In economics the period of "long-run" often signifies the length of time within which transient fluctuations disappear, and a system comes back to an equilibrium state (or path). Among some interesting cases of long run analysis, the concept of cointegration is a relatively new concept of the long run equilibrium. This paper discusses how to determine the length of the long-run period for a cointegration relation. In an application to a consumption-income relation for three countries. U.S., Germany and Japan, we found that the length of the long-run period for the relation for these countries is about two to three years

Kim, JaeYoung

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Measuring the LeJlgth of Period for the Long-Run E~quilibrium in a Cointegration Relation Jae-Young Kim* In economics the period of “ long-run" often signifies the length of time within which transient fluctuations disappear, and a system comes back to ;페1 equilibrium state (01' path). Among some interesting cases of long run anaJysis , the concept of cointegration is a relatively new concept of the long run equilibrium. This paper discusses how to determine the length of the long-run period for a cointegration relation. In an applica디on to a consumption-income relation for three countries , U.S. ‘ Germany and Japan , we found that the length of the long-run period for the relation for these countries is about two to three years Keywords: Long run equilibrium , Cointegration , Consump디on-income relation. JEL Classifìcation: C 1, C22 , C!", 1. Introduction The concept of cointegration defined by Engle and Granger (1 987) has become a useful concept for anaψzing many linear dynamic systems in economics. For a set of variables the existence of a cointegration relation implies that there exists a long-run equilib-rium that ties the series together. Thus , although disturbances to *Associate Professor, School of Economics , Seoul National University , San 56 • 1 Shilim-dong Kwanak-gu , Seoul 151-742 , Korea , (Tel) +82-2-880-6390. (E-mail)jykim017@snu.ac.kr. The author thanks the editor and anonymous referees for valuable comments. FinanciaJ supports from Seoul National University Foundation and a BK21 Program of Korea Research Foundation are greatly appreciated. Thanks aJso go to Joo-Hee Oh for assistance in editing the final draft of the paper in a word file. [Seoul Journal of Economics 2003 , Vol. 1ß, No. 1 ) 72 SEOUL ]OURNAL OF ECONOMICS in버띠du려 variable(s) have permanent effects. they have only temporaπ effects on the system as a whole. When a shock is given to a system. the system deviates from the initial equilibrium However. after a while. impacts of the shock are absorbed in the system. 없ld 야le system eventually comes back to the initial equilibrium. An interes디ng ques디on we want to study in this paper is how long it takes for a system to return to the long-run equilibrium after a shock disturbs the system. A dynamic system seldom stays in an equilibrium. if any. even for a very short period of 디me because shocks are given to the system at each period of time and they survive for a while. An equilibrium error is usual1y an accumulation of current and past shocks to the system. Our study in this paper is to find how long it takes for shocks given at one period of time to disappear. that is. how long for shocks given at one period survive. We consider the consump디on-spending model analyzed by Davidson. Hendry. Srba. and Yeo (1 978) as an example of a cointegrated system. For data from three countries. Germany. Japan and U.S .. we find that a shock to the system is shown to survive for nine. ten and twelve quarters. respec디vely. in the consump디on-spending model. This implies that for the system of consump디on-income relation it takes about two to three years to return to the state of long-run equilibrium after a shock disturbs the system. Our discussion in this paper goes as follows. Section II discusses the methodology of our an머ysis. In section III we apply the method to the model of consumption-spen띠ng and discuss implications of our results. Mathematical proofs are provided in the appendix. 