estimate
Fit vector error-correction (VEC) model to data
Syntax
Description
fits the VEC(p – 1) model EstMdl
= estimate(Mdl
,Tbl1
)Mdl
to variables
in the input table or timetable Tbl1
, which contains time
series data, and returns the fully specified, estimated VEC(p
– 1) model EstMdl
. estimate
selects
the variables in Mdl.SeriesNames
or all variables in
Tbl1
. To select different variables in
Tbl1
to fit the model to, use the
ResponseVariables
name-value argument.
[
returns the estimated, asymptotic standard errors of the estimated parameters EstMdl
,EstSE
,logL
,Tbl2
] = estimate(Mdl
,Tbl1
)EstSE
, the optimized loglikelihood objective function value logL
, and the table or timetable Tbl2
of all variables in Tbl1
and residuals corresponding to the response variables to which the model is fit (ResponseVariables
).
[___] = estimate(___,
specifies options using one or more name-value arguments in
addition to any of the input argument combinations in previous syntaxes.
Name,Value
)estimate
returns the output argument combination for the
corresponding input arguments. For example, estimate(Mdl,Y,Model="H1*",X=Exo)
fits the VEC(p – 1) model Mdl
to the
matrix of response data Y
, and specifies the H1* Johansen
form of the deterministic terms and the matrix of exogenous predictor data
Exo
.
Supply all input data using the same data type. Specifically:
If you specify the numeric matrix
Y
, optional data sets must be numeric arrays and you must use the appropriate name-value argument. For example, to specify a presample, set theY0
name-value argument to a numeric matrix of presample data.If you specify the table or timetable
Tbl1
, optional data sets must be tables or timetables, respectively, and you must use the appropriate name-value argument. For example, to specify a presample, set thePresample
name-value argument to a table or timetable of presample data.
Examples
Fit VEC(1) Model to Matrix of Response Data
Fit a VEC(1) model to seven macroeconomic series. Supply the response data as a numeric matrix.
Consider a VEC model for the following macroeconomic series:
Gross domestic product (GDP)
GDP implicit price deflator
Paid compensation of employees
Nonfarm business sector hours of all persons
Effective federal funds rate
Personal consumption expenditures
Gross private domestic investment
Suppose that a cointegrating rank of 4 and one short-run term are appropriate, that is, consider a VEC(1) model.
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
For more information on the data set and variables, enter Description
at the command line.
Determine whether the data needs to be preprocessed by plotting the series on separate plots.
figure tiledlayout(2,2) nexttile plot(FRED.Time,FRED.GDP); title("Gross Domestic Product"); ylabel("Index"); xlabel("Date"); nexttile plot(FRED.Time,FRED.GDPDEF); title("GDP Deflator"); ylabel("Index"); xlabel("Date"); nexttile plot(FRED.Time,FRED.COE); title("Paid Compensation of Employees"); ylabel("Billions of $"); xlabel("Date"); nexttile plot(FRED.Time,FRED.HOANBS); title("Nonfarm Business Sector Hours"); ylabel("Index"); xlabel("Date");
figure tiledlayout(2,2) nexttile plot(FRED.Time,FRED.FEDFUNDS) title("Federal Funds Rate") ylabel("Percent") xlabel("Date") nexttile plot(FRED.Time,FRED.PCEC) title("Consumption Expenditures") ylabel("Billions of $") xlabel("Date") nexttile plot(FRED.Time,FRED.GPDI) title("Gross Private Domestic Investment") ylabel("Billions of $") xlabel("Date")
Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.
FRED.GDP = 100*log(FRED.GDP); FRED.GDPDEF = 100*log(FRED.GDPDEF); FRED.COE = 100*log(FRED.COE); FRED.HOANBS = 100*log(FRED.HOANBS); FRED.PCEC = 100*log(FRED.PCEC); FRED.GPDI = 100*log(FRED.GPDI);
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = FRED.Properties.VariableNames
Mdl = vecm with properties: Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend" SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more NumSeries: 7 Rank: 4 P: 2 Constant: [7×1 vector of NaNs] Adjustment: [7×4 matrix of NaNs] Cointegration: [7×4 matrix of NaNs] Impact: [7×7 matrix of NaNs] CointegrationConstant: [4×1 vector of NaNs] CointegrationTrend: [4×1 vector of NaNs] ShortRun: {7×7 matrix of NaNs} at lag [1] Trend: [7×1 vector of NaNs] Beta: [7×0 matrix] Covariance: [7×7 matrix of NaNs]
Mdl
is a vecm
model object. All properties containing NaN
values correspond to parameters to be estimated given data.
Estimate the model using the entire data set and the default options.
