jcitest
Johansen cointegration test
Syntax
[h,pValue,stat,cValue,mles]
= jcitest(Y)
[h,pValue,stat,cValue,mles]
= jcitest(Y,Name,Value)
Description
Johansen tests assess the null hypothesis H(r) of
cointegration rank less than or equal to r among
the numDims
dimensional time series in Y
against
alternatives H(numDims
) (trace
test)
or H(r+1) (maxeig
test).
The tests also produce maximum likelihood estimates of the parameters
in a vector errorcorrection (VEC) model of the cointegrated series.
[
performs the Johansen
cointegration test on a data matrix h
,pValue
,stat
,cValue
,mles
]
= jcitest(Y
)Y
.
[
performs
the Johansen cointegration test on a data matrix h
,pValue
,stat
,cValue
,mles
]
= jcitest(Y
,Name,Value
)Y
with
additional options specified by one or more Name,Value
pair
arguments.
Input Arguments


NameValue Arguments
Specify optional
commaseparated pairs of Name,Value
arguments. Name
is
the argument name and Value
is the corresponding value.
Name
must appear inside quotes. You can specify several name and value
pair arguments in any order as
Name1,Value1,...,NameN,ValueN
.

Character vector, such as $$\Delta {y}_{t}=C{y}_{t1}+{B}_{1}\Delta {y}_{t1}+\dots +{B}_{q}\Delta {y}_{tq}+DX+{\epsilon}_{t}$$ If r < Values of
Deterministic terms outside of the cointegrating relations, c_{1} and d_{1}, are identified by projecting constant and linear regression coefficients, respectively, onto the orthogonal complement of A.  

Scalar or vector of nonnegative integers indicating the number q of lagged differences in the VEC(q) model of y_{t}. Lagging and differencing a time series reduce the sample size. Absent any presample values, if y_{t} is defined for t = 1:N, then the lagged series y_{t−k} is defined for t = k + 1:N. Differencing reduces the time base to k+2:N. With q lagged differences, the common time base is q+2:N and the effective sample size is T = N−(q+1). Default: 0  

Character vector, such as
 

Scalar or vector of nominal significance levels for the tests. Values must be between 0.001 and 0.999. Default: 0.05  

Character vector, such as
Character vectors values are expanded to the length of any vector value (the number of tests). Vector values must have equal length. 
Output Arguments

Rows of Values of  

Rows of  

Rows of  

Rows of  

Rows of 
Examples
Algorithms
If
jcitest
fails to reject the null of cointegration rank r = 0, the inference is that the errorcorrection coefficient C is zero, and the VEC(q) model reduces to a standard VAR(q) model in first differences. Ifjcitest
rejects all cointegration ranks r less thannumDims
, the inference is that C has full rank, and y_{t} is stationary in levels.The parameters A and B in the reducedrank VEC(q) model are not uniquely identified, though their product C = AB′ is.
jcitest
constructsB
=V
(:,1:r) using the orthonormal eigenvectorsV
returned byeig
, then renormalizes so thatV'*S11*V = I
, as in [3].To test linear constraints on the errorcorrection speeds A and the space of cointegrating relations spanned by B, use
jcontest
.Time series in
Y
might be stationary in levels or first differences (i.e., I(0) or I(1)). Rather than pretesting series for unit roots (using, e.g.,adftest
,pptest
,kpsstest
, orlmctest
), the Johansen procedure formulates the question within the model. An I(0) series is associated with a standard unit vector in the space of cointegrating relations, and its presence can be tested usingjcontest
.To convert VEC(q) model parameters in the
mles
output to VAR(q+1) model parameters, usevec2var
.Deterministic cointegration, where cointegrating relations, perhaps with an intercept, produce stationary series, is the traditional sense of cointegration introduced by Engle and Granger [1] (see
egcitest
). Stochastic cointegration, where cointegrating relations produce trendstationary series (that is,d0
is nonzero), extends the definition of cointegration to accommodate a greater variety of economic series.Unless higherorder trends are actually present in the data, models with fewer restrictions can produce good insample fits, but poor outofsample forecasts.
References
[1] Engle, R. F. and C. W. J. Granger. "CoIntegration and ErrorCorrection: Representation, Estimation, and Testing." Econometrica. v. 55, 1987, pp. 251–276.
[2] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[3] Johansen, S. LikelihoodBased Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.
[4] MacKinnon, J. G., A. A. Haug, and L. Michelis. “Numerical Distribution Functions of Likelihood Ratio Tests for Cointegration.” Journal of Applied Econometrics. v. 14, 1999, pp. 563–577.
[5] Turner, P. M. “Testing for Cointegration Using the Johansen Approach: Are We Using the Correct Critical Values?” Journal of Applied Econometrics. v. 24, 2009, pp. 825–831.