11. Methodology Let x( be an n-vector of 1(1) variables. Suppose that there exists a cointegration relation among the variables. Then. there exists an (n x 1) vector α such that g ’x(=u( (1 ) is a stationary process. A cointegration relation such as (1) is often interpreted as a 10ng-run equilibrium re1ation among the variab1es. MEA8URING THE LENGTH OF PERIOD FOR THE LONG-RUN 73 That is. there exists a stable relation among the variables over a long period of time although there exist short-run f1uctuations around the stable long-run relation. The short-run f1uctuation Ut can be interpreted as an equilibrium error to the system. Usually the errof term Ut contains shocks given in current and past periods Often the stationary error term I1t is assumed to fo11ow an AR process: 1 Ut 二 a+ØlUI → 1 + Ø2Ut-2 + ". + øpUt • p+Et (2) where EI~ i. i ， d ， (O. σ2ε) and E( E(4) < ∞: all roots of 1 - ØlZ _j_ Ø2Z2 + ’ .. +øpzP =o lie outsicle the unit circle. The stationary error term 111 can be written in MA(∞) representation as Ut 二 μ +Et+ ψ1 Et-l+ ψ'2 Et-2+" 씨 { This MA representation implies that Ut is an accumulation of the current and past shocks E ’S. The coefficient ψ~ measures the level of impacts of Et -5 on Ut for s == 0. 1. , ••• ∞ More formally. the coefficient qrs has the interpretation qr<= ~Ut+s_ •_----aEt that is. 봐 evaluates the consequences of a one unit increase in the shocks of the past S period on the current equilibrium error. 1n practice we can get the e딩timate of 11's by conver디ng the estimate of AR coefficient ø’s. Such an estimate of qrs , 딴 만(ø). has the fo11owing sample proper양 where ø is the LS estimator of ø: Lemma 1 Assume that Ut 밍ld Et sa디sfY the conditions above in (2) with all roots of the equa디on 1-- ØlZ+ øzz2 + … 얘PZ!' = 0 lying outside the unit circle. Then , the estimator of ψs. ψ~~=Ws( ø). is such I For example. in the augmented Dickey-Fuller test for cointegration we assume that the error term is a stationary AR process. 74 SEOUL ]OURNAL OF ECONOMICS that 、/π1fs -ψ~)으~N(O ， σ2 (Il1S)) (4) where 02(ψ~) =Gs( 02ε 0Q-l)Gs ’ ， for Q=E( l/t νt’)， l/t’= (1 Ut-lUt-2'" Ur -p) and {Gs,5 = 1 ,2 ,"'} is obtained by iterating on Gs 二 (0만 -1ψs ←2'"만←p) + ø 1Gs-1 + ø2Gs • 2+'" + øpGs •p (5) where Gs = 0 for 5 드 0 ， ψ0= 1 and ψ's=O for 5<0. Now , we are interested in the following hypothesis Ho:ψ~=O for 5> r and ψ~""O for 5 르 r , (6) which implies that the shock E survives for r periods and after that it either disappears or is absorbed into the system.2 We can decide that the hypothesis is true for a given r if V'T‘jls .Jf'l's |一τ=주二=-1 드 Ck for 5> r and 1-τ속二~ 1 >Ck for 5 드 r. 、! 5~(ψ~) 、!5‘{ψ~)(7) for a given value C k , where 52(ψ~) is the OLS estimate of σ2(만) . From (4) we know that Ck can be asymptotically approximated by a critical value of the t distribution. In many cases of practice the disturbance Ur is not observed. When Ur is not observed , one may want to use an estimate of it , for example , Ûr= â ’Xr where â is an estimate of α. Statistical behavior of Ûr , however , is different from that of Ur although 바 converges to U( as s없nple size gets larger. 