gctest
Block-wise Granger causality and block exogeneity tests
Syntax
Description
The gctest
function conducts a block-wise Granger causality test by accepting sets of time series data representing the "cause" and "effect" multivariate response variables in the test. gctest
supports the inclusion of optional endogenous conditioning variables in the model for the test.
To conduct the leave-one-out, exclude-all, and block-wise Granger causality tests on the response variables of a fully specified VAR model (represented by a varm
model object), see gctest
.
returns the test decision from conducting a block-wise Granger causality test for
assessing whether a set of time series variables in the first input numeric array
Granger-causes a distinct set of the time series variables in the second input numeric
array. The h
= gctest(Y1
,Y2
)gctest
function conducts tests in the vector autoregression (VAR)
framework and treats the input arrays as response (endogenous) variables during
testing.
conducts a 1-step Granger causality test for the first two input arrays, conditioned on a
distinct set of time series variables in the third input numeric array. The variable
h
= gctest(Y1
,Y2
,Y3
)Y3
is endogenous in the underlying VAR model, but
gctest
does not consider it a "cause" or an "effect" in the
test.
returns a table containing the test results from conducting a block-wise Granger causality test for
assessing whether all variables, except the last variable, in the input table or timetable
Granger-cause the last variable. (since R2023a)StatTbl
= gctest(Tbl
)
To select different Granger-cause or effect variables for the test, use the
CauseVariables
or EffectVariables
name-value
argument.
___ = gctest(___,
specifies options using one or more name-value arguments. For example,
Name,Value
)gctest(Y1,Y2,Test="f",NumLags=2)
specifies conducting an
F test that compares the residual sum of squares between restricted
and unrestricted VAR(2) models for all response variables.
gctest
returns the output argument combination for the
corresponding input arguments.
Examples
Conduct Granger Causality Test Using Data in Numeric Arrays
Conduct a Granger causality test to assess whether the M1 money supply has an impact on the predictive distribution of the consumer price index (CPI). Supply data as numeric vectors.
Load the US macroeconomic data set Data_USEconModel.mat
.
load Data_USEconModel
The data set includes the MATLAB® timetable DataTimeTable
, which contains 14 variables measured from Q1 1947 through Q1 2009. M1SL
is the table variable containing the M1 money supply, and CPIAUCSL
is the table variable containing the CPI. For more details, enter Description
at the command line.
Visually assess whether the series are stationary by plotting them in the same figure.
figure; yyaxis left plot(DataTimeTable.Time,DataTimeTable.CPIAUCSL) ylabel("CPI"); yyaxis right plot(DataTimeTable.Time,DataTimeTable.M1SL); ylabel("Money Supply");
Both series are nonstationary.
Stabilize the series by converting them to rates.
m1slrate = price2ret(DataTimeTable.M1SL); inflation = price2ret(DataTimeTable.CPIAUCSL);
Assume that a VAR(1) model is an appropriate multivariate model for the rates. Conduct a default Granger causality test to assess whether the M1 money supply rate Granger-causes the inflation rate.
h = gctest(m1slrate,inflation)
h = logical
1
The test decision h
is 1
, which indicates the rejection of the null hypothesis that the M1 money supply rate does not Granger-cause inflation.
Test Series for Feedback
Time series undergo feedback when they Granger-cause each other. Assess whether the US inflation and M1 money supply rates undergo feedback.
Load the US macroeconomic data set Data_USEconModel.mat
. Convert the price series to returns.
load Data_USEconModel
inflation = price2ret(DataTimeTable.CPIAUCSL);
m1slrate = price2ret(DataTimeTable.M1SL);
Conduct a Granger causality test to assess whether the inflation rate Granger-causes the M1 money supply rate. Assume that an underlying VAR(1) model is appropriate for the two series. The default level of significance for the test is 0.05. Because this example conducts two tests, decrease by half for each test to achieve a family-wise level of significance of 0.05.
hIRgcM1 = gctest(inflation,m1slrate,Alpha=0.025)
hIRgcM1 = logical
1
The test decision hIRgcM1
= 1
indicates rejection of the null hypothesis of noncausality. There is enough evidence to suggest that the inflation rate Granger-causes the M1 money supply rate at 0.025 level of significance.