EstMdl = estimate(Mdl,FRED.Variables)
EstMdl = vecm with properties: Description: "7-Dimensional Rank = 4 VEC(1) Model" SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more NumSeries: 7 Rank: 4 P: 2 Constant: [14.1329 8.77841 -7.20359 ... and 4 more]' Adjustment: [7×4 matrix] Cointegration: [7×4 matrix] Impact: [7×7 matrix] CointegrationConstant: [-28.6082 109.555 -77.0912 ... and 1 more]' CointegrationTrend: [4×1 vector of zeros] ShortRun: {7×7 matrix} at lag [1] Trend: [7×1 vector of zeros] Beta: [7×0 matrix] Covariance: [7×7 matrix]
EstMdl
is an estimated vecm
model object. It is fully specified because all parameters have known values. By default, estimate
imposes the constraints of the H1 Johansen VEC model form by removing the cointegrating trend and linear trend terms from the model. Parameter exclusion from estimation is equivalent to imposing equality constraints to zero.
Display a short summary from the estimation.
results = summarize(EstMdl)
results = struct with fields:
Description: "7-Dimensional Rank = 4 VEC(1) Model"
Model: "H1"
SampleSize: 238
NumEstimatedParameters: 112
LogLikelihood: -1.4939e+03
AIC: 3.2118e+03
BIC: 3.6007e+03
Table: [133x4 table]
Covariance: [7x7 double]
Correlation: [7x7 double]
The Table
field of results
is a table of parameter estimates and corresponding statistics.
Specify Presample Values
Consider the model and data in Fit VEC(1) Model to Matrix of Response Data, and suppose that the estimation sample starts at Q1 of 1980.
Load the Data_USEconVECModel
data set and preprocess the data.
load Data_USEconVECModel
FRED.GDP = 100*log(FRED.GDP);
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);
FRED.HOANBS = 100*log(FRED.HOANBS);
FRED.PCEC = 100*log(FRED.PCEC);
FRED.GPDI = 100*log(FRED.GPDI);
Identify the index corresponding to the start of the estimation sample.
estIdx = FRED.Time(2:end) > '1979-12-31';
Create a default VEC(1) model using the shorthand syntax. Assume that the appropriate cointegration rank is 4. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = FRED.Properties.VariableNames;
Estimate the model using the estimation sample. Specify all observations before the estimation sample as presample data. Also, specify estimation of the H Johansen form of the VEC model, which includes all deterministic parameters.
Y0 = FRED{~estIdx,:}; EstMdl = estimate(Mdl,FRED{estIdx,:},'Y0',Y0,'Model',"H")
EstMdl = vecm with properties: Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend" SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more NumSeries: 7 Rank: 4 P: 2 Constant: [17.5698 3.74759 -20.1998 ... and 4 more]' Adjustment: [7×4 matrix] Cointegration: [7×4 matrix] Impact: [7×7 matrix] CointegrationConstant: [85.4825 -57.3569 -81.7344 ... and 1 more]' CointegrationTrend: [-0.0264185 -0.00275396 -0.0249583 ... and 1 more]' ShortRun: {7×7 matrix} at lag [1] Trend: [0.000514564 -0.000291183 0.00179965 ... and 4 more]' Beta: [7×0 matrix] Covariance: [7×7 matrix]
Because the VEC model order p is 2, estimate
uses only the last two observations (rows) in Y0
as a presample.
Fit VEC Model to Response Variables in Timetable
Fit a VEC(1) model to seven macroeconomic series. Supply a timetable of data and specify the series for the fit. This example is based on Fit VEC(1) Model to Matrix of Response Data.
Load and Preprocess Data
Load the Data_USEconVECModel
data set.
load Data_USEconVECModel
head(FRED)
Time GDP GDPDEF COE HOANBS FEDFUNDS PCEC GPDI ___________ _____ ______ _____ ______ ________ _____ ____ 31-Mar-1957 470.6 16.485 260.6 54.756 2.96 282.3 77.7 30-Jun-1957 472.8 16.601 262.5 54.639 3 284.6 77.9 30-Sep-1957 480.3 16.701 265.1 54.375 3.47 289.2 79.3 31-Dec-1957 475.7 16.711 263.7 53.249 2.98 290.8 71 31-Mar-1958 468.4 16.892 260.2 52.043 1.2 290.3 66.7 30-Jun-1958 472.8 16.94 259.9 51.297 0.93 293.2 65.1 30-Sep-1958 486.7 17.043 267.7 51.908 1.76 298.3 72 31-Dec-1958 500.4 17.123 272.7 52.683 2.42 302.2 80
Stabilize all series, except the federal funds rate, by applying the log transform. Scale the resulting series by 100 so that all series are on the same scale.
FRED.GDP = 100*log(FRED.GDP); FRED.GDPDEF = 100*log(FRED.GDPDEF); FRED.COE = 100*log(FRED.COE); FRED.HOANBS = 100*log(FRED.HOANBS); FRED.PCEC = 100*log(FRED.PCEC); FRED.GPDI = 100*log(FRED.GPDI); numobs = height(FRED)
numobs = 240
Prepare Timetable for Estimation
When you plan to supply a timetable directly to estimate, you must ensure it has all the following characteristics:
All selected response variables are numeric and do not contain any missing values.
The timestamps in the
Time
variable are regular, and they are ascending or descending.
Remove all missing values from the table.
DTT = rmmissing(FRED); numobs = height(DTT)
numobs = 240
DTT
does not contain any missing values.
