Autocorrelation and the AR(1) Process

Transcription

1 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 1 Auocorrelaion and he AR(1) Process Hun Myoung Park This documen discusses auocorrelaion (or serial correlaion) in linear regression models wih focus on he firs-order auoregression process, AR(1). This documen is largely based on Greene (003). 1. Defining Auocorrelaion Auocorrelaion occurs in ime-series daa more ofen han in cross-secional daa. Auocorrelaion (or auoregressiion and serial correlaion) is a resul of he violaion of he nonauocorrelaion assumpion ha each disurbance is uncorrelaed wih every oher disurbance 1.1 Saionariy and Auocorrelaion In he presence of auocorrelaion, E( ε X ) = E( ε X ) = 0 s Var( ε X ) = Var( ε s X ) = σ, bu Cov( ε, X ) = 0 for all s. ε s The disribuion of disurbances is said o be covariance saionary or weekly saionary. 1 E( ε ' ε ) σ I, bu = E ( ε ' ε ) σ Ω ha is a full, posiive definie marix wih a consan σ on he diagonal. Since Ωs is a funcion of -s, bu no of or s alone (saionary assumpion), he covariance beween observaions and s is also a finie funcion of -s, he disance apar in ime of he observaions. The auocovariances is defined as Cov ( ε, ε s X ) = Cov( ε + s, ε X ) = γ s = σ Ω, s = σ Ω + s, s and σ Ω = γ = σ Auocorrelaion is he correlaion beween ε and ε s, Cov( ε, ε s X ) γ s Corr( ε, ε s X ) = = = ρs Var( ε X ) Var( ε X ) γ 1. Auoregression and AR(p) s 0, 0 A ypical auoregression model AR(p) is y = μ + φ1 y 1 + φ y... + φ p y p + ε or 1 Srong saionariy requires ha whole join disribuion is he same over he ime periods.

2 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 1 p 1 B φb... φ p B y = μ ε or simply φ B) y = μ + ε (1 φ ) + backward shif operaor. (, where B denoes he The firs-order auoregression AR(1) process is srucured so ha he influence of a given disurbance fades as i recedes ino he more disan pas bu vanishes only asympoically. y μ + φ + ε = 1 y 1 = + φ1 u + φ1 y + ε 1 ) + ε = μ + φ1u + φ1 y + φ1ε 1 ε = ρε 1 + u y μ ( + ε Alernaively, In conras, he firs-order moving-average MA(1) process has a shor memory, ε = u λu 1. Ineresingly, AR(1) can be wrien in MA( ) form, ε ρε + u, where E ( ) = 0, = 1 E( u ) = σ u, and Cov ( u, u s ) = 0 if s. Repeaed subsiuion ends up wih ε = u + ρu 1 + ρ u... Each disurbance embodies he enire pas hisory of us, wih he mos recen observaions receiving greaer weigh han hose in he disan pas. The variance and covariance of disurbances are 4 σ u Var( ε ) = σ u + ρ σ u + ρ σ u +... = = σ ε 1 ρ Cov( ε, ε 1) = E( ε, ε 1) = E( ε 1, ρε 1 + u ) = ρvar( ε ρσ u ) = 1 ρ 1. Causes and Consequences of Auocorrelaion Auocorrelaion may resul from a problem in (linear) funcional form assumpion, omied relevan explanaory variables (ofen lagged dependen variables), or measuremen errors ha could be auocorrelaed. In pracice, he specificaion errors (ignoring relevan variables) appear o be mos criical. Like heeroscedasiciy, auocorrelaion makes esimaed variances of OLS (ordinary leas squares) parameer esimaes asympoically inefficien. Technically speaking, σ is biased (underesimaed). However, OLS parameer esimaes hemselves remain unbiased and consisen. In shor, OLS is no BLUE. 3. Deecing Auocorrelaion This secion considers several es saisics including Breusch-Godfrey LM, Box-Pierce Q, Ljung-Box Q, Durbin-Wason d, and Durbin h. 3.1 Lagrange Muliplier Tes for AR(p) u

