Detecting Structural Change and Testing for the Stability of Structural Coefficients

Transcription

1 Chaper 10 Deecing Srucural Change and Tesing for he Sabiliy of Srucural Coefficiens

2 Secion 10.1 Inroducion

3 By definiion, srucural change refers o non-consan srucural parameers over ime. In essence, he srucural coefficiens are dependen on ime. I is noeworhy o disinguish beween abrup or gradual srucural change. Soluion(s) o he srucural change issue are condiional on he ype of parameer insabiliy. Diagnosics associaed wih srucural change focus exclusively on he se of srucural coefficiens of he model. The issue is wheher or no he parameers of he model are sable over ime. The consequences of ignoring his issue are ha we may obain incorrec parameer esimaes and hence incorrec marginal effecs. As well, our forecasing accuracy may be affeced. 3

4 Secion 10.2 Diagnosic Tess for Srucural Change

5 Diagnosic Tess for Srucural Change Chow Tes Farley-Hinrich-McGuire Tes Recursive Coefficiens Recursive Residuals CUSUM Tes CUSUMSQ Tes 5

6 Sequenial Chow Tess Tes wheher or no a parameer or parameers are unchanged from one daa subsample o anoher. Spli he sample in wo subsamples a various breakpoins. Each subsample mus conain more observaions han he number of coefficiens in he equaion. In pracice, we need he wo-sub-samples o have a comparable number of observaions; oherwise, he es migh rejec he null hypohesis more ofen han needed. Suppose, for example, ha we have 200 observaions and we esimae 3 parameers; if we run wo regressions, one wih 10 observaions and he oher wih he remaining 190 observaions, i is highly possible ha we will ge quie differen resuls, even when he srucural parameers are in fac sable. Reason for his problem: he properies of hese ess are based on asympoic approximaions. When we have very few observaions, he asympoic approximaions break down. 6

7 Chow Tes Tess involving he equaliy of coefficiens of differen regressions ( 1) Yi = β0 + β1x1i + β2x2i βkxki + εi (i = 1,..., N)df (N k 1) ( 2) Yj = σ0 + σ1x1j + σ2x2 j σkxkj + ε j ( J = 1,...,M)df (M k 1) Calculae ESSUR = ESS1 + ESS2 df (N + M 2k 2) H 0 : β0 = σ0, β1 = σ1, β2 = σ2,..., β k = σ k 7 coninued...

8 Pooling Under 0 H ) 1,..., ( = = M N X X X Y k α k ε α α α 8 2) 2 /( 1 ) / ( 2 2 1, + + = + + k M N ESS k ESS ESS F ESS UR UR R k M N k R

9 Sequenial Chow Tess Spli he sample in wo pieces a various breakpoins. Reason for doing so due o he fac ha he exac breakpoin ofen is unknown. Basic es for srucural change Chow, G. Tess of Equaliy Beween Ses of Coefficiens in Two Linear Regressions, Economerica, 28 (1960): Tess for a srucural break of unknown iming are based on he larges Chow saisic when calculaed for all break periods. Tes of Abrup Srucural Change 9

10 Farley, J.V., M.J. Hinrich, and T. McGuire, Some Comparisons of Tess for a Shif in he Slopes of a Mulivariae Linear Time Series Model, Journal of Economerics, 3, (1975): Farley, Hinrich, and McGuire es (1975) Tes of srucural change wherein each parameer or seleced parameers is/are hypohesized o be a linear funcion of ime. Useful for gradual srucural change. Suppose he generic represenaion of he model looks like: Y = B + B X Noice ha he srucural coefficiens of he model have ime subscrips. Farley, Hinrich, and McGuire posi ha B i 1 B = B + γ, where i = i i k X k 0,1,..., + k. 10 is an arificial consruc such ha = 1 for obs 1, = 2 for obs 2, and = T for he las observaion in he daa se.

11 Consequenly, Y = B + γ ) + ( B + γ ) X ( B + γ ) X ( k k k +. Combining erms, Y = B + B X B X + 1 ( [ γ ] 0 + γ Xi ) k ( X k) k k γ In oher words, he original model is augmened by he erms in brackes. These erms are funcions of and cross producs of wih each of he explanaory variables. H 0 : no srucural change is anamoun o H 0 : γ 0 = γ 1 = = γ k = 0. The es saisic follows an F-disribuion wih (k+1) and (T 2(k + 1)) degrees of freedom. 11

12 Recursive Esimaion Begin wih a small sample of daa; esimae he model. Then, add anoher observaion and reesimae he model, coninuing in ha fashion unil he sample is exhaused. Imporance: Sabiliy assessmen of he model parameers. Recursive coefficien esimaes are OLS esimaes based on he firs observaions. For = T, recursive coefficiens are OLS esimaes. 12

