# bayesvarm

Create prior Bayesian vector autoregression (VAR) model object

## Syntax

PriorMdl = bayesvarm(numseries,numlags)
PriorMdl = bayesvarm(numseries,numlags,ModelType=modelType)
PriorMdl = bayesvarm(numseries,numlags,ModelType=modelType,Name=Value)

## Description

To create a Bayesian linear regression model for univariate regression analysis, or to perform Bayesian predictor selection, see bayeslm. To create a non-Bayesian VAR model, see varm.

example

PriorMdl = bayesvarm(numseries,numlags) creates the Bayesian VAR(p) model object PriorMdl, which specifies dimensionalities and prior assumptions for all model coefficients $\Lambda ={\left[\begin{array}{ccccccc}{\Phi }_{1}& {\Phi }_{2}& \cdots & {\Phi }_{p}& c& \delta & Β\end{array}\right]}^{\prime }$ and the innovations covariance Σ, where: numseries is the number of response time series variables.p = numlags is the AR polynomial order.The joint prior distribution of (Λ,Σ) is diffuse. 

example

PriorMdl = bayesvarm(numseries,numlags,ModelType=modelType) specifies the joint prior distribution modelType for Λ and Σ. For this syntax, modelType can be 'conjugate', 'semiconjugate', 'diffuse', or 'normal'. For example, ModelType="semiconjugate" specifies semiconjugate priors for the multivariate normal likelihood—specifically, vec(Λ)|Σ is multivariate normal, Σ is inverse Wishart, and Λ and Σ are independent.

example

PriorMdl = bayesvarm(numseries,numlags,ModelType=modelType,Name=Value) uses additional options specified by one or more name-value arguments. For example, for non-diffuse models, you can specify Minnesota prior regularization options to regularize the coefficients using the Minnesota prior parameter structure.

## Examples

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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 variance $\Sigma$.

Suppose that the AR coefficient matrices ${\Phi }_{1},...,{\Phi }_{4}$, model constant $\mathit{c}$, and innovations covariance matrix $\Sigma$ are random variables, and their prior distributions are unknown. In this case, use the noninformative diffuse prior: the joint prior distribution $\left({\Phi }_{1},...,{\Phi }_{4},\mathit{c},\Sigma \right)$is proportional to ${|\Sigma |}^{-2}$.

Create a diffuse prior model for the 3-D VAR(4) model parameters, which is the default prior model type.

numseries = 3; numlags = 4; PriorMdl = bayesvarm(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 diffusevarm Bayesian VAR model object representing the prior distribution of the AR coefficient matrices, model constant vector, and innovations covariance matrix. bayesvarm displays a summary of the prior distributions at the command line.

• AR — Prior means of the AR coefficient matrices.

• Constant — Prior means of the model constant vector.

• Trend and Beta — Prior means of the linear time trend vector and exogenous regression coefficient matrix, respectively. Because the values are empty arrays, the corresponding parameters are not in the model.

• Covariance — Prior mean of the innovations covariance matrix.

If you have data, then you can estimate characteristics of the posterior distribution by passing PriorMdl and the data to estimate.

Consider the 3-D VAR(4) model in Default Diffuse Prior Model. Assume the following:

• $\left[{\Phi }_{1},...,{\Phi }_{4},c{\right]}^{\prime }|\Sigma \sim {N}_{13×3}\left(M,V,\Sigma \right)$. $M$ is a 13-by-3 matrix of prior coefficient means ($M\left(1:3,1:3\right)$ is the prior mean matrix of ${\Phi }_{1}^{\prime }$, $M\left(4:6,1:3\right)$ is the prior mean matrix of ${\Phi }_{2}^{\prime }$,..., and $M\left(13,1:3\right)$ is the prior mean vector of $c$). $V$ is a 13-by-13 matrix representing the among-coefficient prior covariance matrix within an equation. $\Sigma$ is the 3-by-3 random innovations covariance matrix.

• $\Sigma \sim Inverse\phantom{\rule{0.2777777777777778em}{0ex}}Wishart\left(\Omega ,\nu \right)$. $\Omega$ is the 3-by-3 scale matrix, and $\nu$ is the degrees of freedom of the inverse Wishart distribution.