Determine whether the sampling timestamps have a regular frequency and are sorted.
areTimestampsRegular = isregular(DTT,"quarters")
areTimestampsRegular = logical
0
areTimestampsSorted = issorted(DTT.Time)
areTimestampsSorted = logical
1
areTimestampsRegular = 0
indicates that the timestamps of DTT are irregular. areTimestampsSorted = 1
indicates that the timestamps are sorted. Macroeconomic series in this example are timestamped at the end of the month. This quality induces an irregularly measured series.
Remedy the time irregularity by shifting all dates to the first day of the quarter.
dt = DTT.Time; dt = dateshift(dt,"start","quarter"); DTT.Time = dt; areTimestampsRegular = isregular(DTT,"quarters")
areTimestampsRegular = logical
1
DTT
is regular with respect to time.
Create Model Template for Estimation
Create a VEC(1) model using the shorthand syntax. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = FRED.Properties.VariableNames
Mdl = vecm with properties: Description: "7-Dimensional Rank = 4 VEC(1) Model with Linear Time Trend" SeriesNames: "GDP" "GDPDEF" "COE" ... and 4 more NumSeries: 7 Rank: 4 P: 2 Constant: [7×1 vector of NaNs] Adjustment: [7×4 matrix of NaNs] Cointegration: [7×4 matrix of NaNs] Impact: [7×7 matrix of NaNs] CointegrationConstant: [4×1 vector of NaNs] CointegrationTrend: [4×1 vector of NaNs] ShortRun: {7×7 matrix of NaNs} at lag [1] Trend: [7×1 vector of NaNs] Beta: [7×0 matrix] Covariance: [7×7 matrix of NaNs]
Fit Model to Data
Estimate the model. Pass the entire timetable DTT
. By default, estimate
selects the response variables in Mdl.SeriesNames
to fit to the model. Alternatively, you can use the ResponseVariables
name-value argument.
Return the timetable of residuals and data fit to the model.
[EstMdl,~,~,Tbl2] = estimate(Mdl,DTT);
EstMdl
is an estimated vecm
model object. It is fully specified because all parameters have known values.
Display the head of the table Tbl2
.
head(Tbl2)
Time GDP GDPDEF COE HOANBS FEDFUNDS PCEC GPDI GDP_Residuals GDPDEF_Residuals COE_Residuals HOANBS_Residuals FEDFUNDS_Residuals PCEC_Residuals GPDI_Residuals ___________ ______ ______ ______ ______ ________ ______ ______ _____________ ________________ _____________ ________________ __________________ ______________ ______________ 01-Jul-1957 617.44 281.55 558.01 399.59 3.47 566.71 437.32 0.12076 0.090979 -0.31114 -0.47341 -0.013177 0.14899 1.1764 01-Oct-1957 616.48 281.61 557.48 397.5 2.98 567.26 426.27 -2.4005 -0.39287 -2.1158 -2.1552 -0.86464 -0.89017 -12.289 01-Jan-1958 614.93 282.68 556.15 395.21 1.2 567.09 420.02 -2.0142 0.92195 -1.5874 -1.1852 -1.3247 -0.72797 -4.4964 01-Apr-1958 615.87 282.97 556.03 393.76 0.93 568.09 417.59 0.2131 -0.39586 -0.22658 -0.070487 -0.24993 0.17697 -0.31486 01-Jul-1958 618.76 283.57 558.99 394.95 1.76 569.81 427.67 2.0866 0.45876 2.4738 1.9098 0.98197 1.0195 9.119 01-Oct-1958 621.54 284.04 560.84 396.43 2.42 571.11 438.2 0.68671 0.053454 0.48556 0.63518 0.23659 -0.21548 4.2428 01-Jan-1959 623.66 284.31 563.55 398.35 2.8 573.62 442.12 0.39546 -0.066055 0.97292 1.0224 -0.054929 0.86153 0.68805 01-Apr-1959 626.19 284.46 565.91 400.24 3.39 575.54 449.31 0.24314 -0.22217 0.33889 0.4216 -0.20457 0.26963 -0.15985
Because Mdl.P
is 2
, estimation requires two presample observations. Consequently, estimate
uses the first two rows (first two quarters of 1957) of DTT
as a presample, fits the model to the remaining observations, and returns only those observations used in estimation in Tbl2
.
Plot the residuals.
varnames = Tbl2.Properties.VariableNames; resnames = varnames(contains(Tbl2.Properties.VariableNames,"_Residuals")); figure tiledlayout(3,3) for j = 1:7 nexttile plot(Tbl2.Time,Tbl2{:,resnames(j)}) title(resnames(j),Interpreter="none") grid on end
Include Exogenous Predictor Variables
Consider the model and data in Fit VEC(1) Model to Matrix of Response Data.
