CDS M Phil Econometrics

Transcription

1 6//9 OLS Volaton of Assmptons an Plla N Assmpton of Sphercal Dstrbances Var( E( T I n E( T E( E( E( n E( E( E( n E( n E( n E( n I n Therefore the reqrement for sphercal dstrbances s ( Var( E(,..., n homoskedastcty and ( Cov(, E(, j j j No atocorrelaton

2 6//9 Heteroscedastcty CDS M Phl Econometrcs 3 Heteroskedastcty: Defnton Heteroskedastcty s a problem where the error terms do not have a constant varance. E( That s, they may have a larger varance when vales of some X (or the Y s themselves are large (or small. 4

3 6//9 Example of Heteroskedastcty f(y x... E(y x = b + b x x x x 3 x 5 Heteroskedastcty: Defnton Ths often gves the plots of the resdals by the dependent varable or approprate ndependent varables a characterstc fan or fnnel shape Seres 6 3

4 6//9 Heteroskedastcty: Defnton 7 Resdal Analyss for Eqal Varance Y Y x x resdals Non-constant varance CDS M Phl Econometrcs x resdals Constant varance 8 x 4

5 6//9 5 9 Heteroskedastcty Heteroskedastcty Wth nonsphercal errors (e.g. heteroskedastcty and/or atocorrelaton no longer apples. I ( Var( E ( E ( ( ( ( ( ( ( ( ( ( n n n n n T E E E E E E E E E E n nn Heteroskedastcty Heteroskedastcty

6 6//9 6 Atocorrelaton Atocorrelaton ( ( ( ( ( ( ( ( ( ( n n n n n T E E E E E E E E E E n n n n Heteroskedastcty Heteroskedastcty Gven or model, y = X + where X s a non-stochastc matrx wth fll colmn rank E( = and E( The OLS estmator of s X (X X ˆ E(ˆ So OLSE s stll nbased

7 6//9 The varance matrx s Var(ˆ E{(ˆ (ˆ } E(XX Therefore any nference based on Heteroskedastcty (XX XX(XX XE( X(XX (XX X (XX s X(XX (XX(XX (XX wll be ncorrect. s 3 may be a based estmator of Heteroskedastcty: Cases It may be cased by: Model msspecfcaton - omtted varable or mproper fnctonal form. Learnng behavors across tme Changes n data collecton or defntons. Otlers or breakdown n model. Freqently observed n cross sectonal data sets where demographcs are nvolved (poplaton, GNP, etc. 4 7

8 6//9 Heteroskedastcty: Implcatons The regresson s are nbased /consstent. Bt they are no longer the best estmator. They are not BLUE (not mnmm varance - hence not effcent. 5 Heteroskedastcty: Implcatons (cont. The estmator varances are not asymptotcally effcent, and they are based. So confdence ntervals are nvald. Wrong nference 6 8

9 6//9 Heteroskedastcty: Implcatons (cont. Types of Heteroskedastcty There are a nmber of types of heteroskedastcty. Addtve Mltplcatve ARCH (Atoregressve condtonal heteroskedastc - a tme seres problem. 7 Testng for Heteroskedastcty A nmber of formal tests : Ramsey RESET test Park test Glejser test Goldfeld-Qandt test Bresch-Pagan test Whte test 8 9

10 6//9 Testng for Heteroskedastcty Essentally want to test H : Var( x, x,, x k = s, eqvalent to H : E( x, x,, x k = E( = s If assme the relatonshp between and x j s lnear, can test H as a lnear restrcton So, for = d + d x + + d k x k + v ths means testng»h : d = d = = d k = 9 The Bresch-Pagan Test Estmate the resdals from the OLS regresson Get z û / that s the resdals sqared dvded by û /n Regress z on all of the xs. can have 3 tests:

11 6//9 The Bresch-Pagan Test can have 3 tests:. = ½ RSS, where RSS = regresson sm of sqares from regressng z on all of the xs ; (k df. The Bresch-Pagan Test. The F statstc s jst the reported F statstc for overall sgnfcance of the regresson, F = [R /k] / [( R /(n k ], whch s dstrbted F k, n k

12 6//9 The Bresch-Pagan Test 3. The (Bresch-Pagan-Godfrey LM statstc s LM = nr, whch s dstrbted as k- 3 Consmpton $ Income $ The Bresch-Pagan Test : An Example In Stata Statstcs: Lnear models and related Regresson dagnostcs Specfcaton tests, etc. 4