• The coefficients and the innovations covariance matrix are dependent.

• Prior coefficient variances among the equations are proportional.

These assumptions and the data likelihood imply a matrix-normal-inverse-Wishart conjugate model.

Create a matrix-normal-inverse-Wishart conjugate prior model for the VAR model parameters.

numseries = 3; numlags = 4; PriorMdl = bayesvarm(numseries,numlags,ModelType="conjugate")
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 matrix. bayesvarm displays a summary of the prior distributions at the command line; it returns the prior mean matrix in vectorized form.

The model contains many estimable parameters. To achieve a parsimonious model, bayesvarm applies the Minnesota prior regularization method to the AR coefficients, by default. Inspect the default prior means (centers of shrinkage) of the AR coefficient matrices.

AR1 = PriorMdl.AR{1}
AR1 = 3×3 0.5000 0 0 0 0.5000 0 0 0 0.5000 
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 

Each series is an AR(1) model with AR coefficient 0.5, a priori.

The tightness on shrinkage of the coefficients is proportional among the equations. Inspect the default tightness values by displaying a heatmap chart of the property V of PriorMdl, which contains a matrix of the scaled tightness on shrinkage of the coefficients for one equation (the unscaled shrinkage is $\Sigma \otimes V$ = kron(PriorMdl.Covariance,PriorMdl.V)). Omit the final row and column, which correspond to the model constant.

% 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_{11,"+r+"}" "\phi_{12,"+r+"}" "\phi_{13,"+r+"}"]; end heatmap(arcoeffnames,arcoeffnames,PriorMdl.V(1:end-1,1:end-1)); The tightness values decrease with lag, which suggests (a priori) that the means of the corresponding greater-lagged coefficients are more tightly locked around their center of 0.

Display the tightness of the model constant vector.

PriorMdl.V(end,end)
ans = 10000 

The center of the model constant vector is 0 but has a large variance, which allows the estimation procedure to defer more to the data than the prior for the posterior mean of the constant vector.

You can specify alternative values after you create a model by using dot notation. For example, increase the tightness of all coefficients by a factor of 100.

PriorMdl.V = 100*PriorMdl.V;

Consider the 3-D VAR(4) model in Default Diffuse Prior Model. Assume these prior distributions, as presented in :

• $vec\left({\left[{\Phi }_{1},...,{\Phi }_{4},c\right]}^{\prime }\right)|\Sigma \sim {N}_{39}\left(\mu ,V\right)$. $\mu$ is a 39-by-1 vector of prior coefficient means (the model has 39 individual coefficients), and $V$ is a 39-by-39 prior coefficient covariance matrix.

• The innovations covariance $\Sigma$ is a fixed matrix.

Suppose econometric theory dictates that

$\Sigma =\left[\begin{array}{ccc}{10}^{-5}& 0& {10}^{-4}\\ 0& 0.1& -0.2\\ {10}^{-4}& -0.2& 1.6\end{array}\right].$

Create a normal conjugate prior model for the VAR model coefficients. Specify the value of $\Sigma$ by using the Sigma name-value argument.

numseries = 3; numlags = 4; Sigma = [10e-5 0 10e-4; 0 0.1 -0.2; 10e-4 -0.2 1.6]; PriorMdl = bayesvarm(numseries,numlags,ModelType="normal", ... Sigma=Sigma)
PriorMdl = normalbvarm 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: [39x39 double] Sigma: [3x3 double] AR: {[3x3 double] [3x3 double] [3x3 double] [3x3 double]} Constant: [3x1 double] Trend: [3x0 double] Beta: [3x0 double] Covariance: [3x3 double] 

PriorMdl is a normalbvarm Bayesian VAR model object representing the prior distribution of the coefficients. Because $\Sigma$ is fixed for normalbvarm prior models, PriorMdl.Sigma and PriorMdl.Covariance are equal.

