Johansen Test
A Johansen Test is a statistical test for testing cointegration of time series based on Dickey-Fuller Test.
- AKA: Johansen Cointegration Test.
- …
- Counter-Example(s)
- See: Error Correction Model, Statistics, Econometrics, Søren Johansen, Cointegration, Order of Integration, Time Series, Engle–Granger Test, Dickey–Fuller Test, Augmented Dickey–Fuller Test, Unit Root, Eigenvalue.
References
2016
- (Wikipedia, 2016) ⇒ https://en.wikipedia.org/wiki/Johansen_test Retrieved:2016-12-17.
- In statistics, the Johansen test, named after Søren Johansen, is a procedure for testing cointegration of several, say k, I(1) time series. This test permits more than one cointegrating relationship so is more generally applicable than the Engle–Granger test which is based on the Dickey–Fuller (or the augmented) test for unit roots in the residuals from a single (estimated) cointegrating relationship.
There are two types of Johansen test, either with trace or with eigenvalue, and the inferences might be a little bit different. The null hypothesis for the trace test is that the number of cointegration vectors is r=r*<k, vs. the alternative that r=k. Testing proceeds sequentially for r*=1,2,etc. and the first non-rejection of the null is taken as an estimate of r. The null hypothesis for the "maximum eigenvalue" test is as for the trace test but the alternative is r=r*+1 and, again, testing proceeds sequentially for r*=1,2,etc., with the first non-rejection used as an estimator for r.
Just like a unit root test, there can be a constant term, a trend term, both, or neither in the model. For a general VAR(p) model: : [math]\displaystyle{ X_{t}=\mu+\Phi D_{t}+\Pi_{p}X_{t-p}+\cdots+\Pi_{1}X_{t-1}+e_{t},\quad t=1,\dots,T }[/math] There are two possible specifications for error correction: that is, two VECM (vector error correction models):
1. The longrun VECM:
:: [math]\displaystyle{ \Delta X_t =\mu+\Phi D_{t}+\Pi X_{t-p}+\Gamma_{p-1}\Delta X_{t-p+1}+\cdots+\Gamma_{1}\Delta X_{t-1}+\varepsilon_t,\quad t=1,\dots,T }[/math] :where
:: [math]\displaystyle{ \Gamma_i = \Pi_1 + \cdots + \Pi_i - I,\quad i=1,\dots,p-1. \, }[/math] 2. The transitory VECM:
:: [math]\displaystyle{ \Delta X_{t}=\mu+\Phi D_{t}-\Gamma_{p-1}\Delta X_{t-p+1}-\cdots-\Gamma_{1}\Delta X_{t-1}+\Pi X_{t-1}+\varepsilon_{t},\quad t=1,\cdots,T }[/math] :where
:: [math]\displaystyle{ \Gamma_i = \left(\Pi_{i+1}+\cdots+\Pi_p\right),\quad i=1,\dots,p-1. \, }[/math] Be aware that the two are the same. In both VECM (Vector Error Correction Model), : [math]\displaystyle{ \Pi=\Pi_{1}+\cdots+\Pi_{p}-I. \, }[/math] Inferences are drawn on Π, and they will be the same, so is the explanatory power.
- In statistics, the Johansen test, named after Søren Johansen, is a procedure for testing cointegration of several, say k, I(1) time series. This test permits more than one cointegrating relationship so is more generally applicable than the Engle–Granger test which is based on the Dickey–Fuller (or the augmented) test for unit roots in the residuals from a single (estimated) cointegrating relationship.
2007
- (Osterholm & Hjalmarsson, 2007) ⇒ Österholm, Pär, and Erik Hjalmarsson. Testing for cointegration using the Johansen methodology when variables are near-integrated. No. 7-141. International Monetary Fund, 2007.
- ABSTRACT: We investigate the properties of Johansen's (1988, 1991) maximum eigenvalue and trace tests for cointegration under the empirically relevant situation of near-integrated variables. Using Monte Carlo techniques, we show that in a system with near-integrated variables, the probability of reaching an erroneous conclusion regarding the cointegrating rank of the system is generally substantially higher than the nominal size. The risk of concluding that completely unrelated series are cointegrated is therefore non-negligible. The spurious rejection rate can be reduced by performing additional tests of restrictions on the cointegrating vector(s), although it is still substantially larger than the nominal size.
1988
- (Johansen, 1988 ) ⇒ Søren Johansen (1988). “Statistical analysis of cointegration vectors.” In: Journal of economic dynamics and control, 12.2 (1988): 231-254. doi:10.1016/0165-1889(88)90041-3
- We consider a nonstationary vector autoregressive process which is integrated of order 1, and generated by i.i.d. Gaussian errors. We then derive the maximum likelihood estimator of the space of cointegration vectors and the likelihood ratio test of the hypothesis that it has a given number of dimensions. Further we test linear hypotheses about the cointegration vectors.
- The asymptotic distribution of these test statistics are found and the first is described by a natural multivariate version of the usual test for unit root in an autoregressive process, and the other is a χ2 test.