13 Recursive Leas Squares In recursive leas squares, he equaion is esimaed repeaedly, using ever larger subses of he sample daa. If here are k coefficiens o be esimaed, hen he firs k observaions are used o form he iniial se of recursive parameers of he respecive explanaory variables esimaes. The nex observaion is hen added o he daa se and k + 1 observaions are used o compue he second se of recursive parameer esimaes. This process is repeaed unil all he T sample poins have been used, yielding T k + 1 recursive parameer esimaes of he explanaory variables. We may race he evoluion of esimaes for any srucural coefficien via plos over ime. We may also show wo sandard error bands around he esimaed coefficiens. A each sep, he recursive parameer esimaes can be used o predic he nex value of he dependen variable. The one-sep ahead forecas error resuling from his predicion, suiably scaled, is defined o be a recursive residual. 13 Key Quesion: Do he coefficien esimaes sabilize, wander around, drif, or break sharply a one or more poins?

14 Recursive Residuals The recursive residuals w are compued as v e w = , where b denoes he vecor of recursive coefficiens a ime. i i i x x x x v 1 1 = + = b x y e =

15 Recursive residuals are sandardized one-sep ahead predicion errors. Unlike OLS residuals, recursive residuals are homoscedasic (because hey are sandardized), and hey are independen of one anoher. These aracive properies have made hem a popular alernaive o OLS residuals for use in calculaing a variey of regression diagnosics. We may plo he recursive residuals, ogeher wih plus and minus wo sandard error bands. Residuals ouside he sandard error bands sugges insabiliy in he srucural parameers. 15

16 Tesing he Sabiliy of Regression Coefficiens Y = B + B X B k X k + H 0 : Coefficien Sabiliy B j = B js, s = 1, 2,, T CUSUM Tes CUSUMSQ Tes } Brown, Durbin, and Evans (1975) Brown, R., J. Durbin, and J. Evans, Techniques for Tesing he Consancy of Regression Relaionships over Time, Journal of The Royal Saisical Sociey, B37 (1975):

17 The CUSUM es is based on a plo of he sum of he recursive residuals. The CUSUMSQ es is similar o he CUSUM es, bu plos he cumulaive sum of squared recursive residuals. If eiher he CUSUM or CUSUMSQ values is ouside a criical bound, one concludes ha here was a srucural break a he poin a which eiher saisic began is movemen oward he bound. Edgeron and Wells (1994) provide criical values of he CUSUMSQ saisic. (Oxford Bullein of Economics and Saisics, 56 (1994): ) 17

18 The CUSUM and CUSUMSQ saisics are compued using he recursive residuals. where w i are he recursive residuals, + = = k i w i w CUSUM 1 σ + = + = = T k i i k i i w w CUSUMSQ i and k is he number of explanaory variables in he model., 1) ( ˆ ) ( 2 1 = + = k T w w i T k i w σ + = = T k i w i k T w ˆ

19 The upper and lower criical values for CUSUM are where a = for a significance level of.01, for.05, and for.10. These criical values are oupu by he CUSUMLB= and + ± 2 1 ) ( ) ( 2 k T k k T a 19 CUSUMUB= opions for he significance level specified by he ALPHACSM= opion in SAS. The upper and lower criical values of CUSUMSQ are given by k T k a + ± ) (

20 Where he value of a is obained from he able by Durbin (1969) if ½(T k) Edgeron and Wells (1994) provide he mehod of obaining he value of a for large samples. These criical values are oupu by he CUSUMSQLB= and CUSUMSQUB= opions for he significance level specified by he ALPHACSM= opion in SAS. 20

21 Secion 10.3 Example: U.S. Gasoline Consumpion

22 EXAMPLE: U.S. Gasoline Consumpion (p. 947 Greene (2003)) G PcG Y Pg Pnc Puc Pp Pd Ps Pn oal U.S. gasoline consumpion per capial U.S. gasoline consumpion per capia disposable income price index for gasoline price index for new cars price index for used cars price index for public ransporaion price index for consumer durables price index for consumer services price index for consumer non-durables 22 PcG = f(y,pg,pnc,puc,pp,pd,ps,pn)

23 Secion 10.4 Sequenial Chow Tess Associaed wih he U.S. Gasoline Consumpion Problem

24 The AUTOREG Procedure Dependen Variable lnpcg Ordinary Leas Squares Esimaes SSE DFE 27 MSE Roo MSE SBC AIC Regress R-Square Toal R-Square Durbin-Wason

25 Srucural Change Tes Break Tes Poin Num DF Den DF F Value Pr > F F-ess Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Chow Larges F-saisic 25