Load the Data_USEconVECModel
data set and preprocess the data.
load Data_USEconVECModel
FRED.GDP = 100*log(FRED.GDP);
FRED.GDPDEF = 100*log(FRED.GDPDEF);
FRED.COE = 100*log(FRED.COE);
FRED.HOANBS = 100*log(FRED.HOANBS);
FRED.PCEC = 100*log(FRED.PCEC);
FRED.GPDI = 100*log(FRED.GPDI);
The Data_Recessions
data set contains the beginning and ending serial dates of recessions. Load this data set. Convert the matrix of date serial numbers to a datetime array.
load Data_Recessions dtrec = datetime(Recessions,'ConvertFrom','datenum');
Create a dummy variable that identifies periods in which the U.S. was in a recession or worse. Specifically, the variable should be 1
if FRED.Time
occurs during a recession, and 0
otherwise.
isin = @(x)(any(dtrec(:,1) <= x & x <= dtrec(:,2))); isrecession = double(arrayfun(isin,FRED.Time));
Create a VEC(1) model using the shorthand syntax. Assume that the appropriate cointegration rank is 4. You do not have to specify the presence of a regression component when creating the model. Specify the variable names.
Mdl = vecm(7,4,1); Mdl.SeriesNames = FRED.Properties.VariableNames;
Estimate the model using the entire sample. Specify the predictor identifying whether the observation was measured during a recession. Return the standard errors.
[EstMdl,EstSE] = estimate(Mdl,FRED.Variables,'X',isrecession);
Display the regression coefficient for each equation and the corresponding standard errors.
EstMdl.Beta
ans = 7×1
-1.1975
-0.0187
-0.7530
-0.7094
-0.5932
-0.6835
-4.4839
EstSE.Beta
ans = 7×1
0.1547
0.0581
0.1507
0.1278
0.2471
0.1311
0.7150
EstMdl.Beta
and EstSE.Beta
are 7-by-1 vectors. Rows correspond to response variables in EstMdl.SeriesNames
and columns correspond to predictors.
To check whether the effects of recessions are significant, obtain summary statistics from summarize
, and then display the results for Beta
.
results = summarize(EstMdl);
isbeta = contains(results.Table.Properties.RowNames,'Beta');
betaresults = results.Table(isbeta,:)
betaresults=7×4 table
Value StandardError TStatistic PValue
_________ _____________ __________ __________
Beta(1,1) -1.1975 0.15469 -7.7411 9.8569e-15
Beta(2,1) -0.018738 0.05806 -0.32273 0.7469
Beta(3,1) -0.75305 0.15071 -4.9966 5.8341e-07
Beta(4,1) -0.70936 0.12776 -5.5521 2.8221e-08
Beta(5,1) -0.5932 0.24712 -2.4004 0.016377
Beta(6,1) -0.68353 0.13107 -5.2151 1.837e-07
Beta(7,1) -4.4839 0.715 -6.2712 3.5822e-10
whichsig = EstMdl.SeriesNames(betaresults.PValue < 0.05)
whichsig = 1x6 string
"GDP" "COE" "HOANBS" "FEDFUNDS" "PCEC" "GPDI"
All series except GDPDEF
appear to have a significant recessions effect.
Input Arguments
Mdl
— VEC model
vecm
model object
VEC model containing unknown parameter values, specified as a vecm
model object returned by vecm
.
NaN
-valued elements in properties indicate unknown, estimable parameters. Specified elements indicate equality constraints on parameters in model estimation. The innovations covariance matrix Mdl.Covariance
cannot contain a mix of NaN
values and real numbers; you must fully specify the covariance or it must be completely unknown (NaN(Mdl.NumSeries)
).
Y
— Observed multivariate response series
numeric matrix
Observed multivariate response series to which estimate
fits the
model, specified as a numobs
-by-numseries
numeric
matrix.
numobs
is the sample size. numseries
is the
number of response variables (Mdl.NumSeries
).
Rows correspond to observations, and the last row contains the latest observation.
Columns correspond to individual response variables.
Y
represents the continuation of the presample response series in
Y0
.
Data Types: double
Tbl1
— Time series data
table | timetable
Time series data, to which estimate
fits the model, specified
as a table or timetable with numvars
variables and
numobs
rows.
Each variable is a numeric vector representing a single path of
numobs
observations. You can optionally specify
numseries
response variables to fit to the model by using the
ResponseVariables
name-value argument, and you can specify
numpreds
predictor variables for the exogenous regression
component by using the PredictorVariables
name-value argument.
Each row is an observation, and measurements in each row occur simultaneously.
If Tbl1
is a timetable, it must represent a sample with a regular
datetime time step (see isregular
), and the datetime vector
Tbl1.Time
must be strictly ascending or descending.
If Tbl1
is a table, the following conditions hold:
The last row contains the latest observation.
Tbl1.Properties.RowsNames
must be empty.
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: esimtate(Mdl,Y,Model="H1*",X=Exo)
fits the
VEC(p – 1) model Mdl
to the matrix of
response data Y
, and specifies the H1* Johansen form of the
deterministic terms and the matrix of exogenous predictor data
Exo
.
ResponseVariables
— Variables to select from Tbl1
to treat as response variables y_{t}
string vector | cell vector of character vectors | vector of integers | logical vector
Variables to select from Tbl1
to treat as response variables
y_{t}, specified as one of the following
data types:
String vector or cell vector of character vectors containing
numseries
variable names inTbl1.Properties.VariableNames
A length
numseries
vector of unique indices (integers) of variables to select fromTbl1.Properties.VariableNames
A length
numvars
logical vector, whereResponseVariables(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
, andsum(ResponseVariables)
isnumseries
The selected variables must be numeric vectors and cannot contain missing values
(NaN
).
