conjugatebvarm

Bayesian vector autoregression (VAR) model with conjugate prior for data likelihood

Since R2020a

Description

The Bayesian VAR model object conjugatebvarm specifies the joint prior or posterior distribution of the array of model coefficients Λ and the innovations covariance matrix Σ of an m-D VAR(p) model. The joint prior distribution (Λ,Σ) is the dependent, matrix-normal-inverse-Wishart conjugate model.

In general, when you create a Bayesian VAR model object, it specifies the joint prior distribution and characteristics of the VARX model only. That is, the model object is a template intended for further use. Specifically, to incorporate data into the model for posterior distribution analysis, pass the model object and data to the appropriate object function.

Creation

Syntax

PriorMdl = conjugatebvarm(numseries,numlags)

PriorMdl = conjugatebvarm(numseries,numlags,Name,Value)

Description

To create a conjugatebvarm object, use either the conjugatebvarm function (described here) or the bayesvarm function. The syntaxes for each function are similar, but the options differ. bayesvarm enables you to set prior hyperparameter values for Minnesota prior[1] regularization easily, whereas conjugatebvarm requires the entire specification of prior distribution hyperparameters.

PriorMdl = conjugatebvarm(numseries,numlags) creates a numseries-D Bayesian VAR(numlags) model object PriorMdl, which specifies dimensionalities and prior assumptions for all model coefficients $$\Lambda ={\left[\begin{array}{ccccccc}{\Phi }_1& {\Phi }_2& \cdots & {\Phi }_p& c& \delta & {\rm B}\end{array}\right]}^{\prime }$$ and the innovations covariance Σ, where:

• numseries = m, the number of response time series variables.

• numlags = p, the AR polynomial order.

• The joint prior distribution of (Λ,Σ) is the dependent, matrix-normal-inverse-Wishart conjugate model.

PriorMdl = conjugatebvarm(numseries,numlags,Name,Value) sets writable properties (except NumSeries and P) using name-value pair arguments. Enclose each property name in quotes. For example, conjugatebvarm(3,2,'SeriesNames',["UnemploymentRate" "CPI" "FEDFUNDS"]) specifies the names of the three response variables in the Bayesian VAR(2) model.

Input Arguments

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 in each equation of y_t, specified as a nonnegative integer. The resulting model is a VAR(numlags) model; each lag has a numseries-by-numseries coefficient matrix.

numlags sets the P property.

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 3-D Bayesian VAR(1) model and label the first through third response variables, and then include a linear time trend term, enter:

PriorMdl = conjugatebvarm(3,1,'SeriesNames',["UnemploymentRate" "CPI" "FEDFUNDS"]);
PriorMdl.IncludeTrend = true;

Model Characteristics and Dimensionality

Description — Model description

string scalar | character vector

Model description, specified as a string scalar or character vector. The default value describes the model dimensionality, for example '2-Dimensional VAR(3) Model'.

Example: "Model 1"

Data Types: string | char

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.

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

Data Types: double

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']. conjugatebvarm stores SeriesNames as a string vector.

Example: ["UnemploymentRate" "CPI" "FEDFUNDS"]

Data Types: string

IncludeConstant — Flag for including model constant c

true (default) | false

Flag for including a model constant c, specified as a value in this table.

Value	Description
false	Response equations do not include a model constant.
true	All response equations contain a model constant.

Data Types: logical

IncludeTrend — Flag for including linear time trend term δt

false (default) | true

Flag for including a linear time trend term δt, specified as a value in this table.

Value	Description
false	Response equations do not include a linear time trend term.
true	All response equations contain a linear time trend term.

Data Types: logical

NumPredictors — Number of exogenous predictor variables in model regression component

0 (default) | nonnegative integer

Number of exogenous predictor variables in the model regression component, specified as a nonnegative integer. conjugatebvarm includes all predictor variables symmetrically in each response equation.

Distribution Hyperparameters

Mu — Mean of vectorized matrix normal prior on Λ

zeros(NumSeries*(NumSeries*P + IncludeIntercept + IncludeTrend + NumPredictors),1) (default) | numeric vector

Mean of the vectorized matrix normal prior on Λ, specified as a NumSeries*k-by-1 numeric vector, where k = NumSeries*P + IncludeIntercept + IncludeTrend + NumPredictors (the number of coefficients in a response equation).

Mu(1:k) corresponds to all coefficients in the equation of response variable SeriesNames(1), Mu((k + 1):(2*k)) corresponds to all coefficients in the equation of response variable SeriesNames(2), and so on. For a set of indices corresponding to an equation:

• Elements 1 through NumSeries correspond to the lag 1 AR coefficients of the response variables ordered by SeriesNames.

• Elements NumSeries + 1 through 2*NumSeries correspond to the lag 2 AR coefficients of the response variables ordered by SeriesNames.