26 The AUTOREG Procedure Sandard Approx Variable DF Esimae Error Value Pr > Inercep <.0001 lny <.0001 lnpg <.0001 lnpnc lnpuc lnpp lnpd lnps <.0001 lnpn <.0001 Esimaes of he pooled model Because of he exisense of abrup srucural change, we may wish o augmen ou model wih eiher an inercep shifer variable D = 1 if year > 1980, D = 0 if year < 1980, and/or seleced slope shifer variables. 26

27 Secion 10.5 Farley, Hinrich, and McGuire Tes for Srucural Change U.S. Gasoline Consumpion Problem

28 The REG Procedure Dependen Variable: lnpcg Number of Observaions Read 36 Number of Observaions Used 36 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Correced Toal Roo MSE R-Square Dependen Mean Adj R-Sq Coeff Var

29 Parameer Esimaes Parameer Sandard Variable DF Esimae Error Value Pr > Inercep <.0001 lny <.0001 lnpg <.0001 lnpnc lnpuc lnpp lnpd lnps <.0001 lnpn < OLS esimaes Full sample of observaions These coefficiens represen elasiciies which are consan over he enire ime period of 1960 o 1995.

30 The REG Procedure Dependen Variable: lnpcg Durbin-Wason D Number of Observaions 36 1s Order Auocorrelaion Number of Observaions Read 36 Number of Observaions Used 36 Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Correced Toal Roo MSE R-Square Dependen Mean Adj R-Sq Coeff Var

31 Parameer Esimaes Parameer Sandard Variance Variable DF Esimae Error Value Pr > Inflaion Inercep lny lnpg lnpnc lnpuc lnpp lnpd lnps lnpn T inac inac inac inac inac inac inac inac

32 Collineariy Diagnosics Condiion Proporion of Variaion Number Eigenvalue Index Inercep lny lnpg lnpnc lnpuc lnpp lnpd E E E E E E E E E E E E E E E E E E E E E E E

33 Collineariy Diagnosics Condiion Proporion of Variaion Number Eigenvalue Index Inercep lny lnpg lnpnc lnpuc lnpp lnpd E E Collineariy Diagnosics Proporion of Variaion Number lnps lnpn T inac1 inac2 inac3 inac4 inac5 inac E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E E

34 Collineariy Diagnosics --Proporion of Variaion- Number inac7 inac E E E E Durbin-Wason D Number of Observaions 36 1s Order Auocorrelaion

35 The REG Procedure Tes 1 Resuls for Dependen Variable lnpcg Mean Source DF Square F Value Pr > F Numeraor Denominaor F-es based on he join hypohesis ha he coefficiens associaed wih and all ineracion erms are joinly equal o zero. We rejec his hypohesis and conclude ha srucural change is eviden based on he Farley-Hinrich-McGuire es. Again, we use he augmened model o derive he appropriae marginal effecs and o produce forecass of gasoline consumpion. 35

36 Secion 10.6 Illusraion of Recursive Coefficiens, Recursive Residuals, CUSUM, and CUSUMSQ Tess for he U.S. Gasoline Consumpion Problem

37 The AUTOREG Procedure Dependen Variable lnpcg Ordinary Leas Squares Esimaes SSE DFE 27 MSE Roo MSE SBC AIC Regress R-Square Toal R-Square Durbin-Wason Sandard Approx Variable DF Esimae Error Value Pr > 37 Inercep <.0001 lny <.0001 lnpg <.0001 lnpnc lnpuc lnpp lnpd lnps <.0001 lnpn <.0001

38 Recursive Coefficiens inercep lnpd lnpg 38 lnpn lnpnc lnppt

39 Recursive Coefficiens lnps lnpuc lny Is srucural change eviden from he graphs of recursive coefficiens? 39

40 Obs Year lnpcg recres Obs Year lnpcg recres

41 41 Obs Year csum csumlowerb csumupperb The CUSUM saisic exceeds he bounds. Thus, srucural change is eviden.

42 42 Obs Year csumsq csumsqlowerb csumsqupperb From he CUSUMSQ saisic, srucural change is eviden a observaion 25.

43 Secion 10.7 Commenary

44 44 Commenary Wih he use of ime-series daa, i is imporan o check on he sabiliy of srucural coefficiens in economeric models. Essenially, we look for parameer esimaes which vary over ime; srucural change hen is defined as he nonconsancy of parameers in economeric models. Sequenial Chow ess are useful o examine he exisence of abrup srucural change. The Farley-Hinrich-McGuire es is useful o examine gradual srucural change. Recursive Esimaion deals exclusively wih he sabiliy assessmen of he model parameers. Formal ess of he sabiliy of regression coefficiens include he CUSUM es and he CUSUMSQ es. If srucural change exiss, hen adjusmens in he economeric model are necessary. One may need o esimae alernaive models depending on he ime period, or creae criical consrucs o accoun for he srucural change. A CAUTIONARY NOTE: A breakpoin found a he very beginning or very end of he daase, where one of he wo sub-samples is very small, should be considered carefully.