3 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 3 Breusch (1978) and Godfrey (1978) develop a Lagrange muliplier es ha can be applied o he ph order auoregression models. Thus, his es is more general han D-W d and Durbin h. The null hypohesis is a model wihou lagged dependen variables, ρ ρ =... = ρ 0. 1 = p = The LM es consiss of several seps. Firs, regress Y on Xs o ge residuals. Compue lagged residuals up o ph order. Replace missing values for lagged residuals wih zeros. Regress ε on Xs and e -1, e -, and e -p o ge R. Finally compue LM saisic using he R and he number of observaions T used in he model. 3 LM = TR ~ χ ( p) This saisic follows he chi-squared disribuion wih p degrees of freedom. This Breusch- Godfrey LM is preferred o oher es saisics. 3. Q and Q Tes for AR(p) Box and Pierce (1970) develop he Q es ha is asympoically equivalen o he Breusch- Godfrey LM es. Box-Pierce Q has a chi-squared disribuion wih p degrees of freedom. The Q saisic is Q = T P j = 1 r j ~ χ ( p), where r T = p+ 1 p = T = 1 e e e p Firs, regress Y on Xs o ge residuals and compue lagged residuals up o ph order. Compue individual rp s using e and ee p. Finally, plug rp s in he formula o compue Box-Pierce Q. Ljung and Box (1979) refine he Box-Pierce Q es o ge Q. You may use informaion obained above. Ljung-Box Q also follows he chi-squared disribuion wih p degrees of freedom. r Q' = T ( T + ) ~ χ ( p). j P j j = 1 T 3.3 Durbin-Wason d for AR(1) The Durbin-Wason (D-W) es is based on he principle ha if he rue disurbances are auocorrelaed, his fac will be revealed hrough he auocorrelaions of he leas squares residuals (Durbin and Wason 1950, 1951, 1971). The null hypohesis is ha disurbances are no auocorrelaed, ρ = 0. The es saisic is This model is viewed as a resriced model, whereas he full or unresriced model has p lagged dependen variables. 3 Since missing values in lagged residuals are filled wih zero, he number of observaions used in he model is he same as ha in he original model.

4 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 4 d T = = T ( e e 1) = 1 e From he Durbin-Wason saisic able (T and k), we ge he following decision crieria. 4 0 d * d d L U U * d 4 L Rejec H 0 Inconclusive Do no rejec H 0 Inconclusive Rejec H 0 ρ > 0 (uncerain) H 0 : ρ = 0 (uncerain) ρ < 0 D-W d ranges from zero (perfec posiive auocorrelaion) o 4 (perfec negaive auocorrelaion). Noe ha here are wo inconclusive areas where he null hypohesis canno be esed properly. The upper and lower limis reflec ha he sequence of disurbances depends no only on he sequence of residuals bu also on he sequence of values of independen variables (Derick Boyd s memo). The presence of inconclusive regions, which will large in a small sample, implies shorcomings in pracice (Greene 003). The relaionship beween D-W d and auocorrelaion coefficien is known as e1 + et DW = (1 r) T e = 1 When he sample is large, he las erm above will be negligible. Thus, DW (1 ˆ ρ) or ˆ ρ 1 DW The Durbin-Wason es has been found o be quie powerful when compared wih ohers for AR(1) processes. However, he es is no likely o be valid when here is a lagged dependen variable in he equaion (Greene 003). 3.4 Durbin h for AR(1) wih a Lagged Dependen Variable D-W d is ofen biased oward a finding of no auocorrelaion (DW=) (Greene 003). Durbin (1970) proposes a Lagrange muliplier saisic o es auocorrelaion in he presence of a lagged dependen variable. D-W d and Durbin h are known as asympoically equivalen. T d T h r 1 =, where Ts lag < 1. 1 Tslag 1 Tslag T is he number of observaions used in he model wih a lagged dependen variable. Compared o D-W d, Durbin h loses one observaion in compuaion. slag is he esimaed variance of he OLS coefficien of he lagged dependen variable Y -1. Simply, his variance is he squared 4 k is he number of regressors excluding he inercep.

5 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 5 s lag is also an elemen in he diagonal of he variance- sandard error of he parameer esimaor. covariance marix. The h saisic is approximaely normally disribued wih zero mean and uni variance. Noe ha his h es is he one-ailed es. 3.5 Sofware Issue The Table 3.1 summarizes procedures and commands ha conduc AR(1) es. STATA.durbina command produces a chi-squared saisic, which is differen from he z score. Table 3.1 Comparison of Compuing Tes Saisics SAS 9.3 STATA 9. LIMDEP 8.0 B-G LM AUTOREG /GODFREY=1.bgodfrey,lags(p) - D-W d REG /DW AUTOREG /DW=1.dwsa (.esa dwason) Regress D-W h AUTOREG /LAGDV= DW=1.durbina (.esa durbinal) - B-P Q L-B Q * Box-Pierce Q and Ljung-Box Q are no suppored by saisical sofware. 4. Correcing Auocorrelaion The auoregressive error model correcs auocorrelaion. If Ω is known, you may ake he generalized leas squares (GLS) mehod. Oherwise, you have o esimae he feasible generalized leas squares (FGLS). If an auoregressive error model also suffers from heeroscedasiciy, you may ry he generalized auoregressive condiional heeroscedasiciy (GARCH) model. 4.1 Generalized Leas Squares If Ω is known, he generalized leas squares esimaor is ˆ 1 1 β = ( X ' Ω X ) X ' Ω 1 1 consisen. The variance of parameer esimaes is Var( ˆ) β = σ ( X ' Ω X ). ε 1 y, which is GLS needs ransformaion of dependen variable, independen variables, and he inercep. See he following. 1 ρ Since ρ ρ P = in AR(1), ρ 1 1 ρ y 1 1 ρ 1 ρ x ρ xk1 * y ρy1 Y = *, 1 ρ x1 ρx11... xk ρxk1 X = y T ρy T 1 1 ρ x1, T ρx1, T 1... xk, T ρxk, T 1