PriorMdl.Sigma
ans = 3×3 0.0001 0 0.0010 0 0.1000 -0.2000 0.0010 -0.2000 1.6000 
PriorMdl.Covariance
ans = 3×3 0.0001 0 0.0010 0 0.1000 -0.2000 0.0010 -0.2000 1.6000 

Consider the 3-D VAR(4) model in Default Diffuse Prior Model. Assume the following:

• $vec\left({\left[{\Phi }_{1},...,{\Phi }_{4},c\right]}^{\prime }\right)|\Sigma \sim {N}_{39}\left(\mu ,V\right)$. $\mu$ is a 39-by-1 vector of prior coefficient means (the model has 39 individual coefficients), and $V$ is a 39-by-39 prior coefficient covariance matrix.

• $\Sigma \sim InverseWishart\left(\Omega ,\nu \right)$. $\Omega$ is the 3-by-3 scale matrix, and $\nu$ is the degrees of freedom of the inverse Wishart distribution.

• The coefficients and the innovations covariance matrix are independent.

These assumptions and the data likelihood imply a normal-inverse-Wishart semiconjugate model.

The model contains many estimable parameters. To achieve a parsimonious model, bayesvarm enables you to regularize the coefficients by using the Minnesota prior regularization method, rather than specifying each prior mean and variance.

Create a normal-inverse-Wishart semiconjugate prior model for the VAR model parameters. Specify the following:

• All series are AR(1) models, a priori, with AR coefficient 0.9. Set the Center name-value argument to a 3-by-1 vector composed of 0.9.

• The tightness around self lags in ${\Phi }_{1}$ is 1. Set the SelfLag name-value argument to 1.

• The tightness around cross lags in ${\Phi }_{1}$ is 0.5. Set the CrossLag name-value argument to 0.5.

• All tightness values decay by a factor of the lag degree squared. Set the Decay name-value argument to 2.

numseries = 3; numlags = 4; center = 0.9*ones(numseries,1); PriorMdl = bayesvarm(numseries,numlags,ModelType="semiconjugate", ... Center=center,SelfLag=1,CrossLag=0.5,Decay=2)
PriorMdl = semiconjugatebvarm 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: [39x39 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 semiconjugatebvarm Bayesian VAR model object representing the prior distribution of the coefficients and innovations covariance matrix. bayesvarm displays a summary of the prior distributions at the command line; it returns the prior mean matrix in vectorized form.

Display the prior means of the AR coefficient matrices.

AR1 = PriorMdl.AR{1}
AR1 = 3×3 0.9000 0 0 0 0.9000 0 0 0 0.9000 
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 

Each series is an AR(1) model, a priori.

The property V of PriorMdl contains a matrix of the tightness on shrinkage of the coefficients. The rows and columns of V correspond to the elements of the Mu property of PriorMdl.

• Elements 1 through 3 correspond to the lag 1 AR coefficients in the first equation ordered by response variable, that is, ${\varphi }_{1,11}$, ${\varphi }_{1,12}$, and ${\varphi }_{1,13}$.

• Elements 4 through 6 correspond to the lag 2 AR coefficients in the first equation.

• Elements 7 through 9 correspond to the lag 3 AR coefficients in the first equation.

• Elements 10 through 12 correspond to the lag 4 AR coefficients in the first equation.

• Element 13 is the model constant in the first equation.

MATLAB® repeats the pattern for each equation.

In this example, the tightness of shrinkage is the same for all equations. Display a heatmap chart of the property V of PriorMdl for the tightness values of the AR coefficients in the first 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:numARCoeffMats,1:numARCoeffMats)); The tightness values decrease with lag, which suggests (a priori) that the means of the corresponding greater-lagged coefficients are more tightly locked around their center of 0. By default, AR coefficients are uncorrelated.

Display the tightness of the model constant vector.

PriorMdl.V(numARCoeffMats + 1,numARCoeffMats + 1)
ans = 10000 

The center of the model constant vector is 0 but has a large variance, which allows the estimation procedure to defer more to the data than the prior for the posterior mean of the constant vector.

You can specify alternative values after you create a model by using dot notation. For example, increase the tightness of all coefficients by a factor of 100.