Conduct another Granger causality test to assess whether the M1 money supply rate Granger-causes the inflation rate.
hM1gcIR = gctest(m1slrate,inflation,Alpha=0.025)
hM1gcIR = logical
0
The test decision hM1gcIR
= 0
indicates that the null hypothesis of noncausality should not be rejected. There is not enough evidence to suggest that the M1 money supply rate Granger-causes the inflation rate at 0.025 level of significance.
Because not enough evidence exists to suggest that the inflation rate Granger-causes the M1 money supply rate, the two series do not undergo feedback.
Conduct 1-Step Granger Causality Test Conditioned on Variable
Assess whether the US gross domestic product (GDP) Granger-causes CPI conditioned on the M1 money supply.
Load the US macroeconomic data set Data_USEconModel.mat
.
load Data_USEconModel
The variables GDP
and GDPDEF
of DataTimeTable
are the US GDP and its deflator with respect to year 2000 dollars, respectively. Both series are nonstationary.
Convert the M1 money supply and CPI to rates. Convert the US GDP to the real GDP rate.
m1slrate = price2ret(DataTimeTable.M1SL); inflation = price2ret(DataTimeTable.CPIAUCSL); rgdprate = price2ret(DataTimeTable.GDP./DataTimeTable.GDPDEF);
Assume that a VAR(1) model is an appropriate multivariate model for the rates. Conduct a Granger causality test to assess whether the real GDP rate has an impact on the predictive distribution of the inflation rate, conditioned on the M1 money supply. The inclusion of a conditional variable forces gctest
to conduct a 1-step Granger causality test.
h = gctest(rgdprate,inflation,m1slrate)
h = logical
0
The test decision h
is 0
, which indicates failure to reject the null hypothesis that the real GDP rate is not a 1-step Granger-cause of inflation when you account for the M1 money supply rate.
gctest
includes the M1 money supply rate as a response variable in the underlying VAR(1) model, but it does not include the M1 money supply in the computation of the test statistics.
Conduct the test again, but without conditioning on the M1 money supply rate.
h = gctest(rgdprate,inflation)
h = logical
0
The test result is the same as before, suggesting that the real GDP rate does not Granger-cause inflation at all periods in a forecast horizon and regardless of whether you account for the M1 money supply rate in the underlying VAR(1) model.
Adjust Number of Lags for Underlying VAR Model
By default, gctest
assumes an underlying VAR(1) model for all specified response variables. However, a VAR(1) model might be an inappropriate representation of the data. For example, the model might not capture all the serial correlation present in the variables.
To specify a more complex underlying VAR model, you can increase the number of lags by specifying the NumLags
name-value argument of gctest
.
Consider the Granger causality tests conducted in Conduct 1-Step Granger Causality Test Conditioned on Variable. Load the US macroeconomic data set Data_USEconModel.mat
. Convert the M1 money supply and CPI to rates. Convert the US GDP to the real GDP rate.
load Data_USEconModel
m1slrate = price2ret(DataTimeTable.M1SL);
inflation = price2ret(DataTimeTable.CPIAUCSL);
rgdprate = price2ret(DataTimeTable.GDP./DataTimeTable.GDPDEF);
Preprocess the data by removing all missing observations (indicated by NaN
).
idx = sum(isnan([m1slrate inflation rgdprate]),2) < 1;
m1slrate = m1slrate(idx);
inflation = inflation(idx);
rgdprate = rgdprate(idx);
T = numel(m1slrate); % Total sample size
Fit VAR models, with lags ranging from 1 to 4, to the real GDP and inflation rate series. Initialize each fit by specifying the first four observations. Store the Akaike information criteria (AIC) of the fits.
numseries = 2; 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(1:nummdls) = varm(numseries,0); aic = zeros(nummdls,1); % Fit VAR models to data. Y0 = [rgdprate(idxpre) inflation(idxpre)]; % Presample Y = [rgdprate(idxest) inflation(idxest)]; % Estimation sample for j = 1:numel(numlags) Mdl = varm(numseries,numlags(j)); Mdl.SeriesNames = ["rGDP" "Inflation"]; EstMdl(j) = estimate(Mdl,Y,Y0=Y0); results = summarize(EstMdl(j)); aic(j) = results.AIC; end p = numlags(aic == min(aic))
p = 3
A VAR(3) model yields the best fit.