• In general, elements (q – 1)*NumSeries + 1 through q*NumSeries corresponds to the lag q AR coefficients of the response variables ordered by SeriesNames.

• If IncludeConstant is true, element NumSeries*P + 1 is the model constant.

• If IncludeTrend is true, element NumSeries*P + 2 is the linear time trend coefficient.

• If NumPredictors > 0, elements NumSeries*P + 3 through k constitute the vector of regression coefficients of the exogenous variables.

This figure shows the structure of the transpose of Mu for a 2-D VAR(3) model that contains a constant vector and four exogenous predictors:

$$[\stackrel{y_{1,t}}{\overbrace{\begin{array}{ccccccccccc}{\varphi }_{1,11}& {\varphi }_{1,12}& {\varphi }_{2,11}& {\varphi }_{2,12}& {\varphi }_{3,11}& {\varphi }_{3,12}& c_1& {\beta }_{11}& {\beta }_{12}& {\beta }_{13}& {\beta }_{14}\end{array}}}\stackrel{y_{2,t}}{\overbrace{\begin{array}{ccccccccccc}{\varphi }_{1,21}& {\varphi }_{1,22}& {\varphi }_{2,21}& {\varphi }_{2,22}& {\varphi }_{3,21}& {\varphi }_{3,22}& c_2& {\beta }_{21}& {\beta }_{22}& {\beta }_{23}& {\beta }_{24}\end{array}}}],$$

where

• ϕ_q,jk is element (j,k) of the lag q AR coefficient matrix.

• c_j is the model constant in the equation of response variable j.

• B_ju is the regression coefficient of the exogenous variable u in the equation of response variable j.

Tip

bayesvarm enables you to specify Mu easily by using the Minnesota regularization method. To specify Mu directly:

1. Set separate variables for the prior mean of each coefficient matrix and vector.

2. Horizontally concatenate all coefficient means in this order:

   $$Coeff=\left[\begin{array}{ccccccc}{\Phi }_1& {\Phi }_2& \cdots & {\Phi }_p& c& \delta & {\rm B}\end{array}\right].$$

3. Vectorize the transpose of the coefficient mean matrix.

   Mu = Coeff.';
   Mu = Mu(:);

Data Types: double

V — Scaled conditional covariance matrix of vectorized matrix normal prior on Λ

eye(NumSeries*P + IncludeIntercept + IncludeTrend + NumPredictors) (default) | positive definite numeric matrix

Scaled conditional covariance matrix of vectorized matrix normal prior on Λ, specified as a k-by-k symmetric, positive definite matrix, where k = NumSeries*P + IncludeIntercept + IncludeTrend + NumPredictors (the number of coefficients in a response equation).

Row and column indices correspond to all model coefficients relative to the coefficients in the equation of the first response variable y_1,t (for more details, see Algorithms).

• Elements 1 through NumSeries correspond to the lag 1 AR coefficients of the response variables ordered by SeriesNames.

• Elements NumSeries + 1 through 2*NumSeries correspond to the lag 2 AR coefficients of the response variables ordered by SeriesNames.

• In general, elements (q – 1)*NumSeries + 1 through q*NumSeries correspond to the lag q AR coefficients of the response variables ordered by SeriesNames.

• Element NumSeries*P + 1 is the model constant.

• Element NumSeries*P + 2 is the linear time trend coefficient.

• Element NumSeries*P + 3 through k constitute the vector of regression coefficients of the exogenous variables.

For example, consider a 3-D VAR(2) model containing a constant and four exogenous variables.

• V(1,1) is Var(ϕ_1,11), Var(ϕ_1,21), and Var(ϕ_1,31).

• V(1,4) is Cov(ϕ_1,11,ϕ_2,11), Cov(ϕ_1,21,ϕ_2,21), and Cov(ϕ_1,31,ϕ_2,31).

• V(8,9) is Cov(β₁₁,β₁₂), Cov(β₂₁,β₂₂), and Cov(β₃₁,β₃₂), which are the covariances of the regression coefficients of the first and second exogenous variables for all equations.

Tip

bayesvarm enables you to create any Bayesian VAR prior model and specify V easily by using the Minnesota regularization method.

Data Types: double

Omega — Inverse Wishart scale matrix

eye(numseries) (default) | positive definite numeric matrix

Inverse Wishart scale matrix, specified as a NumSeries-by-NumSeries positive definite numeric matrix.

Data Types: double

DoF — Inverse Wishart degrees of freedom

numseries + 10 (default) | positive numeric scalar

Inverse Wishart degrees of freedom, specified as a positive numeric scalar.

For a proper distribution, specify a value that is greater than numseries – 1. For a distribution with a finite mean, specify a value that is greater than numseries + 1.

Data Types: double

VAR Model Parameters Derived from Distribution Hyperparameters

AR — Distribution mean of autoregressive coefficient matrices Φ₁,…,Φ_p

cell vector of numeric matrices

This property is read-only.

