Distribution of Estimates

Transcription

1 Disribuion of Esimaes From Economerics (40) Linear Regression Model Assume (y,x ) is iid and E(x e )0 Esimaion Consisency y α + βx + he esimaes approach he rue values as he sample size increases Esimaion variance decreases as he sample size increases e

2 Illusraion of Consisency ake random sample of U.S. whie men Esimae linear regression of log(wages) on educaion oal sample 089 Sar wih 00 observaions, sequenially increase o 089

3 Sequence of Slope Coefficiens

4 Asympoic Normaliy y α + βx + e ˆ β a ˆ β ~ N σ ( ) β, σ ˆ β var( x ) e [ var( x )]

5 Illusraion of Asympoic Normaliy

6 ime Series Do hese resuls apply o ime series daa? Consisency Asympoic normaliy Variance formula ime series models AR models, i.e., x y rend and seasonal models One sep and muli sep forecasing

7 Derivaion of Variance Formula For simpliciy Assume he variables have zero mean he regression has no inercep Model wih no inercep: y βx + e

8 Model wih no inercep OLS minimizes he sum of squares he firs order condiion is e x y + β ( ) + x y x y x y β β β + x y x ˆ 0 β

9 Soluion Now subsiue x y x x y x ˆβ e x y + β ( ) + + x e x x e x x ˆ β β β

10 We have he denominaor is he sample variance (when x has mean zero), so + x e x ˆ β β ( ) a x x ~ var

11 hen where Since hen E ˆ a β ~ v var + β x ( ˆ) β e a ~ var v ( x ) ( v ) E( x e ) 0 var v [ var( x )]

12 From he covariance formula var When he observaions are independen, he covariances are zero. And since v ( ) var v + cov( v, v j ) j ( v ) var( x e ) var we obain var v var ( x e )

13 We have found var ( ) a ˆ var( x ) e β ~ [ var( x )] as saed a he beginning var var ( x ) e [ ( x )]

14 Exension o ime Series he only place in his argumen where we used he assumpion of he independence of observaions was o show ha v x e has zero covariance wih v j x j e j his is saying ha v is no auocorrelaed. When does his happen in ime series?

15 Unforecasable one sep errors Claim: In one sep ahead forecasing, if he regression error is unforecasable hen v is no auocorrelaed In his case, he variance formula for he leassquares esimae is he same as in regression var ( ) a ˆ var( x ) e β ~ [ var( x )]

16 Why is he claim rue? he error is unforecasable if E(e Ω )0 For simpliciy suppose x hen for j ( e e ) 0 cov( v, v ) E j j

17 Summary In one sep ahead ime series models, if he error is unforecasable, hen leas squares esimaes saisfy he asympoic (approximae) disribuion ˆ β a ˆ β ~ N σ ( ) β, σ ˆ β var( x ) e [ var( x )] As he sample size is in he denominaor, he variance decreases as he sample size increases. his means ha leas squares is consisen

18 Variance Formula he variance formula for he leas squares esimae akes he form σ ˆ β var( x ) e [ var( x )] his formula is valid in ime series regression when he error is unforecasable

19 Classical Variance Formula If we make he simplifying assumpion hen ( x e ) ( x ) var( e ) var var var( x ) e [ var( x )] σ ˆ β his can be a useful simplificaion var var ( e ) ( x )

20 Homoskedasiciy he variance simplificaion is valid under condiional homoskedasiciy E E ( e Ω ) ( ) e Ω σ his is a simplifying assumpion made o make calculaions easier, and is a convenional assumpion in inroducory economerics courses I is no used in serious economerics 0

21 Variance Formula : AR() Model ake he AR() model wih unforecasable homoskedasic errors y α + βy + e hen he variance of he OLS esimae is σ ˆ β since x y in his model E E ( e Ω ) ( ) e Ω σ var var ( e ) ( x ) 0 var var ( e ) ( y )

22 AR() Asympoic Variance We know ha So σ ˆ β var ( y ) ( e ) ( y ) var β var ( e ) var β he asympoic disribuion is very simple ˆ β ~ N β, β a

23 ˆ β ~ N β, β a he variance is a funcion of he unknown rue value of β As β increases, he variance decreases, so he OLS esimae is acually more precise

24 Disribuion of Leas Squares In classic regression, if he errors are iid normal, and independen of he regressors, hen he leas squares esimaes have an exac normal disribuion, no jus asympoic his is no rue in mos ime series regressions.