6 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 6 Then, regress Y * on X *. In SAS and STATA, he inercep should be suppressed. 4. Feasible Generalized Leas Squares In he real world Ω is ofen unknown. So he feasible generalized leas squares (FGLS) seems o be more plausible han GLS. FGLS begins wih esimaing ρ. The following mehods are commonly used. r 1 T = e e 1 = e r1 1 DW Theil s (1971) adjused esimaor r adj r *( T K) /( T 1). 1 Once you compue r, again perform daa ransformaion. Do no forge o ransform he inercep erm. The Prais-Winsen (1954) and Cochrane-Orcu (1949) FGLS can be boh wo-sepped and ieraive. The Prais-Winsen esimaor uses all observaions, while he Cochrane-Orcu FGLS ignores he firs observaion. The Prais-Winsen FGLS is ofen called he Yule-Walker mehod in SAS (METHOD=YW). You may ierae he procedure o ge more saisfacory oupus. SAS provides wo-sep, ieraive wo-sep, and maximum likelihood mehods, while STATA suppors he firs wo mehods. Insead, STATA allows researchers o use various ρ esimaors. In addiion o wo-sep, ieraive wo-sep, and maximum likelihood mehods, LIMDEP suppors he Haanaka s (1974) model for auocorrelaion wih a lagged dependen variable, which is asympoically equivalen o he maximum likelihood model. 4.3 Sofware Issue The Table 4.1 summarizes esimaion mehods suppored in each saisical sofware. Table 4.1 Comparison Esimaion Mehods SAS 9.3 STATA 9. LIMDEP 8.0 OLS REG.regress Regress -sep P-W AUTOREG /YW.prais, wosep Regress;AR1;Maxi=1;Rho= -sep C-O -.prais, corc wosep Regress;AR1;Maxi=1;Alg=C;Rho= -sep P-W (dw) -.prais, rhoype(dw) wosep Regress;AR1;Maxi=1 -sep C-O (dw) -.prais, rho(dw) corc wosep Regress;AR1;Alg=Corc;Maxi=1 Ieraive P-W AUTOREG /ITYW.prais Regress;AR1; Ieraive C-O -.prais, corc Regress;AR1;Alg=Corc MLE AUTOREG /ML - Regress;AR1;Alg=MLE Two-sage (IV) - - SLS;Ins;AR1;Haanaka GARCH AUTOREG /GARCH.arch, garch(p) Regress;Model=Garch(p,q,1) * The defaul ypes of rho are auocorrelaion coefficien in SAS, residual regression-based rho in STATA, and he D-W d-based rho in LIMDEP.

7 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 7 5. Example: Deecing Auocorrelaion This secion illusraes how o deec auocorrelaion using STATA and SAS. 5.1 STATA STATA has he.dwsa command for D-W d and.durbina for he Durbin h saisic. These commands are posesimaion commands of he.regress. The.durbina produces a chisquared saisic insead of a z score and reurns a slighly differen p-value Daa Preparaion The daa are downloaded from Greene s webpage a hp://pages.sern.nyu.edu/~wgreene. This daa se for he U.S. Gasoline Marke, , is drawn from he Economic Repor of he Presiden: 1996, Council of Economic Advisors, The variables included are G = Toal U.S. gasoline consumpion, compued as oal expendiure divided by price index. Pg = Price index for gasoline, Y = Per capia disposable income, Pnc = Price index for new cars, Puc = Price index for used cars, Pp = Price index for public ransporaion, Pd = Aggregae price index for consumer durables, Pn = Aggregae price index for consumer nondurables, Ps = Aggregae price index for consumer services, Pop = U.S. oal populaion in millions.. infile Year G Pg Y Pnc Puc Pp Pd Pn Ps Pop /// using hp://pages.sern.nyu.edu/~wgreene/tex/ables/tablef-.x, clear. drop if Year==.. sse Year ime variable: Year, 1960 o gen lng=ln(g/pop). gen lnpg=ln(pg). gen lni=ln(y). gen lnpnc=ln(pnc). gen lnpuc=ln(puc). sum ln* Variable Obs Mean Sd. Dev. Min Max lng lnpg lni lnpnc lnpuc global OLS "lng lnpg lni" // OLS. global OLS "lng l1.lng lnpg lni" // OLS for Durbin h. global K=5