Assess whether the real GDP rate Granger-causes inflation. gctest
removes observations from the beginning of the input data to initialize the underlying VAR() model for estimation. Prepend only the required = 3 presample observations to the estimation sample. Specify the concatenated series as input data. Return the -value of the test.
rgdprate3 = [Y0((end - p + 1):end,1); Y(:,1)]; inflation3 = [Y0((end - p + 1):end,2); Y(:,2)]; [h,pvalue] = gctest(rgdprate3,inflation3,NumLags=p)
h = logical
1
pvalue = 7.7741e-04
The -value is approximately 0.0008
, indicating the existence of strong evidence to reject the null hypothesis of noncausality, that is, that the three real GDP rate lags in the inflation rate equation are jointly zero. Given the VAR(3) model, there is enough evidence to suggest that the real GDP rate Granger-causes at least one future value of the inflation rate.
Alternatively, you can conduct the same test by passing the estimated VAR(3) model (represented by the varm
model object in EstMdl(3)
), to the object function gctest
. Specify a block-wise test and the "cause" and "effect" series names.
h = gctest(EstMdl(3),Type="block-wise", ... Cause="rGDP",Effect="Inflation")
H0 Decision Distribution Statistic PValue CriticalValue ____________________________________________ ___________ ____________ _________ __________ _____________ "Exclude lagged rGDP in Inflation equations" "Reject H0" "Chi2(3)" 16.799 0.00077741 7.8147
h = logical
1
Address Integrated Series During Testing
If you are testing integrated series for Granger causality, then the Wald test statistic does not follow a or distribution, and test results can be unreliable. However, you can implement the Granger causality test in [5] by specifying the maximum integration order among all the variables in the system using the Integration
name-value argument.
Consider the Granger causality tests conducted in Conduct 1-Step Granger Causality Test Conditioned on Variable. Load the US macroeconomic data set Data_USEconModel.mat
and take the log of real GDP and CPI.
load Data_USEconModel
cpi = log(DataTimeTable.CPIAUCSL);
rgdp = log(DataTimeTable.GDP./DataTimeTable.GDPDEF);
Assess whether the real GDP Granger-causes CPI. Assume the series are , or order-1 integrated. Also, specify an underlying VAR(3) model and the test. Return the test statistic and -value.
[h,pvalue,stat] = gctest(rgdp,cpi,NumLags=3, ... Integration=1,Test="f")
h = logical
1
pvalue = 0.0031
stat = 4.7557
The -value = 0.0031
, indicating the existence of strong evidence to reject the null hypothesis of noncausality, that is, that the three real GDP lags in the CPI equation are jointly zero. Given the VAR(3) model, there is enough evidence to suggest that real GDP Granger-causes at least one future value of the CPI.
In this case, the test augments the VAR(3) model with an additional lag. In other words, the model is a VAR(4) model. However, gctest
tests only whether the first three lags are 0.
Test for Block Exogeneity Using Data in Timetable
Since R2023a
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. Provide data in a timetable.
Load the US macroeconomic data set Data_USEconModel.mat
. Convert the price series to returns; specify durations between sampling times relative to years.
load Data_USEconModel DataTimeTable.RGDP = DataTimeTable.GDP./DataTimeTable.GDPDEF; EffectVariables = ["CPIAUCSL" "RGDP" "PCEC"]; DTTRet = price2ret(DataTimeTable,DataVariables=EffectVariables, ... Units="years");
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 test.