25 Non Classical Disribuions Esimaes in auoregressive models Biased downwards Skewed hick ails Especially When auoregressive coefficiens are large Sample sizes are small hese issues diminish in large samples

26 Example ake he AR() model wih inercep y + α + βy e e N(0,) 00, 500, 000 β0.0, β0.5, β0.9, Numerically calculae disribuion of leassquares esimae of β

27 Disribuion, β0.0

28 Disribuion, β0.5

29 Disribuion, β0.9

30 Inerpreaion Esimaes of auoregressive parameers are random Even if regression error is normal, he parameer esimaes are no normally disribued Disribuions are less normal when AR coefficien is large Disribuions are more concenraed and normal when sample size is large

31 Asympoic Sandard Deviaion he leas squares esimae is asympoically (approximaely) normally disribued In he simple model hen ˆ β a ˆ β ~ N σ ( ) β, σ y βx + ˆ β var( x ) e [ var( x )] he sandard deviaion measures he precision of he esimae, bu i is unknown. e

32 Sandard Errors Esimaes of he sandard deviaions are called sandard errors, and are repored in regression oupu hey are used o measure esimaion precision.

33 Classical sandard errors A classic sandard error is an esimae of he sandard deviaion from he formula ˆ σ β var n var ( e ) ( x ) his formula is valid under condiional homoskedasiciy E E ( e Ω ) ( ) e Ω σ 0

34 Robus sandard errors Robus sandard errors are esimaes of σ β n var( xe ) [ var( x )] hese are he convenional sandard errors for regression analysis Also known as Whie sandard errors

35 Halber Whie Professor Hal Whie, UCSD (950 0) Leading conribuor o economeric mehods, especially ime series analysis Inroduced robus sandard errors ino economerics (980) Mos referenced paper in economics Founded Baes Whie consuling firm, a leader in economic policy analysis

36 Have you seen robus sandard errors? If you ook an economerics course oher han 40, you may no be familiar wih robus sandard errors If you are currenly aking 40, you won cover robus sandard errors unil laer in he course Wooldridge uses he homoskedasiciy assumpion in he early par of his ex Sock Wason use robus sandard errors hroughou

37 Does he Choice Maer? Classic sandard errors are for he assumpion of condiional homoskedasiciy E ( ) σ e Ω his is unforecasabiliy in he variance his is no implied by convenional unforecasabiliy I may be a convenien approximaion for macro daa I is a bad assumpion (quie false) in financial daa

38 Example: Sock Reurns, AR() he robus sandard error on he AR() coefficien is almos wice as large as he convenional sandard error

39 Compuaion In SAA, he defaul is convenional sandard errors. hey are auomaically repored wih he regress (reg) command For robus sandard errors, use he r opion.reg y x, r

40 Example: Real GDP Growh

41 Wih Robus s. errors

42 Robus s. errors Wih he r opion.reg y x, r You ge robus Sandard errors saisics and p values es saisics

43 Annoyance In SAA, wih he r opion, SAA omis sum of squared error able Ye his can be useful So boh commands may be useful.reg y x.reg y x, r

44 Inerpreaion of sandard errors he sandard errors measure precision of he esimae Forecass use esimaed coefficiens. Small sandard errors mean he esimae is precise Good for forecasing Large sandard errors mean he esimae is no precise Bad for forecasing Inaccurae esimaes leads o inaccurae forecass

45 Inerpreaion of saisics is he coefficien esimae divided by he sandard error. I is used o es if he coefficien is zero P> is he p value of he saisic If p<.05 you rejec he hypohesis of a zero coefficien Hypohesis ess are useful for assessing economic heories Bu are less useful for picking good forecasing models

46 Inerpreaion of Confidence Inerval he 95% inerval is he coefficien esimae plus and minus.96 imes he sandard error Helps gauge possible values for he rue coefficien Useful ool

47 Summary In one sep ahead forecas regressions wih unforecasable errors Robus sandard errors generally appropriae Classical sandard errors appropriae under condiional homoskedasiciy