Distribution mean of the autoregressive coefficient matrices Φ₁,…,Φ_p associated with the lagged responses, specified as a P-D cell vector of NumSeries-by-NumSeries numeric matrices.

AR{j} is Φ_j, the coefficient matrix of lag j. Rows correspond to equations and columns correspond to lagged response variables; SeriesNames determines the order of response variables and equations. Coefficient signs are those of the VAR model expressed in difference-equation notation.

If P = 0, AR is an empty cell. Otherwise, AR is the collection of AR coefficient means extracted from Mu.

Data Types: cell

Constant — Distribution mean of model constant c

numeric vector

This property is read-only.

Distribution mean of the model constant c (or intercept), specified as a NumSeries-by-1 numeric vector. Constant(j) is the constant in equation j; SeriesNames determines the order of equations.

If IncludeConstant = false, Constant is an empty array. Otherwise, Constant is the model constant vector mean extracted from Mu.

Data Types: double

Trend — Distribution mean of linear time trend δ

numeric vector

This property is read-only.

Distribution mean of the linear time trend δ, specified as a NumSeries-by-1 numeric vector. Trend(j) is the linear time trend in equation j; SeriesNames determines the order of equations.

If IncludeTrend = false (the default), Trend is an empty array. Otherwise, Trend is the linear time trend coefficient mean extracted from Mu.

Data Types: double

Beta — Distribution mean of regression coefficient matrix Β

numeric matrix

This property is read-only.

Distribution mean of the regression coefficient matrix B associated with the exogenous predictor variables, specified as a NumSeries-by-NumPredictors numeric matrix.

Beta(j,:) contains the regression coefficients of each predictor in the equation of response variable j y_j,t. Beta(:,k) contains the regression coefficient in each equation of predictor x_k. By default, all predictor variables are in the regression component of all response equations. You can down-weight a predictor from an equation by specifying, for the corresponding coefficient, a prior mean of 0 in Mu and a small variance in V.

When you create a model, the predictor variables are hypothetical. You specify predictor data when you operate on the model (for example, when you estimate the posterior by using estimate). Columns of the predictor data determine the order of the columns of Beta.

Data Types: double

Covariance — Distribution mean of innovations covariance matrix Σ

positive definite numeric matrix

This property is read-only.

Distribution mean of the innovations covariance matrix Σ of the NumSeries innovations at each time t = 1,...,T, specified as a NumSeries-by-NumSeries positive definite numeric matrix. Rows and columns correspond to innovations in the equations of the response variables ordered by SeriesNames.

Data Types: double

Object Functions

estimate	Estimate posterior distribution of Bayesian vector autoregression (VAR) model parameters
forecast	Forecast responses from Bayesian vector autoregression (VAR) model
simsmooth	Simulation smoother of Bayesian vector autoregression (VAR) model
simulate	Simulate coefficients and innovations covariance matrix of Bayesian vector autoregression (VAR) model
summarize	Distribution summary statistics of Bayesian vector autoregression (VAR) model

Examples

Create Matrix-Normal-Inverse-Wishart Conjugate Prior Model

Consider the 3-D VAR(4) model for the US inflation (INFL), unemployment (UNRATE), and federal funds (FEDFUNDS) rates.

$$\left[\begin{array}{l}{\text{INFL}}_t\\ {\text{UNRATE}}_t\\ {\text{FEDFUNDS}}_t\end{array}\right]=c+\sum _{j=1}^4{\Phi }_j\left[\begin{array}{l}{\text{INFL}}_{t-j}\\ {\text{UNRATE}}_{t-j}\\ {\text{FEDFUNDS}}_{t-j}\end{array}\right]+\left[\begin{array}{c}{\epsilon }_{1,t}\\ {\epsilon }_{2,t}\\ {\epsilon }_{3,t}\end{array}\right].$$

For all $$t$$, $${\epsilon }_t$$ is a series of independent 3-D normal innovations with a mean of 0 and covariance $\Sigma$. Assume the following prior distributions:

• ${\left[{\Phi }_1,...,{\Phi }_4,\mathit{c}\right]}^{\prime }|\Sigma \sim {{\rm N}}_{13\times 3}\left({\rm M},\mathit{V},\Sigma \right)$, where M is a 13-by-3 matrix of means and $\mathit{V}$ is the 13-by-13 among-coefficient scale matrix. Equivalently, $\mathrm{vec}\left({\left[{\Phi }_1,...,{\Phi }_4,\mathit{c}\right]}^{\prime }\right)|\Sigma \sim {{\rm N}}_{39}\left(\mathrm{vec}\left({\rm M}\right),\Sigma \otimes \mathit{V}\right)$.

• $$\Sigma \sim InverseWishart(\Omega ,\nu )$$, where $\Omega$ is the 3-by-3 scale matrix and $\nu$ is the degrees of freedom.