StatTblADFTest = adftest(DataTimeTable,DataVariable="FEDFUNDS",Model="ard")
StatTblADFTest=1×8 table
h pValue stat cValue Lags Alpha Model Test
_____ ________ _______ _______ ____ _____ _______ ______
Test 1 false 0.071419 -2.7257 -2.8751 0 0.05 {'ARD'} {'T1'}
The test decision h
= 0
indicates that the null hypothesis that the series has a unit root should not be rejected, at 0.05 significance level.
To stabilize the federal funds rate series, apply the first difference to it.
DTTRet.DFEDFUNDS = diff(DataTimeTable.FEDFUNDS);
Assume a 4-D VAR(3) model for the four series. Assess whether the federal funds rate is block exogenous with respect to the real GDP, personal consumption expenditures, and inflation rates. Supply the data in the timetable DTTRet
, and conduct an -based Wald test.
StatTblGCTEST = gctest(DTTRet,CauseVariables="DFEDFUNDS", ... EffectVariables=EffectVariables,NumLags=2,Test="f")
StatTblGCTEST=1×6 table
h pValue stat cValue alpha test
_____ __________ ______ ______ _____ ____
Test 1 true 3.9311e-10 10.465 2.1426 0.05 "f"
The test decision hgc
= 1
indicates that the null hypothesis that the federal funds rate is block exogenous should be rejected. This result suggests that the federal funds rate Granger-causes at least one of the other variables in the system.
To determine which variables the federal funds rate Granger-causes, you can run a leave-one-out test. For more details, see gctest
function of the VAR model object varm
.
Input Arguments
Y1
— Data for response variables representing Granger-causes
numeric vector | numeric matrix
Data for the response variables representing the Granger-causes in the test, specified as a numobs1
-by-1 numeric vector or a numobs1
-by-numseries1
numeric matrix. numobs1
is the number of observations and numseries1
is the number of time series variables.
Y1
must have enough rows to initialize and estimate the
underlying VAR model. gctest
uses the first
NumLags
observations to initialize the model for
estimation.
Columns correspond to distinct time series variables.
Data Types: double
| single
Y2
— Data for response variables affected by Granger-causes
numeric vector | numeric matrix
Data for response variables affected by the Granger-causes in the test, specified as a numobs2
-by-1 numeric vector or a numobs2
-by-numseries2
numeric matrix. numobs2
is the number of observations in the data and numseries2
is the number of time series variables.
Y2
must have enough rows to initialize and estimate the
underlying VAR model. gctest
uses the first
NumLags
observations to initialize the model for
estimation.
Columns correspond to distinct time series variables.
Data Types: double
| single
Y3
— Data for conditioning response variables
numeric vector | numeric matrix
Data for conditioning response variables, specified as a numobs3
-by-1 numeric vector or a numobs3
-by-numseries3
numeric matrix. numobs3
is the number of observations in the data and numseries3
is the number of time series variables.
Y3
must have enough rows to initialize and estimate the
underlying VAR model. gctest
uses the first
NumLags
observations to initialize the model for
estimation.
Columns correspond to distinct time series variables.
If you specify Y3
, then Y1
,
Y2
, and Y3
represent the response variables in
the underlying VAR model. gctest
assesses whether
Y1
is a 1-step Granger-cause of Y2
, given the
presence of Y3
.
Data Types: double
| single
Tbl
— Time series data
table | timetable
Since R2023a
Time series data for the Granger-cause
y1,t, effect
y2,t, conditioning
y3,t, and predictor
xt variables of the test, specified as a
table or timetable with numseries
variables and
numobs
rows.
Row t contains the observation in time t, and
the last row contains the latest observation. Tbl
must have enough
rows to initialize and estimate the underlying VAR model.
gctest
uses the first NumLags
observations as a presample to initialize the model for estimation.
You can optionally select Granger-cause, effects, conditioning, or predictor
variables by using the CauseVariables
,
EffectVariables
, ConditionVariables
, or
PredictorVariables
name-value argument, respectively.
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: gctest(Y1,Y2,Alpha=0.10,NumLags=2)
specifies a
0.10
significance level for the test and uses an underlying VAR(2)
model for all response variables.