PriorMdl.V = 100*PriorMdl.V;

## Input Arguments

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

Data Types: double

Number of lagged responses p to include in the VAR model, specified as a nonnegative integer. bayesvarm includes lags 1 through numlags.

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.

Example: IncludeTrend=true,NumPredictors=3 specifies a linear trend term and a linear regression term for three exogenous variables in all response equations.

Model Options

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Joint prior distribution of (Λ,Σ), specified as a value in the following table. In the table:

• λ = vec(Λ).

• The inverse Wishart hyperparameters Ω and ν correspond to the name-value arguments and output model properties Omega and DoF, respectively. You can adjust their values by specifying the name-value arguments or by using dot notation after bayesvarm returns PriorMdl.

ValueDescription
'conjugate'

Matrix-normal-inverse-Wishart conjugate model. The priors are

$\begin{array}{c}\Lambda |\Sigma ~{N}_{\left(mp+d\right)×m}\left(Μ,V,\Sigma \right)\\ ~\text{​}{N}_{m\left(mp+d\right)}\left(\mu ,\Sigma \otimes V\right)\\ \Sigma ~\text{Inverse}\text{\hspace{0.17em}}\text{Wishart}\left(\Omega ,\nu \right),\end{array}$

where Λ and Σ are dependent.

'semiconjugate'

Normal-inverse-Wishart semiconjugate model. The priors are

$\begin{array}{l}\lambda |\Sigma ~{N}_{m\left(mp+d\right)}\left(\mu ,V\right)\\ \Sigma ~\text{Inverse}\text{\hspace{0.17em}}\text{Wishart}\left(\Omega ,\nu \right),\end{array}$

where Λ and Σ are independent.

'diffuse'

Diffuse prior distributions. The joint prior pdf is

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

Regularization options do not apply to diffuse priors.

'normal'

Normal conjugate prior model. The prior is

$\lambda ~{\text{N}}_{m\left(mp+d\right)}\left(\mu ,V\right).$

Σ is known and fixed, and it corresponds to the property Sigma of PriorMdl. After bayesvarm returns PriorMdl, you can adjust the value of Σ by using dot notation.

Note

• The multivariate normal hyperparameters μ and V correspond to the Mu and V properties of PriorMdl, respectively. The Minnesota prior regularization options  enable you to specify μ and V for coefficient shrinkage and tightness completely and easily. You can also display or adjust their values directly by using dot notation after bayesvarm returns PriorMdl.

• The prior model type that you choose depends on your assumptions about the joint distribution of the parameters. Your choice can affect posterior estimates and inferences. For more details, see Implement Bayesian Linear Regression.

Example: ModelType="conjugate"

Data Types: char | string

Response series names for display, specified as a length m string vector or cell vector of character vectors. The default is ["Y1" "Y2" ... "Ym"].

Example: SeriesNames=["CPI" "Unemployment"]

Data Types: string | char

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

ValueDescription
falseResponse equations do not include a model constant.
trueAll response equations contain a model constant.

Example: IncludeConstant=false

Data Types: logical

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

ValueDescription
falseResponse equations do not include a linear time trend term.
trueAll response equations contain a linear time trend term.

Example: IncludeTrend=true

Data Types: logical

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

Example: NumPredictors=3

Model description, specified as a string scalar or character vector. 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

Minnesota Prior Regularization Options for Nondiffuse Priors

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Center of shrinkage for lag 1 self lags, or the prior expectation on the diagonal elements of Φ1, specified as a numseries-by-1 numeric vector. Center(j) is the prior mean of ${\varphi }_{1,jj}$.

Each element can be any real number, but typical values are in the interval [0,1]. This table describes the prior model of an individual response series for the specified value.

ValuePrior Model
0White noise process
In the interval (0,1)Stationary AR(1)
1Random walk

bayesvarm sets the prior means of the following variables to 0:

• Off-diagonal elements of Φ1

• All elements of Φq, q > 1

• Model constants c

• Linear time trend coefficients δ

• Exogenous predictor coefficients Β

For more details, see Minnesota Prior.

