Johansen constraint test
[h,pValue,stat,cValue,mles]
= jcontest(Y,r,test,Cons)
[h,pValue,stat,cValue,mles] = jcontest(Y,r,test,Cons,Name,Value)
jcontest
tests linear constraints on either
the errorcorrection speeds A or the cointegration
space spanned by B in the reducedrank VEC(q)
model of y_{t}:
$$\Delta {y}_{t}=A{B}^{\prime}{y}_{t1}+{B}_{1}\Delta {y}_{t1}+\dots +{B}_{q}\Delta {y}_{tq}+DX+{\epsilon}_{t}.$$
Null hypotheses specifying constraints on A or B are tested against the alternative H(r) of cointegration rank less than or equal to r, without the constraints. The tests also produce maximum likelihood estimates of the parameters in the VEC(q) model, subject to the constraints.
[
performs
the Johansen constraint test on a data matrix h
,pValue
,stat
,cValue
,mles
]
= jcontest(Y
,r
,test
,Cons
)Y
.
[
performs
the Johansen constraint test on a data matrix h
,pValue
,stat
,cValue
,mles
] = jcontest(Y
,r
,test
,Cons
,Name,Value
)Y
with
additional options specified by one or more Name,Value
pair
arguments.

 

Scalar or vector of integers between 1 and  

Character vector, such as
 

Matrix or cell vector of matrices specifying test constraints.
For constraints on B, the number of rows in each
matrix,

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
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  

Scalar or vector of nominal significance levels for the tests.
Values must be greater than zero and less than one. The default value
is 
Singleelement values for inputs are expanded to the length of any vector value (the number of tests). Vector values must have equal length. If any value is a row vector, all outputs are row vectors.

Vector of Boolean decisions for the tests, with length equal
to the number of tests. Values of  

Vector of righttail probabilities of the test statistics, with length equal to the number of tests.  

Vector of test statistics, with length equal to the number of tests. Statistics are likelihood ratios determined by the test.  

Critical values for righttail probabilities, with length equal to the number of tests. The asymptotic distributions of the test statistics are chisquare, with the degreeoffreedom parameter determined by the test.  

Structure of maximum likelihood estimates associated with the VEC(q) model of y_{t}, subject to the constraints. Each structure has the following fields:

The parameters A and B in
the reducedrank VEC(q) model are not uniquely
identified. jcontest
identifies B using
the methods in [3], depending on the test.
When constructing constraints, interpret the rows
and columns of the numDims
byr matrices A and B as
follows:
Row i of A contains the adjustment speeds of variable y_{i} to disequilibrium in each of the r cointegrating relations.
Column j of A contains
the adjustment speeds of each of the numDims
variables
to disequilibrium in the jth
cointegrating relation.
Row i of B contains the coefficients of variable y_{i} in each of the r cointegrating relations.
Column j of B contains
the coefficients of each numDims
variable in the jth
cointegrating relation.
Tests on B answer questions about the space of cointegrating relations. Tests on A answer questions about common driving forces in the system. For example, an allzero row in A indicates a variable that is weakly exogenous with respect to the coefficients in B. Such a variable might affect other variables, but it does not adjust to disequilibrium in the cointegrating relations. Similarly, a standard unit vector column in A indicates a variable that is exclusively adjusting to disequilibrium in a particular cointegrating relation.
Constraints matrices R
satisfying R′A = 0 or R′B = 0 are equivalent to A = Hφ or B = Hφ,
where H is the orthogonal complement of R (null(R')
)
and φ is a vector of free parameters.
jcontest
compares finitesample
statistics to asymptotic critical values, and tests can show significant
size distortions for small samples. See [2]. Larger
samples lead to more reliable inferences.
To convert VEC(q) model parameters
in the mles
output to vector autoregressive (VAR)
model parameters, use the utility vec2var
.
[1] Hamilton, J. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[2] Haug, A. “Testing Linear Restrictions on Cointegrating Vectors: Sizes and Powers of Wald Tests in Finite Samples.” Econometric Theory. v. 18, 2002, pp. 505–524.
[3] Johansen, S. LikelihoodBased Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.
[4] Juselius, K. The Cointegrated VAR Model. Oxford: Oxford University Press, 2006.
[5] Morin, N. “Likelihood Ratio Tests on Cointegrating Vectors, Disequilibrium Adjustment Vectors, and their Orthogonal Complements.” European Journal of Pure and Applied Mathematics. v. 3, 2010, pp. 541–571.