13 6//9 The Bresch-Pagan Test : An Example ( = 3.84, = 5% ( = 6.63, = % F (, 8 = 4., = 5% F (, 8 = 7.56, = % 5 The Whte Test: Whte s Generalzed Heteroskedastcty test The Bresch-Pagan test wll detect any lnear forms of heteroskedastcty The Whte test allows for nonlneartes by sng sqares and crossprodcts of all the xs sng an F or LM to test whether all the x j, x j, and x j x k are jontly sgnfcant can get to be nweldy 6 3

14 6//9 The Whte Test: Whte s Generalzed Heteroskedastcty test The test proceeds as follows: Step : Estmate the orgnal eqaton by least sqares and obtan the resdals Step : Regress the sqared resdals on a constant, all the regressors, the regressors sqared and ther cross-prodcts (nteractons. For example, wth two explanatory varables x x x x x x where x represents the constant term 3 3 x3 7 The Whte Test: Whte s Generalzed Heteroskedastcty test Step 3: The test statstc s H : Constant varance ~ nr (k If nr > then we have an sse wth heteroskedastcty. 8 4

15 6//9 The Whte Test: Whte s Generalzed Heteroskedastcty test: An Example 9 The Whte Test: Whte s Generalzed Heteroskedastcty test: An Example Generate varables n Stata 3 5

16 6//9 The Whte Test: An Example NR = 5 x.357 = dstrbton wth 5 df =.75, = 5% Conclson? nr < 3 homoskedastcty Alternatve form of the Whte test Consder that the ftted vales from OLS, ŷ, are a fncton of all the xs Ths, ŷ wll be a fncton of the sqares and crossprodcts and ŷ and ŷ can proxy for all of the x j, x j, and x j x k ; so Regress the resdals sqared on ŷ and ŷ and se the R to form an F or LM statstc 3 6

17 6//9 Heteroskedastcty: Tests (cont. Park test An exploratory test, log the resdals sqared and regress them on the logged vales of the sspected ndependent varable. ln ln B ln X v a B ln X v If the B s sgnfcant, then heteroskedastcty may be a problem. 33 Heteroskedastcty: Tests (cont. Glejser Test Smlar to the park test, except that t ses the absolte vales of the resdals, and a varety of transformed X s. B B X v B B v X B B X v B B X v B B v X B B X v A sgnfcant B ndcated Heteroskedastcty. 34 7

18 6//9 Heteroskedastcty: Tests (cont. Goldfeld-Qandt test Rank the n cases of the X that yo thnk s correlated wth e n descendng order Drop a secton of c cases ot of the mddle (one-ffth s a reasonable nmber. 35 Heteroskedastcty: Tests (cont. Goldfeld-Qandt test Rn separate regressons on both pper and lower (eqal samples of /(n - m observatons (where n = sample sze and m = mddle observatons

19 6//9 Heteroskedastcty: Tests (cont. Goldfeld-Qandt test If the dstrbances are homoskedastc then Var (U shold be the same for both sbsamples..e., the rato of the two resdal sms of sqares shold be approxmately eqal to nty. 37 Heteroskedastcty: Tests (cont. Goldfeld-Qandt test Do F-test for dfference n error varances H F has (n - c - k/ degrees of freedom for each H : : 38 9

20 6//9 Remedes for Heteroskedastcty Ths depends on the form heteroskedastcty takes. Indrect: Re-specfy the model; Use heteroscedastc-consstent SEs Drect: GLS (WLS adjst the varance-covarance matrx 39 Heteroskedastc Consstent SEs OLS estmate: nbased and consstent. Bt Var(ˆ (XX (XX(XX Ths can be re-wrtten as Var(ˆ (XX XDag( X(XX where Dag( Dag(,,..., n.e., we need to estmate all the ' s - whch s mpossble. 4 n

21 6//9 Heteroskedastc Consstent SEs Whte (98 arges that all we really need s an estmate of X X Under very general condtons, t can be shown that n n X X x x e x x Therefore the adjsted varance s est.asym.var(ˆ (XX n e x x (XX 4 Heteroskedastc Consstent SEs A consstent estmate of the varance, the sqare root can be sed as a standard error for nference Typcally known as robst standard errors Sometmes the estmated varance s corrected for degrees of freedom by mltplyng by n/(n k As n t s all the same, thogh 4