Create a conjugate prior model for the 3-D VAR(4) model parameters.

numseries = 3;
numlags = 4;
PriorMdl = conjugatebvarm(numseries,numlags)

PriorMdl = 
  conjugatebvarm with properties:

        Description: "3-Dimensional VAR(4) Model"
          NumSeries: 3
                  P: 4
        SeriesNames: ["Y1"    "Y2"    "Y3"]
    IncludeConstant: 1
       IncludeTrend: 0
      NumPredictors: 0
                 Mu: [39x1 double]
                  V: [13x13 double]
              Omega: [3x3 double]
                DoF: 13
                 AR: {[3x3 double]  [3x3 double]  [3x3 double]  [3x3 double]}
           Constant: [3x1 double]
              Trend: [3x0 double]
               Beta: [3x0 double]
         Covariance: [3x3 double]

PriorMdl is a conjugatebvarm Bayesian VAR model object representing the prior distribution of the coefficients and innovations covariance of the 3-D VAR(4) model. The command line display shows properties of the model. You can display properties by using dot notation.

Display the prior mean matrices of the four AR coefficients by setting each matrix in the cell to a variable.

AR1 = PriorMdl.AR{1}

AR1 = 3×3

     0     0     0
     0     0     0
     0     0     0

AR2 = PriorMdl.AR{2}

AR2 = 3×3

     0     0     0
     0     0     0
     0     0     0

AR3 = PriorMdl.AR{3}

AR3 = 3×3

     0     0     0
     0     0     0
     0     0     0

AR4 = PriorMdl.AR{4}

AR4 = 3×3

     0     0     0
     0     0     0
     0     0     0

conjugatebvarm centers all AR coefficients at 0 by default. The AR property is read-only, but it is derived from the writeable property Mu.

Create Conjugate Bayesian AR(2) Model

Consider a 1-D Bayesian AR(2) model for the daily NASDAQ returns from January 2, 1990 through December 31, 2001.

$$y_t=c+{\varphi }_1{y}_{t-1}+{\varphi }_2{y}_{t-1}+{\epsilon }_t.$$

The priors are:

• $$[\begin{array}{ccc}{\varphi }_1& {\varphi }_2& c\end{array}{]}^{\prime }|{\sigma }^2\sim N_3\left(\mu ,{\sigma }^{2}V\right)$$, where $\mu$ is a 3-by-1 vector of coefficient means and $\mathit{V}$ is a 3-by-3 scaled covariance matrix.

• ${\sigma }^2\sim \mathrm{IG}\left(\alpha ,\beta \right)$, where $\alpha =\frac{\nu }{2}$ is the degrees of freedom and $\beta =\frac{\Omega }{2}$ is the scale.

Create a conjugate prior model for the AR(2) model parameters.

numseries = 1;
numlags = 2;
PriorMdl = conjugatebvarm(numseries,numlags)

PriorMdl = 
  conjugatebvarm with properties:

        Description: "1-Dimensional VAR(2) Model"
          NumSeries: 1
                  P: 2
        SeriesNames: "Y1"
    IncludeConstant: 1
       IncludeTrend: 0
      NumPredictors: 0
                 Mu: [3x1 double]
                  V: [3x3 double]
              Omega: 1
                DoF: 11
                 AR: {[0]  [0]}
           Constant: 0
              Trend: [1x0 double]
               Beta: [1x0 double]
         Covariance: 0.1111

conjugatebvarm interprets the innovations covariance matrix as an inverse Wishart random variable. Because the scales and degrees of freedom hyperparameters among the inverse Wishart and inverse gamma distributions are not equal, you can adjust them by using dot notation. For example, to achieve 10 degrees of freedom for the inverse gamma interpretation, set the inverse Wishart degrees of freedom to 20.

PriorMdl.DoF = 20

PriorMdl = 
  conjugatebvarm with properties:

        Description: "1-Dimensional VAR(2) Model"
          NumSeries: 1
                  P: 2
        SeriesNames: "Y1"
    IncludeConstant: 1
       IncludeTrend: 0
      NumPredictors: 0
                 Mu: [3x1 double]
                  V: [3x3 double]
              Omega: 1
                DoF: 20
                 AR: {[0]  [0]}
           Constant: 0
              Trend: [1x0 double]
               Beta: [1x0 double]
         Covariance: 0.0556

Specify High Lag Coefficient Tightness and Response Names

In the 3-D VAR(4) model of Create Matrix-Normal-Inverse-Wishart Conjugate Prior Model, consider excluding lags 2 and 3 from the model.

You cannot exclude coefficient matrices from models, but you can specify high prior tightness on zero for coefficients that you want to exclude.

Create a conjugate prior model for the 3-D VAR(4) model parameters. Specify response variable names.