If the number of variables in Tbl1
matches
Mdl.NumSeries
, the default specifies all variables in
Tbl1
. If the number of variables in Tbl1
exceeds Mdl.NumSeries
, the default matches variables in
Tbl1
to names in Mdl.SeriesNames
.
Example: ResponseVariables=["GDP" "CPI"]
Example: ResponseVariables=[true false true false]
or
ResponseVariable=[1 3]
selects the first and third table
variables as the response variables.
Data Types: double
| logical
| char
| cell
| string
Y0
— Presample response observations
numeric matrix
Presample response observations to initialize the model for estimation, specified as a
numpreobs
-by-numseries
numeric matrix.
numpreobs
is the number of presample observations. Use
Y0
only when you supply a matrix of response data
Y
.
Rows correspond to presample observations, and the last row contains the latest
observation. Y0
must have at least Mdl.P
rows. If
you supply more rows than necessary, estimate
uses the latest
Mdl.P
observations only.
Columns must correspond to the numseries
response variables in
Y
.
By default, estimate
uses Y(1:Mdl.P,:)
as
presample observations, and then fits the model to Y((Mdl.P +
1):end,:)
. This action reduces the effective sample size.
Data Types: double
Presample
— Presample data
table | timetable
Presample data to initialize the model for estimation, specified as a table or
timetable, the same type as Tbl1
, with
numprevars
variables and numpreobs
rows. Use
Presample
only when you supply a table or timetable of data
Tbl1
.
Each variable is a single path of numpreobs
observations
representing the presample of the corresponding variable in
Tbl1
.
Each row is a presample observation, and measurements in each row occur
simultaneously. numpreobs
must be at least Mdl.P
.
If you supply more rows than necessary, estimate
uses the latest
Mdl.P
observations only.
If Presample
is a timetable, all the following conditions must be true:
Presample
must represent a sample with a regular datetime time step (seeisregular
).The inputs
Tbl1
andPresample
must be consistent in time such thatPresample
immediately precedesTbl1
with respect to the sampling frequency and order.The datetime vector of sample timestamps
Presample.Time
must be ascending or descending.
If Presample
is a table, the following conditions hold:
The last row contains the latest presample observation.
Presample.Properties.RowsNames
must be empty.
By default, estimate
uses the first or earliest
Mdl.P
observations in Tbl1
as a presample,
and then it fits the model to the remaining numobs - Mdl.P
observations. This action reduces the effective sample size.
PresampleResponseVariables
— Variables to select from Presample
to use for presample response data
string vector | cell vector of character vectors | vector of integers | logical vector
Variables to select from Presample
to use for presample data,
specified as one of the following data types:
String vector or cell vector of character vectors containing
numseries
variable names inPresample.Properties.VariableNames
A length
numseries
vector of unique indices (integers) of variables to select fromPresample.Properties.VariableNames
A length
numvars
logical vector, wherePresampleResponseVariables(
selects variablej
) = true
fromj
Presample.Properties.VariableNames
, andsum(PresampleResponseVariables)
isnumseries
The selected variables must be numeric vectors and cannot contain missing values
(NaN
).
PresampleResponseNames
does not need to contain the same names as
in Tbl1
; estimate
uses the data in selected
variable PresampleResponseVariables(
as
a presample for
j
)ResponseVariables(
.j
)
The default specifies the same response variables as those selected from
Tbl1
, see ResponseVariables
.
Example: PresampleResponseVariables=["GDP" "CPI"]
Example: PresampleResponseVariables=[true false true false]
or
PresampleResponseVariable=[1 3]
selects the first and third table
variables for presample data.
Data Types: double
| logical
| char
| cell
| string
X
— Predictor data
numeric matrix
Predictor data for the regression component in the model, specified as a numeric
matrix containing numpreds
columns. Use X
only
when you supply a matrix of response data Y
.
numpreds
is the number of predictor variables.
Rows correspond to observations, and the last row contains the latest observation.
estimate
does not use the regression component in the
presample period. X
must have at least as many observations as are
used after the presample period:
If you specify
Y0
,X
must have at leastnumobs
rows (seeY
).Otherwise,
X
must have at leastnumobs
–Mdl.P
observations to account for the presample removal.
In either case, if you supply more rows than is necessary,
estimate
uses the latest observations only.
Columns correspond to individual predictor variables. All predictor variables are present in the regression component of each response equation.
By default, estimate
excludes the regression component,
regardless of its presence in Mdl
.
Data Types: double
PredictorVariables
— Variables to select from Tbl1
to treat as exogenous predictor variables x_{t}
string vector | cell vector of character vectors | vector of integers | logical vector
Variables to select from Tbl1
to treat as exogenous predictor variables
x_{t}, specified as one of the following data types:
String vector or cell vector of character vectors containing
numpreds
variable names inTbl1.Properties.VariableNames
A length
numpreds
vector of unique indices (integers) of variables to select fromTbl1.Properties.VariableNames
A length
numvars
logical vector, wherePredictorVariables(
selects variablej
) = true
fromj
Tbl1.Properties.VariableNames
, andsum(PredictorVariables)
isnumpreds
The selected variables must be numeric vectors and cannot contain missing values
(NaN
).
