Johansen and Juselius's Tests for Cointegrating Relationships. In 1990,
econometricians Soren Johansen and Katarina Juselius of the University.
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Section 18.8 Multiple Cointegrating Relationships
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f there are multiple cointegrating relationships, there may be multiple error correction terms in the error correction specification. As noted earlier, multiple cointegrating relationships require more advanced techniques. We can extend the drunk and her dog example to illustrate multiple cointegrating relationships. In that example, there were error correction models for both the woman’s and the dog’s cointegrated meanderings, each containing the same cointegrating error correction variable. The relationships were Yt - Yt-1 = et + g1(Yt-1 - Zt-1) and Zt - Zt-1 = wt + g2(Yt-1 - Zt-1). Let’s add an inebriated husband to the story, along with a third stochastically trending variable, H1, that records the location of the husband. One way to cointegrate the locations of all three wanderers is to posit that the husband calls to the dog and listens for the dog’s bark, and that he and the dog adjust their locations to partially close the distance between them. In this scenario, the spouses do not communicate with one another. The error correction relationships then become Yt - Yt-1 = et + g1(Yt-1 - Zt-1), and Zt - Zt-1 = wt + g2(Yt-1 - Zt-1) + g3(Zt-1 - Ht-1), and Ht - Ht-1 = tt + g4(Zt-1 - Ht-1). EXT 7-11
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In general, there is no reason for one equation, here the dog’s, to adjust to deviations from both cointegrating relationships, whereas the other equations adjust to only deviations from one of the cointegrating relationships. Also, in general, the two cointegrating relationships may both involve all three variables. Thus, a more general statement of error correction equations among three variables with two cointegrating relationships would be Yt - Yt-1 = et + g1(Yt-1 - b Z1Zt-1 - b H1Ht-1) + g2(Yt-1 - b Z2Zt-1 - b H2Ht-1),
18.9
Zt - Zt-1 = wt + g3(Yt-1 - b Z1Zt-1 - b H1Ht-1) + g4(Yt-1 - b Z2Zt-1 - b H2Ht-1),
18.10
Ht - Ht-1 = tt + g5(Yt-1 - b Z1Zt-1 - b H1Ht-1) + g6(Yt-1 - b Z2Zt-1 - b H2Ht-1).
18.11
and
and
Notice that the same two cointegrating relationships appear in all the error correction equations. Notice, too, that the adjustment coefficients can differ from one equation to the next. In particular, some adjustment coefficients can be zero, which could lead, for example, to the specific arrangement of adjustments noted earlier for the woman, her dog, and her husband. A mathematical requirement for the two cointegrating relationships is that they be linearly independent of one another—they must be distinct relationships. It is possible for a single cointegrating relationship to hold all three variables together. In such a case, the cointegrating relationship must involve all three variables. A yet more general specification of the error correction model allows the dependent variables in Equations 18.9 through 18.11 to depend on the lagged changes of the dependent variables, to account for additional serial correlation, much as in the augmented Dickey–Fuller regression: Yt - Yt-1 = et + g1(Yt-1 - b Z1Zt-1 - b H1Ht-1) + g2(Yt-1 - b Z2Zt-1 - b H2Ht-1) + tYY ¢Yt-1 + tYZ ¢Zt-1 + tYH ¢Ht-1, and Zt - Zt-1 = wt + g3(Yti-1 - b Z1Zt-1 - b H1Ht-1) + g4(Yt-1 - b Z2Zt-1 - b H2Ht-1) + tZY ¢Yt-1 + tZZ ¢Zt-1 + tZH ¢Ht-1,
Multiple Cointegrating Relationships
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and Ht - Ht-1 = tt + g5(Yt-1 - b Z1Zt-1 - b H1Ht-1) + g6(Yt-1 - b Z2Zt-1 - b H2Ht-1) + tHY ¢Yt + tfZ ¢Zt + tHH ¢Ht-1. We can use these error correction equations to estimate the short-run dynamics of cointegrated variables. OLS consistently estimates these equations. Our usual OLS-based test statistics are valid asymptotically for inferences about the error correction equation.
Johansen and Juselius’s Tests for Cointegrating Relationships In 1990, econometricians Soren Johansen and Katarina Juselius of the University of Copenhagen devised an estimation and testing procedure for models with one or more cointegrating relationships.1 Their approach estimates all three error correction equations together, obtaining estimates of the long-run and short-run coefficients in one pass. The usual computer output for Johansen and Juselius’s approach provides tests of hypotheses about the number of cointegrating relationships. When there are three stochastically trending variables in the cointegrated regression, Johansen and Juselius’s method tests three hypotheses about the cointegrating relationships: 1. There are no cointegrating relationships; the regression is spurious. 2. There is at most one cointegrating relationship. 3. There are at most two cointegrating relationships. The number of such hypotheses tested corresponds directly to the number of cointegrating variables. The Johansen and Juselius strategy is to ask whether one estimated cointegrating relationship is a multiple of another or is a linear combination of some others. Johansen and Juselius offer two test statistics for each hypothesis. They call the first the trace statistic; they call the second the maximum eigen-value statistic. Both are usually reported by econometrics software packages that implement the Johansen and Juselius procedure. There is not much reason to prefer one over the other. Fortunately, they frequently lead to the same conclusion. Econometricians commonly attend to these tests sequentially. If none of the three hypotheses are rejected, we must worry that the regression is spurious. If we reject the first hypothesis only, we proceed assuming that there is only one cointegrating relationship. If we reject the first and second hypotheses, we proceed assuming that there are two cointegrating relationships. If we reject all three hypotheses, we conclude that none of the variables contain stochastic trends after all, because that is the only way there could be as many cointegrating relationships as variables.
