Stationary Vector Autoregressive Representation of Error Correction Models

Theoretical Economics Letters, 2012, 2, 152-156 http://dx.doi.org/10.4236/tel.2012.22027 Published Online May 2012 (http://www.SciRP.org/journal/tel)

Yun-Yeong Kim Department of International Trade, Dankook University, Yong-In, Korea Email: [email protected]

Received December 16, 2011; revised January 20, 2012; accepted January 28, 2012

ABSTRACT The paper introduces a stationary vector autoregressive (VAR) representation of the error correction model (ECM). This representation explicitly regards the cointegration error a dependent variable, making the direct implementation of standard dynamic analyses using standard VAR models possible, particularly with respect to the cointegration error. Of course, an ECM does not have an explicit VAR form, and thus, it is not convenient for conducting such dynamic analyses. In this regard, we transform the original nonstationary VAR model into a VAR model with the cointegration error and stationary variables. Finally, we employ the model to dynamically analyze the real exchange rate between the US dollar and the Japanese yen.

Keywords: VAR Model; Cointegration; Error Correction Model; Stationary VAR

1. Introduction the real exchange rate between the US dollar and the Japanese yen. The dynamic analysis of the cointegration error and sta- The rest of this paper is organized as follows. Section tionary variables in the short run is important as the long- 2 introduces the model and assumptions. Section 3 dis- run equilibrium for practitioners and policy makers. Of cusses on the stationary VAR model representation of course, such work is possible through the classical error ECMs. Section 4 presents the empirical results, and Sec- correction model (ECM), which was popularized by tion 5 concludes. Engle and Granger (1987). Furthermore, the persistence profiles of Pesaran and Shin (1996) and Hansen (2005) 2. Model and Assumptions are also useful alternatives for this purpose. Another possible option is to follow the vector autore- We consider the  -dimensionaland integrated of one gressive (VAR) approach of Sims (1980), which is now a VAR(p) process of zt given by standard method. In particular, such work may be possi- zztt 11 2 z t 2 ptp z t (1) ble if we transform the ECM into a VAR form of the or cointegration error and stationary variables, which would zz p1  z  , allow the more direct exploitation of the rich tools of tt1 i1 itit (2) VAR analyses (i.e., the Granger causality test, impulse where response analysis, variance decomposition, and optimal p  ,, I forecasting). Obviously, an ECM does not have an ex- i1 i  , plicit VAR form, and thus, it is not convenient for con- iii 12   p ducting such dynamic analyses.   Then the question is whether we can transform the and t is an 1 vector of an independently and iden- ECM into a finite-order VAR model of the cointegration tically distributed disturbance term with a finite variance   0 , where I denotes an  -dimensional identity error and stationary variables. In this regard, this paper  matrix and zz z. derives a finite-order VAR model with the cointegration ttt1 Further we assume the cointegration of Model (1) (e.g., error that is conformable to the ECM for the short-run Johansen, 1995) as: adjustment. In the VAR model, the cointegration error is regarded as a dependent variable, and VAR-type dy- Assumption namic analyses may be conducted directly. Finally, we employ the model to dynamically analyze We suppose    , where  and  are  r

matrices of the full-column rank r . * Ir 1 Note that Model (2) may be written in an ECM as  0'  Ir p1 (3) zutt 1 i1 i zti t  where 1 is   rr , 2 is rr , and   12,  . under Assumption 2.1, where uztt  . Proof of Proposition 3.1. We first write from th e Model (3) obviously consists of all stationary variables definition of z and u under Assumption 2.1. We are now in- *1 t t T T terested in the dynamic interaction between these variables z and u . In this regard, we may apply the IIrr0 ππ11 12 0 t t   (6) results of Pesaran and Shin (1996) and Hansen (2005). IIrrππ21 22  However Sims’ (1980) VAR approach is sometimes ππ   π convenient for practical reasons. For instance, the opti-  11 12 12  mal forecasting of the k-th-period-ahead cointegration ππ11 21 π 12  π 22 ππ12  22 error utk is executed mechanically in a VAR system. by using Thus, we now show that Model (3) may be represented I 0 as a stationary VAR model of a part of variables z 1 r t T     I and ut when there is an ECM. r 3. Stationary VAR Representation of ECM for the first equality, where π11 π12  To obtain a stationary VAR representation, assume that a . ππ21 22 given cointegration vector is normalized as   , Ir  of the rank r , where  is   r r. Then a confirm- We now represent the submatrices of  in the third able non-singular square matrix can be defined as equality of (6) by using  and  , where   

