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egcitest

Engle-Granger cointegration test

Syntax


              
               h = egcitest(Y)


              
               [h,pValue,stat,cValue]
= egcitest(Y)


              
               StatTbl = egcitest(Tbl)


              
               [
               
                ___
               
               ] = egcitest(
               
                ___
               
               ,Name=Value)


              
               [
               
                ___
               
               ,reg1,reg2] = egcitest(
               
                ___
               
               )

Description

example

h = egcitest( Y ) returns the rejection decision from conducting the Engle-Granger cointegration test for assessing the null hypothesis of no cointegration among the variables in the input multivariate time series data. egcitest forms test statistics by regressing the response data Y(:,1) onto the predictor data Y(:,2:end) , and then it tests the residuals for a unit root.

example

[ h , pValue , stat , cValue ] = egcitest( Y ) also returns the p -value pValue , test statistic stat , and critical value cValue of the test.

example

StatTbl = egcitest( Tbl ) returns a table containing variables for the test results, statistics, and settings from conducting the Engle-Granger cointegration test on the variables of the input table or timetable.

The response variable in the regression is the first table variable, and all other variables are the predictor variables. To select a different response variable for the regression, use the ResponseVariable name-value argument. To select different predictor variables, use the PredictorNames name-value argument.

example

[ ___ ] = egcitest( ___ , Name=Value ) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. egcitest returns the output argument combination for the corresponding input arguments.

Some options control the number of tests to conduct. The following conditions apply when egcitest conducts multiple tests:

egcitest treats each test as separate from all other tests.
If you specify Y , all outputs are vectors.
If you specify Tbl , each row of StatTbl contains the results of the corresponding test.

For example, egcitest(Tbl,ResponseVariable="GDP",Alpha=0.025,Lags=[0 1]) chooses GDP as the response variable from the table Tbl and conducts two tests at a level of significance of 0.025. The first test includes 0 lag in the residual regression, and the second test includes 1 lag in the residual regression.

example

[ ___ , reg1 , reg2 ] = egcitest( ___ ) additionally returns the following structures of regression statistics, which are required to form the test statistic:

reg1 – Statistics resulting from the cointegrating regression of the specified response variable ResponseVariable onto the specified predictor variables PredictorVariables
reg2 – Statistics resulting from the residual regression implemented by the specified unit root test RReg

Examples

collapse all

Conduct Engle-Granger Cointegration Test on Matrix of Data

Open Live Script

Test a multivariate time series for cointegration using the default values of the Engle-Granger cointegration test. Input the time series data as a numeric matrix.

Load data of Canadian inflation and interest rates Data_Canada.mat , which contains the series in the matrix Data .

load Data_Canada
series'

ans = 5x1 cell
    {'(INF_C) Inflation rate (CPI-based)'         }
    {'(INF_G) Inflation rate (GDP deflator-based)'}
    {'(INT_S) Interest rate (short-term)'         }
    {'(INT_M) Interest rate (medium-term)'        }
    {'(INT_L) Interest rate (long-term)'          }

Test the interest rate series for cointegration by using the Engle-Granger cointegration test. Use default options and return the rejection decision and $p$ -value.

h = egcitest(Data(:,3:end))

h = logical

egcitest uses the $τ$ test by default, and it fails to reject the null hypothesis ( h = 0 ) of no cointegration among the interest rate series.

Return Test p -Value and Decision Statistics

Open Live Script

Load data of Canadian inflation and interest rates Data_Canada.mat .

load Data_Canada

Test the interest rate series for cointegration by using the Engle-Granger cointegration test. Use default options and return the rejection decision, $p$ -value, $τ$ -test statistic, and critical value.

[h,pValue,stat,cValue] = egcitest(Data(:,3:end))

h = logical

pValue = 0.0526

stat = -3.9321

cValue = -3.9563

Conduct Default Engle-Granger Cointegration Test on Table Variables

Open Live Script

Conduct the Engle-Granger cointegration test on a multivariate time series using default options, which use the first table variable as the response, all other table variables as predictors, and includes a constant term in the cointegrating regression. Return a table of test results.

Load data of Canadian inflation and interest rates Data_Canada.mat . Convert the table DataTable to a timetable.

load Data_Canada
dates = datetime(dates,12,31);
TT = table2timetable(DataTable,RowTimes=dates);
TT.Observations = [];

Conduct the Engel-Granger cointegration test by passing the timetable to egcitest and using default options. For the cointegrating regression, egcitest uses the CPI-based inflation rate as the response variable and all other variables in the timetable as predictors.

