varm

Create vector autoregression (VAR) model

Description

The varm function returns a varm object specifying the functional form and storing the parameter values of a p-order, stationary, multivariate vector autoregression model (VAR(p)) model.

The key components of a varm object include the number of time series (response-variable dimensionality) and the order of the multivariate autoregressive polynomial (p) because they completely specify the model structure. Other model components include a regression component to associate the same exogenous predictor variables to each response series, and constant and time trend terms. Given the response-variable dimensionality and p, all coefficient matrices and innovation-distribution parameters are unknown and estimable unless you specify their values.

To estimate models containing unknown parameter values, pass the model and data to estimate. To work with an estimated or fully specified varm model object, pass it to an object function. Alternatively, you can create and work with varm model objects interactively by using Econometric Modeler.

Creation

Syntax

Mdl = varm

Mdl = varm(numseries,numlags)

Mdl = varm(Name,Value)

Description

Mdl = varm creates a VAR(0) model composed of one response series.

Mdl = varm(numseries,numlags) creates a VAR(numlags) model composed of numseries response series. The maximum nonzero lag is numlags. All lags have numseries-by-numseries coefficient matrices composed of NaN values.

This shorthand syntax allows for easy model template creation. The model template is suited for unrestricted parameter estimation, that is, estimation without parameter equality constraints. After you create a model, you can alter property values using dot notation.

Mdl = varm(Name,Value) sets properties or additional options using name-value pair arguments. Enclose each name in quotes. For example, 'Lags',[1 4],'AR',AR specifies the two autoregressive coefficient matrices in AR at lags 1 and 4.

This longhand syntax allows for creating more flexible models. varm infers the number of series (NumSeries) and autoregressive polynomial degree (P) from the properties that you set. Therefore, property values that correspond to the number of series or autoregressive polynomial degree must be consistent with each other.

Input Arguments

The shorthand syntax provides an easy way for you to create model templates that are suitable for unrestricted parameter estimation. For example, to create a VAR(2) model composed of three response series, enter:

Mdl = varm(3,2);

numseries — Number of time series m

1 (default) | positive integer

Number of time series m, specified as a positive integer. numseries specifies the dimensionality of the multivariate response variable y_t and innovation ε_t.

numseries sets the NumSeries property.

Data Types: double

numlags — Number of lagged responses

nonnegative integer

Number of lagged responses to include in the model, specified as a nonnegative integer. The resulting model is a VAR(numlags) model. All lags have numseries-by-numseries coefficient matrices composed of NaN values.

Data Types: double

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.

The longhand syntax enables you to create models in which some or all coefficients are known. During estimation, estimate imposes equality constraints on any known parameters.

Example: 'Lags',[4 8] specifies a VAR(8) model with nonzero autoregressive coefficient matrices at lags 4 and 8.

To set values for writable properties, use Name,Value pair argument syntax. For example, 'Constant',[1; 2],'AR',{[0.1 -0.2; -0.3 0.5]} sets Constant to [1; 2] and AR to {[0.1 -0.2; -0.3 0.5]}.

Lags — Autoregressive polynomial lags

1:P (default) | numeric vector of unique positive integers

Autoregressive polynomial lags, specified as the comma-separated pair consisting of 'Lags' and a numeric vector containing at most P elements of unique positive integers.

The lengths of Lags and AR must be equal. Lags(j) is the lag corresponding to the coefficient matrix AR{j}.

Example: 'Lags',[1 4]

Data Types: double

Properties

You can set writable property values when you create the model object by using name-value argument syntax, or after you create the model object by using dot notation. For example, to create a VAR(1) model composed of two response series, and then specify an unknown time trend term, enter:

Mdl = varm('AR',{NaN(2)});
Mdl.Trend = NaN;

NumSeries — Number of time series m

positive integer

This property is read-only.

Number of time series m, specified as a positive integer. NumSeries specifies the dimensionality of the multivariate response variable y_t and innovation ε_t.