Ir 0 under Assumption 2.1. For this, note that T  .   Ir  ππ11 12   I Noteworthy is that the above lower triangular matrix ππ21 22  (7) T transforms the VAR variable zxyttt ,  into a 11  Ir  variable wt of xt and cointegration error ut .      I  . 22r Tzttt x, u wt because the matrix Φ can be written as Note that xt is the explanatory variable of yt in a   I (8) cointegration relation as yxt  tut.We then transform  the above VAR model (1) of z into a VAR model of t from the coefficient definition of (2) for the second the variable w . In particular, we multiply the T on the t equality. left of Model (1) and modify the VAR coefficients to get Then the third equality in Equation (7) directly implies the following VAR model: that ** wwtt11   ptp w et (4) π11  1   Ir (9) *1 where iiTT  for ip1, 2,, and eTtt  . π   (10) Note that Model (4) is an observationally equivalent 12 1 π     (11) form of Model (1) or (3) if t has a Gaussian distribu- 21 2 tion. However, we cannot determine the correct model by π    I . (12) simply analyzing observed data. 22 2 r Then it is helpful to write Model (4) as Consequently, if we plug the submatrices (9)-(12) into **p1 the last term in Equation (6), then we get following: wwtt1 i1  iti w et (5) 1) ππ11 12 1 IIrr 1 where *** * and **p . iii12   p i1 i 2) π12  1 * Then we may simply arrange  in Equation (5) as 3) ππ π  π follows: 11 21 12 22   1212             0

3.1. Proposition 4) ππ12 22 1  2 IIrr . Suppose that Assumption 2.1 holds. Then The above results 1)-4), together with the equality in

(6), proves the claimed result as VAR model of wt : I   ww  w  we. (16) ■ * r 1 tt11 2 t 2 ptpt  0   Ir Note the restrictions from (6). ■  p1  0 (17) Note that both the dependent and explanatory variables 1 are imposed on  from Theorem 3.2, and x does in (5), have the nonstationary variables xt and xt1 in p tp not appear in Equati on (13). If these restrictions (1 7) are wt and wt1 . Thus, we show how we may transform not exploited and the coefficients  p1 is estimated with Model (5) of wt into a VAR model of a purely station- 1  errors, then the efficiency of dynamic analyses by using ary variable wxuttt, . these estimated coefficients may be reduced. For instance, 3.2. Theorem if we provide forecasts using a VAR model, then forecasting mean squared error may be increased not by us- Suppose that Assumption 2.1 holds. Then ing restrictions (17). Campbell and Shiller (1987, Equa-

wwtt11  2 w t 2   p wtpet (13) tion (5)) used the system (13) without referring to the VAR model (1) of level variable z , the rank deficiency where wxu,  , t ti ti  ti  of matrix    in Assumption 2.1, and the restric-  tion of Theorem 3.2 as  p1  0 of the coefficient  . * ii for i,,p 1 2 , , 1 p i 12 Further, note that the vector m oving average model of ()-r  r 1 wt is defined as wLett from (13) if  L 11 ii i1 11,,  2 i  122 ,    is invertible, where L denotes a time-lag operator and p pp11 LI 1 L p L. This is conformable to the for i = 2, 3,···, p − 1 and p 12,   and p1 result in Hansen (2005, Corollary1). 1  0 . In the following example, we transform an ECM into a Proof of Theorem 3.2. Using Proposition 3.1, we first VAR model of stationary variables. rewrite Model (5) as

Ir 1 p1 * 3.3. Example wwtti1 i1 wtiet 0'  Ir For a VAR (2), an ECM representation can be given as or zutt 111 z tt (18) x  p1 t 1 * under Assumption 2.1. The second-order VAR represen- uwtiti1 i1 et. (14) utr  I  tation of wt is also possible as

The right hand side of (14) may be written as xxtt1 * 1   uett11    (19) p1 ii uxuue (15) uutt  Ir  1  ttititi1121i1 t defining from Model (14) by using Proposition 3.1. The right- hand side of (19) may be written as 1   . uxuutt11121   tt   2 e t (20)   Ir defining *  ,  conformably and Finally, the term in (15) may be written as 112 1 1 p1 i   . 21uxtti1  1i   Ir p1 ii11p i2 22uuti2 t pe t Finally, the term in (20) may be written as x 11t1 211122uxuetttt   12,  ut1 xx  (, )tt12 0, e . x 12 2 t p1 ii i1 ti uutt12  12,  2 i2 u ti Thus, Model (19) may be rewritten as a second order

p1 xtp VAR model of wt : 0,  e . 2 u t tp wtt 11w 2 w t 2et (21)

Thus, Model (14) may be rewritten as a p-th order where 11,,  2 and 220,   confor-

  ˆ mably, where wxttt,u. Thus xt2 does not wxuˆti ti , ˆ ti   where uˆti z ti. Thus we may appear in Equation (21). Thus, the ECM (18) is written consistently estimate coefficients i in Model (13) as  as a stationary VAR (2) model, as in (21). ■ i with variables wˆ ti by using the OLS, where the