By default, estimate
excludes the regression component, regardless of its presence in Mdl
.
Example: PredictorVariables=["M1SL" "TB3MS" "UNRATE"]
Example: PredictorVariables=[true false true false]
or
PredictorVariable=[1 3]
selects
the first and third table variables as the response
variables.
Data Types: double
| logical
| char
| cell
| string
Model
— Johansen form of VEC(p – 1) model deterministic terms
"H1"
(default) | "H2"
| "H1*"
| "H*"
| "H"
Johansen form of the VEC(p – 1) model deterministic terms [2], specified as a value in this table (for variable definitions, see Vector Error-Correction Model).
Value | Error-Correction Term | Description |
---|---|---|
"H2" | AB´y_{t − 1} | No intercepts or trends are present in the cointegrating relations, and no deterministic trends are present in the levels of the data. Specify this model only when all response series have a mean of zero. |
"H1*" | A(B´y_{t−1}+c_{0}) | Intercepts are present in the cointegrating relations, and no deterministic trends are present in the levels of the data. |
"H1" | A(B´y_{t−1}+c_{0})+c_{1} | Intercepts are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data. |
"H*" | A(B´y_{t−1}+c_{0}+d_{0}t)+c_{1} | Intercepts and linear trends are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data. |
"H" | A(B´y_{t−1}+c_{0}+d_{0}t)+c_{1}+d_{1}t | Intercepts and linear trends are present in the cointegrating relations, and deterministic quadratic trends are present in the levels of the data. If quadratic trends are not present in the data, this model can produce good in-sample fits but poor out-of-sample forecasts. |
During estimation, if the overall model constant, overall linear trend, cointegrating constant, or cointegrating linear trend parameters are not in the model, then estimate
constrains them to zero. If you specify a different equality constraint, that is, if the properties corresponding to those deterministic terms being constrained to zero have a value other than a vector of NaN
values or zeros, then estimate
issues an error. To enforce supported equality constraints, choose the Johansen model containing the deterministic term that you want to constrain.
Example: Model="H1*"
Data Types: string
| char
Display
— Estimation information display type
"off"
(default) | "table"
| "full"
| character vector
Estimation information display type, specified as a value in this table.
Value | Description |
---|---|
"off" | estimate does not display estimation
information at the command line. |
"table" | estimate displays a table of estimation
information. Rows correspond to parameters, and columns correspond to
estimates, standard errors, t statistics, and
p values. |
"full" | In addition to a table of summary statistics,
estimate displays the estimated innovations
covariance and correlation matrices, loglikelihood value, Akaike
Information Criterion (AIC), Bayesian Information Criterion (BIC), and
other estimation information. |
Example: Display="full"
Data Types: string
| char
MaxIterations
— Maximum number of solver iterations allowed
1000
(default) | positive numeric scalar
Maximum number of solver iterations allowed, specified as a positive numeric scalar.
estimate
dispatches
MaxIterations
to mvregress
.
Example: MaxIterations=2000
Data Types: double
Note
NaN
values inY
,Y0
, andX
indicate missing values.estimate
removes missing values from the data by list-wise deletion.For the presample,
estimate
removes any row containing at least oneNaN
.For the estimation sample,
estimate
removes any row of the concatenated data matrix[Y X]
containing at least oneNaN
.
This type of data reduction reduces the effective sample size.
estimate
issues an error when any table or timetable input contains missing values.
Output Arguments
EstMdl
— Estimated VEC(p – 1) model
vecm
model object
Estimated VEC(p – 1) model, returned as a vecm
model object. EstMdl
is a fully specified vecm
model.
estimate
uses mvregress
to implement multivariate normal, maximum likelihood estimation. For more details, see Estimation of Multivariate Regression Models.
EstSE
— Estimated, asymptotic standard errors of estimated parameters
structure array
Estimated, asymptotic standard errors of the estimated parameters, returned as a structure array containing the fields in this table.
Field | Description |
---|---|
Constant | Standard errors of the overall model constants (c) corresponding to the estimates in EstMdl.Constant , an Mdl.NumSeries -by-1 numeric vector |
Adjustment | Standard errors of the adjustment speeds (A) corresponding to the estimates in EstMdl.Adjustment , an Mdl.NumSeries -by-Mdl.Rank numeric vector |
Impact | Standard errors of the impact coefficient (Π) corresponding to the estimates in EstMdl.Impact , an Mdl.NumSeries -by-Mdl.NumSeries numeric vector |
ShortRun | Standard errors of the short-run coefficients (Φ) corresponding to estimates in EstMdl.ShortRun , a cell vector with elements corresponding to EstMdl.ShortRun |
Beta | Standard errors of regression coefficients (β) corresponding to the estimates in EstMdl.Beta , an Mdl.NumSeries -by-numpreds numeric matrix |
Trend | Standard errors of the overall linear time trends (d) corresponding to the estimates in EstMdl.Trend , an Mdl.NumSeries -by-1 numeric vector |
If estimate
applies equality constraints during estimation by fixing any parameters to a value, then corresponding standard errors of those parameters are 0
.
estimate
extracts all standard errors from the inverse of the expected Fisher information matrix returned by mvregress
(see Standard Errors).
logL
— Optimized loglikelihood objective function value
numeric scalar
Optimized loglikelihood objective function value, returned as a numeric scalar.