8 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 8. regress $OLS Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons predic e, residuals. gen e_=e^. gen ee1=e*l.e. gen e1_=l.e^. gen e_e1_=(e-l.e)^. lis e l.e e_ ee1 e1_ e_e1_ in 1/ e L.e e_ ee1 e1_ e_e1_ absa e e_ ee1 e1_ e_e1_, sa(n sum mean) save sas e e_ ee1 e1_ e_e1_ N sum -3.73e mean -1.03e marix sum=r(satoal). local s_e_ = sum[,] // local s_ee1 = sum[,3] // local s_e1_ = sum[,4] // local s_e_e1_ = sum[,5] // global T = sum[1,1] // Breusch-Godfrey LM Tes Unlike he Durbin-Wason d es, he Breusch-Godfrey Lagrange muliplier es can be applied o general AR(p) processes. The STATA.bgodfrey, a posesimaion command, compues he saisic up o he ph order. Compare he four LM saisics wih hose in quiely regress $OLS (oupu is skipped). bgodfrey,lags(1) Breusch-Godfrey LM es for auocorrelaion lags(p) chi df Prob > chi

9 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: H0: no serial correlaion. bgodfrey,lags() Breusch-Godfrey LM es for auocorrelaion lags(p) chi df Prob > chi H0: no serial correlaion. bgodfrey,lags(3) Breusch-Godfrey LM es for auocorrelaion lags(p) chi df Prob > chi H0: no serial correlaion. bgodfrey,lags($lag) // equivalen o.esa bgodfrey,lags() Breusch-Godfrey LM es for auocorrelaion lags(p) chi df Prob > chi H0: no serial correlaion In order o manually compue he LM saisic, creae p lagged residuals and regress residuals on all independen variables and all lagged residuals. Do no forge o fill missing in he lagged residuals wih zero.. gen e1=l1.e. replace e1=0 if e1==. Now run he OLS o ge R.. regress e lnpg lni e1 Source SS df MS Number of obs = F( 3, 3) = 9. Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.038 e Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni e _cons The LM saisic is T*R, which follows a chi-squared disribuion wih p degrees of freedom.. local r=e(r). local lm = ($T)*`r'. disp `lm' // LM saisic

10 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: disp chiail(1, `lm') // p-value Q and Q Tes Now, consider Box-Pierce Q and Ljung-Box Q saisics. Firs compue individual auocorrelaion coefficiens up o he ph order.. regress lnpg lni lnpnc lnpuc e1 // for AR(1) Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.181 lnpg Coef. Sd. Err. P> [95% Conf. Inerval] lni lnpnc lnpuc e _cons gen ee=e^. gen ee1=e*e1. absa ee ee1, sa(n sum mean) save // for AR(1) sas ee ee N sum mean marix sum=r(satoal). local r1=sum[,]/sum[,1]. local Q = $T*(`r1'^) // for AR(1). disp `Q' disp chiail(1,`q') // for AR(1) local Q1 = $T*($T+)*(`r1'^/($T-1)) // for AR(1). disp `Q1' disp chiail(1,`q1') // for AR(1) AS and STATA do no have opion or command o compue Q or Q Durbin-Wason d Tes

11 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 11 Firs, le us compue D-W d manually o make sure i is idenical o he saisic provided by STATA. Noe ha.dwsa and.esa dwason are equivalen.. local dw= `s_e_e1_'/`s_e_' // dwsa // equivalen o.esa dwason Durbin-Wason d-saisic( 5, 36) = local rho=1-`dw'/ // DW based rho: rhoype(dw) local rho=`s_ee1'/`s_e_' // Auocorrelaion rho: rhoype(scorr) Durbin h Tes Finally, le us compue he Durbin h for a model wih a lagged dependen variable. Noe ha Durbin h is no a wo-ailed es, bu a one-ailed es.. regress $OLS Source SS df MS Number of obs = F( 5, 9) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.018 lng Coef. Sd. Err. P> [95% Conf. Inerval] lng L lnpg lni lnpnc lnpuc _cons marix lis e(v) symmeric e(v)[6,6] L. lng lnpg lni lnpnc lnpuc _cons L.lnG lnpg lni lnpnc lnpuc _cons marix V = e(v). local v_lag = V[1,1] // variance of coefficien of he lagged DV. predic e, residuals. gen e_=e^. gen ee1=e*l.e. lis e l.e e_ ee1 in 1/ e L.e e_ ee e