Data Types: double

P — Multivariate autoregressive polynomial order

nonnegative integer

This property is read-only.

Multivariate autoregressive polynomial order, specified as a nonnegative integer. P is the maximum lag that has a nonzero coefficient matrix. Lags that are less than P can have coefficient matrices composed entirely of zeros.

P specifies the number of presample observations required to initialize the model.

Data Types: double

Constant — Model intercepts

NaN(NumSeries,1) (default) | numeric vector

Model intercepts (or constants), specified as a NumSeries-by-1 numeric vector.

Example: 'Constant',[1; 2]

Data Types: double

AR — Autoregressive coefficient matrices

cell vector of numeric matrices

Autoregressive coefficient matrices associated with the lagged responses, specified as a cell vector of NumSeries-by-NumSeries numeric matrices.

Specify coefficient signs corresponding to those coefficients in the VAR model expressed in difference-equation notation.

• If you set the Lags name-value pair argument to Lags, then the following conditions apply.

  • The lengths of AR and Lags are equal.

  • AR{j} is the coefficient matrix of lag Lags(j).

  • By default, AR is a numel(Lags)-by-1 cell vector of matrices composed of NaN values.

• Otherwise, the following conditions apply.

  • The length of AR is P.

  • AR{j} is the coefficient matrix of lag j.

  • By default, AR is a P-by-1 cell vector of matrices composed of NaN values.

Example: 'AR',{[0.5 -0.1; 0.1 0.2]}

Data Types: cell

Trend — Linear time trend

zeros(NumSeries,1) (default) | numeric vector

Linear time trend term, specified as a NumSeries-by-1 numeric vector. The default value specifies no linear time trend in the model.

Example: 'Trend',[0.1; 0.2]

Data Types: double

Beta — Regression coefficient matrix

NumSeries-by-0 empty matrix (default) | numeric matrix

Regression coefficient matrix associated with the predictor variables, specified as a NumSeries-by-NumPreds numeric matrix. NumPreds is the number of predictor variables, that is, the number of columns in the predictor data.

Beta(j,:) contains the regression coefficients for each predictor in the equation of response y_j,t. Beta(:,k) contains the regression coefficient in each response equation for predictor x_k. By default, all predictor variables are in the regression component of all response equations. You can exclude certain predictors from certain equations by specifying equality constraints to 0.

Example: In a model that includes 3 responses and 4 predictor variables, to exclude the second predictor from the third equation and leave the others unrestricted, specify [NaN NaN NaN NaN; NaN NaN NaN NaN; NaN 0 NaN NaN].

The default value specifies no regression coefficient in the model. However, if you specify predictor data when you estimate the model using estimate, then MATLAB^® sets Beta to an appropriately sized matrix of NaN values.

Example: 'Beta',[2 3 -1 2; 0.5 -1 -6 0.1]

Data Types: double

Covariance — Innovations covariance matrix

NaN(NumSeries) (default) | numeric, positive definite matrix

Innovations covariance matrix of the NumSeries innovations at each time t = 1,...,T, specified as a NumSeries-by-NumSeries numeric, positive definite matrix.

Example: 'Covariance',eye(2)

Data Types: double

Description — Model description

string scalar | character vector

Model description, specified as a string scalar or character vector. varm stores the value as a string scalar. The default value describes the parametric form of the model, for example "2-Dimensional VAR(3) Model".

Example: 'Description','Model 1'

Data Types: string | char

SeriesNames — Response series names

string vector | cell array of character vectors

Response series names, specified as a NumSeries length string vector. The default is ['Y1' 'Y2' ... 'YNumSeries'].

Example: 'SeriesNames',{'CPI' 'Unemployment'}

Data Types: string

Note

NaN-valued elements in properties indicate unknown, estimable parameters. Specified elements indicate equality constraints on parameters in model estimation. The innovations covariance matrix Covariance cannot contain a mix of NaN values and real numbers; you must fully specify the covariance or it must be completely unknown (NaN(NumSeries)).