StatTbl = egcitest(TT)

StatTbl=1×9 table
                h       pValue       stat      cValue     Lags    Alpha     Test     CReg      RReg  
              _____    _________    _______    _______    ____    _____    ______    _____    _______
    Test 1    true     0.0023851    -6.2491    -4.7673     0      0.05     {'t1'}    {'c'}    {'ADF'}

StatTbl is a table of test results. The rows correspond to variables in the input timetable TT , and the columns correspond to the rejection decision, and corresponding $p$ -value, decision statistics, and specified test options. In this case, the test rejects the null hypothesis in favor of the alternative of cointegration among all the table variables.

By default, egcitest includes all input table variables in the cointegration test. To select a response variable for the cointegrating regression, set the ResponseVariable option. To select predictor variables, set the PredictorVariables option.

Select Test Statistics and Plot Cointegrating Relation

Open Live Script

Load data of Canadian inflation and interest rates Data_Canada.mat . Convert the table DataTable to a timetable of the interest rate series only.

load Data_Canada
dates = datetime(dates,12,31);
idxINT = contains(DataTable.Properties.VariableNames,"INT");
TT = table2timetable(DataTable(:,idxINT),RowTimes=dates);
TT.Observations = [];

Plot the interest rate series.

figure
plot(TT.Time,TT.Variables)
legend(series(idxINT),Location="northwest")
grid on

Reproduce row 1 of Table II in [3] by testing for cointegration, specifying the default variable assignments for the cointegrating regression and deterministic terms (response variable $y_{1}$ is INT_S , the other interest rates $y_{2}$ and $y_{3}$ are predictors, and the model has a constant $c$ ), and specifying the $τ$ and $z$ tests. Return the cointegrating regression statistics.

[StatTbl,reg] = egcitest(TT,Test=["t1" "t2"]);
StatTbl

StatTbl=2×9 table
                h       pValue      stat      cValue     Lags    Alpha     Test     CReg      RReg  
              _____    ________    _______    _______    ____    _____    ______    _____    _______
    Test 1    false    0.052627    -3.9321    -3.9563     0      0.05     {'t1'}    {'c'}    {'ADF'}
    Test 2    true     0.020157    -25.454    -22.115     0      0.05     {'t2'}    {'c'}    {'ADF'}

The $τ$ test ( Test 1 ) fails to reject the null hypothesis, but the $z$ test ( Test 2 ) rejects the null hypothesis in favor of the presence of cointegration.

Plot the estimated cointegrating relation using the regression statistics from the $z$ test $y_{1} - [\begin{array}{c} y_{2} & y_{3} \end{array}] [\begin{array}{c} b_{1} \\ b_{2} \end{array}] - Xa$ , where $Xa = c$ .

c = reg(2).coeff(1);
b = reg(2).coeff(2:3);
figure
plot(TT.Time,TT.Variables*[1; -b] - c)
grid on

Input Arguments

collapse all

`Y` — Data representing observations of multivariate time series y _t
numeric matrix

Data representing observations of a multivariate time series y _t , specified as a numObs -by- numDims numeric matrix. Each column of Y corresponds to a variable, and each row corresponds to an observation. The test regresses the response variable Y(:,1) on the predictor variables Y(:,2:end) .

Data Types: double

`Tbl` — Data representing observations of multivariate time series y _t
table | timetable

Data representing observations of a multivariate time series y _t , specified as a table or timetable with numObs rows. Each row of Tbl is an observation.

The test regresses the response variable, which is the first variable in Tbl , on the predictor variables, which are all other variables in Tbl . To select a different response variable for the regression, use the ResponseVariable name-value argument. To select different predictor variables, use the PredictorNames name-value argument. The selected variables must be numeric.

Note

egcitest removes, from the specified data, all observations containing at least one missing observation, represented by a NaN value.

Name-Value Arguments

Specify optional pairs of arguments as Name1=Value1,...,NameN=ValueN , where Name is the argument name and Value is the corresponding value. Name-value arguments must appear after other arguments, but the order of the pairs does not matter. Before R2021a, use commas to separate each name and value, and enclose Name in quotes.