By default, AR coefficient prior means are zero. Specify high tightness values for lags 2 and 3 by setting their prior variances to 1e-6. Leave all other coefficient tightness values at their defaults:

• 1 for AR coefficient variances

• 1e3 for constant vector variances

• 0 for all coefficient covariances

Also, for conjugate Bayesian VAR models only, MATLAB® assumes that coefficient variances are proportional across response equations. Therefore, specify variances relative to the first equation.

numseries = 3;
numlags = 4;
seriesnames = ["INFL"; "UNRATE"; "FEDFUNDS"];
vPhi1 = ones(1,numseries);
vPhi2 = 1e-6*ones(1,numseries);
vPhi3 = 1e-6*ones(1,numseries);
vPhi4 = ones(1,numseries);
vc = 1e3;
V = diag([vPhi1 vPhi2 vPhi3 vPhi4 vc]);
PriorMdl = conjugatebvarm(numseries,numlags,'SeriesNames',seriesnames,...
    'V',V)

PriorMdl = 
  conjugatebvarm with properties:

        Description: "3-Dimensional VAR(4) Model"
          NumSeries: 3
                  P: 4
        SeriesNames: ["INFL"    "UNRATE"    "FEDFUNDS"]
    IncludeConstant: 1
       IncludeTrend: 0
      NumPredictors: 0
                 Mu: [39x1 double]
                  V: [13x13 double]
              Omega: [3x3 double]
                DoF: 13
                 AR: {[3x3 double]  [3x3 double]  [3x3 double]  [3x3 double]}
           Constant: [3x1 double]
              Trend: [3x0 double]
               Beta: [3x0 double]
         Covariance: [3x3 double]

Prepare Prior for Exogenous Predictor Variables

Consider the 2-D VARX(1) model for the US real GDP (RGDP) and investment (GCE) rates that treats the personal consumption (PCEC) rate as exogenous:

$$\left[\begin{array}{l}{\text{RGDP}}_t\\ {\text{GCE}}_t\end{array}\right]=c+\Phi \left[\begin{array}{l}{\text{RGDP}}_{t-1}\\ {\text{GCE}}_{t-1}\end{array}\right]+{\text{PCEC}}_t\beta +{\epsilon }_t.$$

For all $$t$$, $${\epsilon }_t$$ is a series of independent 2-D normal innovations with a mean of 0 and covariance $\Sigma$. Assume the following prior distributions:

• ${\left[\begin{array}{ccc}\Phi & \mathit{c}& \beta \end{array}\right]}^{\prime }|\Sigma \sim {{\rm N}}_{4\times 2}\left({\rm M},\mathit{V},\Sigma \right)$, where M is a 4-by-2 matrix of means and $\mathit{V}$ is the 4-by-4 among-coefficient scale matrix. Equivalently, $\mathrm{vec}\left({\left[\begin{array}{ccc}\Phi & \mathit{c}& \beta \end{array}\right]}^{\prime }\right)|\Sigma \sim {{\rm N}}_8\left(\mathrm{vec}\left({\rm M}\right),\Sigma \otimes \mathit{V}\right)$.

• $$\Sigma \sim InverseWishart(\Omega ,\nu )$$, where Ω is the 2-by-2 scale matrix and $\nu$ is the degrees of freedom.

Create a conjugate prior model for the 2-D VARX(1) model parameters.

numseries = 2;
numlags = 1;
numpredictors = 1;
PriorMdl = conjugatebvarm(numseries,numlags,'NumPredictors',numpredictors)

PriorMdl = 
  conjugatebvarm with properties:

        Description: "2-Dimensional VAR(1) Model"
          NumSeries: 2
                  P: 1
        SeriesNames: ["Y1"    "Y2"]
    IncludeConstant: 1
       IncludeTrend: 0
      NumPredictors: 1
                 Mu: [8x1 double]
                  V: [4x4 double]
              Omega: [2x2 double]
                DoF: 12
                 AR: {[2x2 double]}
           Constant: [2x1 double]
              Trend: [2x0 double]
               Beta: [2x1 double]
         Covariance: [2x2 double]

Display the prior mean of the coefficients Mu with the corresponding coefficients.

coeffnames = ["phi(11)"; "phi(12)"; "c(1)"; "beta(1)"; "phi(21)"; "phi(22)"; "c(2)"; "beta(2)"];
array2table(PriorMdl.Mu,'VariableNames',{'PriorMean'},'RowNames',coeffnames)

ans=8×1 table
               PriorMean
               _________

    phi(11)        0    
    phi(12)        0    
    c(1)           0    
    beta(1)        0    
    phi(21)        0    
    phi(22)        0    
    c(2)           0    
    beta(2)        0    

Set Prior Hyperparameters for Minnesota Regularization

conjugatebvarm options enable you to specify prior hyperparameter values directly, but bayesvarm options are well suited for tuning hyperparameters following the Minnesota regularization method.