NumLags
— Number of lagged responses
1
(default) | nonnegative integer
Number of lagged responses to include in the underlying VAR model for all response
variables, specified as a nonnegative integer. The resulting underlying model is a
VAR(NumLags
) model.
Example: NumLags=2
Data Types: double
| single
Integration
— Maximum order of integration
0
(default) | nonnegative integer
Maximum order of integration among all response variables, specified as a nonnegative integer.
To address integration, gctest
augments the VAR(NumLags
) model by adding additional lagged responses beyond NumLags
to all equations during estimation. For more details, see [5] and [3].
Example: Integration=1
Data Types: double
| single
Constant
— Flag indicating inclusion of model intercepts
true
(default) | false
Flag indicating the inclusion of model intercepts (constants) in the underlying VAR model, specified as a value in this table.
Value | Description |
---|---|
true | All equations in the underlying VAR model have an intercept.
gctest estimates the intercepts with all other
estimable parameters. |
false | All underlying VAR model equations do not have an intercept.
gctest sets all intercepts to 0. |
Example: Constant=false
Data Types: logical
Trend
— Flag indicating inclusion of linear time trends
false
(default) | true
Flag indicating the inclusion of linear time trends in the underlying VAR model, specified as a value in this table.
Value | Description |
---|---|
true | All equations in the underlying VAR model have a linear time trend.
gctest estimates the linear time trend
coefficients with all other estimable parameters. |
false | All underlying VAR model equations do not have a linear time trend. |
Example: Trend=false
Data Types: logical
X
— Predictor data for exogenous predictor variables xt
numeric matrix
Predictor data for exogenous predictor variables
xt in the underlying VAR model,
specified as a numeric matrix containing numpreds
columns.
numpreds
is the number of predictor variables.
Row t contains the observation in time t,
and the last row contains the latest observation. gctest
does not use the regression component in the presample period. X
must have at least as many observations as the number of observations used by
gctest
after the presample period. Specifically,
X
must have at least numobs
–
Mdl.P
observations, where numobs
=
min([numobs1 numobs2 numobs3])
. If you supply more rows than
necessary, gctest
uses the latest observations only.
Columns correspond to individual predictor variables.
gctest
treats predictors as exogenous. All predictor
variables are present in the regression component of each response equation.
By default, gctest
excludes a regression component from
all equations.
Example: X=x
applies the numeric vector of data
x
to the 1-D regression component of each response equation of in
the underlying VAR model.
Data Types: double
Alpha
— Significance level
0.05
(default) | numeric scalar in (0,1)
Significance level for the test, specified as a numeric scalar in (0,1).
Example: Alpha=0.1
Data Types: double
| single
Test
— Test statistic distribution under the null hypothesis
"chi-square"
(default) | "f"
Test statistic distribution under the null hypothesis, specified as 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
CauseVariables
— Variables to select from Tbl
to treat as Granger-cause variables y1,t
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2023a
Variables to select from Tbl
to treat as the Granger-cause
variables y1,t, specified
as one of the following data types:
String vector or cell vector of character vectors of variable names in
Tbl.Properties.VariableNames
A vector of unique indices (positive integers) of variables to select from
Tbl.Properties.VariableNames
A logical vector, where
CauseVariables(
selects variablej
) = true
fromj
Tbl.Properties.VariableNames
The selected variables must be numeric vectors and cannot
contain missing values (NaN
s). The variables in
CauseVariables
, EffectVariables
, and
ConditionVariables
, and PredictorVariables
must be
mutually exclusive.
The default specifies all variables in Tbl
except for the
variable specified by the EffectVariables
name-value
argument.
Example: CauseVariables=["GDP" "CPI"]
Example: CauseVariables=[true false true false]
or
CauseVariable=[1 3]
selects the first and third table variables
as the Granger-cause variables.
