gctest

Granger causality and block exogeneity tests for vector autoregression (VAR) models

Syntax

h = gctest(Mdl)

h = gctest(Mdl,Name,Value)

[h,Summary] = gctest(___)

Description

The gctest object function can conduct leave-one-out, exclude-all, and block-wise Granger causality tests for the response variables of a fully specified vector autoregression (VAR) model (represented by a varm model object).

To conduct a block-wise Granger causality test from specified sets of time series data representing "cause" and "effect" multivariate response variables, or to address possibly integrated series for the test, see the gctest function.

h = gctest(Mdl) returns the test decision h from conducting leave-one-out Granger causality tests on all response variables that compose the VAR(p) model Mdl.

h = gctest(Mdl,Name,Value) uses additional options specified by one or more name-value pair arguments. For example, 'Type',"block-wise",'Cause',1:2,'Effect',3:5 specifies conducting a block-wise test to assess whether the response variables Mdl.SeriesNames(1:2) Granger-cause the response variables Mdl.SeriesNames(3:5) conditioned on all other variables in the model.

[h,Summary] = gctest(___) additionally returns a table that summarizes all test results, using any of the input argument combinations in the previous syntaxes. Results include test decisions and p-values.

Examples

Conduct Leave-One-Out Granger Causality Test

Conduct a leave-one-out Granger causality test to assess whether each variable in a 3-D VAR model Granger-causes another variable, given the third variable. The variables in the VAR model are the M1 money supply, consumer price index (CPI), and US gross domestic product (GDP).

Load the US macroeconomic data set Data_USEconModel.mat.

load Data_USEconModel

The data set includes the MATLAB® timetable DataTable, which contains 14 variables measured from Q1 1947 through Q1 2009.

• M1SL is the table variable containing the M1 money supply.

• CPIAUCSL is the table variable containing the CPI.

• GDP is the table variable containing the US GDP.

For more details, enter Description at the command line.

Visually assess whether the series are stationary.

plot(DataTable.Time,DataTable.CPIAUCSL)
ylabel("Money Supply");

plot(DataTable.Time,DataTable.M1SL)
ylabel("CPI");

plot(DataTable.Time,DataTable.GDP)
ylabel("GDP")

All series are nonstationary.

Stabilize the series.

• Convert the M1 money supply prices to returns.

• Convert the CPI to the inflation rate.

• Convert the GDP to the real GDP rate with respect to year 2000 dollars.

m1slrate = price2ret(DataTable.M1SL);
inflation = price2ret(DataTable.CPIAUCSL);
rgdprate = price2ret(DataTable.GDP./DataTable.GDPDEF);

Preprocess the data by removing all missing observations (indicated by NaN).

tbl = table(m1slrate,inflation,rgdprate);
tbl = rmmissing(tbl);
T = size(tbl,1); % Total sample size

Fit VAR models, with lags ranging from 1 to 4, to the series. Initialize each fit by specifying the first four observations. Store the Akaike information criteria (AIC) of the fits.

numseries = 3;
numlags = (1:4)';
nummdls = numel(numlags);

% Partition time base.
maxp = max(numlags); % Maximum number of required presample responses
idxpre = 1:maxp;
idxest = (maxp + 1):T;

% Preallocation
EstMdl(nummdls) = varm(numseries,0);
aic = zeros(nummdls,1);

% Fit VAR models to data.
Y0 = tbl{idxpre,:}; % Presample
Y = tbl{idxest,:};  % Estimation sample
for j = 1:numel(numlags)
    Mdl = varm(numseries,numlags(j));
    Mdl.SeriesNames = tbl.Properties.VariableNames;
    EstMdl(j) = estimate(Mdl,Y,'Y0',Y0);
    results = summarize(EstMdl(j));
    aic(j) = results.AIC;
end

[~,bestidx] = min(aic);
p = numlags(bestidx)

p = 3

BestMdl = EstMdl(bestidx);

A VAR(3) model yields the best fit.