Now any standard dynamic analysis is possible by us- variable xtp should be excluded from the regression.  ing Model (13). Note that avariable wxuti ti , ti  Finally, we may conduct standard dynamic analyses by for ip 0,1,2, , is known as long as the cointegration using the VAR model vector  (and thus uz  ) is known. Thus we ti ti   wwˆˆtt 11 2 wˆ t 2 ptp wˆ et may consistently estimate coeff icients i in Model ˆ (13) as i with variables wti , using ord inary least where square (OLS ) method. However re mind that the variable  p1 p 0, 2 . xtp should be excluded from this regression.   Then we may conduct standard dynamic analyses by In this case, we may still conduct standard dynamic using the VAR model analyses and draw inferences from stationary VAR ˆˆ ˆ analyses because ˆ may be considered as known  wwtt11  2wt2 pw tp  eˆt because of its rapid convergence. See Hamilton (1994) where for a nice review on this issue. ˆ p1 p 0, ˆ2 . 4. Application: An Impulse Response Suppose cointegration vector  , Ir is not known ˆ Analysis of the Real Exchange Rate and estimated as  ˆ, Ir with super-c onsistency as 1/2 n ˆp 0 [c.f., Johansen (1995)] where n is a The US is highly concerned about the potential exchange sample number. Then wti should be replaced by rate manipulation by East Asian countries for their trade

0.010 0.010 0.010 0.008 0.008 0.008

0.006 0.006 0.006

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-0.002 -0.002 -0.002

-0.004 -0.004 -0.004 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 * pttp pttp uptt 0.004 0.004 0.004 0.003 0.003 0.003

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-0.002 -0.002 -0.002 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 p p* p**p * tt tt uptt  0.05 0.05 0.05 0.04 0.04 0.04 0.03 0.03 0.03 0.02 0.02 0.02 0.01 0.01 0.01 0.00 0.00 0.00 -0.01 -0.01 -0.01

-0.02 -0.02 -0.02

-0.03 -0.03 -0.03 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 *  up tt  up tt  uu tt

Figure 1. Impulse response analysis for USA-JAPAN PPP (response to cholesky one S.D. innovations ± 2 S.E.).

surplus. Krugman and Baldwin (1987) provide a classic 5. Conclusion study of this issue. See also Clarida and J. Gali (1994). The ECM is clearly one of the most powerful methods in The logic behind this manipulation can be explained by econometrics. However, it does not have an explicit the law of one price (LOOP). Let p , p* and f de- t t t VAR form with the cointegration error as a dependent note the log of the price level in the US ( P ), the lo g of * variable. This paper introduces a stationary VAR repre- the Japanese price level ( P ), and the nomin al yen/dollar sentation of the ECM that explicitly regards the cointera- exchange rate ( F ), respect ively. The LOOP states that * tion error as a dependent variable, making the direct im- FP P 1 and th at the same goods should be sold at the plementation of standard dynamic analyses using sta- same prices both at home and in foreign countries, where tionary VAR models possible, particularly with respect prices are converted into local prices at prevailing ex- to the cointegration error. change rates. Here if the exchange rate F is manipu lated as F through state intervention, then i t is possible 6. Acknowledgements that FP* P 1 in the short run, where FPP* is the relative price o f the foreign good. Then the cheap er price I am grateful to Joon Y. Park, Byeongseon Seo and the of the home country good may induce a trade surplus. anonymous referee for their invaluable comments. All Then the question is whether any exchange rate or remaining errors are the author’s own. price shock would have a statistically meaningful effect on the real exchange rate (a cointegration error if the real REFERENCES exchange rate is I(0)). To address this question, we con- [1] J. Y. Campbell and R. J. Shiller, “Cointegration and Tests *  of Present Value Models,” Journal of Political Economy, struct a VAR model of wpputi t,,  t t in (13), Vol. 95, No. 5, 1987, pp. 1062-1088. *  doi:10.1086/261502 where uppftttt and  1, 1, 1 . Finally, we conduct an impul se response analysis on [2] R. Clarida and J. Gali, “Sources of Real Exchange Rate Fluctuations: How Important Are Nominal Shocks?” ut by using the Model (13). An ECM (3) does not have NBER Working Papers No. 4658, 1994. an explicit VAR form, and thus, it is not convenient for [3] R. F. Engle and C. W. J. Granger, “Co-Integration and conducting such impulse response analysis. For this, we Error Correction: Representation, Estimation, and Test- employ monthly data for the period from January 1998 to ing,” Econometrica, Vol. 55, No. 2, 1987, pp. 251-276. December 2008. We excluded the recent global financial doi:10.2307/1913236 crisis because of a possible structural break. [4] J. D. Hamilton, “Time Series Analysis,” Princeton Uni- * We conduct unit root tests for the variables pt , pt , versity Press, Princeton, 1994.

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