# estimate

Fit vector error-correction (VEC) model to data

## Description

example

EstMdl = estimate(Mdl,Y) returns a fully specified VEC(p – 1) model. This model stores the estimated parameter values resulting from fitting the VEC(p – 1) model Mdl to all variables (columns) of the matrix of observed multivariate response series Y using maximum likelihood.

example

[EstMdl,EstSE,logL,E] = estimate(Mdl,Y) returns the estimated, asymptotic standard errors of the estimated parameters EstSE, optimized loglikelihood objective function value logL, and the multivariate residuals E.

example

EstMdl = estimate(Mdl,Tbl1) fits the VEC(p – 1) model 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.

example

[EstMdl,EstSE,logL,Tbl2] = estimate(Mdl,Tbl1) returns the estimated, asymptotic standard errors of the estimated parameters 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).

example

[___] = estimate(___,Name,Value) specifies options using one or more name-value arguments in addition to any of the input argument combinations in previous syntaxes. 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 the Y0 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 the Presample name-value argument to a table or timetable of presample data.

## Examples

collapse all

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.

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);
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]
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]'
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.

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.

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]'
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 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.

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]
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.

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

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.

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.

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

collapse all

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)).

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

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.

Variables to select from Tbl1 to treat as response variables yt, specified as one of the following data types:

• String vector or cell vector of character vectors containing numseries variable names in Tbl1.Properties.VariableNames

• A length numseries vector of unique indices (integers) of variables to select from Tbl1.Properties.VariableNames

• A length numvars logical vector, where ResponseVariables(j) = true selects variable j from Tbl1.Properties.VariableNames, and sum(ResponseVariables) is numseries

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

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 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 (see isregular).

• The inputs Tbl1 and Presample must be consistent in time such that Presample immediately precedes Tbl1 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.

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 in Presample.Properties.VariableNames

• A length numseries vector of unique indices (integers) of variables to select from Presample.Properties.VariableNames

• A length numvars logical vector, where PresampleResponseVariables(j) = true selects variable j from Presample.Properties.VariableNames, and sum(PresampleResponseVariables) is numseries

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(j) as a presample for 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

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 least numobs rows (see Y).

• Otherwise, X must have at least numobsMdl.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

Variables to select from Tbl1 to treat as exogenous predictor variables xt, specified as one of the following data types:

• String vector or cell vector of character vectors containing numpreds variable names in Tbl1.Properties.VariableNames

• A length numpreds vector of unique indices (integers) of variables to select from Tbl1.Properties.VariableNames

• A length numvars logical vector, where PredictorVariables(j) = true selects variable j from Tbl1.Properties.VariableNames, and sum(PredictorVariables) is numpreds

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

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).

ValueError-Correction TermDescription
"H2"

AB´yt − 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´yt−1+c0)

Intercepts are present in the cointegrating relations, and no deterministic trends are present in the levels of the data.

"H1"

A(B´yt−1+c0)+c1

Intercepts are present in the cointegrating relations, and deterministic linear trends are present in the levels of the data.

"H*"A(B´yt−1+c0+d0t)+c1

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´yt−1+c0+d0t)+c1+d1t

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

Estimation information display type, specified as a value in this table.

ValueDescription
"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

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 in Y, Y0, and X indicate missing values. estimate removes missing values from the data by list-wise deletion.

• For the presample, estimate removes any row containing at least one NaN.

• For the estimation sample, estimate removes any row of the concatenated data matrix [Y X] containing at least one NaN.

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

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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.

Estimated, asymptotic standard errors of the estimated parameters, returned as a structure array containing the fields in this table.

FieldDescription
ConstantStandard errors of the overall model constants (c) corresponding to the estimates in EstMdl.Constant, an Mdl.NumSeries-by-1 numeric vector
AdjustmentStandard errors of the adjustment speeds (A) corresponding to the estimates in EstMdl.Adjustment, an Mdl.NumSeries-by-Mdl.Rank numeric vector
ImpactStandard errors of the impact coefficient (Π) corresponding to the estimates in EstMdl.Impact, an Mdl.NumSeries-by-Mdl.NumSeries numeric vector
ShortRunStandard errors of the short-run coefficients (Φ) corresponding to estimates in EstMdl.ShortRun, a cell vector with elements corresponding to EstMdl.ShortRun
BetaStandard errors of regression coefficients (β) corresponding to the estimates in EstMdl.Beta, an Mdl.NumSeries-by-numpreds numeric matrix
TrendStandard 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).

Optimized loglikelihood objective function value, returned as a numeric scalar.

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, then E has numobs rows (see Y).

• Otherwise, E has numobsMdl.P rows to account for the presample removal.