Example: Center=0.01*ones(3,1)

Data Types: double

Tightness of shrinkage on all self lags of Φ1, specified as a positive numeric scalar.

SelfLag contributes to the prior variances of all self-lag coefficients in the model (property V of the output model PriorMdl).

Tip

Relatively small tightness values indicate strong belief in prior assumptions during estimation (that is, relatively small values tightly lock self lags around their prior mean). Relatively large values place more weight on information in the data during estimation.

For more details, see Minnesota Prior.

Example: SelfLag=0.5

Data Types: double

Tightness on all cross-variable lag coefficients of Φ1, specified as a positive numeric scalar. For conjugate prior models, bayesvarm sets 'CrossLag' to the value of the SelfLag name-value argument.

CrossLag contributes to the prior variances of all cross-variable lag coefficients in the model (property V of the output model PriorMdl).

Tip

Relatively small tightness values indicate strong belief in prior assumptions during estimation (that is, relatively small values tightly lock cross lags around their prior mean). Relatively large values place more weight on information in the data during estimation.

For more details, see Minnesota Prior.

Example: CrossLag=0.05

Data Types: double

Speed of the prior variance decay with increasing lag, specified as a positive numeric scalar.

Decay contributes to the prior variance of all lag coefficient matrices greater than lag 1 (property V of the output model PriorMdl).

Tip

Relatively large values cause lag coefficient variances to decay more quickly, which tightly locks higher-order lag coefficients to their prior means.

Example: Decay=2

Data Types: double

Response variable variances for the cross-variable lag coefficient tightness CrossLag, specified as a numseries-by-1 positive numeric vector. Elements correspond to the response variables. For conjugate prior models, bayesvarm ignores Scale.

Scale contributes to the prior variances of all cross-variable lag coefficients in the model (property V of the output model PriorMdl), but does not directly contribute to the innovations covariance matrix stored in the property Sigma.

Tip

Specify 'Scale' when response variable scales are unbalanced.

Example: Scale=[2 1]

Data Types: double

Prior variance of exogenous coefficients, specified as a positive numeric scalar. VarianceX sets the prior variances of all exogenous variables, including the model constant c, linear time trend term δ, and exogenous predictor coefficients Β.

VarianceX contributes to the value of the prior coefficient variance (property V of the output model PriorMdl).

Tip

Relatively small tightness values indicate strong belief in prior assumptions during estimation (that is, relatively small values tightly lock coefficients of exogenous variables to their prior means). Relatively large values place more weight on information in the data during estimation.

Example: VarianceX=100

Data Types: double

Innovations Covariance Hyperparameter Options

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Fixed innovations covariance matrix for the normal prior model, specified as a numseries-by-numseries positive definite numeric matrix.

If you specify ModelType="normal", you must specify Sigma. For other prior models, Σ is a random variable, so Sigma does not apply.

Example: Sigma=eye(2)

Data Types: double

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

Example: Omega=eye(numseries)

Data Types: double

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.

Example: DoF=8

Data Types: double

## Output Arguments

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Bayesian VAR model storing prior model assumptions, returned as one of the model objects listed in this table.

Value of ModelTypeReturned Bayesian VAR Model Object
'conjugate'conjugatebvarm
'semiconjugate'semiconjugatebvarm
'diffuse'diffusebvarm
'normal'normalbvarm

PriorMdl specifies the joint prior distribution and characteristics of the VAR model only. The model object is a template intended for further use. To incorporate data into the model for posterior distribution analysis, pass the model object and data to the appropriate object function, for example, estimate or simulate.

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### 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.

ModelEquation
Reduced-form VAR(p) in difference-equation notation
${y}_{t}={\Phi }_{1}{y}_{t-1}+...+{\Phi }_{p}{y}_{t-p}+c+\delta t+Β{x}_{t}+{\epsilon }_{t}.$
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:

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

• Φ1,…,Φp 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 xt, 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 Zt 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 0z is a 1-by-(mp + r + 2) vector of zeros.