For each variable and equation in the system, conduct a leave-one-out Granger causality test to assess whether a variable in the fitted VAR(3) model is the 1-step Granger-cause of another variable, given the third variable.

h = gctest(BestMdl)

                          H0                                Decision         Distribution    Statistic     PValue      CriticalValue
    _______________________________________________    __________________    ____________    _________    _________    _____________

    "Exclude lagged inflation in m1slrate equation"    "Cannot reject H0"     "Chi2(3)"       7.0674       0.069782       7.8147    
    "Exclude lagged rgdprate in m1slrate equation"     "Cannot reject H0"     "Chi2(3)"       2.5585         0.4648       7.8147    
    "Exclude lagged m1slrate in inflation equation"    "Cannot reject H0"     "Chi2(3)"       2.7025         0.4398       7.8147    
    "Exclude lagged rgdprate in inflation equation"    "Reject H0"            "Chi2(3)"       14.338      0.0024796       7.8147    
    "Exclude lagged m1slrate in rgdprate equation"     "Cannot reject H0"     "Chi2(3)"       7.0352       0.070785       7.8147    
    "Exclude lagged inflation in rgdprate equation"    "Reject H0"            "Chi2(3)"       12.006      0.0073619       7.8147    

h = 6x1 logical array

   0
   0
   0
   1
   0
   1

gctest conducts six tests, so h is a 6-by-1 logical vector of test decisions corresponding to the rows of the test summary table. The results indicate the following decisions, at a 5% level of significance:

• Do not reject the claim that the inflation rate is not the 1-step Granger-cause of the M1 money supply rate, given the real GDP rate (h(1) = 0).

• Do not reject the claim that the real GDP rate is not the 1-step Granger-cause of the M1 money supply rate, given the inflation rate (h(2) = 0).

• Do not reject the claim that the M1 money supply rate is not the 1-step Granger-cause of the inflation rate, given the real GDP rate (h(3) = 0).

• Reject the claim that the real GDP rate is not the 1-step Granger-cause of the inflation rate, given the M1 money supply rate (h(4) = 1).

• Do not reject the claim that the M1 money supply rate is not the 1-step Granger-cause of the real GDP rate, given the inflation rate (h(5) = 0).

• Reject the claim that the inflation rate is not the 1-step Granger-cause of the real GDP rate, given the M1 money supply rate (h(6) = 1).

Because the inflation and real GDP rates are 1-step Granger-causes of each other, they constitute a feedback loop.

Conduct Exclude-All Granger Causality Test

Consider the 3-D VAR(3) model and leave-one-out Granger causality test in Conduct Leave-One-Out Granger Causality Test.

Load the US macroeconomic data set Data_USEconModel.mat. Preprocess the data. Fit a VAR(3) model to the preprocessed data.

load Data_USEconModel

m1slrate = price2ret(DataTable.M1SL);
inflation = price2ret(DataTable.CPIAUCSL);
rgdprate = price2ret(DataTable.GDP./DataTable.GDPDEF);

tbl = table(m1slrate,inflation,rgdprate);
tbl = rmmissing(tbl);

Mdl = varm(3,3);
Mdl.SeriesNames = tbl.Properties.VariableNames;
EstMdl = estimate(Mdl,tbl{5:end,:},'Y0',tbl{2:4,:});

Conduct an exclude-all Granger causality test on the variables of the fitted model.

h = gctest(EstMdl,'Type',"exclude-all");

                               H0                                    Decision         Distribution    Statistic     PValue      CriticalValue
    ________________________________________________________    __________________    ____________    _________    _________    _____________

    "Exclude all but lagged m1slrate in m1slrate equation"      "Cannot reject H0"     "Chi2(6)"        9.477        0.14847       12.592    
    "Exclude all but lagged inflation in inflation equation"    "Reject H0"            "Chi2(6)"       19.475      0.0034327       12.592    
    "Exclude all but lagged rgdprate in rgdprate equation"      "Reject H0"            "Chi2(6)"        19.16      0.0039014       12.592    

gctest conducts numtests = 3 tests. The results indicate the following decisions, each at a 5% level of significance:

• Do not reject the claim that the inflation and real GDP rates do not Granger-cause the M1 money supply rate.

• Reject the claim that the M1 money supply and real GDP rates do not Granger-cause the inflation rate.

• Reject the claim that the M1 money supply and inflation rates do not Granger-cause the real GDP rate.

Adjust Significance Level for Multiple Tests

The false discovery rate increases with the number of simultaneous hypothesis tests you conduct. To combat the increase, decrease the level of significance per test by using the 'Alpha' name-value pair argument. Consider the 3-D VAR(3) model and leave-one-out Granger causality test in Conduct Leave-One-Out Granger Causality Test.