Data Types: double
| logical
| char
| cell
| string
EffectVariables
— Variables to select from Tbl
to treat as effect variables y2,t
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2023a
Variables to select from Tbl
to treat as the effect variables
y2,t, specified as one
of the following data types:
String vector or cell vector of character vectors of variable names in
Tbl.Properties.VariableNames
A vector of unique indices (positive integers) of variables to select from
Tbl.Properties.VariableNames
A logical vector, where
EffectVariables(
selects variablej
) = true
fromj
Tbl.Properties.VariableNames
The selected variables must be numeric vectors and cannot
contain missing values (NaN
s). The variables in
CauseVariables
, EffectVariables
, and
ConditionVariables
, and PredictorVariables
must be
mutually exclusive.
The default specifies the last variable in Tbl
,
Tbl.Properties.VariableNames(end)
.
Example: EffectVariables="COE"
Example: EffectVariables=[false true false false]
or
EffectVariable=2
selects the second variable as the effect
variable.
Data Types: double
| logical
| char
| cell
| string
ConditionVariables
— Variables to select from Tbl
to treat as conditioning variables y3,t
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2023a
Variables to select from Tbl
to treat as the conditioning
variables y3,t, specified
as one of the following data types:
String vector or cell vector of character vectors of variable names in
Tbl.Properties.VariableNames
A vector of unique indices (positive integers) of variables to select from
Tbl.Properties.VariableNames
A logical vector, where
ConditionVariables(
selects variablej
) = true
fromj
Tbl.Properties.VariableNames
The selected variables must be numeric vectors and cannot
contain missing values (NaN
s). The variables in
CauseVariables
, EffectVariables
, and
ConditionVariables
, and PredictorVariables
must be
mutually exclusive.
If you specify ConditionVariables
, the following conditions apply:
You must specify
CauseVariables
andEffectVariables
.The
CauseVariables
,EffectVariables
, andConditionVariables
represent the response variables in the underlying VAR model.gctest
assesses whetherCauseVariables
is a 1-step Granger-cause ofEffectVariables
, given the presence ofConditionVariables
.
By default, gctest
does not apply conditioning
variables.
Example: ConditionVariables=["GDP" "CPI"]
Example: ConditionVariables=[true true true false]
or
ConditionVariable=4
selects the fourth table variable as the
conditioning variable.
Data Types: double
| logical
| char
| cell
| string
PredictorVariables
— Variables to select from Tbl
to treat as exogenous predictor variables xt
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2023b
Variables to select from Tbl
to treat as exogenous predictor
variables xt in the underlying VAR model,
specified as one of the following data types:
String vector or cell vector of character vectors of variable names in
Tbl.Properties.VariableNames
A vector of unique indices (positive integers) of variables to select from
Tbl.Properties.VariableNames
A logical vector, where
PredictorVariables(
selects variablej
) = true
fromj
Tbl.Properties.VariableNames
The selected variables must be numeric vectors and cannot
contain missing values (NaN
s). The variables in
CauseVariables
, EffectVariables
, and
ConditionVariables
, and PredictorVariables
must be
mutually exclusive.
gctest
does not apply the regression component in the
presample period. All selected variables are present in the regression component of
each response equation.
If you specify PredictorVariables
, you must also specify
CauseVariables
and EffectVariables
.
By default, gctest
excludes a regression component from
all equations.
Example: PredictorVariables=["M1SL" "TB3MS" "UNRATE"]
selects
the variables M1SL
, TB3MS
, and
UNRATE
.
Example: PredictorVariables=[true false true false]
or
=[1 3]
selects the first and third table variables to supply the
predictor data.
Data Types: double
| single
| logical
| char
| cell
| string
Note
gctest
assumes you synchronize the input data sets and
arrange those inputs by increasing sampling time, regardless of data type.