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 ResponseJ in Tbl1 ResponseJ_Residuals. For example, if one of the selected response variables for estimation in Tbl1 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.

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### 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.

$\begin{array}{c}\Delta {y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\epsilon }_{t}\\ =c+dt+AB\prime {y}_{t-1}+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\epsilon }_{t}.\end{array}$

The cointegrating relations are B'yt – 1 + c0 + d0t and the error-correction term is A(B'yt – 1 + c0 + d0t).

• 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.

$\begin{array}{c}\Delta {y}_{t}=\Pi {y}_{t-1}+A\left({c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\epsilon }_{t}\\ =c+dt+\Pi {y}_{t-1}+{\Phi }_{1}\Delta {y}_{t-1}+...+{\Phi }_{p-1}\Delta {y}_{t-\left(p-1\right)}+\beta {x}_{t}+{\epsilon }_{t}.\end{array}$

In the equations:

• yt is an m-by-1 vector of values corresponding to m response variables at time t, where t = 1,...,T.

• Δyt = ytyt – 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.

• c0 is an r-by-1 vector of constants (intercepts) in the cointegrating relations.

• d0 is an r-by-1 vector of linear time trends in the cointegrating relations.

• c1 is an m-by-1 vector of constants (deterministic linear trends in yt).

• d1 is an m-by-1 vector of linear time-trend values (deterministic quadratic trends in yt).

• c = Ac0 + c1 and is the overall constant.

• d = Ad0 + d1 and is the overall time-trend coefficient.

• Φj is an m-by-m matrix of short-run coefficients, where j = 1,...,p – 1 and Φp – 1 is not a matrix containing only zeros.

• xt is a k-by-1 vector of values corresponding to k exogenous predictor variables.

• β is an m-by-k matrix of regression coefficients.

• εt 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 ts, εt and εs are independent.

Condensed and in lag operator notation, the system is

$\begin{array}{c}\Phi \left(L\right)\left(1-L\right){y}_{t}=A\left(B\prime {y}_{t-1}+{c}_{0}+{d}_{0}t\right)+{c}_{1}+{d}_{1}t+\beta {x}_{t}+{\epsilon }_{t}\\ =c+dt+AB\prime {y}_{t-1}+\beta {x}_{t}+{\epsilon }_{t}\end{array}$

where $\Phi \left(L\right)=I-{\Phi }_{1}-{\Phi }_{2}-...-{\Phi }_{p-1}$, I is the m-by-m identity matrix, and Lyt = yt – 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.

FormError-Correction TermDeterministic CoefficientsEquality Constraints
H2

AB´yt − 1

c = 0 (Constant).

d = 0 (Trend).

c0 = 0 (CointegrationConstant).

d0 = 0 (CointegrationTrend).

You can fully specify B.

All deterministic coefficients are zero.

H1*

A(B´yt−1+c0)

c = Ac0.

d = 0.

d0 = 0.

If you fully specify either B or c0, then you must fully specify the other.

MATLAB® derives the value of c from c0 and A.

All deterministic trends are zero.

H1

A(B´yt−1 + c0) + c1

c = Ac0 + c1.

d = 0.

d0 = 0.

You can fully specify B.

You can specify a mixture of NaN and numeric values for c.

MATLAB derives the value of c0 from c and A.

All deterministic trends are zero.

H*

A(B´yt−1 + c0 + d0t) + c1

c = Ac0 + c1.

If you fully specify either B or d0, then you must fully specify the other.

You can specify a mixture of NaN and numeric values for c.

MATLAB derives the value of c0 from c and A.

MATLAB derives the value of d from A and d0.

H

A(B´yt−1+c0+d0t)+c1+d1t

c = Ac0 + c1.

d = A.d0 + d1.

You can fully specify B.

You can specify a mixture of NaN and numeric values for c and d.

MATLAB derives the values of c0 and d0 from c, d, and A.

## Algorithms

• If 1 ≤ Mdl.RankMdl.NumSeries1, as with most VEC models, then estimate performs parameter estimation in two steps.

1. 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, see jcitest and jcontest.

• 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 = V1:r, where V1:r is the matrix composed of all rows and the first r columns of the eigenvector matrix V. V is normalized so that Vʹ*S11*V = I. For more details, see [2].

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, and lmctest. As an alternative, you can test for standard unit vectors in the context of the full model by using jcontest.

• If 1Mdl.RankMdl.NumSeries1, the asymptotic error covariances of the parameters in the cointegrating relations (which include B, c0, and d0 corresponding to the Cointegration, CointegrationConstant, and CointegrationTrend 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*c0 and A*d0corresponding to the Constant and Trend 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