estimate
Fit vector autoregression (VAR) model to data
Syntax
Description
fits the VAR(p) 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
VAR(p) 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. (since R2022b)
[
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
). (since R2022b)
[___] = 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,Y0=PS,X=Exo)
fits
the VAR(p) model Mdl
to the matrix of
response data Y
, and specifies the matrix of presample
response data PS
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 VAR(4) Model to Matrix of Response Data
Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate series. Supply the response series as a numeric matrix.
Load the Data_USEconModel
data set.
load Data_USEconModel
Plot the two series on separate plots.
figure; plot(DataTimeTable.Time,DataTimeTable.CPIAUCSL); title('Consumer Price Index') ylabel('Index') xlabel('Date')
figure; plot(DataTimeTable.Time,DataTimeTable.UNRATE); title('Unemployment Rate'); ylabel('Percent'); xlabel('Date');
Stabilize the CPI by converting it to a series of growth rates. Synchronize the two series by removing the first observation from the unemployment rate series.
rcpi = price2ret(DataTimeTable.CPIAUCSL); unrate = DataTimeTable.UNRATE(2:end);
Create a default VAR(4) model by using the shorthand syntax.
Mdl = varm(2,4)
Mdl = varm with properties: Description: "2-Dimensional VAR(4) Model" SeriesNames: "Y1" "Y2" NumSeries: 2 P: 4 Constant: [2×1 vector of NaNs] AR: {2×2 matrices of NaNs} at lags [1 2 3 ... and 1 more] Trend: [2×1 vector of zeros] Beta: [2×0 matrix] Covariance: [2×2 matrix of NaNs]
Mdl
is a varm
model object. All properties containing NaN
values correspond to parameters to be estimated given data.
Estimate the model using the entire data set.
EstMdl = estimate(Mdl,[rcpi unrate])
EstMdl = varm with properties: Description: "AR-Stationary 2-Dimensional VAR(4) Model" SeriesNames: "Y1" "Y2" NumSeries: 2 P: 4 Constant: [0.00171639 0.316255]' AR: {2×2 matrices} at lags [1 2 3 ... and 1 more] Trend: [2×1 vector of zeros] Beta: [2×0 matrix] Covariance: [2×2 matrix]
EstMdl
is an estimated varm
model object. It is fully specified because all parameters have known values. The description indicates that the autoregressive polynomial is stationary.
Display summary statistics from the estimation.
summarize(EstMdl)
AR-Stationary 2-Dimensional VAR(4) Model Effective Sample Size: 241 Number of Estimated Parameters: 18 LogLikelihood: 811.361 AIC: -1586.72 BIC: -1524 Value StandardError TStatistic PValue ___________ _____________ __________ __________ Constant(1) 0.0017164 0.0015988 1.0735 0.28303 Constant(2) 0.31626 0.091961 3.439 0.0005838 AR{1}(1,1) 0.30899 0.063356 4.877 1.0772e-06 AR{1}(2,1) -4.4834 3.6441 -1.2303 0.21857 AR{1}(1,2) -0.0031796 0.0011306 -2.8122 0.004921 AR{1}(2,2) 1.3433 0.065032 20.656 8.546e-95 AR{2}(1,1) 0.22433 0.069631 3.2217 0.0012741 AR{2}(2,1) 7.1896 4.005 1.7951 0.072631 AR{2}(1,2) 0.0012375 0.0018631 0.6642 0.50656 AR{2}(2,2) -0.26817 0.10716 -2.5025 0.012331 AR{3}(1,1) 0.35333 0.068287 5.1742 2.2887e-07 AR{3}(2,1) 1.487 3.9277 0.37858 0.705 AR{3}(1,2) 0.0028594 0.0018621 1.5355 0.12465 AR{3}(2,2) -0.22709 0.1071 -2.1202 0.033986 AR{4}(1,1) -0.047563 0.069026 -0.68906 0.49079 AR{4}(2,1) 8.6379 3.9702 2.1757 0.029579 AR{4}(1,2) -0.00096323 0.0011142 -0.86448 0.38733 AR{4}(2,2) 0.076725 0.064088 1.1972 0.23123 Innovations Covariance Matrix: 0.0000 -0.0002 -0.0002 0.1167 Innovations Correlation Matrix: 1.0000 -0.0925 -0.0925 1.0000
Specify Presample Values
Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate data. The estimation sample starts at Q1 of 1980.
