COV. Violation of constant variance of ε i s but they are still independent. The error term (ε) is said to be heteroscedastic.

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1 c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty olato of costat varace of s but they are stll depedet. C,, he error term s sad to be heteroscedastc. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 3 What s Heteroscedastcty? Weghted east Square WS ests for arace equato Remedes Geeralzed Heteroscedastcty c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty What happeed to S estmators? > S s stll UE but ot BUE. > arge sample propertes??? Weghted east Square WS w Y Y where w β w + β w β w + w β + β β + ν Y w Y c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 4 k w, k,..., k ν w

2 atrx form Y Y,, ν c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 5 Explaatory varables observato dex some or all the s codtoal mea of Y gve varable Z s ot the model lagged varables to be dscussed me Seres Aalyss c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 7 Note that Y β + s CNR ad ν s homoscedastc. WS estmator Note that, gve, the estmator s BUE. ν,,..., ν I Y ˆ β c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 6 Detecto squared resdual plots Whte s est ther tests c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 8

3 squared resduals agast the suspected explaatory varable S test agast oly oe varable at a tme fuctoal forms, e.g., lear, log-l, log-log, quadratc, polyomal, recprocal, etc. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 9 Fuctoal form of the varace equato must be assumed most tests. For example, Breusch-Paga s test assumes a lear fucto of suspected varables Gleser s test sets the form of stadard devato stead of the varace Whte s test adds squared ad cross terms to the varace equato of BP test. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty ˆ ˆ S S ear γ S Recprocal c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty l γ S or log-l γ S Harvey-Godfrey s test assumes a log-l or 3 exp γ S S +.. S 3 P P Note that HG assures postve ftted varaces whle others do ot. Park test assumes double log form. Goldfeld-Quadt does ot assume the form of the varace fucto. Istead, t checks for equalty of the varaces betwee the hgh group ad the low group usg varace rato testf-test. See text. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty

4 Whte s Geeral Heteroscedastcty est Cocept: he varables that are suspected to cause heteroscedastcty are the s, ther squared terms ad ther cross terms. γ c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 3 Accept H > No heteroscedastcty If reect H, o specfc remedy ca be doe. Wthout ay remedy, must be re-calculated sce the old oe s cosstet wrog. Whte s Heteroscedastcty Cosstet Covarace s [ ] ˆ [ ] ˆ β k where s the th observato of. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 5 Ru S o the auxlary regresso of the squared resduals ad all the suspected varables above. Assume for all. Perform the overall F-test or test for the auxlary regresso ru o the followg : γ H H : γ hypothess. γ... γ 3 γ... γ 3 γ γ Eews also provde a quck soluto for Whte s test. γ γ γ... γ... γ... γ c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 4 Feasble Geeralzed east Squares FGS Estmate the varace equato usg auxlary regresso of squared resduals o the explaatory varables for the varace equato, e.g., ˆ + ξ γ + ξ S where ξ s the error term for the varace equato c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 6

5 Calculate the ftted value of squared resduals or estmate ˆ ˆ γ ˆ S Use the weght WS w FGS s based but cosstet. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 7 Example S ˆ S + ξ where s a postve costat. Apply WS wth w S Note that ν for all,, > WS estmator s ubased. No FGS eeded. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 9 arace equato wth sgle parameter, e.g., Example or ˆ + ξ where S S s a postve costat. Apply WS wth w S Note that ν for all,, > WS estmator s ubased. No FGS eeded c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 8 olato of costat varace ad/or depedece of s C, for some,, he error term s sad to be geeralzed heteroscedastc. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty,

6 S or WS estmators are stll UE but ot BUE. Geeralzed east Square GS Y WY, W, ν W where W s a x symmetrc matrx such that WW c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty If s kow up to a proporto, Ω, where s ukow but Ω matrx s kow., c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 3 Note that Y β + ν s CNR ad ν s homoscedastc., GS estmator ν,..., ν I Y ˆ β Note that f s kow, W ca be calculated. So s. >the estmator s BUE. c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty Choose the weghtg matrx W such that WW Ω Defe Y WY, W, ν W Note that Y β + ν s CNR ad ν s homoscedastc or > GS estmator s BUE ν I c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 4

7 c Pogsa Porchawseskul, Faculty of Ecoomcs, Chulalogkor Uversty 5 s restrcted parameters. For example,,,, Ρ Ρ

Recall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I

Recall MLR 5 Homskedasticity error u has the same variance given any values of the explanatory variables Var(u x1,...,xk) = 2 or E(UU ) = 2 I Chapter 8 Heterosedastcty Recall MLR 5 Homsedastcty error u has the same varace gve ay values of the eplaatory varables Varu,..., = or EUU = I Suppose other GM assumptos hold but have heterosedastcty.