12 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: absa e e_ ee1, sa(n sum mean) save sas e e_ ee N sum -1.86e mean -5.3e marix sum=r(satoal). local s_e_ = sum[,] // local s_ee1 = sum[,3] // global T = sum[1,1] // 35. local rho=`s_ee1'/`s_e_' // Auocorrelaion rho: rhoype(scorr). disp `rho' disp $T*`v_lag' // o check if Ts < local h = `rho'*sqr($t/(1-$t*`v_lag')). disp `h' disp 1-norm(`h') SAS AUTOREG procedure reurns he same Durbin h.9575 (See 5..). Using he alernaive erm ˆ ρ 1 DW will give you a quie differen saisic largely because his sample is no large sufficienly.. dwsa // equivalen o esa dwason Durbin-Wason d-saisic( 6, 35) = marix dw = r(dw). local dw = dw[1,1] // dw. local h = (1-`dw'/)*sqr($T/(1-$T*`v_lag')). disp `h' disp 1-norm(`h') Le us run eiher.durbina or.esa durbinal o conduc he Durbin s alernaive es, which produces a chi-squared saisic whose p-value is differen from ha of h above.. durbina // equivalen o.esa durbinal Durbin's alernaive es for auocorrelaion lags(p) chi df Prob > chi H0: no serial correlaion 5. SAS REG and AUTOREG Procedure

13 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 13 In SAS, you may use he REG procedure of SAS/STAT and he AUTOREG procedure of SAS/ETS. REG compues he D-W d saisic, while AUTORE produces boh D-W d and Durbin h saisics SAS REG Procedure The /DW opion in he REG procedure compues he D-W d saisic. PROC REG DATA=masil.gasoline; MODEL lng = lnpg lni lnpnc lnpuc /DW; RUN; The REG Procedure Model: MODEL1 Dependen Variable: lng 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 Parameer Esimaes Parameer Sandard Variable DF Esimae Error Value Pr > Inercep <.0001 lnpg lni <.0001 lnpnc lnpuc Durbin-Wason D Number of Observaions 36 1s Order Auocorrelaion The firs order auocorrelaion in he REG procedure is compued using ee 1 and for he compuaion in he STATA. 5.. SAS AUTOREG Procedure e See 5.1. The AUTOREG procedure conducs D-W d and Breusch-Godfrey LM ess. You may specify he order in he DW= and GODFREY= opions. For example, he GODFREY=4 produces LM saisics up o 4 h order auoregression.

14 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 14 PROC AUTOREG DATA=masil.gasoline; MODEL lng = lnpg lni lnpnc lnpuc /DW=1 GODFREY=4 ; RUN; The AUTOREG Procedure Dependen Variable lng Ordinary Leas Squares Esimaes SSE DFE 31 MSE Roo MSE SBC AIC Regress R-Square Toal R-Square Durbin-Wason Godfrey's Serial Correlaion Tes Alernaive LM Pr > LM AR(1) <.0001 AR() <.0001 AR(3) AR(4) Sandard Approx Variable DF Esimae Error Value Pr > Inercep <.0001 lnpg lni <.0001 lnpnc lnpuc The AUTOREG procedure also compue Durbin h saisics wih he /LAGDV and DW opions. The /COVB opion reurns he variance and covariance marix of parameer esimaes. The variance of he coefficien of he lagged dependen variable is =.1139^. PROC AUTOREG DATA=masil.gasoline; MODEL lng = lng1 lnpg lni lnpnc lnpuc /LAGDV=lnG1 DW=1 DWPROB COVB; RUN; The AUTOREG Procedure Dependen Variable lng Ordinary Leas Squares Esimaes SSE DFE 9 MSE Roo MSE SBC AIC Regress R-Square Toal R-Square Durbin h Pr > h Durbin-Wason Sandard Approx Variable DF Esimae Error Value Pr > Inercep

15 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 15 lng <.0001 lnpg lni lnpnc lnpuc Covariance of Parameer Esimaes Inercep lng1 lnpg lni lnpnc lnpuc Inercep lng lnpg lni lnpnc lnpuc The AUTOREG procedure esimaes he linear regression model wih auocorrelaion correced using he /NLAG=1 opion indicaing he AR(1) process.

16 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: Correcing Auocorrelaion: Feasible Generalized Leas Squares This secion illusraes mehods o correc auocorrelaion using he Prais-Winsen s FGLS (Feasible Generalized Leas Squares) and he Cochrane-Orcu FGLS. If Ω is known, you may ry GLS (Generalized Leas Squares). 6.1 FGLS in STATA This secion illusraes how o esimae FGLS in STATA. See secion 5 for he descripion of he daa se used Daa Preparaion. infile Year G Pg Y Pnc Puc Pp Pd Pn Ps Pop /// using hp://pages.sern.nyu.edu/~wgreene/tex/ables/tablef-.x, clear. drop if Year==.. sse Year ime variable: Year, 1960 o gen lng=ln(g/pop). gen lnpg=ln(pg). gen lni=ln(y). gen lnpnc=ln(pnc). gen lnpuc=ln(puc). sum ln* // summary saisics Variable Obs Mean Sd. Dev. Min Max lng lnpg lni lnpnc lnpuc global OLS "lng lnpg lni lnpnc lnpuc". regress $OLS Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons predic e, residuals. gen e1=l.e. gen e_=e^. gen ee1=e*e1. gen e1_=e1^. gen e_e1_=(e-e1)^