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Table 18.15 contains the pertinent EViews output from conducting a Johansen analysis of the deficit and interest rate data in the file deficit.*** on this textbook’s Web site (www.aw-bc.com/murray), in which there are four stochastically trending variables. Both the trace test and the eigen-value test reject the hypothesis that there are no cointegrating relationships among these variables. Both tests also fail to reject the hypothesis that there is at most one cointegrating relationship. We can therefore conclude there is a single cointegrating relationship among these variables. The convention in Johansen and Juselius’s procedure is to report the cointegrating coefficients in a particular form. Equation 18.9 expresses the first cointegrating relationship as Yt - b YZZi - b YHHi = ei,
18.12
reflecting the fact that we have previously estimated the error correction variable by the residuals from OLS or dynamic OLS. Johansen and Juselius would report the corresponding cointegrating relationship as Yt + uYZZi + uYHHi = ei.
18.13
Thus, what we have treated as the dependent variable in the cointegrating relationship (Y), Johansen and Juselius assign a coefficient of 1.0. For the other variables, their u’s are the negative of our b ’s. The choice of which variable in a cointegrating relationship gets a coefficient of one is arbitrary. Multiplying both sides of Equation 18.12 or Equation 18.13 would result in a nontrending variable on both the left- and right-hand sides, and would represent the same underlying nontrending relationship among the variables. Conventional practice assigns a coefficient of one to some variable. Table 18.16 contains the estimates of the cointegrating relationship and the estimated standard errors on the coefficients that we obtain with the Johansen and Juselius estimator, with the signs changed for comparison with the regression output, along with the estimated coefficients from dynamic OLS. The estimates are quite similar. Is the comparison of estimates affected by the fact that the regression analysis included the change in per capita income and the Johansen and Juselius procedure did not? Not if our sample is large enough. A striking feature of the estimation of cointegrating relationships is that the estimates are unaffected asymptotically by omitting nontrending explanators. They are also unaffected asymptotically by measurement error bias. The unbounded variance of the stochastically trending variables swamps those biases in large samples.
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Table 18.15 Interest Rates and Deficits Revisited with the Methods of Johansen
and Juselius Sample(adjusted): 1956 1998 Included observations: 43 after adjusting endpoints Trend assumption: Linear deterministic trend Series: FYGT10 FYGT1 INFL USDEF Lags interval (in first differences): 1 to 2 Unrestricted Cointegration Rank Test
Hypothesized No. of CE(s) None** At most 1 At most 2 At most 3
Trace Statistic
5 Percent Critical Value
1 Percent Critical Value
72.87506 27.16235 11.29286 3.207041
47.21 29.68 15.41 3.76
54.46 35.65 20.04 6.65
* (**) denotes rejection of the hypothesis at the 5% (1%) level Trace test indicates 1 cointegrating equation(s) at both 5% and 1% levels
Hypothesized No. of CE(s)
Max-Eigen Statistic
5 Percent Critical Value
1 Percent Critical Value
45.71270 15.86949 8.085821 3.207041
27.07 20.97 14.07 3.76
32.24 25.52 18.63 6.65
None** At most 1 At most 2 At most 3
* (**) denotes rejection of the hypothesis at the 5% (1%) level Max-eigenvalue test indicates 1 cointegrating equation(s) at both 5% and 1% levels 1 Cointegrating Equation(s): Normalized cointegrating coefficients (std. err. in parentheses)
FYGT10 1.000000
FYGT1 0.814075 (0.02152)
INFL 0.059832 (0.02342)
Adjustment coefficients (std. err. in parentheses)
D(FYGT10) D(FYGT1) D(INFL) D(USDEF)
1.788307 (0.53989) 0.970915 (0.93561) 1.010442 (0.71246) 9.441568 (39.3954)
USDEF 0.004307 (0.00043)
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Table 18.16 Coefficient Estimates from Johansen and Juselius and from Dynamic OLS Johansen and Juselius
FYGT1 INFL USDEF
Coefficient
Standard Error
0.814 0.060 0.004
0.0215 0.0234 0.00004
Coefficient
Standard Error
0.813292 0.092467 0.003175
0.035775 0.038396 0.000568
Dynamic OLS
FYGT1 INFL USDEF
Concepts for Review Maximum eigen-value statistic Trace statistic
ENDNOTES 1. Soren Johansen and Katarina Juselius, “Maximum Likelihood Estimation and Inference on Cointegration—with Applications to the Demand for Money,” Oxford Bulletin of Economics and Statistics 52 (1990): 169–210.