3 Thus , the above asymptotic result may not be a good approximate for finite sample analysis when Ûr is used in place of Ur. In such a case we may apply a re-sampling method such as the bootstrap method to get a ιThe hypotheses in (6) can be generalized to Ho: ['I's [ 르 E for s> r and ['I's[ > E for s 드 r for a smal1 E >0. In particular. it would be more approp디ate to employ this generεùized setup when one considers conversion of the AR coefficients to MA coefficients to get 만. Our hypotheses in (6) appro잉mate this generalized setup with an infinitesimal value of E. This idea came out of a comment from one of referees. 1 appreciate the referee for it ‘ 3Note that û， 二 u，+( δ- α)'x， 밍ld â- α =Op(T 1), x ,=Op(e I2 ) for a coin-tegrating vector α and a vector of I( l) variables x, without drift. MEASURING THE LENGTH OF PERIOD FOR THE LONG-RUN 75 critical value Ck. 111. An Example: A Consumption-Income Relation We now consider an example of a cointegration relation to find how long it takes for the system to retum to its inìtial equilibrium after a shock disturbs the system. The example we consider in this section is a cointegration regression formed by two variables. consump디on and income , denoted by c and y , respec디vely: c(= β0+βlν(+U( (8) The regression (8) is often studied in the literature of cointegration , for example , Davidson , Hendrγ， Srb장， and Yeo (1 978). We consider data from three countries , U.S. , Japan. and Germany. Data from U .S. are quarterly time series on real personal consumption (c) and real persomù disposable income (y) from 1947: 1 through 1994: 1. Data from the other countries are quarterly observations on real private consump디on and real national income. By an augmented Dickey-Fuller test the unit root null is not rejected at 5% level for both y 없ld c for data of all three countries. 4 Also. for both an augmented Dickey-Fuller test and Phillips and Ouliaris (199이 tests of cointegration we reject 납le null of no cointegra디on at 5% level for the consumption-income relation. Tables 3.1 (a)-(c) provide values of the statistic t(s)~. I νTψJ 、 S2(WS ) I for 5 = 1. 2 , ’ ‘ 16 for data of the three countries. Figures ~3.1 (a)-(c) exhibit the information in Tables 3.1 (a)-(c) graphically From Table 3.1(a) and Figure 3.1(a) we find that for U.S. data the period r with 11 < r < 12 satisfies the condition (7) with the 5% critical value 1.96. This result implies that for the U.S. case it takes about 12 quarters for the consumption-income relation 10 return to its initial equilibríum after a shock disturbs the system. Also. from Tables 3.1 (b)-(c) and Fïgures 3.1 (b)-(c) we find that the periods for the relation to return to its ínitial equilibríum are nine and ten quarters , respectively. for Germany and Japan. 'The res비ts of tests for a unit root ancl cointegration are available from the author upon request. 9 8 7 6 !(S)§ 3 2 0 | rττ 鐵 |1 3 5 7 9 11 13 15 s FIGURE 3.1삐 MEASURING TIIE LENGTII OF PERIOD FOR THE LONG-RUN 77 TABLE 3.1(c) t(S)-VALUES. JAPAN s 2 3 4 5 6 7 8 t(s) 7.67 4.44 4.79 4.72 3.67 3.21 2.76 2.36 S 9 10 11 12 13 14 15 16 t(s) 2.10 1.87 1.68 1.53 1.40 1.29 1.20 1. 12 9 8 7 6 I ( s)핵ν\ |--t(인 | - - - 5% c.V 3 2 0 3 5 7 9 11 13 15 s FIGURE 3 . l(c) It would be interesting to explore economic reasons why there exists such difference across countries in the length of the period for the long-run equilib디um. 5 π1is ìs 없1 impor떠nt subject to study. However. it is beyond the scope of 암11s paper and is set aside for future research. The difTerence might have something to do with clifferences in economic fundamentals or economic policies across the countries. Appendix A: Proof of Lemma 1: The [1이lowing proof ìs based on several lemmas and results summarized in Hamilton (1994). Some of those lemmas or results are 려so available elsewhere. for example White (1984). Lemma A.l: Assume that Ul satisfies conditions in Lemma 1. Le t π= (c‘ q) l ..... q)p) 없1d 7rr be the OLS estimator of π based on the sample 이‘ size T . Also. let δ2E =T- 1εLl ε? where εt is the OU‘3 5A referee suggested to include this issue to better motivate the work (lf this paper. 