diffusebvarm

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

Since R2020a

Description

The Bayesian VAR model object diffusebvarm specifies the joint prior 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 diffuse model.

A diffuse prior model does not enable you to specify hyperparameter values for coefficient sparsity; all AR lags in the model are weighted equally. To implement Minnesota regularization, create a conjugate, semiconjugate, or normal prior model by using bayesvarm.

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 = diffusebvarm(numseries,numlags)

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

Description

To create a diffusebvarm object, use either the diffusebvarm function (described here) or the bayesvarm function.

PriorMdl = diffusebvarm(numseries,numlags) creates a numseries-D Bayesian VAR(numlags) model object PriorMdl, which specifies dimensionalities and prior assumptions for all model coefficients $$\lambda =\text{vec}\left(\Lambda \right)=\text{vec}\left({\left[\begin{array}{ccccccc}{\Phi }_1& {\Phi }_2& \cdots & {\Phi }_p& c& \delta & {\rm B}\end{array}\right]}^{\prime }\right)$$ 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 diffuse model.

PriorMdl = diffusebvarm(numseries,numlags,Name,Value) sets writable properties (except NumSeries and P) using name-value pair arguments. Enclose each property name in quotes. For example, diffusebvarm(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 = diffusebvarm(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']. diffusebvarm 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. diffusebvarm includes all predictor variables symmetrically in each response equation.

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

nan(NumSeries,NumSeries)

Distribution mean of the innovations covariance matrix Σ of the NumSeries innovations at each time t = 1,...,T, specified as a NumSeries-by-NumSeries matrix of NaN. Because the prior model is diffuse, the mean of Σ is unknown, a priori.

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 Diffuse 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 that the joint prior distribution of the VAR model parameters $\left({\left[{\Phi }_1,...,{\Phi }_4,\mathit{c}\right]}^{\prime },\Sigma \right)$ is diffuse.

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

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

PriorMdl = 
  diffusebvarm with properties:

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

PriorMdl is a diffusebvarm 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 covariance 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

diffusebvarm centers all AR coefficients at 0 by default. Because the model is diffuse, the data informs the posterior distribution.

Create Diffuse 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 joint prior is diffuse.

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

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

PriorMdl = 
  diffusebvarm with properties:

        Description: "1-Dimensional VAR(2) Model"
          NumSeries: 1
                  P: 2
        SeriesNames: "Y1"
    IncludeConstant: 1
       IncludeTrend: 0
      NumPredictors: 0
                 AR: {[0]  [0]}
           Constant: 0
              Trend: [1x0 double]
               Beta: [1x0 double]
         Covariance: NaN

Specify Response Names and Include Linear Time Trend

Consider adding a linear time trend term to the 3-D VAR(4) model of Create Diffuse Prior Model:

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

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

numseries = 3;
numlags = 4;
seriesnames = ["INFL"; "UNRATE"; "FEDFUNDS"];
PriorMdl = diffusebvarm(numseries,numlags,'SeriesNames',seriesnames,...
    'IncludeTrend',true)

PriorMdl = 
  diffusebvarm with properties:

        Description: "3-Dimensional VAR(4) Model"
          NumSeries: 3
                  P: 4
        SeriesNames: ["INFL"    "UNRATE"    "FEDFUNDS"]
    IncludeConstant: 1
       IncludeTrend: 1
      NumPredictors: 0
                 AR: {[3x3 double]  [3x3 double]  [3x3 double]  [3x3 double]}
           Constant: [3x1 double]
              Trend: [3x1 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 that the joint prior distribution is diffuse.

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

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

PriorMdl = 
  diffusebvarm with properties:

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

Work with Prior and Posterior Distributions

Consider the 3-D VAR(4) model of Create Diffuse 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 diffuse Bayesian VAR(4) prior model for the three response series. Specify the response variable names.

numseries = numel(seriesnames);
numlags = 4;

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

Estimate Posterior Distribution

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

rng(1); % For reproducibility
PosteriorMdl = estimate(PriorMdl,rmDataTimeTable{:,seriesnames},'Display','equation');

Bayesian VAR under diffuse 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      |  0.1241     -0.4809        0.1005      0.3236     -0.0503        0.0450      0.4272      0.2738        0.0523      0.0167     -0.1830        0.0067      0.1007  
           | (0.0762)    (0.1536)      (0.0390)    (0.0868)    (0.1647)      (0.0413)    (0.0860)    (0.1620)      (0.0428)    (0.0901)    (0.1520)      (0.0395)    (0.0832) 
 DUNRATE   | -0.0219      0.4716        0.0391      0.0913      0.2414        0.0536     -0.0389      0.0552        0.0008      0.0285     -0.1795        0.0088     -0.0499  
           | (0.0413)    (0.0831)      (0.0211)    (0.0469)    (0.0891)      (0.0223)    (0.0465)    (0.0876)      (0.0232)    (0.0488)    (0.0822)      (0.0214)    (0.0450) 
 DFEDFUNDS | -0.1586     -1.4368       -0.2905      0.3403     -0.2968       -0.3117      0.2848     -0.7401        0.0028     -0.0690      0.1494       -0.1372     -0.4221  
           | (0.1632)    (0.3287)      (0.0835)    (0.1857)    (0.3526)      (0.0883)    (0.1841)    (0.3466)      (0.0917)    (0.1928)    (0.3253)      (0.0845)    (0.1781) 
 
       Innovations Covariance Matrix       
           |   INFL     DUNRATE  DFEDFUNDS 
-------------------------------------------
 INFL      |  0.3028   -0.0217     0.1579  
           | (0.0321)  (0.0124)   (0.0499) 
 DUNRATE   | -0.0217    0.0887    -0.1435  
           | (0.0124)  (0.0094)   (0.0283) 
 DFEDFUNDS |  0.1579   -0.1435     1.3872  
           | (0.0499)  (0.0283)   (0.1470) 

PosteriorMdl is a conjugatebvarm model object; the posterior is analytically tractable. 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.

Diffuse Model

The diffuse model is an m-D Bayesian VAR model that has the noninformative joint prior distribution

$$\left(\Lambda ,\Sigma \right)\propto {\left|\Sigma \right|}^{-\frac{m+1}{2}}.$$

The diffuse model is the limiting case of the conjugate prior model (see conjugatebvarm) when Μ → 0, V^-1 → 0, Ω → 0, and ν → -k, where:

• k = mp + r + 1_c + 1_δ, the number of coefficients per response equation.

• r = NumPredictors.

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

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

If the sample size is large enough to satisfy least-squares estimation, the posterior distributions are proper and analytically tractable.

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

where:

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

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

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

• $$\overline{\nu }=T+k.$$

Algorithms

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

Version History

Introduced in R2020a