Load the US macroeconomic data set Data_USEconModel.mat. Preprocess the data. Fit a VAR(3) model to the preprocessed data.

load Data_USEconModel

m1slrate = price2ret(DataTable.M1SL);
inflation = price2ret(DataTable.CPIAUCSL);
rgdprate = price2ret(DataTable.GDP./DataTable.GDPDEF);
tbl = table(m1slrate,inflation,rgdprate);
tbl = rmmissing(tbl);

Mdl = varm(3,3);
Mdl.SeriesNames = tbl.Properties.VariableNames;
EstMdl = estimate(Mdl,tbl{5:end,:},'Y0',tbl{2:4,:});

A leave-one-out Granger causality test on the variables in the model results in numtests = 6 simultaneous tests. Conduct the tests, but specify a family-wise significance level of 0.05 by specifying a level of significance of alpha = 0.05/numtests for each test.

numtests = 6;
alpha = 0.05/numtests

alpha = 0.0083

gctest(EstMdl,'Alpha',alpha);

                          H0                                Decision         Distribution    Statistic     PValue      CriticalValue
    _______________________________________________    __________________    ____________    _________    _________    _____________

    "Exclude lagged inflation in m1slrate equation"    "Cannot reject H0"     "Chi2(3)"       7.0674       0.069782       11.739    
    "Exclude lagged rgdprate in m1slrate equation"     "Cannot reject H0"     "Chi2(3)"       2.5585         0.4648       11.739    
    "Exclude lagged m1slrate in inflation equation"    "Cannot reject H0"     "Chi2(3)"       2.7025         0.4398       11.739    
    "Exclude lagged rgdprate in inflation equation"    "Reject H0"            "Chi2(3)"       14.338      0.0024796       11.739    
    "Exclude lagged m1slrate in rgdprate equation"     "Cannot reject H0"     "Chi2(3)"       7.0352       0.070785       11.739    
    "Exclude lagged inflation in rgdprate equation"    "Reject H0"            "Chi2(3)"       12.006      0.0073619       11.739    

The test decisions of these more conservative tests are the same as the tests decisions in Conduct Leave-One-Out Granger Causality Test. However, the conclusions of the conservative tests hold simultaneously at a 5% level of significance.

Access Test p-Values

Consider the 3-D VAR(3) model and leave-one-out Granger causality test in Conduct Leave-One-Out Granger Causality Test.

Load the US macroeconomic data set Data_USEconModel.mat. Preprocess the data. Fit a VAR(3) model to the preprocessed data.

load Data_USEconModel

m1slrate = price2ret(DataTable.M1SL);
inflation = price2ret(DataTable.CPIAUCSL);
rgdprate = price2ret(DataTable.GDP./DataTable.GDPDEF);

tbl = table(m1slrate,inflation,rgdprate);
tbl = rmmissing(tbl);

Mdl = varm(3,3);
Mdl.SeriesNames = tbl.Properties.VariableNames;
EstMdl = estimate(Mdl,tbl{5:end,:},'Y0',tbl{2:4,:});

Conduct a leave-one-out Granger causality test on the variables of the fitted model. Return the test result summary table and suppress the test results display.

[~,Summary] = gctest(EstMdl,'Display',false)

Summary=6×6 table
                          H0                                Decision         Distribution    Statistic     PValue      CriticalValue
    _______________________________________________    __________________    ____________    _________    _________    _____________

    "Exclude lagged inflation in m1slrate equation"    "Cannot reject H0"     "Chi2(3)"       7.0674       0.069782       7.8147    
    "Exclude lagged rgdprate in m1slrate equation"     "Cannot reject H0"     "Chi2(3)"       2.5585         0.4648       7.8147    
    "Exclude lagged m1slrate in inflation equation"    "Cannot reject H0"     "Chi2(3)"       2.7025         0.4398       7.8147    
    "Exclude lagged rgdprate in inflation equation"    "Reject H0"            "Chi2(3)"       14.338      0.0024796       7.8147    
    "Exclude lagged m1slrate in rgdprate equation"     "Cannot reject H0"     "Chi2(3)"       7.0352       0.070785       7.8147    
    "Exclude lagged inflation in rgdprate equation"    "Reject H0"            "Chi2(3)"       12.006      0.0073619       7.8147    

Summary is a MATLAB table containing numtests = 6 rows. The rows contain the results of each test. The columns are table variables containing characteristics of the tests.