Example: egcitest(Tbl,ResponseVariable="GDP",Alpha=0.025,Lags=[0 1]) chooses GDP as the response variable from the table Tbl and conducts two tests at a level of significance of 0.025. The first test includes 0 lag in the residual regression, and the second test includes 1 lag in the residual regression.

`CReg` — Cointegrating regression form
`"c"` (default) | `"nc"` | `"ct"` | `"ctt"` | character vector | string vector | cell vector of character vectors

Cointegrating regression form, specified as the name of a form, or a string vector or cell vector of form names.

In general, cointegrating regression is

$y_{1} = X a + Y_{2} b + ε$

where y ₁ is the response variable, Y ₂ contains the predictor variables, and X is a design matrix for optional deterministic coefficients a , including a constant, linear time trend, and quadratic time trend. This table contains the supported forms and their names.

Form Name	Description
`"nc"`	The regression does not include X ; no constant or trends.
`"c"`	X contains a variable for the constant, but not for the trends.
`"ct"`	X contains variables for the constant and the linear time trend.
`"ctt"`	X contains variables for the constant, linear time trend, and quadratic time trend.

egcitest conducts a separate test for each form name in CReg .

Example: CReg=["ct" "ctt"] includes a constant and linear time trend terms in the cointegrating regression for the first test, and then includes all three deterministic terms in the cointegrating regression for the second test.

Data Types: char | string | cell

`CVec` — Cointegrating-regression coefficient equality constraints
numeric vector (default) | cell vector of numeric vectors

Cointegrating-regression coefficient equality constraints, specified as the numeric vector [ a ; b ] or cell vector of such numeric vectors.

a contains the equality constraints of the deterministic terms in the cointegrating regression. The length of a depends on the corresponding value of the CReg name-value argument, one of 0, 1, 2, or 3. For coefficients in the regression, their order in a is constant, linear trend, and quadratic trend.

b contains the numDims − 1 equality constraints for the coefficient of the corresponding predictor variable in Y ₂ .

Specify NaN entries to estimate the corresponding coefficient in the regression.

When CVec is completely specified (does not contain any NaN values), egcitest does not perform the cointegrating regression.

By default, CVec is a completely unspecified cointegrating vector (completely composed of NaN values). Consequently, egcitest estimates all coefficients.

egcitest conducts a separate test for each set of equality constraints in CVec .

Example: egcitest(Tbl,CVec=[2 NaN NaN]) fixes the constant in the cointegrating regression to 2 and estimates the coefficients of the two predictor variables in Tbl .

Example: egcitest(Tbl,CVec={[2 NaN NaN]; nan(3,1)) , for the first test, fixes the constant in the cointegrating regression to 2 and estimates the coefficients of the two predictor variables in Tbl , and for the second test, estimates all coefficients.

Example: egcitest(Tbl,CReg="ctt",CVec=[2 0.5 0.25 NaN NaN]) fixes the constant to 2 , the linear trend to 0.5 , and the quadratic trend to 0.25 , and estimates the coefficients of the two predictor variables in Tbl .

Data Types: double | cell

`RReg` — Residual regression form
`"pp"` (default) | `"adf"` | character vector | string vector | cell vector of character vectors

Residual regression form, specified as the name of a form, or a string vector or cell vector of form names.

Form Name	Description
`"adf"`	Augmented Dickey-Fuller test ( `adftest` ) of residuals from the cointegrating regression
`"pp"`	Phillips-Perron test ( `pptest` ) of residuals from the cointegrating regression

egcitest computes test statistics by calling adftest and pptest with the setting Model="AR" . This setting requires residuals from appropriately demeaned and detrended data, which is specified by the cointegrating-regression form CReg .

egcitest conducts a separate test for each form name in RReg .

Example: CReg=["adf" "pp"] performs the augmented Dickey-Fuller test for the residual regression of the first test, and then performs the Phillips-Perron test for the residual regression of the second test.

Data Types: char | string | cell

`Lags` — Number of lags in residual regression
`0` (default) | nonnegative integer | vector of nonnegative integers

Number of lags in the residual regression, specified as a nonnegative integer or vector of nonnegative integers. The meaning of Lags depends on the value of the RReg name-value argument. For more details, see the Lags argument of the adftest and pptest functions.

egcitest conducts a separate test for each element in Lags .