Object Functions

estimate	Fit vector autoregression (VAR) model to data
fevd	Generate vector autoregression (VAR) model forecast error variance decomposition (FEVD)
filter	Filter disturbances through vector autoregression (VAR) model
forecast	Forecast vector autoregression (VAR) model responses
gctest	Granger causality and block exogeneity tests for vector autoregression (VAR) models
infer	Infer vector autoregression model (VAR) innovations
irf	Generate vector autoregression (VAR) model impulse responses
simulate	Monte Carlo simulation of vector autoregression (VAR) model
summarize	Display estimation results of vector autoregression (VAR) model
vecm	Convert vector autoregression (VAR) model to vector error-correction (VEC) model

Examples

Create and Modify Default Model

Create a zero-degree VAR model composed of one response series.

Mdl = varm

Mdl = 
  varm with properties:

     Description: "1-Dimensional VAR(0) Model"
     SeriesNames: "Y" 
       NumSeries: 1
               P: 0
        Constant: NaN
              AR: {}
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: NaN

Mdl is a varm model object. It contains one response series, an unknown constant, and an unknown innovation variance. Properties of the model appear at the command line.

Suppose your problem has an autoregressive coefficient at lag 1. To create such a model, set the autoregressive coefficient property (AR) to a cell containing a NaN value using dot notation.

Mdl.AR = {NaN}

Mdl = 
  varm with properties:

     Description: "1-Dimensional VAR(1) Model"
     SeriesNames: "Y" 
       NumSeries: 1
               P: 1
        Constant: NaN
              AR: {NaN} at lag [1]
           Trend: 0
            Beta: [1×0 matrix]
      Covariance: NaN

If your problem contains multiple response series, then use a different varm syntax for model creation.

Create VAR(4) Model Template for Parameter Estimation

Create a VAR(4) model for the consumer price index (CPI) and unemployment rate.

Load the Data_USEconModel data set. Declare variables for the CPI (CPI) and unemployment rate (UNRATE) series.

load Data_USEconModel
cpi = DataTimeTable.CPIAUCSL;
unrate = DataTimeTable.UNRATE;

Create a default VAR(4) model 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. It serves as a template for model estimation. MATLAB® considers any NaN values as unknown parameter values to be estimated. For example, the Constant property is a 2-by-1 vector of NaN values. Therefore, model constants are active model parameters to be estimated.

Include an unknown linear time trend term by setting the Trend property to NaN using dot notation.

Mdl.Trend = NaN

Mdl = 
  varm with properties:

     Description: "2-Dimensional VAR(4) Model with Linear Time Trend"
     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 NaNs]
            Beta: [2×0 matrix]
      Covariance: [2×2 matrix of NaNs]

MATLAB expands NaN to the appropriate length, that is, a 2-by-1 vector of NaN values.

Specify All Parameter Values of VAR Model

Create a VAR model for three arbitrary response series. Specify the parameter values in this system of equations.

$$\begin{array}{l}{y}_{1,t}=1+0.2y_{1,t-1}-0.1y_{2,t-1}+0.5y_{3,t-1}+1.5t+{\epsilon }_{1,t}\\ y_{2,t}=1-0.4y_{1,t-1}+0.5y_{2,t-1}+2t+{\epsilon }_{2,t}\\ y_{3,t}=-0.1y_{1,t-1}+0.2y_{2,t-1}+0.3y_{3,t-1}+{\epsilon }_{3,t}.\end{array}$$

Assume the innovations are multivariate Gaussian with a mean of 0 and the covariance matrix

$$\Sigma =\left[\begin{array}{ccc}0.1& 0.01& 0.3\\ 0.01& 0.5& 0\\ 0.3& 0& 1\end{array}\right].$$

Create variables for the parameter values.

c = [1; 1; 0];
Phi1 = {[0.2 -0.1 0.5; -0.4 0.2 0; -0.1 0.2 0.3]}; 
delta = [1.5; 2; 0];
Sigma = [0.1 0.01 0.3; 0.01 0.5 0; 0.3 0 1];

Create a VAR(1) model object representing the system of dynamic equations using the appropriate name-value pair arguments.