17 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 17. absa e e1 e_ ee1 e1_ e_e1_, sa(n sum mean) save sas e e1 e_ ee1 e1_ e_e1_ N sum -3.73e mean -1.03e marix sum=r(satoal). local s_e_ = sum[,3] // local s_ee1 = sum[,4] // local s_e1_ = sum[,5] // local s_e_e1_ = sum[,6] // local T = sum[1,1] // Compuing Auocorrelaion Coefficien There are various ways of esimaing he auocorrelaion parameer ρ. Auoregressive effor models o correc auocorrelaion depend on he ρ esimaor and esimaion mehods such as ieraive and marix likelihood mehods. The ρ is ofen esimaed using auocorrelaion formula and Durbin-Wason d.. local rho=`s_ee1'/`s_e_' // Auocorrelaion rho: rhoype(scorr) dwsa // DW d Durbin-Wason d-saisic( 5, 36) = local dw= `s_e_e1_'/`s_e_' // DW d local rho = 1-`dw'/ // DW based rho: rhoype(dw) In addiion o scorr (auocorrelaion coefficien) and dw (D-W d-based rho), STATA provides heil (adjusmen of auocorrelaion coefficien), nagar (adjusmen of D-W d-based coefficien), regress (defaul opion, he coefficien of regression e on e -1 wihou inercep), and freg (he coefficien of regression e on e +1 wihou inercep) opions. local rho=`s_ee1'/`s_e_'*($t-$k)/($t) //Theil rho: rhoype(heil) local rho = ((1-`dw'/)*$T^+$K^)/($T^-$K^) // Nagar regress e e1, noc // for he rho based on regression on he lagged residuals Source SS df MS Number of obs = F( 1, 34) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.043 e Coef. Sd. Err. P> [95% Conf. Inerval] e marix b1 = e(b). local rho = b1[1,1]. disp `rho' // gen e0=e[_n+1] (1 missing value generaed)

18 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 18. quiely regress e e0, noc // rho based on regression wih he leaded residuals. marix b = e(b). local rho = b[1,1] // Once a ρ esimaor is deermined, variables and he inercep need o be ransformed using he esimaor.. foreach var of global OLS {. gen T`var' = sqr(1-`rho'^)*`var' if (_n==1) 3. replace T`var' = `var'-`rho'*`var'[_n-1] if (_n!=1) 4. }. gen Inercep = sqr(1-`rho'^) if (_n==1). replace Inercep = 1 -`rho' if (_n!=1) Alernaively, you may explicily ransform daa variable by variable as follows.. gen TlnG = sqr(1-`rho'^)*lng if (_n==1). replace TlnG = lng-`rho'*lng[_n-1] if (_n!=1). gen TlnPg = sqr(1-`rho'^)*lnpg if (_n==1). replace TlnPg = lnpg-`rho'*lnpg[_n-1] if (_n!=1). gen TlnI = sqr(1-`rho'^)*lni if (_n==1). replace TlnI = lni-`rho'*lni[_n-1] if (_n!=1). gen TlnPnc = sqr(1-`rho'^)*lnpnc if (_n==1). replace TlnPnc = lnpnc-`rho'*lnpnc[_n-1] if (_n!=1). gen TlnPuc = sqr(1-`rho'^)*lnpuc if (_n==1). replace TlnPuc = lnpuc-`rho'*lnpuc[_n-1] if (_n!=1) Prais-Winsen FGLS Now, you are ready o fis he Prais-Winsen FGLS wih he ransformed daa. The inercep should be suppressed in he OLS. Thus, he F es and R are no reliable. Le us firs uses he ρ esimaor compued from he auocorrelaion formula.. local rho=`s_ee1'/`s_e_' // Auocorrelaion rho: rhoype(scorr) (daa ransformaion is skipped). regress TlnG TlnPg TlnI TlnPnc TlnPuc Inercep, nocons // Prais-Winsen FGLS Source SS df MS Number of obs = F( 5, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0159 TlnG Coef. Sd. Err. P> [95% Conf. Inerval] TlnPg TlnI TlnPnc TlnPuc Inercep