Load the Data_USEconModel
data set.
load Data_USEconModel
Stabilize the CPI by converting it to a series of growth rates. Synchronize the two series by removing the first observation from the unemployment rate series.
rcpi = price2ret(DataTimeTable.CPIAUCSL); unrate = DataTimeTable.UNRATE(2:end);
Identify the index corresponding to the start of the estimation sample.
estIdx = DataTimeTable.Time(2:end) > '1979-12-31';
Create a default VAR(4) model by using the shorthand syntax.
Mdl = varm(2,4);
Estimate the model using the estimation sample. Specify all observations before the estimation sample as presample data. Display a full estimation summary.
Y0 = [rcpi(~estIdx) unrate(~estIdx)]; EstMdl = estimate(Mdl,[rcpi(estIdx) unrate(estIdx)],'Y0',Y0,'Display',"full");
AR-Stationary 2-Dimensional VAR(4) Model Effective Sample Size: 117 Number of Estimated Parameters: 18 LogLikelihood: 419.837 AIC: -803.674 BIC: -753.955 Value StandardError TStatistic PValue __________ _____________ __________ __________ Constant(1) 0.003564 0.0024697 1.4431 0.14898 Constant(2) 0.29922 0.11882 2.5182 0.011795 AR{1}(1,1) 0.022379 0.092458 0.24204 0.80875 AR{1}(2,1) -2.6318 4.4484 -0.59163 0.5541 AR{1}(1,2) -0.0082357 0.0020373 -4.0425 5.2884e-05 AR{1}(2,2) 1.2567 0.09802 12.82 1.2601e-37 AR{2}(1,1) 0.20954 0.10182 2.0581 0.039584 AR{2}(2,1) 10.106 4.8987 2.063 0.039117 AR{2}(1,2) 0.0058667 0.003194 1.8368 0.066236 AR{2}(2,2) -0.14226 0.15367 -0.92571 0.35459 AR{3}(1,1) 0.56095 0.098691 5.6839 1.3167e-08 AR{3}(2,1) 0.44406 4.7483 0.093518 0.92549 AR{3}(1,2) 0.0049062 0.003227 1.5204 0.12841 AR{3}(2,2) -0.040037 0.15526 -0.25787 0.7965 AR{4}(1,1) 0.046125 0.11163 0.41321 0.67945 AR{4}(2,1) 6.758 5.3707 1.2583 0.20827 AR{4}(1,2) -0.0030032 0.002018 -1.4882 0.1367 AR{4}(2,2) -0.14412 0.097094 -1.4843 0.13773 Innovations Covariance Matrix: 0.0000 -0.0003 -0.0003 0.0790 Innovations Correlation Matrix: 1.0000 -0.1686 -0.1686 1.0000
Because the VAR model degree p is 4, estimate
uses only the last four observations in Y0
as a presample.
Fit VAR(4) Model to Response Variables in Timetable
Since R2022b
Fit a VAR(4) model to the consumer price index (CPI) and unemployment rate series. Supply a timetable of data and specify the series for the fit.
Load and Preprocess Data
Load the Data_USEconModel
data set. Compute the CPI growth rate. Because the growth rate calculation consumes the earliest observation, include the rate variable in the timetable by prepending the series with NaN
.
load Data_USEconModel
DataTimeTable.RCPI = [NaN; price2ret(DataTimeTable.CPIAUCSL)];
numobs = height(DataTimeTable)
numobs = 249
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, relative to the CPI rate (RCPI
) and unemployment rate (UNRATE
) series.
varnames = ["RCPI" "UNRATE"]; DTT = rmmissing(DataTimeTable,DataVariables=varnames); numobs = height(DTT)
numobs = 245
rmmissing
removes the four initial missing observations from the DataTimeTable
to create a sub-table DTT
. The variables RCPI
and UNRATE
of DTT
do not have any missing observations.
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 default VAR(4) model by using the shorthand syntax. Specify the response variable names.
Mdl = varm(2,4); Mdl.SeriesNames = varnames
Mdl = varm with properties: Description: "2-Dimensional VAR(4) Model" SeriesNames: "RCPI" "UNRATE" NumSeries: 2 P: 4 Constant: [2×1 vector of NaNs] AR: {2×2 matrices of NaNs} at lags [1 2 3 ... and 1 more] Trend: [2×1 vector of zeros] Beta: [2×0 matrix] Covariance: [2×2 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. Summarize the estimated model.
