastcty Estmaton by FGLS ( ) ( ) cgs = log ncome.94log cgprc 0.463educ (7.80) (0.44) (4.46) (0.0) 0.48age age 3.46restaurn (0.097) (0.0009) (0.80) n = 807, R = 0.34 Dscusson The ncome elastcty s now statstcally sgnfcant; other coeffcents are also more precsely estmated (wthout changng qualtatve results)

13 EC ON 48 / WH Hong Heteroskedastcty What f the assumed heter oskedastcty functon s wrong? If the heteroskedastcty functon s msspecfed, WLS s stll unbased and consstent under MLR. through MLR.4, but not effcent However, f we estmate ˆ( ) h x wth a msspecfed functonal form, then FGLS s based and nconsstent. In contrast, OLS s always unbased n the presence of heteroskedastcty. Practcally, OLS estmaton wth robust standard error adjustment would be a safe way. 3