# Documentation

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# jcontest

Johansen constraint test

## Syntax

[h,pValue,stat,cValue,mles] = jcontest(Y,r,test,Cons)
[h,pValue,stat,cValue,mles] = jcontest(Y,r,test,Cons,Name,Value)

## Description

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 yt:

$\Delta {y}_{t}=A{B}^{\prime }{y}_{t-1}+{B}_{1}\Delta {y}_{t-1}+\dots +{B}_{q}\Delta {y}_{t-q}+DX+{\epsilon }_{t}.$

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.

[h,pValue,stat,cValue,mles] = jcontest(Y,r,test,Cons) performs the Johansen constraint test on a data matrix Y.

[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 yt, 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

Character vector, such as 'ACon', or cell vector of character vectors 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'

Character vector, such as 'H2', or cell vector of character vectors specifying the form of the deterministic components of the VEC(q) model of yt. Values of model are those considered by Johansen [3]:

ValueForm of Cyt−1 + DX
'H2'

AB´yt−1. There are no intercepts or trends in the cointegrated series and there are no deterministic trends in the levels of the data.

'H1*'

A(B´yt−1+c0). There are intercepts in the cointegrated series and there are no deterministic trends in the levels of the data.

'H1'

A(B´yt−1+c0)+c1. There are intercepts in the cointegrated series and there are deterministic linear trends in the levels of the data. This is the default value.

'H*'A(B´yt−1+c0+d0t)+c1. There are intercepts and linear trends in the cointegrated series and there are deterministic linear trends in the levels of the data.
'H'A(B´yt−1+c0+d0t)+c1+d1t. There are intercepts and linear trends in the cointegrated series and there are deterministic quadratic trends in the levels of the data.

Deterministic terms outside of the cointegrating relations, c1 and d1, 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 yt.

Lagging and differencing a time series reduce the sample size. Absent any presample values, if yt is defined for t  = 1:N, then the lagged series ytk 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 yt, subject to the constraints. Each structure has the following fields:

 paramNames Cell vector of parameter names, of the form:{A, B, B1,...,Bq, c0, d0, c1, d1}Elements depend on the values of lags and model. paramVals Structure of parameter estimates with field names corresponding to the parameter names in paramNames. res T-by-numDims matrix of residuals, where T is the effective sample size, obtained by fitting the VEC(q) model of y(t) to the input data. EstCov Estimated covariance Q of the innovations process εt. rLL Restricted loglikelihood of Y under the null. uLL Unrestricted loglikelihood of Y under the alternative. dof Degrees of freedom of the asymptotic chi-square distribution of the test statistic.

## Examples

collapse all

Load data on Australian and U.S. prices:

p1 = DataTable.PAU; % Log Australian Consumer Price Index
p2 = DataTable.PUS; % Log U.S. Consumer Price Index
s12 = DataTable.EXCH; % Log AUD/USD Exchange Rate
Y = [p1 p2 s12];
plot(dates,Y)
datetick('x','yyyy')
legend(series(1:3),'Location','Best')
grid on

Pretest the individual series for stationarity:

[h0,pValue0] = jcontest(Y,1,'BVec',{[1 0 0]',[0 1 0]',[0 0 1]'})
h0 =

1x3 logical array

1   1   0

pValue0 =

0.0000    0.0000    0.0657

Test for cointegration:

[h1,pValue1] = jcitest(Y)
************************
Results Summary (Test 1)

Data: Y
Effective sample size: 76
Model: H1
Lags: 0
Statistic: trace
Significance level: 0.05

r  h  stat      cValue   pValue   eigVal
----------------------------------------
0  1  60.3393   29.7976  0.0010   0.4687
1  0  12.2749   15.4948  0.1446   0.1157
2  0  2.9315    3.8415   0.0869   0.0378

h1 =

1x3 table

r0       r1       r2
_____    _____    _____

t1    true     false    false

pValue1 =

1x3 table

r0        r1          r2
_____    _______    ________

t1    0.001    0.14455    0.086906

Test for purchasing power parity ():

[h2,pValue2] = jcontest(Y,1,'BCon',[1 -1 -1]')
h2 =

logical

0

pValue2 =

0.0540

## Algorithms

• 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 yi 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 yi 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 RA =  0 or RB =  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 vec2var.

## References

[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.