[EstMdl,~,~,Tbl2] = estimate(Mdl,DTT); summarize(EstMdl)
AR-Stationary 2-Dimensional VAR(4) Model Effective Sample Size: 241 Number of Estimated Parameters: 18 LogLikelihood: 811.361 AIC: -1586.72 BIC: -1524 Value StandardError TStatistic PValue ___________ _____________ __________ __________ Constant(1) 0.0017164 0.0015988 1.0735 0.28303 Constant(2) 0.31626 0.091961 3.439 0.0005838 AR{1}(1,1) 0.30899 0.063356 4.877 1.0772e-06 AR{1}(2,1) -4.4834 3.6441 -1.2303 0.21857 AR{1}(1,2) -0.0031796 0.0011306 -2.8122 0.004921 AR{1}(2,2) 1.3433 0.065032 20.656 8.546e-95 AR{2}(1,1) 0.22433 0.069631 3.2217 0.0012741 AR{2}(2,1) 7.1896 4.005 1.7951 0.072631 AR{2}(1,2) 0.0012375 0.0018631 0.6642 0.50656 AR{2}(2,2) -0.26817 0.10716 -2.5025 0.012331 AR{3}(1,1) 0.35333 0.068287 5.1742 2.2887e-07 AR{3}(2,1) 1.487 3.9277 0.37858 0.705 AR{3}(1,2) 0.0028594 0.0018621 1.5355 0.12465 AR{3}(2,2) -0.22709 0.1071 -2.1202 0.033986 AR{4}(1,1) -0.047563 0.069026 -0.68906 0.49079 AR{4}(2,1) 8.6379 3.9702 2.1757 0.029579 AR{4}(1,2) -0.00096323 0.0011142 -0.86448 0.38733 AR{4}(2,2) 0.076725 0.064088 1.1972 0.23123 Innovations Covariance Matrix: 0.0000 -0.0002 -0.0002 0.1167 Innovations Correlation Matrix: 1.0000 -0.0925 -0.0925 1.0000
EstMdl
is an estimated varm
model object. It is fully specified because all parameters have known values.
Display the head of the table Tbl2
.
head(Tbl2)
Time COE CPIAUCSL FEDFUNDS GCE GDP GDPDEF GPDI GS10 HOANBS M1SL M2SL PCEC TB3MS UNRATE RCPI RCPI_Residuals UNRATE_Residuals _____ _____ ________ ________ ____ _____ ______ ____ ____ ______ ____ ____ _____ _____ ______ __________ ______________ ________________ Q1-49 144.1 23.91 NaN 45.6 270 16.531 40.9 NaN 53.961 NaN NaN 177 1.17 5 -0.0058382 -0.013422 0.64674 Q2-49 141.9 23.92 NaN 47.3 266.2 16.35 34 NaN 53.058 NaN NaN 178.6 1.17 6.2 0.00041815 0.0051673 0.6439 Q3-49 141 23.75 NaN 47.2 267.7 16.256 37.3 NaN 52.501 NaN NaN 178 1.07 6.6 -0.0071324 0.0030175 -0.099092 Q4-49 140.5 23.61 NaN 46.6 265.2 16.272 35.2 NaN 52.291 NaN NaN 180.4 1.1 6.6 -0.0059122 -0.001196 -0.0066535 Q1-50 144.6 23.64 NaN 45.6 275.2 16.222 44.4 NaN 52.696 NaN NaN 183.1 1.12 6.3 0.0012698 0.0024607 -0.013354 Q2-50 150.6 23.88 NaN 46.1 284.6 16.286 49.9 NaN 53.997 NaN NaN 187 1.15 5.4 0.010101 0.010823 -0.53098 Q3-50 159 24.34 NaN 45.9 302 16.63 56.1 NaN 55.7 NaN NaN 200.7 1.3 4.4 0.01908 0.012566 -0.38177 Q4-50 166.9 24.98 NaN 49.5 313.4 16.95 65.9 NaN 56.213 NaN NaN 198.1 1.34 4.3 0.025954 0.010998 0.50761
Because the VAR model has degree of 4, estimation requires four presample observations. Consequently, estimate
uses the first four rows (all quarters of 1948) 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.
figure tiledlayout(2,1) nexttile plot(Tbl2.Time,Tbl2.RCPI_Residuals) hold on yline(0,"r--"); hold off title("CPI Rate Residuals") nexttile plot(Tbl2.Time,Tbl2.UNRATE_Residuals) hold on yline(0,"r--"); hold off title("Unemployment Rate Residuals")
Include Exogenous Predictor Variables
Estimate a VAR(4) model of the consumer price index (CPI), unemployment rate, and real gross domestic product (GDP). Include a linear regression component containing the current quarter and the last four quarters of government consumption expenditures and investment (GCE).
