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Engle-Granger cointegration test
[h,pValue,stat,cValue,reg1,reg2]
= egcitest(Y)
[h,pValue,stat,cValue,reg1,reg2]
= egcitest(Y,Name,Value)
Engle-Granger tests assess the null hypothesis of no cointegration among the time series in Y. The test regresses Y(:,1) on Y(:,2:end), then tests the residuals for a unit root.
[h,pValue,stat,cValue,reg1,reg2] = egcitest(Y) performs the Engle-Granger test on a data matrix Y.
[h,pValue,stat,cValue,reg1,reg2] = egcitest(Y,Name,Value) performs the Engle-Granger test on a data matrix Y with additional options specified by one or more 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.
'creg' |
String or cell vector of strings indicating the form of the cointegrating regression, where y_{1} = Y(:,1) is regressed on Y_{2} = Y(:,2:end)and optional deterministic terms in X: y_{1} = Xa + Y_{2}b + ε Values are
Default: c |
'cvec' |
Vector or cell vector of vectors containing coefficients [a;b] to be held fixed in the cointegrating regression. The length of a is 0, 1, 2 or 3, depending on creg, with coefficient order: constant, linear trend, quadratic trend. The length of b is numDims − 1. It is assumed that the coefficient of y_{1} = Y(:,1) has been normalized to 1. NaN values indicate coefficients to be estimated. If cvec is completely specified (no NaN values), no cointegrating regression is performed. Default: Completely unspecified cointegrating vector (all NaN values). |
'rreg' |
String or cell vector of strings indicating the form of the residual regression. Values are ADF, for an augmented Dickey-Fuller test of residuals from the cointegrating regression, or PP, for a Phillips-Perron test. Test statistics are computed by calling adftest and pptest with the model parameter set to AR, assuming data have been demeaned or detrended, as necessary, in the cointegrating regression. Default: ADF |
'lags' |
Scalar or vector of nonnegative integers indicating the number of lags used in the residual regression. The meaning of the parameter depends on the value of rreg (see the documentation for the lags parameter in adftest and pptest). Default: 0 |
'test' |
String or cell vector of strings indicating the type of test statistic computed from the residual regression. Values are t1 (a "τ test") or t2 (a "z test"). The meaning of the parameter depends on the value of rreg (see the documentation for the test parameter in adftest and pptest). Default: t1 |
'alpha' |
Scalar or vector of nominal significance levels for the tests. Values must be between 0.001 and 0.999. Default: 0.05 |
Single-element parameter values 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.
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 in favor of the alternative of cointegration. Values of h equal to 0 (false) indicate a failure to reject the null. | ||||||||||||||||||||||||||||||||||||||||||||||||
pValue |
Vector of p-values of the test statistics, with length equal to the number of tests. p-values are left-tail probabilities. | ||||||||||||||||||||||||||||||||||||||||||||||||
stat |
Vector of test statistics, with length equal to the number of tests. The statistic depends on the rreg and test values (see the documentation for adftest and pptest). | ||||||||||||||||||||||||||||||||||||||||||||||||
cValue |
Vector of critical values for the tests, with length equal to the number of tests. Values are for left-tail probabilities. Since residuals are estimated rather than observed, critical values are different from those used in adftest or pptest (unless the cointegrating vector is completely specified by cvec). egcitest loads tables of critical values from the file Data_EGCITest.mat, then linearly interpolates test values from the tables. Critical values in the tables were computed using methods described in [3]. | ||||||||||||||||||||||||||||||||||||||||||||||||
reg1 |
Structure of regression statistics from the cointegrating regression. | ||||||||||||||||||||||||||||||||||||||||||||||||
reg2 |
Structure of regression statistics from the residual regression. The number of records in reg1 and reg2 equals the number of tests. Each record has the following fields:
*Lagging and differencing a time series reduces 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 p lagged differences, the common time base is p+2:N and the effective sample size is N−(p+1). |
Load data on term structure of interest rates in Canada:
load Data_Canada Y = Data(:,3:end); names = series(3:end); plot(dates,Y) legend(names,'location','NW') grid on
Test for cointegration (and reproduce row 1 of Table II in [3]):
[h,pValue,stat,cValue,reg] = egcitest(Y,'test',... {'t1','t2'}); h,pValue
h = 0 1 pValue = 0.0526 0.0202
Plot the estimated cointegrating relation y_{1}−Y_{2}b−Xa:
a = reg(2).coeff(1);
b = reg(2).coeff(2:3);
plot(dates,Y*[1;-b]-a)
grid on
[1] Engle, R. F. and C. W. J. Granger. "Co-Integration and Error-Correction: 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] MacKinnon, J. G. "Numerical Distribution Functions for Unit Root and Cointegration Tests." Journal of Applied Econometrics. v. 11, 1996, pp. 601–618.