22 6//9 Heteroskedastc Consstent SEs: Robst SEs Important to remember: Robst standard errors only have asymptotc jstfcaton wth small sample szes t statstcs formed wth robst standard errors wll not have a dstrbton close to the t, and nferences wll not be correct In Stata, Lnear regresson: SE/Robst: (select robst defalt 43 44

23 6//9 Generalzed Least Sqares It s always possble to estmate robst standard errors for OLS estmates, Bt f we know somethng the specfc form of the heteroskedastcty, we can obtan more effcent estmates than OLS The basc dea s gong to be to transform the model nto one that has homoskedastc errors called generalzed least sqares 45 Generalzed Least Sqares Gven matrx. E( a postve defnte Any postve defnte matrx can be expressed n the form: PP, where P s nonsnglar: = PP, so that P P = I and P P = 46 3

24 6//9 Generalzed Least Sqares Now premltply the model y = X + by P to get y * = X * + * Where y * = P y ; X * = P X ; * = P 47 Generalzed Least Sqares Gven P P = I and E( y * = X * + * Where y * = P y ; X * = P X ; * = P Now E( * * = E(P P = (P P = ( P P = I : Homoscedastc OLS assmptons satsfed 48 4

25 6//9 Generalzed Least Sqares y * = X * + * OLS estmate of s: b = (X * X * X * y * = (X X X y b s the Generalzed Least Sqares (GLS or Atken estmator of A BLUE of wth Var(b = (X * X * = (X X An nbased estmate of s: Where e = (y Xb ˆ e n k e 49 Generalzed Least Sqares If s normally dstrbted, so s * Ths b s a ML estmator So has mn var n the class of all nbased estmators. 5 5

26 6//9 Generalzed Least Sqares: Weghted Least sqares GLS s a weghted least sqares (WLS procedre where each sqared resdal s weghted by the nverse of Var( x 5 Weghted Least Sqares Let heteroskedastcty be modeled as Var( x = s h(x, where h(x h to be specfed. Now E( / h x =, becase h s only a fncton of x, and Var( / h x = s, becase we know Var( x = s h So dvde the whole eqaton by h and we have a model wth homoskedastc error 5 6

27 6//9 Weghted Least Sqares For example, A common specfcaton: var( to one of the regressors or ts sqare: x k h(x x k ; h x k The weghted (transformed LS regresson model: y x k k x x k x x k... x k E( / x k =, and Var( / x k = s Homoscedastc WLS mnmzes the weghted sm of sqares (weghted by /h 53 Feasble GLS More typcal s the case where we don t know the form of the heteroskedastcty In ths case, need to estmate h(x Ths s the case of FGLS 54 7

28 6//9 Feasble GLS Rn the orgnal OLS model, save the resdals, û, sqare them Regress û on all of the ndependent varables and get the ftted vales, ê Do WLS sng /ê as the weght 55 FGLS: Stata 56 8

29 6//9 FGLS: Stata 57 FGLS: Stata 58 9

30 6//9 FGLS: Stata Also Download wls, sng 59 Resdals Prce (Rs. Resdals Advertsng (Rs s 3

31 6//9 FGLS: Stata Other weght types abse and loge and sqared ftted vales (xb. 6 FGLS: Stata Other weght types abse and loge and sqared ftted vales (xb. 6 3

32 6//9 FGLS: Stata Compare wth the FGLS done by steps 63 Atocorrelaton CDS M Phl Econometrcs 64 3

33 6//9 Atocorrelaton s correlaton of the errors (resdals over tme Volates the regresson assmpton that resdals are random and ndependent Atocorrelaton: Defnton Here, resdals show a cyclc pattern, not random. Cyclcal patterns are a sgn of postve atocorrelaton Resdals Tme (t Resdal Plot Tme (t 65 Atocorrelaton: Defnton The assmpton volated s E( j Ths the Pearson s r between the resdals from OLS and the same resdals lagged on perod s non-zero. E ( t t 66 33

34 6//9 Atocorrelaton: Defnton Types of Atocorrelaton Atoregressve (AR processes Movng Average (MA processes 67 Atocorrelaton: Defnton Atoregressve processes AR(p The resdals are related to ther precedng vales. t t Ths s classc st order atocorrelaton: AR( process t 68 34

35 6//9 Atocorrelaton: Defnton Atoregressve processes (cont. In nd order atocorrelaton the resdals are related to ther t- vales as well AR(: t t t t Larger order processes may occr as well: AR(p t t t... p t p t 69 Atocorrelaton: Defnton Movng Average Processes MA(q The error term s a fncton of some random error and a porton of the prevos random error. MA( process t t t 7 35