Extract the $\mathit{p}$-values of the tests.

pvalues = Summary.PValue

pvalues = 6×1

    0.0698
    0.4648
    0.4398
    0.0025
    0.0708
    0.0074

Test for Block Exogeneity

Time series are block exogenous if they do not Granger-cause any other variables in a multivariate system. Test whether the effective federal funds rate is block exogenous with respect to the real GDP, personal consumption expenditures, and inflation rates.

Load the US macroeconomic data set Data_USEconModel.mat. Convert the price series to returns.

load Data_USEconModel
inflation = price2ret(DataTable.CPIAUCSL);
rgdprate = price2ret(DataTable.GDP./DataTable.GDPDEF);
pcerate = price2ret(DataTable.PCEC);

Test whether the federal funds rate is nonstationary by conducting an augmented Dickey-Fuller test. Specify that the alternative model has a drift term and an $\mathit{F}$ test.

h = adftest(DataTable.FEDFUNDS,'Model',"ard")

h = logical
   0

The test decision h = 0 indicates that the null hypothesis that the series has a unit root should not be rejected.

To stabilize the federal funds rate series, apply the first difference to it.

dfedfunds = diff(DataTable.FEDFUNDS);

Preprocess the data by removing all missing observations (indicated by NaN).

tbl = table(inflation,pcerate,rgdprate,dfedfunds);
tbl = rmmissing(tbl);
T = size(tbl,1); % Total sample size

Assume a 4-D VAR(3) model for the four series. Initialize the model by using the first three observations, and fit the model to the rest of the data. Assign names to the series in the model.

Mdl = varm(4,3);
Mdl.SeriesNames = tbl.Properties.VariableNames;
EstMdl = estimate(Mdl,tbl{4:end,:},'Y0',tbl{1:3,:});

Assess whether the federal funds rate is block exogenous with respect to the real GDP, personal consumption expenditures, and inflation rates. Conduct an $\mathit{F}$-based Wald test, and return the test decision and summary table. Suppress the test results display.

cause = "dfedfunds";
effects = ["inflation" "pcerate" "rgdprate"];
[h,Summary] = gctest(EstMdl,'Type',"blockwise",...
    'Cause',cause,'Effect',effects,'Test',"f",...
    'Display',false);

gctest conducts one test. h = 1 indicates, at a 5% level of significance, rejection of the null hypothesis that the federal funds rate is block exogenous with respect to the other variables in the VAR model. This result suggests that the federal funds rate Granger-causes at least one of the other variables in the system.

Alternatively, you can conduct the same blockwise Granger causality test by passing the data to the gctest function.

causedata = tbl.dfedfunds;
EffectsData = tbl{:,effects};
[hgc,pvalue,stat,cvalue] = gctest(causedata,EffectsData,...
    'NumLags',3,'Test',"f")

hgc = logical
   1

pvalue = 9.0805e-09

stat = 6.9869

cvalue = 1.9265

To determine which variables are Granger-caused by the federal funds rate, conduct a leave-one-out test and specify the "cause" and "effects."

gctest(EstMdl,'Cause',cause,'Effect',effects);

                           H0                            Decision      Distribution    Statistic      PValue      CriticalValue
    ________________________________________________    ___________    ____________    _________    __________    _____________

    "Exclude lagged dfedfunds in inflation equation"    "Reject H0"     "Chi2(3)"       26.157      8.8433e-06       7.8147    
    "Exclude lagged dfedfunds in pcerate equation"      "Reject H0"     "Chi2(3)"       10.151        0.017325       7.8147    
    "Exclude lagged dfedfunds in rgdprate equation"     "Reject H0"     "Chi2(3)"       10.651        0.013772       7.8147    

The test results suggest the following decisions, each at a 5% level of significance:

• Reject the claim that the federal funds rate is not a 1-step Granger-cause of the inflation rate, given all other variables in the VAR model.

• Reject the claim that the federal funds rate is not a 1-step Granger-cause of the personal consumption expenditures rate, given all other variables in the VAR model.

