jcontest tests linear constraints on either
the error-correction speeds A or the cointegration
space spanned by B in the reduced-rank VEC(q)
model of y_{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.
[h,pValue,stat,cValue,mles] = jcontest(Y,r,test,Cons,Name,Value) performs
the Johansen constraint test on a data matrix Y with
additional options specified by one or more Name,Value pair
arguments.
Input Arguments
Y
numObs-by-numDims matrix
representing numObs observations of a numDims-dimensional
time series y_{t},
with the last observation the most recent. Observations containing NaN values
are removed. Initial values for lagged variables in VEC model estimation
are taken from the beginning of the data.
r
Scalar or vector of integers between 1 and numDims−1,
inclusive, specifying the common rank of A and B,
as inferred by jcitest.
test
String or cell vector of strings specifying the type of tests
to be performed. Values are:
ACon
Test linear constraints on A.
AVec
Test specific vectors in A.
BCon
Test linear constraints on B.
BVec
Test specific vectors in B.
Cons
Matrix or cell vector of matrices specifying test constraints.
For constraints on B, the number of rows in each
matrix, numDims1, is the number of dimensions in
the data, numDims, unless model is H*or H1*,
in which case numDims1 = numDims +
1 and constraints include the restricted deterministic term in the
model.
Test
Cons
ACon
numDims-by-numCons matrix R specifying numCons constraints
on A given by R'*A = 0. numCons must not exceed numDims − r.
AVec
numDims-by-numCons matrix
specifying numCons of the error-correction speed
vectors in A. numCons must not
exceed r.
BCon
numDims1-by-numCons matrix R specifying numCons constraints
on B given by R'*B = 0. numCons must not exceed numDims − r.
BVec
numDims1-by-numCons matrix
specifying numCons of the cointegrating vectors
in B. numCons must not exceed r.
Name-Value Pair Arguments
Specify optional comma-separated pairs of Name,Value arguments.
Name is the argument
name and Value is the corresponding
value. Name must appear
inside single quotes (' ').
You can specify several name and value pair
arguments in any order as Name1,Value1,...,NameN,ValueN.
'model'
String or cell vector of strings specifying the form of the
deterministic components of the VEC(q) model of y_{t}.
Values of model are those considered by Johansen [3]:
Value
Form of AB′y_{t−1} + DX
H2
AB′y_{t−1}.
There are no intercepts or trends in the cointegrating relations and
there are no trends in the data. This model is only appropriate if
all series have zero mean.
H1*
A(B′y_{t−1} + c_{0}).
There are intercepts in the cointegrating relations and there are
no trends in the data. This model is appropriate for nontrending data
with nonzero mean.
H1
A(B′y_{t−1} + c_{0})
+ c_{1}. There are intercepts
in the cointegrating relations and there are linear trends in the
data. This is a model of deterministic cointegration,
where the relations eliminate both stochastic and deterministic trends
in the data. This is the default value.
H*
A(B′y_{t−1} + c_{0} + d_{0}t)
+ c_{1}. There are intercepts
and linear trends in the cointegrating relations and there are linear
trends in the data. This is a model of stochastic cointegration,
where the relations eliminate stochastic but not deterministic trends
in the data.
H
A(B′y_{t−1} + c_{0} + d_{0}t)
+ c_{1} + d_{1}t.
There are intercepts and linear trends in the cointegrating relations
and there are quadratic trends in the data. Unless quadratic trends
are actually present in the data, this model may produce good in-sample
fits but poor out-of-sample forecasts.
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.
'lags'
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
'alpha'
Scalar or vector of nominal significance levels for the tests.
Values must be greater than zero and less than one. The default value
is 0.05.
Single-element 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.
Output Arguments
h
Vector of Boolean decisions for the tests, with length equal
to the number of tests. Values of h equal to 1 (true)
indicate rejection of the null that the constraints hold in favor
of the alternative that they do not. Values of h equal
to 0 (false) indicate a failure
to reject the null.
pValue
Vector of right-tail probabilities of the test statistics,
with length equal to the number of tests.
stat
Vector of test statistics, with length equal to the number of
tests. Statistics are likelihood ratios determined by the test.
cValue
Critical values for right-tail probabilities, with length equal
to the number of tests. The asymptotic distributions of the test statistics
are chi-square, with the degree-of-freedom parameter determined by
the test.
mles
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 reduced-rank 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-by-r 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 all-zero 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 finite-sample
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 vectovar.
[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. Likelihood-Based
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.