Consider the 3-D VAR(4) model of Create Matrix-Normal-Inverse-Wishart Conjugate Prior Model. The model contains 39 coefficients. For coefficient sparsity, create a conjugate Bayesian VAR model by using bayesvarm. Specify the following, a priori:

• Each response is an AR(1) model, on average, with lag 1 coefficient 0.75.

• Prior scaled coefficient covariances decay with increasing lag at a rate of 2 (that is, lower lags are more important than higher lags).

numseries = 3;
numlags = 4;
PriorMdl = bayesvarm(numseries,numlags,'ModelType','conjugate',...
    'Center',0.75,'Decay',2)

PriorMdl = 
  conjugatebvarm with properties:

        Description: "3-Dimensional VAR(4) Model"
          NumSeries: 3
                  P: 4
        SeriesNames: ["Y1"    "Y2"    "Y3"]
    IncludeConstant: 1
       IncludeTrend: 0
      NumPredictors: 0
                 Mu: [39x1 double]
                  V: [13x13 double]
              Omega: [3x3 double]
                DoF: 13
                 AR: {[3x3 double]  [3x3 double]  [3x3 double]  [3x3 double]}
           Constant: [3x1 double]
              Trend: [3x0 double]
               Beta: [3x0 double]
         Covariance: [3x3 double]

Display the prior coefficient means in the equation of the first response.

Phi1 = PriorMdl.AR{1}

Phi1 = 3×3

    0.7500         0         0
         0    0.7500         0
         0         0    0.7500

Phi2 = PriorMdl.AR{2}

Phi2 = 3×3

     0     0     0
     0     0     0
     0     0     0

Phi3 = PriorMdl.AR{3}

Phi3 = 3×3

     0     0     0
     0     0     0
     0     0     0

Phi4 = PriorMdl.AR{4}

Phi4 = 3×3

     0     0     0
     0     0     0
     0     0     0

Display a heatmap of the prior scaled covariances of the coefficients in the first response equation.

% Create labels for the chart.
numARCoeffMats = PriorMdl.NumSeries*PriorMdl.P;
arcoeffnames = strings(numARCoeffMats,1);
for r = numlags:-1:1
    arcoeffnames(((r-1)*numseries+1):(numseries*r)) = ["\phi_{"+r+",11}" "\phi_{"+r+",12}" "\phi_{"+r+",13}"];
end

heatmap(arcoeffnames,arcoeffnames,PriorMdl.V(1:end-1,1:end-1));

For conjugate Bayesian VAR models, scaled covariances are proportional among equations.

Work with Prior and Posterior Distributions

Consider the 3-D VAR(4) model of Create Matrix-Normal-Inverse-Wishart Conjugate Prior Model. Estimate the posterior distribution, and generate forecasts from the corresponding posterior predictive distribution.

Load and Preprocess Data

Load the US macroeconomic data set. Compute the inflation rate. Plot all response series.

load Data_USEconModel
seriesnames = ["INFL" "UNRATE" "FEDFUNDS"];
DataTimeTable.INFL = 100*[NaN; price2ret(DataTimeTable.CPIAUCSL)];

figure
plot(DataTimeTable.Time,DataTimeTable{:,seriesnames})
legend(seriesnames)

Stabilize the unemployment and federal funds rates by applying the first difference to each series.

DataTimeTable.DUNRATE = [NaN; diff(DataTimeTable.UNRATE)];
DataTimeTable.DFEDFUNDS = [NaN; diff(DataTimeTable.FEDFUNDS)];
seriesnames(2:3) = "D" + seriesnames(2:3);

Remove all missing values from the data.

rmDataTimeTable = rmmissing(DataTimeTable);

Create Prior Model

Create a conjugate Bayesian VAR(4) prior model for the three response series. Specify the response variable names.

numseries = numel(seriesnames);
numlags = 4;

PriorMdl = conjugatebvarm(numseries,numlags,'SeriesNames',seriesnames);

Estimate Posterior Distribution

Estimate the posterior distribution by passing the prior model and entire data series to estimate.

PosteriorMdl = estimate(PriorMdl,rmDataTimeTable{:,seriesnames},'Display','equation');