E
— Multivariate residuals
numeric matrix
Multivariate residuals from the fitted model EstMdl
, returned as
a numeric matrix containing numseries
columns.
estimate
returns E
only when you supply a
matrix of response data Y
.
If you specify
Y0
, thenE
hasnumobs
rows (seeY
).Otherwise,
E
hasnumobs
–Mdl.P
rows to account for the presample removal.
Tbl2
— Multivariate residuals and estimation data
table | timetable
Multivariate residuals and estimation data, returned as a table or timetable, the same data type as Tbl1
. estimate
returns Tbl2
only when you supply the input Tbl1
.
Tbl2
contains the residuals E
from the model fit to the selected variables in Tbl1
, and it contains all variables in Tbl1
. estimate
names the residuals corresponding to variable
in ResponseJ
Tbl1
. For example, if one of the selected response variables for estimation in ResponseJ
_ResidualsTbl1
is GDP
, Tbl2
contains a variable for the residuals in the response equation of GDP
with the name GDP_Residuals
.
If you specify presample response data, Tbl2
and Tbl1
have the same number of rows, and their rows correspond. Otherwise, because estimate
removes initial observations from Tbl1
for the required presample by default, Tbl2
has numobs - Mdl.P
rows to account for that removal.
If Tbl1
is a timetable, Tbl1
and Tbl2
have the same row order, either ascending or descending.
More About
Vector Error-Correction Model
A vector error-correction (VEC) model is a
multivariate, stochastic time series model consisting of a system of m =
numseries
equations of m distinct, differenced
response variables. Equations in the system can include an error-correction
term, which is a linear function of the responses in levels used to
stabilize the system. The cointegrating rank
r is the number of cointegrating relations that
exist in the system.
Each response equation can include an autoregressive polynomial composed of first differences of the response series (short-run polynomial of degree p – 1), a constant, a time trend, exogenous predictor variables, and a constant and time trend in the error-correction term.
A VEC(p – 1) model in difference-equation notation and in reduced form can be expressed in two ways:
This equation is the component form of a VEC model, where the cointegration adjustment speeds and cointegration matrix are explicit, whereas the impact matrix is implied.
$$\begin{array}{c}\Delta {y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}\\ =c+dt+AB\prime {y}_{t-1}+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}.\end{array}$$
The cointegrating relations are B'y_{t – 1} + c_{0} + d_{0}t and the error-correction term is A(B'y_{t – 1} + c_{0} + d_{0}t).
This equation is the impact form of a VEC model, where the impact matrix is explicit, whereas the cointegration adjustment speeds and cointegration matrix are implied.
$$\begin{array}{c}\Delta {y}_{t}=\Pi {y}_{t-1}+A\left({c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}\\ =c+dt+\Pi {y}_{t-1}+{\Phi}_{1}\Delta {y}_{t-1}+\mathrm{...}+{\Phi}_{p-1}\Delta {y}_{t-(p-1)}+\beta {x}_{t}+{\epsilon}_{t}.\end{array}$$
In the equations:
y_{t} is an m-by-1 vector of values corresponding to m response variables at time t, where t = 1,...,T.
Δy_{t} = y_{t} – y_{t – 1}. The structural coefficient is the identity matrix.
r is the number of cointegrating relations and, in general, 0 < r < m.
A is an m-by-r matrix of adjustment speeds.
B is an m-by-r cointegration matrix.
Π is an m-by-m impact matrix with a rank of r.
c_{0} is an r-by-1 vector of constants (intercepts) in the cointegrating relations.
d_{0} is an r-by-1 vector of linear time trends in the cointegrating relations.
c_{1} is an m-by-1 vector of constants (deterministic linear trends in y_{t}).
d_{1} is an m-by-1 vector of linear time-trend values (deterministic quadratic trends in y_{t}).
c = Ac_{0} + c_{1} and is the overall constant.
d = Ad_{0} + d_{1} and is the overall time-trend coefficient.
Φ_{j} is an m-by-m matrix of short-run coefficients, where j = 1,...,p – 1 and Φ_{p – 1} is not a matrix containing only zeros.
x_{t} is a k-by-1 vector of values corresponding to k exogenous predictor variables.
β is an m-by-k matrix of regression coefficients.
ε_{t} is an m-by-1 vector of random Gaussian innovations, each with a mean of 0 and collectively an m-by-m covariance matrix Σ. For t ≠ s, ε_{t} and ε_{s} are independent.
Condensed and in lag operator notation, the system is
$$\begin{array}{c}\Phi (L)(1-L){y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+\beta {x}_{t}+{\epsilon}_{t}\\ =c+dt+AB\prime {y}_{t-1}+\beta {x}_{t}+{\epsilon}_{t}\end{array}$$
where $$\Phi (L)=I-{\Phi}_{1}-{\Phi}_{2}-\mathrm{...}-{\Phi}_{p-1}$$, I is the m-by-m identity matrix, and Ly_{t} = y_{t – 1}.