Mdl = varm('Constant',c,'AR',Phi1,'Trend',delta,'Covariance',Sigma)

Mdl = 
  varm with properties:

     Description: "AR-Stationary 3-Dimensional VAR(1) Model with Linear Time Trend"
     SeriesNames: "Y1"  "Y2"  "Y3" 
       NumSeries: 3
               P: 1
        Constant: [1 1 0]'
              AR: {3×3 matrix} at lag [1]
           Trend: [1.5 2 0]'
            Beta: [3×0 matrix]
      Covariance: [3×3 matrix]

Mdl is a fully specified varm model object. By default, varm attributes the autoregressive coefficient to the first lag.

You can adjust model properties using dot notation. For example, consider another VAR model that attributes the autoregressive coefficient matrix Phi1 to the second lag term, specifies a matrix of zeros for the first lag coefficient, and treats all else as being equal to Mdl. Create this VAR(2) model.

Mdl2 = Mdl;
Phi = [zeros(3,3) Phi1];
Mdl2.AR = Phi

Mdl2 = 
  varm with properties:

     Description: "AR-Stationary 3-Dimensional VAR(2) Model with Linear Time Trend"
     SeriesNames: "Y1"  "Y2"  "Y3" 
       NumSeries: 3
               P: 2
        Constant: [1 1 0]'
              AR: {3×3 matrix} at lag [2]
           Trend: [1.5 2 0]'
            Beta: [3×0 matrix]
      Covariance: [3×3 matrix]

Alternatively, you can create another model object using varm and the same syntax as for Mdl, but additionally specify 'Lags',2.

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

Forecast Responses from VAR(4) Model

This example follows from Fit VAR(4) Model to Matrix of Response Data.

Create and estimate a VAR(4) model for the CPI growth rate and unemployment rates. Treat the last ten periods as the forecast horizon.

load Data_USEconModel
cpi = DataTimeTable.CPIAUCSL;
unrate = DataTimeTable.UNRATE;

rcpi = price2ret(cpi);
unrate = unrate(2:end);
Y = [rcpi unrate];

Mdl = varm(2,4);
EstMdl = estimate(Mdl,Y(1:(end-10),:));

Forecast 10 responses using the estimated model and in-sample data as presample observations.

YF = forecast(EstMdl,10,Y(1:(end-10),:));

Plot the part of the series with their forecasted values on separate plots.

figure
plot(DataTimeTable.Time(end - 50:end),rcpi(end - 50:end));
hold on
plot(DataTimeTable.Time((end - 9):end),YF(:,1))
h = gca;
fill(DataTimeTable.Time([end - 9 end end end - 9]),h.YLim([1,1,2,2]),'k',...
    'FaceAlpha',0.1,'EdgeColor','none');
legend('True CPI growth rate','Forecasted CPI growth rate',...
    'Location','NW')
title('Quarterly CPI Growth Rate: 1947 - 2009')
ylabel('CPI Growth Rate')
xlabel('Year')
hold off

figure
plot(DataTimeTable.Time(end - 50:end),unrate(end - 50:end));
hold on
plot(DataTimeTable.Time((end - 9):end),YF(:,2))
h = gca;
fill(DataTimeTable.Time([end - 9 end end end - 9]),h.YLim([1,1,2,2]),'k',...
    'FaceAlpha',0.1,'EdgeColor','none');
legend('True unemployment rate','Forecasted unemployment rate',...
    'Location','NW')
title('Quarterly Unemployment Rate: 1947 - 2009')
ylabel('Unemployment Rate')
xlabel('Year')
hold off

More About

Vector Autoregression Model

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

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

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

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

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

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

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

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

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

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

Condensed and in lag operator notation, the system is

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

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

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

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

Version History

Introduced in R2017a

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.