Bayesian VAR under conjugate priors
Effective Sample Size:          197
Number of equations:            3
Number of estimated Parameters: 39
                                                                                 VAR Equations                                                                                
           | INFL(-1)  DUNRATE(-1)  DFEDFUNDS(-1)  INFL(-2)  DUNRATE(-2)  DFEDFUNDS(-2)  INFL(-3)  DUNRATE(-3)  DFEDFUNDS(-3)  INFL(-4)  DUNRATE(-4)  DFEDFUNDS(-4)  Constant 
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
 INFL      |  0.1260     -0.4400        0.1049      0.3176     -0.0545        0.0440      0.4173      0.2421        0.0515      0.0247     -0.1639        0.0080      0.1064  
           | (0.0713)    (0.1395)      (0.0366)    (0.0810)    (0.1490)      (0.0386)    (0.0802)    (0.1467)      (0.0400)    (0.0838)    (0.1385)      (0.0369)    (0.0774) 
 DUNRATE   | -0.0236      0.4440        0.0350      0.0900      0.2295        0.0520     -0.0330      0.0567        0.0010      0.0298     -0.1665        0.0104     -0.0536  
           | (0.0396)    (0.0774)      (0.0203)    (0.0449)    (0.0827)      (0.0214)    (0.0445)    (0.0814)      (0.0222)    (0.0465)    (0.0768)      (0.0205)    (0.0430) 
 DFEDFUNDS | -0.1514     -1.3408       -0.2762      0.3275     -0.2971       -0.3041      0.2609     -0.6971        0.0130     -0.0692      0.1392       -0.1341     -0.3902  
           | (0.1517)    (0.2967)      (0.0777)    (0.1722)    (0.3168)      (0.0820)    (0.1705)    (0.3120)      (0.0851)    (0.1782)    (0.2944)      (0.0785)    (0.1646) 
 
       Innovations Covariance Matrix       
           |   INFL     DUNRATE  DFEDFUNDS 
-------------------------------------------
 INFL      |  0.2725   -0.0197     0.1407  
           | (0.0270)  (0.0106)   (0.0417) 
 DUNRATE   | -0.0197    0.0839    -0.1290  
           | (0.0106)  (0.0083)   (0.0242) 
 DFEDFUNDS |  0.1407   -0.1290     1.2322  
           | (0.0417)  (0.0242)   (0.1220) 

Because the prior is conjugate for the data likelihood, the posterior is a conjugatebvarm object. By default, estimate uses the first four observations as a presample to initialize the model.

Generate Forecasts from Posterior Predictive Distribution

From the posterior predictive distribution, generate forecasts over a two-year horizon. Because sampling from the posterior predictive distribution requires the entire data set, specify the prior model in forecast instead of the posterior.

fh = 8;
FY = forecast(PriorMdl,fh,rmDataTimeTable{:,seriesnames});

FY is an 8-by-3 matrix of forecasts.

Plot the end of the data set and the forecasts.

fp = rmDataTimeTable.Time(end) + calquarters(1:fh);
figure
plotdata = [rmDataTimeTable{end - 10:end,seriesnames}; FY];
plot([rmDataTimeTable.Time(end - 10:end); fp'],plotdata)
hold on
plot([fp(1) fp(1)],ylim,'k-.')
legend(seriesnames)
title('Data and Forecasts')
hold off

Compute Impulse Responses

Plot impulse response functions by passing posterior estimations to armairf.

armairf(PosteriorMdl.AR,[],'InnovCov',PosteriorMdl.Covariance)

More About

Bayesian Vector Autoregression (VAR) Model

A Bayesian VAR model treats all coefficients and the innovations covariance matrix as random variables in the m-dimensional, stationary VARX(p) model. The model has one of the three forms described in this table.

Model	Equation
Reduced-form VAR(p) in difference-equation notation	$$y_t={\Phi }_1{y}_{t-1}+\mathrm{...}+{\Phi }_p{y}_{t-p}+c+\delta t+{\rm B}{x}_t+{\epsilon }_t.$$
Multivariate regression	$$y_t=Z_t\lambda +{\epsilon }_t.$$
Matrix regression	$$y_t={\Lambda }^{\prime }{z}_t^{\prime }+{\epsilon }_t.$$

For each time t = 1,...,T:

• y_t is the m-dimensional observed response vector, where m = numseries.

• Φ₁,…,Φ_p are the m-by-m AR coefficient matrices of lags 1 through p, where p = numlags.

• c is the m-by-1 vector of model constants if IncludeConstant is true.

• δ is the m-by-1 vector of linear time trend coefficients if IncludeTrend is true.

• Β is the m-by-r matrix of regression coefficients of the r-by-1 vector of observed exogenous predictors x_t, where r = NumPredictors. All predictor variables appear in each equation.

• $$z_t=\left[\begin{array}{ccccccc}{y}_{t-1}^{\prime }& y_{t-2}^{\prime }& \cdots & y_{t-p}^{\prime }& 1& t& x_t^{\prime }\end{array}\right],$$ which is a 1-by-(mp + r + 2) vector, and Z_t is the m-by-m(mp + r + 2) block diagonal matrix

  $$\left[\begin{array}{cccc}{z}_t& 0_z& \cdots & 0_z\\ 0_z& z_t& \cdots & 0_z\\ ⋮& ⋮& \ddots & ⋮\\ 0_z& 0_z& 0_z& z_t\end{array}\right],$$

  where 0_z is a 1-by-(mp + r + 2) vector of zeros.