1 appreciate the referee for it. 78 SEOUL ]OURNAL OF ECONOMICS estimate of εt. Then. it is true that (a) y- l 2: ;~1 l/[ νII휴Q; (b) 차「4π; (c) (j 2" J!. σ2ε (d) .Jn 7rr- π)므~N(O. σ2ε 0Q-l). Proof of Lemma A.l: (a) 깐le condition of 려1 roots of 1- Ø1Z+ØzZ2+ ... +øpZ" =0 lying outside the unit circle ensures that the MA(∞) representation is absolutely summab1e. Then , it follows that u[ ìs ergo이c for the fìrst moment and for the second moment. Then , the conclusion of (a) fo11ows. (b) 1t can be shown that Y t 三 ν[Et satisfìes condìtions for the fo11O\찌ng law of 1arge numbers for L1-mi잉ngales by Andrews (1988): Lemma A.2: Let IYtl be an L 1-mixinga1e. 1f IYtl is uniformly integrable and there exists a choice for ICt) such that !imT-1 2:;~1 c[ < ∞， men T I ZLlKAo l →、iBy Lemma A.2 the conclusion of (b) follows since 7r 二 π + (T- 1 2: ;~1νt νt ，)-l(T-l 2: ;~l l/t Et). (c) It follows from the conclusion of (b) and a law of large numbers. (d) We will use the fl이lowing result: Lemma A.3: (Cor이1ary 5.25 of White (1 984 , p. 13이). Let IYtι1 be a martingale difference sequence with Yt 二 (1 /n 2:;~lYt. Suppose that (a) E(Y?) 二 σr2>O 뼈th (1 /n 2:εl σ12→ σ2>0 (b) EIYtl r< ∞ for some r>2 and 머1 t, and (c) (1 /n 2:εlyt2→ (52. Then 、/TY，→N(O ， (5 2). Notice that 、/피 7[- π)=(T 1당~I l/[ νt’)← I(T←l/2zr=lν[ Etl. Also , notice that Y[ = l/t E[ is a mar디ngale difference sequence with fìnite fourth moments and E(YtYt') = σ2，， 0Q. It is not hard to show that MEASURING THE LENGTH OF PERIOD FOR THE LONG-RUN ì9 T lZLlYtY! ’E, σ2ε 0Q. Then , it follows from Lemma A.3 that T→ 1/2 L:J~IY(요N(O ，( σZε0Qll. Then, since TlzLlνt “LQ as is 잉hown in (a) , the conclusion of (d) follows. Now , as the final step we utilíze the following result: Lemma A.4: (Corollaη 5.25 of White (1 984, p. 13이). Let {Xtl be a sequence of random (n x 1) vectors such that 、l T(xT-c)으~X and let g:Rn• Rm have continuous first derivatives with G denoting the (m >( n) matrix of derivatives evaluated at c:G 二 얘/ àX'’ Ix~c Then , 、1 T(g(XT)-- g(c))므~GX. Now, let XT = 1cT , c 二 π ， 없ld X-N(O , σ2， 0Q-l). Also , let gs 二 ψ~ so that Gs = à만( π)/ à π’. Then we have 、/πIj/s ← ψ~)14N(O ， Gs( σ2E@Q l)Gs ’). The equiv벼ence of Gs= àψ~( π)/ àπ ’ and Gs in (5) can be shown by a long but straightf，야ward algebra. See Hamilton (1994 , pp. 344-8). (Received 22 November 2003: Revi.s ed 20 Februarν 2004) References Davidson. James E. H .. Hendry, David F. , Srba. Frank. and Yeo , Stephen. “ Econometric Modelling of the Aggregate Time-Series Relationship between Consumer's Expenditure and Income in the United Kingdom." The Economic Journal 88 (No. 352 1978): 661-92 Engle. Robert F .. and Granger. C. W. J. “ Cointegration and Error Correction: Representation , Estimation. and Testing." Eco-nometrica 55 (No. 2 1987): 251-76 Hami1ton , J. Time Series Analysis. Princeton. NJ: Princeton U. Press. 1994. Phillips , P. C. B. , and Ouliaris. “Asympto디c Proper디es of Residual Based Tests for Cointegration." Econometrica 58 (No. 1 1990): 80 SEOUL }OURNAL OF ECONOMICS 165-93. White. H. Asymptotics Theory Jor Econometπcians. Academic Press 1984. 

Measuring the Length of Period for the Long-Run Equilibrium in a Cointegration Relation

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Measuring the Length of Period for the Long-Run Equilibrium in a Cointegration Relation

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