If m = r, then the VEC model is a stable VAR(p) model in the levels of the responses. If r = 0, then the error-correction term is a matrix of zeros, and the VEC(p – 1) model is a stable VAR(p – 1) model in the first differences of the responses.
Johansen Form
The Johansen forms of a VEC Model differ with respect to the presence of deterministic terms. As detailed in [2], the estimation procedure differs among the forms. Consequently, allowable equality constraints on the deterministic terms during estimation differ among forms. For more details, see The Role of Deterministic Terms.
This table describes the five Johansen forms and supported equality constraints.
Form | Error-Correction Term | Deterministic Coefficients | Equality Constraints |
---|---|---|---|
H2 | AB´y_{t − 1} | c = 0 (Constant). d = 0 (Trend). c_{0} = 0 (CointegrationConstant). d_{0} = 0 (CointegrationTrend). | You can fully specify B. All deterministic coefficients are zero. |
H1* | A(B´y_{t−1}+c_{0}) | c = Ac_{0}. d = 0. d_{0} = 0. | If you fully specify either B or c_{0}, then you must fully specify the other. MATLAB^{®} derives the value of c from c_{0} and A. All deterministic trends are zero. |
H1 | A(B´y_{t−1} + c_{0}) + c_{1} | c = Ac_{0} + c_{1}. d = 0. d_{0} = 0. | You can fully specify B. You can specify a mixture of MATLAB derives the value of c_{0} from c and A. All deterministic trends are zero. |
H* | A(B´y_{t−1} + c_{0} + d_{0}t) + c_{1} | c = Ac_{0} + c_{1}. d = Ad_{0}. | If you fully specify either B or d_{0}, then you must fully specify the other. You can specify a mixture of MATLAB derives the value of c_{0} from c and A. MATLAB derives the value of d from A and d_{0}. |
H | A(B´y_{t−1}+c_{0}+d_{0}t)+c_{1}+d_{1}t | c = Ac_{0} + c_{1}. d = A.d_{0} + d_{1}. | You can fully specify B. You can specify a mixture of MATLAB derives the values of c_{0} and d_{0} from c, d, and A. |
Algorithms
If 1 ≤
Mdl.Rank
≤Mdl.NumSeries
–1
, as with most VEC models, thenestimate
performs parameter estimation in two steps.estimate
estimates the parameters of the cointegrating relations, including any restricted intercepts and time trends, by the Johansen method [2].The form of the cointegrating relations corresponds to one of the five parametric forms considered by Johansen in [2] (see
'Model'
). For more details, seejcitest
andjcontest
.The adjustment speed parameter (A) and the cointegration matrix (B) in the VEC(p – 1) model cannot be uniquely identified. However, the product Π = A*Bʹ is identifiable. In this estimation step, B = V_{1:r}, where V_{1:r} is the matrix composed of all rows and the first r columns of the eigenvector matrix V. V is normalized so that Vʹ*S_{11}*V = I. For more details, see [2].
estimate
constructs the error-correction terms from the estimated cointegrating relations. Then,estimate
estimates the remaining terms in the VEC model by constructing a vector autoregression (VAR) model in first differences and including the error-correction terms as predictors. For models without cointegrating relations (Mdl.Rank
= 0) or with a cointegrating matrix of full rank (Mdl.Rank
=Mdl.Numseries
),estimate
performs this VAR estimation step only.
You can remove stationary series, which are associated with standard unit vectors in the space of cointegrating relations, from cointegration analysis. To pretest individual series for stationarity, use
adftest
,pptest
,kpsstest
, andlmctest
. As an alternative, you can test for standard unit vectors in the context of the full model by usingjcontest
.If
1
≤Mdl.Rank
≤Mdl.NumSeries
–1
, the asymptotic error covariances of the parameters in the cointegrating relations (which include B, c_{0}, and d_{0} corresponding to theCointegration
,CointegrationConstant
, andCointegrationTrend
properties, respectively) are generally non-Gaussian. Therefore,estimate
does not estimate or return corresponding standard errors.In contrast, the error covariances of the composite impact matrix, which is defined as the product A*Bʹ, are asymptotically Gaussian. Therefore,
estimate
estimates and returns its standard errors. Similar caveats hold for the standard errors of the overall constant and linear trend (A*c_{0} and A*d_{0}corresponding to theConstant
andTrend
properties, respectively) of the H1* and H* Johansen forms.
References
[1] Hamilton, James D. Time Series Analysis. Princeton, NJ: Princeton University Press, 1994.
[2] Johansen, S. Likelihood-Based Inference in Cointegrated Vector Autoregressive Models. Oxford: Oxford University Press, 1995.
[3] Juselius, K. The Cointegrated VAR Model. Oxford: Oxford University Press, 2006.
[4] Lütkepohl, H. New Introduction to Multiple Time Series Analysis. Berlin: Springer, 2005.
Version History
Introduced in R2017b
See Also
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