• $$\Lambda ={\left[\begin{array}{ccccccc}{\Phi }_1& {\Phi }_2& \cdots & {\Phi }_p& c& \delta & {\rm B}\end{array}\right]}^{\prime }$$, which is an (mp + r + 2)-by-m random matrix of the coefficients, and the m(mp + r + 2)-by-1 vector λ = vec(Λ).

• ε_t is an m-by-1 vector of random, serially uncorrelated, multivariate normal innovations with the zero vector for the mean and the m-by-m matrix Σ for the covariance. This assumption implies that the data likelihood is

  $$\ell \left(\Lambda ,\Sigma |y,x\right)={\displaystyle \prod _{t=1}^{T}f\left(y_t;\Lambda ,\Sigma ,z_t\right)},$$

  where f is the m-dimensional multivariate normal density with mean z_tΛ and covariance Σ, evaluated at y_t.

Before considering the data, you impose a joint prior distribution assumption on (Λ,Σ), which is governed by the distribution π(Λ,Σ). In a Bayesian analysis, the distribution of the parameters is updated with information about the parameters obtained from the data likelihood. The result is the joint posterior distribution π(Λ,Σ|Y,X,Y₀), where:

• Y is a T-by-m matrix containing the entire response series {y_t}, t = 1,…,T.

• X is a T-by-m matrix containing the entire exogenous series {x_t}, t = 1,…,T.

• Y₀ is a p-by-m matrix of presample data used to initialize the VAR model for estimation.

Dependent, Matrix-Normal-Inverse-Wishart Conjugate Model

The dependent, matrix-normal-inverse-Wishart conjugate model is an m-D Bayesian VAR(p) model in which the conditional prior distribution of Λ|Σ is matrix normal with mean matrix Μ and scale matrices Σ and V. The prior distribution of Σ is inverse Wishart with scale matrix Ω and degrees of freedom ν.

Symbolically:

$$\begin{array}{l}\Lambda |\Sigma ~{\text{N}}_{(mp+r+1_c+1_{\delta })\times m}\left({\rm M},V,\Sigma \right)\\ \Sigma ~\text{Inverse}\text{Wishart}\left(\Omega ,\nu \right),\end{array}$$

which implies, for λ = vec(Λ), $$\lambda |\Sigma ~{\text{N}}_{m(mp+r+1_c+1_{\delta })}\left(\mu ,\Sigma \otimes V\right),$$ where

• μ = vec(Μ) = Mu.

• V = V.

• r = NumPredictors.

• 1_c is 1 if IncludeConstant is true, and 0 otherwise.

• 1_δ is 1 if IncludeTrend is true, and 0 otherwise.

To achieve posterior distributions that are conjugate for the data likelihood, the AR coefficient matrix covariances must be proportional among equations. And, for each equation, self- and cross-lag covariances must be equal.

The posterior distributions are

$$\begin{array}{l}\Lambda |\Sigma ,y_t,x_t~{\text{N}}_{(mp+r+1_c+1_{\delta })\times m}\left(\overline{{\rm M}},\overline{V},\Sigma \right)\\ \Sigma |y_t,x_t~\text{Inverse}\text{Wishart}\left(\overline{\Omega },\overline{\nu }\right),\end{array}$$

where:

• $$\overline{{\rm M}}={\left(V^{-1}+{\displaystyle \sum _{t=1}^T{z}_t^{\prime }{z}_t}\right)}^{-1}\left(V^{-1}{\rm M}+{\displaystyle \sum _{t=1}^T{z}_t^{\prime }{y}_t^{\prime }}\right).$$

• $$\overline{V}={\left(V^{-1}+{\displaystyle \sum _{t=1}^T{z}_t^{\prime }{z}_t}\right)}^{-1}.$$

• $$\overline{\Omega }=\Omega +{\displaystyle \sum _{t=1}^T{y}_t{y}_t^{\prime }}+{\rm M}{V}^{-1}{\rm M}-\overline{{\rm M}}{\overline{V}}^{-1}\overline{{\rm M}}.$$

• $$\overline{\nu }=T+\nu .$$

Algorithms

• If you pass either a conjugatebvarm or diffusebvarm object and data to estimate, MATLAB^® returns a conjugatebvarm object representing the posterior distribution.

• The conditional covariance (unscaled) of the entire vectorized matrix normal prior is Σ⊗V. To achieve conjugacy, these conditions must be true:

  • Prior covariances are assumed to be proportional among all equations. Σ determines the proportionality, and scales V during posterior estimation.

  • For an equation, the covariances between all AR coefficients, self lag and cross lag, are equal.

  conjugatebvarm enforces the first condition, but not the second. Therefore, conjugatebvarm applies elements of V to all coefficients in the model relative to the coefficients in the equation of y_1,t.

References

[1] Litterman, Robert B. "Forecasting with Bayesian Vector Autoregressions: Five Years of Experience." Journal of Business and Economic Statistics 4, no. 1 (January 1986): 25–38. https://doi.org/10.2307/1391384.

Version History

Introduced in R2020a