Vector Error Correction Models

Transcription

1 Vecor Error Correcion Models The vecor auoregressive (VAR) model is a general framework used o describe he dynamic inerrelaionship among saionary variables. So, he firs sep in ime-series analysis should be o deermine wheher he levels of he daa are saionary. If no, ake he firs differences of he series and ry again. Usually, if he levels (or log-levels) of your ime series are no saionary, he firs differences will be. If he ime series are no saionary hen he VAR framework needs o be modified o allow consisen esimaion of he relaionships among he series. The vecor error correcion (VEC) model is jus a special case of he VAR for variables ha are saionary in heir differences (i.e., I(1)). The VEC can also ake ino accoun any coinegraing relaionships among he variables. Consider wo ime-series variables, y and x. Generalizing he discussion abou dynamic relaionships o hese wo inerrelaed variables yields a sysem of equaions: y =β +β y +β x + v y x =β +β y +β x + v x The equaions describe a sysem in which each variable is a funcion of is own lag, and he lag of he oher variable in he sysem. In his case, he sysem conains wo variables y and x. Togeher he equaions consiue a sysem known as a vecor auoregression (VAR). In his example, since he maximum lag is of order one, we have a VAR(1). If y and x are saionary, he sysem can be esimaed using leas squares applied o each equaion. If y and x are no saionary in heir levels, bu saionary in differences (i.e., I(1)), hen ake he differences and esimae: y y =β y +β x + v x =β y +β x + v x using leas squares. If y and x are I(1) and coinegraed, hen he sysem of equaions is modified o allow for he coinegraing relaionship beween he I(1) variables. Inroducing he coinegraing relaionship leads o a model known as he vecor error correcion (VEC) model. ESTIMATING A VEC MODEL In he firs example, daa on he Gross Domesic Produc of Ausralia and he U.S. are used o esimae a VEC model. We decide o use he vecor error correcion model because (1) he ime series are no saionary in heir levels bu are in heir differences (2) he variables are coinegraed. Our iniial impressions are gained from looking a plos of he wo series. To ge sared, change he direcory o he one conaining your daa, open a new log file, and load your daa. In his exercise we ll be using he gdp.da daa. use gdp, clear

2 The daa conain wo quarerly ime series: Ausralian and U.S. GDP from 1970q1 o 2004q4. As usual, creae a sequence of quarerly daes: gen dae = q(1970q1) + _n - 1 forma %q dae sse dae Ploing he levels and differences of he wo GDP series suggess ha he daa are nonsaionary in levels, bu saionary in differences. In his example, we used he sline command wih an opional scheme. A scheme holds saved graph preferences for laer use. You can creae your own or use one of he ones insalled wih Saa. A he command line you can use deermine which schemes are insalled on your compuer by yping sline aus usa, name(levels, replace) sline D.aus D.usa, name(difference, replace) q1 1980q1 1990q1 2000q1 dae real GDP of Ausralia real GDP of USA Neiher series looks saionary in is levels. They appear o have a common rend, an indicaion ha hey may be coinegraed. Uni roo ess are performed using he augmened Dickey-Fuller regressions, which require some judgmen abou specificaion. The user has o decide wheher o include a consan, rend or drif, and lag lenghs for he differences ha augmen he regular Dickey-Fuller regressions. The differences are graphed and his gives some clues abou specificaion. The graph below shows lile evidence of rend or drif.

3 q1 1980q1 1990q1 2000q1 dae real GDP of Ausralia, D real GDP of USA, D Lag lenghs can be chosen using model selecion rules or by saring a a maximum lag lengh, say 4, and eliminaing lags one-by-one unil he -raio on he las lag becomes significan. dfuller aus, regress lags(1) dfuller usa, regress lags(3) Through process of eliminaion he decision is made o include he consan (hough i looks unnecessary) and o include 1 lag for aus and 3 for he usa series. In none of he ADF regressions he auhor esimaed was eiher ADF saisic even close o being significan a he 5% level. Saisfied ha he series are nonsaionary in levels, heir coinegraion is explored. Augmened Dickey-Fuller es for uni roo Number of obs = 122 Inerpolaed Dickey-Fuller Tes 1% Criical 5% Criical 10% Criical Saisic Value Value Value Z() MacKinnon approximae p-value for Z() = dfuller usa, regress lags(3) Augmened Dickey-Fuller es for uni roo Number of obs = 120 Inerpolaed Dickey-Fuller Tes 1% Criical 5% Criical 10% Criical Saisic Value Value Value Z() MacKinnon approximae p-value for Z() = In each case, he null hypohesis of nonsaionariy canno be rejeced a any reasonable level of significance. Nex, esimae he coinegraing equaion using leas squares. Noice ha he coinegraing relaionship does no include a consan.

4 regress aus usa, nocons. reg aus usa, nocons Source SS df MS Number of obs = 124 F( 1, 123) =. Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = aus Coef. Sd. Err. P> [95% Conf. Inerval] usa The residuals are saved in order o conduc an Engle-Granger es of coinegraion and ploed. predic eha, residual sline eha, name(resids, replace) Residuals q1 1980q1 1990q1 2000q1 dae The residuals have an inercep of zero and show lile evidence of rend. Finally, he saved residuals are used in an auxiliary regression The Saa command is: eˆ =φ eˆ + v 1

5 regress D.eha L.eha, noconsan. reg D.eha L.eha, nocons Source SS df MS Number of obs = 123 F( 1, 122) = 8.35 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE =.5985 D.eha Coef. Sd. Err. P> [95% Conf. Inerval] eha L The -raio is equal o The 5% criical value for a coinegraing relaionship wih no inercep is 2.76 and so his falls wihin he rejecion region of he es. The null hypohesis of no coinegraion is rejeced a he 5% level of significance. To measure he one quarer response of GDP o economic shocks we esimae he vecor error correcion model by leas squares. regress D.aus L1.eha regress D.usa L1.eha The VEC model resuls he Ausralian GDP are: Source SS df MS Number of obs = 123 F( 1, 121) = 4.32 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = D.aus Coef. Sd. Err. P> [95% Conf. Inerval] eha L _cons The significan negaive coefficien on eˆ 1 indicaes ha Ausralian GDP responds o disequilibrium beween he U.S. and Ausralia. For he U.S.: Source SS df MS Number of obs = 123 F( 1, 121) = 0.62 Model Prob > F = Residual R-squared = Adj R-squared = Toal Roo MSE = D.usa Coef. Sd. Err. P> [95% Conf. Inerval] eha L _cons

6 The U.S. does no appear o respond o disequilibrium beween he wo economies; he -raio on 1 ˆ e is insignifican. These resuls suppor he idea ha economic condiions in Ausralia depend on hose in he U.S. more han condiions in he U.S. depend on Ausralia. In a simple model of wo economy rade, he U.S. is a large closed economy and Ausralia is a small open economy.