• Reject the claim that the federal funds rate is not a 1-step Granger-cause of the real GDP rate, given all other variables in the VAR model.

Input Arguments

Mdl — VAR model

varm model object

VAR model, specified as a varm model object created by varm or estimate. Mdl must be fully specified.

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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

Example: 'Type',"block-wise",'Cause',1:2,'Effect',3:5 specifies conducting a block-wise test to assess whether the response variables Mdl.SeriesNames(1:2) Granger-cause the response variables Mdl.SeriesNames(3:5) conditioned on all other variables in the model.

'Type' — Granger causality test to conduct

"leave-one-out" (default) | "exclude-all" | "block-wise"

Granger causality test to conduct, specified as the comma-separated pair consisting of 'Type' and a value in this table. Suppose that the VAR model Mdl is m-D (m = Mdl.NumSeries).

Value	Description
"leave-one-out"	Leave-one-out test For j = 1,…,m, k = 1,…,m, and j ≠ k, gctest tests the null hypothesis that variable j does not Granger-cause variable k, conditioned on all other variables in the model. This setting conducts numtests = m(m – 1) tests.
"exclude-all"	Exclude-all test For j = 1,…,m, gctest tests the null hypothesis that all other response variables do not jointly Granger-cause response variable j. This setting conducts numtests = m tests.
"block-wise"	Block-wise test gctest tests the null hypothesis that the response variables specified by Causes do not jointly Granger-cause the response variables specified by Effects. This setting conducts numtests = 1 test. If you do not specify Causes and Effects, then gctest tests the null hypothesis that all lag coefficients, including self lags, are zero.

Example: 'Type',"exclude-all"

Data Types: char | string

'Alpha' — Significance level

0.05 (default) | numeric scalar in (0,1)

Significance level for each conducted test (see Type), specified as the comma-separated pair consisting of 'Alpha' and a numeric scalar in (0,1).

Example: 'Alpha',0.1

Data Types: double | single

'Test' — Test statistic distribution under null hypothesis

"chi-square" (default) | "f"

Test statistic distribution under the null hypothesis, specified as the comma-separated pair consisting of 'Test' and a value in this table.

Value	Description
"chi-square"	gctest derives outputs from conducting a χ2 test.
"f"	gctest derives outputs from conducting an F test.

For test statistic forms, see [4].

Example: 'Test',"f"

Data Types: char | string

'Cause' — VAR model response variables representing Granger-causes

numeric vector of variable indices | vector of variable names

VAR model response variables representing Granger-causes in the 1-step block-wise test, specified as the comma-separated pair consisting of 'Cause' and a numeric vector of variable indices or a vector of variable names.

For either input type, values correspond to the response series names in the SeriesNames property of the input VAR model object Mdl, which you access by using dot notation: Mdl.SeriesNames.

Example: 'Cause',["rGDP" "m1sl"]

Example: 'Cause',1:2

Data Types: single | double | char | string

'Effect' — VAR model response variables affected by Granger-causes

numeric vector of variable indices | vector of variable names

VAR model response variables affected by Granger-causes in the 1-step block-wise test, specified as the comma-separated pair consisting of 'Effect' and a numeric vector of variable indices or a vector of variable names.

For either input type, values correspond to the response series names in the SeriesNames property of the input VAR model object Mdl, which you access by using dot notation: Mdl.SeriesNames.

Example: 'Cause',"inflation"

Example: 'Cause',3

Data Types: single | double | char | string

'Display' — Flag to display test summary table

true (default) | false

Flag to display a test summary table at the command line, specified as the comma-separated pair consisting of 'Display' and a value in this table.

Value	Description
true	Display a test summary table, as returned in Summary, at the command line.
false	Do not display a test summary table.

Example: 'Display',false

Data Types: logical

Output Arguments

h — Granger causality test decisions

logical scalar | logical vector

Granger causality test decisions, returned as a logical scalar or numtests-by-1 logical vector. For j = 1,…,numtests:

• h(j) = 1 indicates that test j rejects the null hypothesis H₀ that the "cause" variables are not 1-step Granger-causes of the "effect" variables. Sufficient evidence exists to support Granger causality and endogeneity of the "cause" variables.

• h(j) = 0 indicates failure to reject H₀.

The number of tests conducted depends on the specified test type (see Type). For more details on the conducted tests, display or return the test summary table (see Display and Summary, respectively).