Example: Lags=[0 1] includes no lags in the residual regression for the first test, and then includes one lag for the residual regression for the second test.

Data Types: double

`Test` — Test statistic type from residual regression
`"t1"` (default) | `"t2"` | character vector | string vector | cell vector of character vectors

Test statistic type from residual regression, specified as test name, or a string vector or cell vector of test names. This table contains the supported test names.

Test Name	Description
`"t1"`	τ test
`"t2"`	z test

For more details, see the Test argument of the adftest and pptest functions.

egcitest conducts a separate test for each element in Test .

Example: Test=["t1" "t2"] computes the τ test from the residual regression for the first test, and then computes the z test from the residual regression for the second test.

Data Types: char | cell | string

`Alpha` — Nominal significance level
`0.05` (default) | numeric scalar | numeric vector

Nominal significance level for the hypothesis test, specified as a numeric scalar between 0.001 and 0.999 or a numeric vector of such values.

egcitest conducts a separate test for each value in Alpha .

Example: Alpha=[0.01 0.05] uses a level of significance of 0.01 for the first test, and then uses a level of significance of 0.05 for the second test.

Data Types: double

`ResponseVariable` — Variable in `Tbl` to use for response
first variable in `Tbl` (default) | string vector | cell vector of character vectors | vector of integers | logical vector

Variable in Tbl to use for response in the cointegrating regression, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames , or an integer or logical vector representing the indices of names. The selected variables must be numeric.

egcitest uses the same specified response variable for all tests.

Example: ResponseVariable="GDP"

Data Types: double | logical | char | cell | string

`PredictorVariables` — Variables in `Tbl` to use for the predictors
string vector | cell vector of character vectors | vector of integers | logical vector

Variables in Tbl to use for the predictors in the cointegrating regression, specified as a string vector or cell vector of character vectors containing variable names in Tbl.Properties.VariableNames , or an integer or logical vector representing the indices of names. The selected variables must be numeric.

egcitest uses the same specified predictors for all tests.

By default, egcitest uses all variables in Tbl that is not specified by the ResponseVariable name-value argument.

Example: DataVariables=["UN" "CPI"]

Example: DataVariables=[true true false false] or DataVariables=[1 2] selects the first and second table variables.

Data Types: double | logical | char | cell | string

Note

When egcitest conducts multiple tests, the function applies all single settings (scalars or character vectors) to each test.
All vector-valued specifications that control the number of tests must have equal length.
If you specify the matrix Y and any value is a row vector, all outputs are row vectors.
A lagged and differenced time series has a reduced sample size. Absent presample values, if the test series y _t is defined for t = 1,…, T , the lagged series y _{t

– k} is defined for t = k +1,…, T . The first difference applied to the lagged series y _{t

– k} further reduces the time base to k +2,…, T . With p lagged differences, the common time base is p +2,…, T and the effective sample size is T –( p +1).

Output Arguments

collapse all

`h` — Test rejection decisions
logical scalar | logical vector

Test rejection decisions, returned as a logical scalar or vector with length equal to the number of tests. egcitest returns h when you supply the input Y .

Values of 1 indicate rejection of the null hypothesis in favor of the alternative of cointegration.
Values of 0 indicate failure to reject the null hypothesis.

`pValue` — Test statistic p -values
numeric scalar | numeric vector

Test statistic p -values, returned as a numeric scalar or vector with length equal to the number of tests. egcitest returns pValue when you supply the input Y .

The p -values are left-tailed probabilities.

`stat` — Test statistics
numeric scalar | numeric vector

Test statistics, returned as a numeric scalar or vector with length equal to the number of tests. egcitest returns stat when you supply the input Y .

The RReg and Test settings of a particular test determine the test statistic. For more details, see adftest and pptest .

`cValue` — Critical values
numeric scalar | numeric vector

Critical values, returned as a numeric scalar or vector with length equal to the number of tests. egcitest returns cValue when you supply the input Y . The critical values are for left-tailed probabilities.

Because egcitest estimates the residuals (that is, residuals are unobserved), critical values are different from those used in adftest or pptest (unless the cointegrating vector is completely specified by the CVec setting). egcitest loads tables of critical values from the file Data_EGCITest.mat , and then linearly interpolates test critical values from the tables. Critical values in the tables derive from methods described in [3] .