• $\Lambda ={\left[\begin{array}{ccccccc}{\Phi }_{1}& {\Phi }_{2}& \cdots & {\Phi }_{p}& c& \delta & Β\end{array}\right]}^{\prime }$, 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).

• εt 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

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

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

Before considering the data, you impose a joint prior distribution assumption on (Λ,Σ) (see the ModelType name-value argument). bayesvarm enables you to adjust hyperparameters by using the Minnesota prior assumptions and parameter structure ; the structure regularizes the coefficients. 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 of (Λ,Σ).

### Minnesota Prior

The Minnesota prior, introduced in , is a hyperparameter structure for the joint prior distribution of (Λ,Σ) used to obtain a parsimonious model by regularizing the endogenous coefficient matrices of a Bayesian VAR(p) model. Minnesota regularization considers a tuning parameter for the center of shrinkage and several tuning parameters for the tightness of shrinkage.

The center of shrinkage is specified by the prior mean of the coefficients (see the Center name-value argument). The Minnesota regularization method sets the prior mean of all coefficients to 0 except the lag 1 self lags (diagonal elements of the AR coefficient matrix Φ1). The prior mean of each lag 1 self lag is a real number, typically in the interval [0,1], where (a priori) response series j is one of the following:

• White noise process if prior mean j is 0

• AR(1) model if prior mean j is in the interval (0,1)

• Random walk if prior mean j is 1

For example, suppose numseries is 2, numlags is 2, NumPredictors is 1, and all other model options have default values. If you specify Center=0.01*ones(2,1), the vectorized prior mean of Λ|Σ is

$\begin{array}{c}\mu =\text{vec}\left(\left[\begin{array}{c}\\ \end{array}\stackrel{{\Phi }_{1}}{\overbrace{\begin{array}{cc}0.01& 0\\ 0& 0.01\end{array}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{{\Phi }_{2}}{\overbrace{\begin{array}{cc}0& 0\\ 0& 0\end{array}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{c}{\overbrace{\begin{array}{c}0\\ 0\end{array}}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\stackrel{Β}{\overbrace{\begin{array}{c}0\\ 0\end{array}}}{\begin{array}{c}\\ \end{array}\right]}^{\prime }\begin{array}{c}\text{​}\text{​}\text{​}\\ \text{​}\text{​}\text{​}\end{array}\right)\\ ={\left[}^{\stackrel{{\varphi }_{1,1:}}{\overbrace{\begin{array}{cc}0.01& 0\end{array}}}},\end{array}$

where ϕq,j: is row j of Φq. MATLAB® stores μ in the Mu property of PriorMdl. You can adjust Mu by using dot notation.

The tightness of shrinkage is specified by the prior variance of the coefficients ϕr,jk. For all prior models except conjugate,

$\text{Var}\left({\varphi }_{q,jk}|\Sigma \right)=\left\{\begin{array}{cc}\frac{{v}_{0}}{{q}^{d}};& j=k\\ \frac{{v}_{×}}{{q}^{d}}\frac{{\sigma }_{j}^{2}}{{\sigma }_{k}^{2}};& j\ne k\end{array},$

where:

• v0 is the tightness on the prior means of all self lags of Φ1 (SelfLag).

• d is the speed of tightness decay (Decay).

• ${\nu }_{×}$ is the tightness on the prior means of all cross-variable lag coefficients of Φ1 (CrossLag).

• ${\sigma }_{j}^{2}$ is the prior response variance (element j of Scale).

For conjugate prior models,

$\text{Var}\left({\varphi }_{q,jk}|\Sigma \right)=\frac{{v}_{0}}{{q}^{d}}\forall j,k.$

## Tips

• Because MATLAB does not adjust input data for variable scales, a best practice is to adjust all series to have a similar magnitude. Consequently, the scales of the coefficients are similar.

• By default, bayesvarm creates Bayesian VAR models by using the Minnesota prior assumptions and parameter structure . After you create a model, you can inspect the effect of coefficient shrinkage by calling summarize(PriorMdl). You can change the prior mean and variance by setting PriorMdl.Mu and PriorMdl.V, respectively.

 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.