Summary — Summary of test results

table

Summary of the test results, returned as a table.

Each row of Summary corresponds to one of the numtests conducted tests. The columns describe the characteristics of the tests.

Column Name	Description	Data Type
H0	Granger causality or block exogeneity test null hypothesis description	String scalar
Decision	Test decisions corresponding to h	String scalar
Distribution	Test statistic distribution under the null hypothesis	String scalar
Statistic	Value of test statistic	Numeric scalar
PValue	Test p-value	Numeric scalar
CriticalValue	Critical value for the significance level Alpha	Numeric scalar

More About

Granger Causality Test

The Granger causality test is a statistical hypothesis test that assesses whether past and present values of a set of m₁ time series variables, called the "cause" variables, affect the predictive distribution of a distinct set of m₂ time series variables, called the "effect" variables. The impact is a reduction in forecast mean squared error (MSE) of the "effect" variables. If past values of the "cause" variables affect the "effect" variables h-steps into the forecast horizon, the "cause" variables are h-step Granger-causes of the "effect" variables. If the "cause" variables are h-step Granger-causes of the "effect" variables for all h ≥ 1, the "cause" variables Granger-cause the "effect" variables.

gctest provides block-wise, leave-one-out, and exclude-all variations of the Granger causality tests (see 'Type') and χ²-based or F-based Wald tests (see 'Test'). For test statistic forms, see [4].

For all test types, assume the following conditions:

• Future values cannot inform past values.

• The "cause" variables uniquely inform the "effect" variables. No other variables have the information to inform the "effect" variables.

Block-Wise Test

Let y_1,t denote the m₁ "cause" variables and y_2,t denote the m₂ "effect" variables. Consider a stationary VAR(p) model for [y_1,t y_2,t]:

$$\left[\begin{array}{c}{y}_{1,t}\\ y_{2,t}\end{array}\right]=c+\delta t+\beta x_t+\left[\begin{array}{cc}{\Phi }_{11,1}& {\Phi }_{12,1}\\ {\Phi }_{21,1}& {\Phi }_{22,1}\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ y_{2,t-1}\end{array}\right]+\mathrm{...}+\left[\begin{array}{cc}{\Phi }_{11,p}& {\Phi }_{12,p}\\ {\Phi }_{21,p}& {\Phi }_{22,p}\end{array}\right]\left[\begin{array}{c}{y}_{1,t-p}\\ y_{2,t-p}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\end{array}\right].$$

If Φ_21,1 = … = Φ_21,p = 0_m1,m2, then y_1,t is not the block-wise Granger-cause of y_{2,t + h}, for all h ≥ 1 and where 0_m2,m1 is an m₂-by-m₁ matrix of zeros. Also, y_1,t is block exogenous with respect to y_2,t. Consequently, the block-wise Granger causality test hypotheses are:

$$\begin{array}{l}{H}_0:{\Phi }_{21,1}=\mathrm{...}={\Phi }_{21,p}=0_{m_2,m_1}\\ H_1:\exists j\in \{1,\mathrm{...},p\}\ni {\Phi }_{21,j}\ne 0_{m_2,m_1}.\end{array}$$

H₁ implies that at least one h ≥ 1 exists such that y_1,t is the h-step Granger-cause of y_2,t.

Distinct endogenous variables in the VAR model that are not "causes" or "effects" in the block-wise test are conditioning variables. If conditioning variables exist in the model, h = 1. In other words, gctest tests the null hypothesis of 1-step noncausality.

Leave-One-Out Test

For each response variable and equation in the VAR model, gctest removes lags of a variable from an equation, except self lags, and tests the null hypothesis of 1-step noncausality. Specifically, consider the m-D VAR(p) model