19 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 19 The.prais command by defaul fis he Prais-Winsen FGLS. Use he rhoype(scorr) opion o specify he ype of ρ esimaor. The wosep opion sops ieraion afer he firs ieraion. The oupu is he same as he above. Compare SSM, he degree of freedom for he model, F o hose of he above.. prais $OLS, rhoype(scorr) wosep // Auocorrelaion Ieraion 0: rho = Ieraion 1: rho = Prais-Winsen AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0159 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) Le us use he D-W d-based ρ esimaor (=1-d/).. local rho = 1-`dw'/ // DW based rho: rhoype(dw) (daa ransformaion is skipped). regress TlnG TlnPg TlnI TlnPnc TlnPuc Inercep, nocons // Prais-Winsen FGLS Source SS df MS Number of obs = F( 5, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.013 TlnG Coef. Sd. Err. P> [95% Conf. Inerval] TlnPg TlnI TlnPnc TlnPuc Inercep The rhoype(dw) opion uses he D-W d-based ρ esimaor when esimaing auoregressive error models.. prais $OLS, rhoype(dw) wosep Ieraion 0: rho = Ieraion 1: rho =

20 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 0 Prais-Winsen AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 31) = 67.4 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.013 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) The following example uses he Theil s ρ esimaor, which adjuss he auocorrelaion coefficien.. local rho=`s_ee1'/`s_e_'*($t-$k)/($t) //Theil rho: rhoype(heil) prais $OLS, rhoype(heil) wosep // Theil rho Ieraion 0: rho = Ieraion 1: rho = Prais-Winsen AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.085 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) The following uses he adjusmen of D-W d-based ρ esimaor.. local rho = ((1-`dw'/)*$T^+$K^)/($T^-$K^) // Nagar prais $OLS, rhoype(nagar) wosep Ieraion 0: rho = Ieraion 1: rho = Prais-Winsen AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 31) = 58.73

21 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 1 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.007 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) The following uses he defaul ype of ρ esimaor, which is obained by regressing e on e -1 wihou he inercep.. prais $OLS, rhoype(regress) wosep // defaul Ieraion 0: rho = Ieraion 1: rho = Prais-Winsen AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0148 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) The following uses he ρ esimaor obained by regressing e on e +1 wihou he inercep.. prais $OLS, rhoype(freg) wosep Ieraion 0: rho = Ieraion 1: rho = Prais-Winsen AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.019 lng Coef. Sd. Err. P> [95% Conf. Inerval]

22 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) Cochrane-Orcu FGLS Like Prais-Winsen FGLS, Cochrane-Orcu FGLS runs OLS wih he ransform daa. Unlike he Prais-Winsen, he Cochrane-Orcu ignores he firs observaion. Le us begin wih Cochrane-Orcu FGLS using he auocorrelaion coefficien.. regress TlnG TlnPg TlnI TlnPnc TlnPuc Inercep if _n > 1, nocons Source SS df MS Number of obs = F( 5, 30) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0183 TlnG Coef. Sd. Err. P> [95% Conf. Inerval] TlnPg TlnI TlnPnc TlnPuc Inercep The.prais command has he corc opion o esimae Cochrane-Orcu FGLS.. prais $OLS, rhoype(scorr) wosep corc Ieraion 0: rho = Ieraion 1: rho = Cochrane-Orcu AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 30) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0183 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original)

23 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 3 Durbin-Wason saisic (ransformed) The following wo oupus use he D-W d-based ρ esimaor.. regress TlnG TlnPg TlnI TlnPnc TlnPuc Inercep if _n > 1, nocons Source SS df MS Number of obs = F( 5, 30) = 8.41 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0159 TlnG Coef. Sd. Err. P> [95% Conf. Inerval] TlnPg TlnI TlnPnc TlnPuc Inercep prais $OLS, rhoype(dw) wosep corc Ieraion 0: rho = Ieraion 1: rho = Cochrane-Orcu AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 30) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0159 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) The followings esimae oher auoregressive error models using oher ρ esimaors such as Theil s esimaor. Pay aenion o he rhoype() opion.. prais $OLS, rhoype(heil) wosep corc Ieraion 0: rho = Ieraion 1: rho = Cochrane-Orcu AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 30) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.083

24 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 4 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) prais $OLS, rhoype(nagar) wosep corc Ieraion 0: rho = Ieraion 1: rho = Cochrane-Orcu AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 30) = 7.31 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0104 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) prais $OLS, rhoype(regress) wosep corc // defaul Ieraion 0: rho = Ieraion 1: rho = Cochrane-Orcu AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 30) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0174 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) prais $OLS, rhoype(freg) wosep corc