`StatTbl` — Test summary
table

Test summary, returned as a table with variables for the outputs h , pValue , stat , and cValue , and with a row for each test. egcitest returns StatTbl when you supply the input Tbl .

StatTbl contains variables for the test settings specified by Lags , Alpha , Test , CReg , and RReg .

`reg1` — Cointegrating regression statistics
structure array

Cointegrating regression statistics, returned as a structure array. The number of records equal to the number of tests.

egcitest regresses the response variable ResponseVariable onto the predictor variables PredictorVariables using the regression form CReg and specified equality constraints CVec .

Each element of reg1 has the fields in this table. You can access a field using dot notation, for example, reg1(3).coeff contains the coefficient estimates of the third test.

`num`	Length of input series with `NaN` s removed
`size`	Effective sample size, adjusted for lags and difference
`names`	Regression coefficient names
`coeff`	Estimated coefficient values
`se`	Estimated coefficient standard errors
`Cov`	Estimated coefficient covariance matrix
`tStats`	t statistics of coefficients and p -values
`FStat`	F statistic and p -value
`yMu`	Mean of the lag-adjusted input series
`ySigma`	Standard deviation of the lag-adjusted input series
`yHat`	Fitted values of the lag-adjusted input series
`res`	Regression residuals
`DWStat`	Durbin-Watson statistic
`SSR`	Regression sum of squares
`SSE`	Error sum of squares
`SST`	Total sum of squares
`MSE`	Mean square error
`RMSE`	Standard error of the regression
`RSq`	R ² statistic
`aRSq`	Adjusted R ² statistic
`LL`	Loglikelihood of data under Gaussian innovations
`AIC`	Akaike information criterion
`BIC`	Bayesian (Schwarz) information criterion
`HQC`	Hannan-Quinn information criterion

`reg2` — Residual regression statistics
structure array

Residual regression statistics, returned as a structure array containing the same fields as reg1 . The number of records equal to the number of tests.


                        egcitest

tests the residuals of the cointegrating regression for a unit root by passing the residuals, and the values of


                         Lags

and


                         Test

, to the test specified by


                         RReg

. The tests form the test statistic by a regression of the residuals using specified options. For more details on the test options and the fields of


                        reg2

, see


                         adftest


                         pptest

Tips

To draw valid inferences from the test, determine a suitable value for Lags . For more details, see the adftest Tips and the pptest Tips .
Samples with less than approximately 20 through 40 observations (depending on the dimension of the data numDims ) can yield unreliable critical values, and therefore unreliable inferences. See [3] .
If a test result suggests that the time series are cointegrated, you can use the residuals as data for the error-correction term in a VEC representation of the variables. Follow this procedure:
1. Extract the residuals from the reg1 output ( reg1.res ).
2. Estimate autoregressive model components using the estimate function of varm , and treat the extracted residual series as exogenous for estimation.

Alternative Functionality

App

The Econometric Modeler app enables you to conduct the Engle-Granger cointegration test.

References

[1] Engle, R. F. and C. W. J. Granger. "Co-Integration and Error-Correction: Representation, Estimation, and Testing." Econometrica . Vol. 55, 1987, pp. 251–276.

[2] Hamilton, James 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 . Vol. 11, 1996, pp. 601–618.

Version History

Introduced in R2011a

egcitest

Syntax

Description

Examples

Conduct Engle-Granger Cointegration Test on Matrix of Data

Return Test p -Value and Decision Statistics

Conduct Default Engle-Granger Cointegration Test on Table Variables

Select Test Statistics and Plot Cointegrating Relation

Input Arguments

`Y` — Data representing observations of multivariate time series y _t
numeric matrix

`Tbl` — Data representing observations of multivariate time series y _t
table | timetable

Name-Value Arguments

`CReg` — Cointegrating regression form
`"c"` (default) | `"nc"` | `"ct"` | `"ctt"` | character vector | string vector | cell vector of character vectors

`CVec` — Cointegrating-regression coefficient equality constraints
numeric vector (default) | cell vector of numeric vectors

`RReg` — Residual regression form
`"pp"` (default) | `"adf"` | character vector | string vector | cell vector of character vectors

`Lags` — Number of lags in residual regression
`0` (default) | nonnegative integer | vector of nonnegative integers

`Test` — Test statistic type from residual regression
`"t1"` (default) | `"t2"` | character vector | string vector | cell vector of character vectors