$$\left[\begin{array}{c}{y}_{j,t}\\ \begin{array}{l}{y}_{k,t}\\ y_{s,t}\end{array}\end{array}\right]=c+\delta t+\beta x_t+\left[\begin{array}{ccc}{\varphi }_{11,1}& {\varphi }_{12,1}& {\varphi }_{13,1}\\ {\varphi }_{21,1}& {\varphi }_{22,1}& {\varphi }_{23,1}\\ {\varphi }_{31,1}& {\varphi }_{32,1}& {\Phi }_{33,1}\end{array}\right]\left[\begin{array}{c}{y}_{1,t-1}\\ \begin{array}{l}{y}_{2,t-1}\\ y_{3,t-1}\end{array}\end{array}\right]+\mathrm{...}+\left[\begin{array}{ccc}{\varphi }_{11,p}& {\varphi }_{12,p}& {\varphi }_{13,p}\\ {\varphi }_{21,p}& {\varphi }_{22,p}& {\varphi }_{23,p}\\ {\varphi }_{31,p}& {\varphi }_{32,p}& {\Phi }_{33,p}\end{array}\right]\left[\begin{array}{c}{y}_{j,t-p}\\ \begin{array}{l}{y}_{k,t-p}\\ y_{s,t-p}\end{array}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{j,t}\\ \begin{array}{l}{\epsilon }_{k,t}\\ {\epsilon }_{s,t}\end{array}\end{array}\right],$$

where:

• y_j,t and y_k,t are 1-D series representing the "cause" and "effect" variables, respectively.

• y_s,t is an (m – 2)-D series of all other endogenous variables; s = {1,…,m} \ {j,k}.

• For ℓ = 1,…,p:

  • ϕ_11,ℓ, ϕ_12,ℓ, ϕ_21,ℓ, and ϕ_22,ℓ are scalar lag coefficients.

  • ϕ_13,ℓ, ϕ_31,ℓ, ϕ_23,ℓ, and ϕ_32,ℓ are (m – 2)-D vectors of lag coefficients.

  • Φ_33,ℓ is an (m – 2)-by-(m – 2) matrix of lag coefficients.

For j = 1,…,m, k = 1,…,m, and j ≠ k, gctest tests the null hypothesis that y_j,t is not a 1-step Granger-cause of y_k,t, given y_s,t:

$$\begin{array}{l}{H}_0:{\varphi }_{21,1}=\mathrm{...}={\varphi }_{21,p}=0\\ H_1:\exists r\in \{1,\mathrm{...},p\}\ni {\varphi }_{21,r}\ne 0.\end{array}$$

gctest conducts m(m – 1) tests.

Exclude-All Test

For each equation in the VAR model, gctest removes all lags from the equation, except self lags, and tests for h-step noncausality. Specifically, consider the m-D VAR(p) model

$$\left[\begin{array}{c}{y}_{k,t}\\ y_{-k,t}\end{array}\right]=c+\delta t+\beta x_t+\left[\begin{array}{cc}{\varphi }_{kk,1}& {\varphi }_{k-k,1}\\ {\varphi }_{-kk,1}& {\Phi }_{-k-k,1}\end{array}\right]\left[\begin{array}{c}{y}_{k,t-1}\\ y_{-k,t-1}\end{array}\right]+\mathrm{...}+\left[\begin{array}{cc}{\varphi }_{kk,p}& {\varphi }_{k-k,p}\\ {\varphi }_{-kk,p}& {\Phi }_{-k-k,p}\end{array}\right]\left[\begin{array}{c}{y}_{k,t-p}\\ y_{-k,t-p}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{k,t}\\ {\epsilon }_{-k,t}\end{array}\right],$$

where:

• y_-k,t is an (m – 1)-D series of all endogenous variables in the VAR model (except y_k,t) representing the "cause" variables.

• y_k,t is the 1-D series representing the "effect" variable.

• For ℓ = 1,…,p:

  • ϕ_kk,ℓ is a scalar lag coefficient.

  • ϕ_k-k,ℓ and ϕ_-kk,ℓ are (m – 1)-D vectors of lag coefficients.

  • Φ_-k-k,ℓ is an (m – 1)-by-(m – 1) matrix of lag coefficients.

For k = 1,…,m, gctest tests the null hypothesis that the variables in y_-k,t are not h-step Granger-causes of y_k,t:

$$\begin{array}{l}{H}_0:{\varphi }_{21,1}=\mathrm{...}={\varphi }_{21,p}=0\\ H_1:\exists r\in \{1,\mathrm{...},p\}\ni {\varphi }_{21,r}\ne 0.\end{array}$$

gctest conducts m tests.

Vector Autoregression Model

A vector autoregression (VAR) model is a stationary multivariate time series model consisting of a system of m equations of m distinct response variables as linear functions of lagged responses and other terms.