Output Arguments
h
— Block-wise Granger causality test decision
logical scalar
Block-wise Granger causality test
decision, returned as a logical scalar. gctest
returns
h
when you supply data as numeric arrays.
h
=1
indicates rejection of H0.If you specify the conditioning response data
Y3
, then sufficient evidence exists to suggest that the response variables represented inY1
are 1-step Granger-causes of the response variables represented inY2
, conditioned on the response variables represented inY3
.Otherwise, sufficient evidence exists to suggest that the variables in
Y1
are h-step Granger-causes of the variables inY2
for some h ≥ 0. In other words,Y1
is block endogenous with respect toY2
.
h
=0
indicates failure to reject H0.If you specify
Y3
, then the variables inY1
are not 1-step Granger-causes of the variables inY2
, conditioned onY3
.Otherwise,
Y1
does not Granger-causeY2
. In other words, there is not enough evidence to reject block exogeneity ofY1
with respect toY2
.
pvalue
— p-value
numeric scalar
p-value, returned as a numeric scalar.
gctest
returns pValue
when you supply
data as numeric arrays.
stat
— Test statistic
numeric scalar
Test statistic, returned as a numeric scalar. gctest
returns stat
when you supply data as numeric arrays.
cvalue
— Critical value
numeric scalar
Critical value for the significance level Alpha
, returned as a
numeric scalar. gctest
returns cValue
when
you supply data as numeric arrays.
StatTbl
— Test summary
table
Test summary, returned as a table with variables for output rejection decisions
h
, p-values pValue
, test
statistics stat
, and critical values cValue
.
gctest
returns StatTbl
when you supply
the input Tbl
.
StatTbl
also contains
variables for the test settings specified by Alpha
and
Test
. (since R2023b)
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 = numseries1
time series variables y1,t, called the "cause" variables, affect the predictive distribution of a distinct set of m2 = numseries2
time series variables y2,t, called the "effect" variables. The impact is a reduction in forecast mean squared error (MSE) of y2,t. If past values of y1,t affect y2,t + h, then y1,t is an h-step Granger-cause of y2,t. In other words, y1,t
Granger-causes
y2,t if y1,t is an h-step Granger-cause of y2,t for all h ≥ 1.
Consider a stationary VAR(p) model for [y1,t y2,t]:
Assume the following conditions:
Future values cannot inform past values.
y1,t uniquely informs y2,t (no other variable has the information to inform y2,t).
If Φ21,1 = … = Φ21,p = 0m1,m2, then y1,t is not the block-wise Granger cause of y2,t + h, for all h ≥ 1 and where 0m2,m1 is an m2-by-m1 matrix of zeros. Also, y1,t is block exogenous with respect to y2,t. Consequently, the block-wise Granger causality test hypotheses are:
H1 implies that at least one h ≥ 1 exists such that y1,t is an h-step Granger-cause of y2,t.
gctest
conducts χ2-based or F-based Wald tests (see 'Test'
). For test statistic forms, see [4].
Distinct conditioning endogenous variables y3,t can be included in the system (see Y3
). In this case, the VAR(p) model is:
gctest
does not test the parameters associated with the conditioning variables. The test assesses only whether y1,t is an 1-step Granger-cause of y2,t.
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
yt is a
numseries
-by-1 vector of values corresponding tonumseries
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.Φj is a
numseries
-by-numseries
matrix of autoregressive coefficients, where j = 1,...,p and Φp is not a matrix containing only zeros.xt is a
numpreds
-by-1 vector of values corresponding tonumpreds
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.εt is a
numseries
-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively anumseries
-by-numseries
covariance matrix Σ. For t ≠ s, εt and εs are independent.
Condensed and in lag operator notation, the system is
where , Φ(L)yt 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:
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, Helmut. 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.
Version History
Introduced in R2019aR2023b: Choose predictor data from input table or timetable and return test settings
The PredictorVariables
name-value argument enables you to select
exogenous predictor variables from the input table or timetable of data for the underlying
VAR model of the Granger-causality test. The output table of test results additionally
contains the level of significance Alpha
and test type
Test
settings of each test.
R2023a: gctest
accepts input data in tables and timetables, and return results in tables and timetables
In addition to accepting input data in numeric arrays,
gctest
accepts input data in tables and timetables. gctest
chooses default series on which to operate, but you can use the following name-value arguments to select variables.
CauseVariables
selects the Granger-cause variables from the input. By default, all variables not specified as Granger-effect variables are Granger-cause variables.EffectVariables
selects the Granger-effect variables from the input. By default, the last variable is the Granger-effect variable.ConditionVariables
selects the conditioning variables from the input. The default is none of the variables.
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