36 6//9 Atocorrelaton: Defnton Hgher order processes for MA(q also exst. t... t t t q tq The error term s a fncton of some random error and some portons of the prevos random errors. 7 Atocorrelaton: Defnton Mxed processes ARMA(p,q t t t t t q t q p t p t The error term s a complex fncton of both atoregressve {AR(p} and movng average {MA(q} processes. 7 36

37 6//9 Atocorrelaton: Defnton AR processes represent shocks to systems that have long-term memory. MA processes are qck shocks to systems, bt have only short term memory. 73 Atocorrelaton: Implcatons Coeffcent estmates are nbased, bt the estmates are not BLUE The varances are often greatly nderestmated (based small Hence hypothess tests are exceptonally sspect

38 6//9 Atocorrelaton: Cases Specfcaton error Omtted varable Wrong fnctonal form Lagged effects Data Transformatons Interpolaton of mssng data dfferencng 75 Atocorrelaton: Tests The Drbn-Watson statstc s sed to test for atocorrelaton H : resdals are not correlated H : postve atocorrelaton s present d n t ( ˆ t n t ˆ ˆ t t The possble range s d 4 d shold be close to f H s tre d < postve atocorrelaton, d > negatve atocorrelaton 76 38

39 6//9 Testng for +ve Atocorrelaton H : postve atocorrelaton does not exst H : postve atocorrelaton s present Calclate the Drbn-Watson test statstc = d (Usng Stata or SPSS Fnd the vales d L and d U from the D-W table (for sample sze, n and nmber of ndependent varables, k Decson rle: reject H f d < d L d U Reject H Inconclsve Do not reject H d L 77 Testng for +ve Atocorrelaton H : postve atocorrelaton does not exst H : postve atocorrelaton s present Decson rle: reject H f d < d L or 4 d L < d < 4 d U d L Do not reject H d 4 d U 4 d L

40 6//9 Drbn-Watson d Test: Decson Rles Nll Hypothess Decson If No + atocorrelaton Reject < d < d L No + atocorrelaton No Decson d L d d U No - atocorrelaton Reject 4 d L < d < 4 No - atocorrelaton No Decson 4 d U d 4 d L No +/- atocorrelaton Do not reject d U < d < 4 d L Testng for +ve Atocorrelaton 79 Testng for +ve Atocorrelaton Sppose we have the followng tme seres data: (contned Sales Is there atocorrelaton? y = x R = Tme 8 4

41 6//9 Testng for +ve Atocorrelaton Example wth n = 5: Drbn-Watson Calclatons Sm of Sqared Dfference of Resdals Sm of Sqared Resdals Drbn-Watson Statstc.494 S ales y = x R = Tme d T t (ˆ t T t ˆ ˆ t t Testng for +ve Atocorrelaton Here, n = 5 and k = : one ndependent varable Usng the Drbn-Watson table, d L =.9 and d U =.45 d =.494 < d L =.9, Therefore the gven lnear model s not the approprate model to forecast sales Decson: reject H snce d =.494 < d L sgnfcant +ve atocorrelaton exsts CDS M Phl Econometrcs Reject H Inconclsve Do not reject H d L =.9 d U =

42 6//9 Atocorrelaton: Tests (cont. Drbn-Watson d (cont. Note that the d s symmetrc abot., so that negatve atocorrelaton wll be ndcated by a d >.. Use the same dstances above. as pper and lower bonds. 83 Atocorrelaton: Tests (cont. Drbn s h Cannot se DW d f there s a lagged endogenos varable n the model h d T TS yt S yt- s the estmated varance of the Y t- term h has a standard normal dstrbton 84 4

43 6//9 Atocorrelaton: Remedes Generalzed Least Sqares Frst dfference method Take st dfferences of yor Xs and Y Regress Y on X Assmes that = + Generalzed dfferences Reqres that be known. 85 Atocorrelaton: Remedes Cochran-Orctt method ( Estmate model sng OLS and obtan the resdals, t. ( Usng the resdals rn the followng regresson. û t ˆû t v t 86 43

44 6//9 Atocorrelaton: Remedes Cochran-Orctt method (cont. (3 sng the obtaned, perform the regresson on the generalzed dfferences ( Yt ˆY t B( ˆ B(Xt ˆX t (t ˆ t (4 Sbsttte the vales of B and B nto the orgnal regresson to obtan new estmates of the resdals. (5 Retrn to step and repeat ntl no longer changes