A VAR(p) model in difference-equation notation and in reduced form is

$$y_t=c+{\Phi }_1{y}_{t-1}+{\Phi }_2{y}_{t-2}+\mathrm{...}+{\Phi }_p{y}_{t-p}+\beta x_t+\delta t+{\epsilon }_t.$$

• y_t is a numseries-by-1 vector of values corresponding to numseries response variables at time t, where t = 1,...,T. The structural coefficient is the identity matrix.

• c is a numseries-by-1 vector of constants.

• Φ_j is a numseries-by-numseries matrix of autoregressive coefficients, where j = 1,...,p and Φ_p is not a matrix containing only zeros.

• x_t is a numpreds-by-1 vector of values corresponding to numpreds exogenous predictor variables.

• β is a numseries-by-numpreds matrix of regression coefficients.

• δ is a numseries-by-1 vector of linear time-trend values.

• ε_t is a numseries-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively a numseries-by-numseries covariance matrix Σ. For t ≠ s, ε_t and ε_s are independent.

Condensed and in lag operator notation, the system is

$$\Phi (L)y_t=c+\beta x_t+\delta t+{\epsilon }_t,$$

where $$\Phi (L)=I-{\Phi }_{1}L-{\Phi }_2{L}^2-\mathrm{...}-{\Phi }_p{L}^p$$, Φ(L)y_t is the multivariate autoregressive polynomial, and I is the numseries-by-numseries identity matrix.

For example, a VAR(1) model containing two response series and three exogenous predictor variables has this form:

$$\begin{array}{l}{y}_{1,t}=c_1+{\varphi }_{11}{y}_{1,t-1}+{\varphi }_{12}{y}_{2,t-1}+{\beta }_{11}{x}_{1,t}+{\beta }_{12}{x}_{2,t}+{\beta }_{13}{x}_{3,t}+{\delta }_{1}t+{\epsilon }_{1,t}\\ y_{2,t}=c_2+{\varphi }_{21}{y}_{1,t-1}+{\varphi }_{22}{y}_{2,t-1}+{\beta }_{21}{x}_{1,t}+{\beta }_{22}{x}_{2,t}+{\beta }_{23}{x}_{3,t}+{\delta }_{2}t+{\epsilon }_{2,t}.\end{array}$$

Tips

• gctest uses the series names in Mdl in test result summaries. To make the output more meaningful for your application, specify series names by setting the SeriesNames property of the VAR model object Mdl by using dot notation before calling gctest. For example, the following code assigns names to the variables in the 3-D VAR model object Mdl:

  Mdl.SeriesNames = ["rGDP" "m1sl" "inflation"];

• The exclude-all and leave-one-out Granger causality tests conduct multiple, simultaneous tests. To control the inevitable increase in the false discovery rate, decrease the level of significance Alpha when you conduct multiple tests. For example, to achieve a family-wise significance level of 0.05, specify 'Alpha',0.05/numtests.

Algorithms

The name-value pair arguments Cause and Effect apply to the block-wise Granger causality test because they specify which equations have lag coefficients set to 0 for the null hypothesis. Because the leave-one-out and exclude-all Granger causality tests cycle through all combinations of variables in the VAR model, the information provided by Cause and Effect is not necessary. However, you can specify a leave-one-out or exclude-all Granger causality test and the Cause and Effect variables to conduct unusual tests such as constraints on self lags. For example, the following code assesses the null hypothesis that the first variable in the VAR model Mdl is not the 1-step Granger-cause of itself:

gctest(Mdl,'Type',"leave-one-out",'Cause',1,'Effect',1);

References

[1] Granger, C. W. J. "Investigating Causal Relations by Econometric Models and Cross-Spectral Methods." Econometrica. Vol. 37, 1969, pp. 424–459.

[2] Hamilton, James. D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.

[3] Dolado, J. J., and H. Lütkepohl. "Making Wald Tests Work for Cointegrated VAR Systems." Econometric Reviews. Vol. 15, 1996, pp. 369–386.

[4] Lütkepohl, H. New Introduction to Multiple Time Series Analysis. New York, NY: Springer-Verlag, 2007.

[5] Toda, H. Y., and T. Yamamoto. "Statistical Inferences in Vector Autoregressions with Possibly Integrated Processes." Journal of Econometrics. Vol. 66, 1995, pp. 225–250.