Load the Data_USEconModel
data set. Compute the real GDP.
load Data_USEconModel
DataTimeTable.RGDP = DataTimeTable.GDP./DataTimeTable.GDPDEF*100;
Plot all variables on separate plots.
figure tiledlayout(2,2) nexttile plot(DataTimeTable.Time,DataTimeTable.CPIAUCSL); ylabel('Index') title('Consumer Price Index') nexttile plot(DataTimeTable.Time,DataTimeTable.UNRATE); ylabel('Percent') title('Unemployment Rate') nexttile plot(DataTimeTable.Time,DataTimeTable.RGDP); ylabel('Output') title('Real Gross Domestic Product') nexttile plot(DataTimeTable.Time,DataTimeTable.GCE); ylabel('Billions of $') title('Government Expenditures')
Stabilize the CPI, GDP, and GCE series by converting each to a series of growth rates. Synchronize the unemployment rate series with the others by removing its first observation.
inputVariables = {'CPIAUCSL' 'RGDP' 'GCE'}; Data = varfun(@price2ret,DataTimeTable,'InputVariables',inputVariables); Data.Properties.VariableNames = inputVariables; Data.UNRATE = DataTimeTable.UNRATE(2:end);
Expand the GCE rate series to a matrix that includes its current value and up through four lagged values. Remove the GCE
variable from Data
.
rgcelag4 = lagmatrix(Data.GCE,0:4); Data.GCE = [];
Create a default VAR(4) model by using the shorthand syntax. You do not have to specify the regression component when creating the model.
Mdl = varm(3,4);
Estimate the model using the entire sample. Specify the GCE rate matrix as data for the regression component. Extract standard errors and the loglikelihood value.
[EstMdl,EstSE,logL] = estimate(Mdl,Data.Variables,'X',rgcelag4);
Display the regression coefficient matrix.
EstMdl.Beta
ans = 3×5
0.0777 -0.0892 -0.0685 -0.0181 0.0330
0.1450 -0.0304 0.0579 -0.0559 0.0185
-2.8138 -0.1636 0.3905 1.1799 -2.3328
EstMdl.Beta
is a 3-by-5 matrix. Rows correspond to response series, and columns correspond to predictors.
Display the matrix of standard errors corresponding to the coefficient estimates.
EstSE.Beta
ans = 3×5
0.0250 0.0272 0.0275 0.0274 0.0243
0.0368 0.0401 0.0405 0.0403 0.0358
1.4552 1.5841 1.6028 1.5918 1.4145
EstSE.Beta
is commensurate with EstMdl.Beta
.
Display the loglikelihood value.
logL
logL = 1.7056e+03
Input Arguments
Mdl
— VAR model
varm
model object
VAR model containing unknown parameter values, specified as a varm
model object returned by varm
.
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
Since R2022b
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 ascending or descending.
If Tbl1
is a table, the last row contains the latest
observation.
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: estimate(Mdl,Y,Y0=Presample,X=Exo)
fits the
VAR(p) model Mdl
to the matrix of response
data Y
, and specifies the matrix of presample response data
Presample
and the matrix of exogenous predictor data
Exo
.
ResponseVariables
— Variables to select from Tbl1
to treat as response variables yt
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2022b
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 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
Since R2022b
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 last row contains the latest
presample observation.
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
Since R2022b
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
numprevars
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 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 xt
string vector | cell vector of character vectors | vector of integers | logical vector
Since R2022b
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 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 to
supply the predictor data.
Data Types: double
| logical
| char
| cell
| string
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 VAR(p) model
varm
model object
Estimated VAR(p) model, returned as a varm
model object. EstMdl
is a fully specified varm
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 model constants corresponding to the estimates in EstMdl.Constant , a numseries -by-1 numeric vector |
AR | Standard errors of the autoregressive coefficients corresponding to estimates in EstMdl.AR , a cell vector with elements corresponding to EstMdl.AR |
Beta | Standard errors of regression coefficients corresponding to the estimates in EstMdl.Beta , a numseries -by-numpreds numeric matrix |
Trend | Standard errors of linear time trends corresponding to the estimates in EstMdl.Trend , a 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
Since R2022b
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.
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 R2017aR2022b: estimate
accepts input data in tables and timetables, and return results in tables and timetables
In addition to accepting input data in numeric arrays,
estimate
accepts input data in tables and timetables. estimate
chooses default series on which to operate, but you can use the following name-value arguments to select variables.
ResponseVariables
specifies the response series names in the input data to which the model is fit.PredictorVariables
specifies the predictor series names in the input data for a model regression component.Presample
specifies the input table or timetable of presample response data.PresampleResponseVariables
specifies the response series names fromPresample
.
R2019b: Equality constraints on innovations covariance matrix
If you specify a positive definite innovations covariance matrix for the
Covariance
property, estimate
treats your
specification as an equality constraint in model estimation.
See Also
Apps
Objects
Functions
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