25 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 5 Ieraion 0: rho = Ieraion 1: rho = Cochrane-Orcu AR(1) regression -- wosep esimaes Source SS df MS Number of obs = F( 4, 30) = 33.8 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0158 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) Ieraive Prais-Winsen and Cochrane-Orcu FGLS STATA provides he ieraive wo-sep esimaion mehod for he Prais-Winsen and Cochrane- Orcu FGLS.. prais $OLS, rhoype(scorr) // Ieraive Prais-Winsen FGLS Ieraion 0: rho = Ieraion 1: rho = Ieraion : rho = Ieraion 3: rho = Ieraion 4: rho = Ieraion 5: rho = Ieraion 6: rho = Ieraion 7: rho = Ieraion 8: rho = Ieraion 9: rho = Ieraion 10: rho = Ieraion 11: rho = Ieraion 1: rho = Ieraion 13: rho = Ieraion 14: rho = Ieraion 15: rho = Ieraion 16: rho = Prais-Winsen AR(1) regression -- ieraed esimaes Source SS df MS Number of obs = F( 4, 31) = Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0189 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons

26 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 6 rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) The following example is he ieraive Cochrane-Orcu FGLS.. prais $OLS, rhoype(scorr)corc // Ieraive Cochrane-Orcu FGLS Ieraion 0: rho = Ieraion 1: rho = Ieraion : rho = Ieraion 3: rho = Ieraion 4: rho = Ieraion 5: rho = Ieraion 6: rho = Ieraion 7: rho = Ieraion 8: rho = Ieraion 9: rho = Ieraion 10: rho = Ieraion 11: rho = Ieraion 1: rho = Ieraion 13: rho = Ieraion 14: rho = Cochrane-Orcu AR(1) regression -- ieraed esimaes Source SS df MS Number of obs = F( 4, 30) = 1.15 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.0188 lng Coef. Sd. Err. P> [95% Conf. Inerval] lnpg lni lnpnc lnpuc _cons rho Durbin-Wason saisic (original) Durbin-Wason saisic (ransformed) FGLS in SAS SAS suppor boh (ieraive) wo-sep Prais-Winen and maximum likelihood algorihms Two-sep Prais-Winsen Esimaion Once variables are ransformed, run OLS wih he inercep suppressed in he REG procedure. SAS by defaul uses he auocorrelaion coefficien as he ρ esimaor. PROC REG DATA=masil.gasoline; MODEL TlnG = Inercep TlnPg TlnI TlnPnc TlnPuc /NOINT; RUN; The REG Procedure Model: MODEL1 Dependen Variable: TlnG

27 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 7 Number of Observaions Read 36 Number of Observaions Used 36 NOTE: No inercep in model. R-Square is redefined. Analysis of Variance Sum of Mean Source DF Squares Square F Value Pr > F Model <.0001 Error Uncorreced Toal Roo MSE R-Square Dependen Mean Adj R-Sq Coeff Var Parameer Esimaes Parameer Sandard Variable DF Esimae Error Value Pr > Inercep <.0001 TlnPg TlnI <.0001 TlnPnc TlnPuc The AUTOREG procedure esimaes auoregressive error models wihou daa ransformaion. The /NLAG=1 specifies he firs-order auocorrelaion. AUTOREG by defaul (METHOD=YW) compues he Yule-Walker (Prais-Winsen) FGLS esimaes. PROC AUTOREG DATA=masil.gasoline; MODEL lng = lnpg lni lnpnc lnpuc /NLAG=1; RUN; The AUTOREG Procedure Dependen Variable lng Ordinary Leas Squares Esimaes SSE DFE 31 MSE Roo MSE SBC AIC Regress R-Square Toal R-Square Durbin-Wason Sandard Approx Variable DF Esimae Error Value Pr > Inercep <.0001 lnpg lni <.0001 lnpnc lnpuc Esimaes of Auocorrelaions

28 Jeeshim and KUCC65 (7/18/006) Auocorrelaion and he AR(1) Process: 8 Lag Covariance Correlaion ******************** ************* Preliminary MSE Esimaes of Auoregressive Parameers Sandard Lag Coefficien Error Value Yule-Walker Esimaes SSE DFE 30 MSE Roo MSE SBC AIC Regress R-Square Toal R-Square Durbin-Wason Sandard Approx Variable DF Esimae Error Value Pr > Inercep <.0001 lnpg lni <.0001 lnpnc lnpuc Ieraive Two-sep Prais-Winsen Esimaion The AUTOREG procedure can correc auocorrelaion using he ieraive Yule-Walker mehod (METHOD=ITYW). PROC AUTOREG DATA=masil.gasoline; MODEL lng = lnpg lni lnpnc lnpuc /NLAG=1 METHOD=ITYW; RUN; The AUTOREG Procedure Dependen Variable lng Ordinary Leas Squares Esimaes SSE DFE 31 MSE Roo MSE SBC AIC Regress R-Square Toal R-Square Durbin-Wason Sandard Approx Variable DF Esimae Error Value Pr > Inercep <.0001 lnpg lni <.0001 lnpnc lnpuc