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
Description
To create a diffusebvarm
object, use either the diffusebvarm
function (described here) or the bayesvarm
function.
creates a PriorMdl
= diffusebvarm(numseries
,numlags
)numseries
-D Bayesian VAR(numlags
) model object PriorMdl
, which specifies dimensionalities and prior assumptions for all model coefficients 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.
sets writable properties (except PriorMdl
= diffusebvarm(numseries
,numlags
,Name,Value
)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 yt and innovation εt.
numseries
sets the NumSeries
property.
Data Types: double
numlags
— Number of lagged responses
nonnegative integer
Number of lagged responses in each equation of yt, 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 yt and innovation εt.
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' ... 'Y
. NumSeries
']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,…,Φp
cell vector of numeric matrices
This property is read-only.
Distribution mean of the autoregressive coefficient matrices Φ1,…,Φp associated with the lagged responses, specified as a P
-D cell vector of NumSeries
-by-NumSeries
numeric matrices.
AR{
is
Φj
}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(
is the constant in equation j
)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(
is the linear time trend in equation j
)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(
contains the regression coefficients of each predictor in the equation of response variable j
yj
,:)j
,t. Beta(:,
contains the regression coefficient in each equation of predictor xk. 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 k
)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.
For all , is a series of independent 3-D normal innovations with a mean of 0 and covariance . Assume that the joint prior distribution of the VAR model parameters 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.
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:
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:
For all , is a series of independent 2-D normal innovations with a mean of 0 and covariance . 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 |
|
Multivariate regression |
|
Matrix regression |
|
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
istrue
.δ is the m-by-1 vector of linear time trend coefficients if
IncludeTrend
istrue
.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.which is a 1-by-(mp + r + 2) vector, and Zt is the m-by-m(mp + r + 2) block diagonal matrix
where 0z is a 1-by-(mp + r + 2) vector of zeros.
, 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
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), 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 {yt}, t = 1,…,T.
X is a T-by-m matrix containing the entire exogenous series {xt}, 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
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 + 1c + 1δ, the number of coefficients per response equation.
r =
NumPredictors
.1c is 1 if
IncludeConstant
is true, and 0 otherwise.1δ 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.
where:
Algorithms
If you pass a
diffusebvarm
object and data toestimate
, MATLAB® returns aconjugatebvarm
object representing the posterior distribution.
Version History
Introduced in R2020a
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