`Alpha` — Nominal significance level
`0.05` (default) | numeric scalar | numeric vector

`ResponseVariable` — Variable in `Tbl` to use for response
first variable in `Tbl` (default) | string vector | cell vector of character vectors | vector of integers | logical vector

`PredictorVariables` — Variables in `Tbl` to use for the predictors
string vector | cell vector of character vectors | vector of integers | logical vector

Output Arguments

`h` — Test rejection decisions
logical scalar | logical vector

`pValue` — Test statistic p -values
numeric scalar | numeric vector

`stat` — Test statistics
numeric scalar | numeric vector

`cValue` — Critical values
numeric scalar | numeric vector

`StatTbl` — Test summary
table

`reg1` — Cointegrating regression statistics
structure array

`reg2` — Residual regression statistics
structure array

Tips

Alternative Functionality

App

References

Version History

See Also

Objects

Functions

Topics

egcitest

Syntax

Description

Examples

Conduct Engle-Granger Cointegration Test on Matrix of Data

Return Test p -Value and Decision Statistics

Conduct Default Engle-Granger Cointegration Test on Table Variables

Select Test Statistics and Plot Cointegrating Relation

Input Arguments

Y — Data representing observations of multivariate time series y t numeric matrix

Tbl — Data representing observations of multivariate time series y t table | timetable

Name-Value Arguments

CReg — Cointegrating regression form "c" (default) | "nc" | "ct" | "ctt" | character vector | string vector | cell vector of character vectors

CVec — Cointegrating-regression coefficient equality constraints numeric vector (default) | cell vector of numeric vectors

RReg — Residual regression form "pp" (default) | "adf" | character vector | string vector | cell vector of character vectors

Lags — Number of lags in residual regression 0 (default) | nonnegative integer | vector of nonnegative integers

Test — Test statistic type from residual regression "t1" (default) | "t2" | character vector | string vector | cell vector of character vectors

Alpha — Nominal significance level 0.05 (default) | numeric scalar | numeric vector

ResponseVariable — Variable in Tbl to use for response first variable in Tbl (default) | string vector | cell vector of character vectors | vector of integers | logical vector

PredictorVariables — Variables in Tbl to use for the predictors string vector | cell vector of character vectors | vector of integers | logical vector

Output Arguments

h — Test rejection decisions logical scalar | logical vector

pValue — Test statistic p -values numeric scalar | numeric vector

stat — Test statistics numeric scalar | numeric vector

cValue — Critical values numeric scalar | numeric vector

StatTbl — Test summary table

reg1 — Cointegrating regression statistics structure array

reg2 — Residual regression statistics structure array

Tips

Alternative Functionality

App

References

Version History

See Also

Objects

Functions

Topics

`Y` — Data representing observations of multivariate time series y _t
numeric matrix

`Tbl` — Data representing observations of multivariate time series y _t
table | timetable

`CReg` — Cointegrating regression form
`"c"` (default) | `"nc"` | `"ct"` | `"ctt"` | character vector | string vector | cell vector of character vectors

`CVec` — Cointegrating-regression coefficient equality constraints
numeric vector (default) | cell vector of numeric vectors

`RReg` — Residual regression form
`"pp"` (default) | `"adf"` | character vector | string vector | cell vector of character vectors

`Lags` — Number of lags in residual regression
`0` (default) | nonnegative integer | vector of nonnegative integers

`Test` — Test statistic type from residual regression
`"t1"` (default) | `"t2"` | character vector | string vector | cell vector of character vectors

`Alpha` — Nominal significance level
`0.05` (default) | numeric scalar | numeric vector

`ResponseVariable` — Variable in `Tbl` to use for response
first variable in `Tbl` (default) | string vector | cell vector of character vectors | vector of integers | logical vector

`PredictorVariables` — Variables in `Tbl` to use for the predictors
string vector | cell vector of character vectors | vector of integers | logical vector

`h` — Test rejection decisions
logical scalar | logical vector

`pValue` — Test statistic p -values
numeric scalar | numeric vector

`stat` — Test statistics
numeric scalar | numeric vector

`cValue` — Critical values
numeric scalar | numeric vector

`StatTbl` — Test summary
table

`reg1` — Cointegrating regression statistics
structure array

`reg2` — Residual regression statistics
structure array