# summarize

Distribution summary statistics of Bayesian vector autoregression (VAR) model

## Syntax

``summarize(Mdl)``
``summarize(Mdl,display)``
``Summary = summarize(Mdl)``

## Description

example

````summarize(Mdl)` displays, at the command line, a tabular summary of the coefficients of the Bayesian VAR(p) model `Mdl`, and the innovations covariance matrix. The summary includes the means and standard deviations of the distribution `Mdl` represents.```

example

``summarize(Mdl,display)` prints the summary using the display style `display`.`

example

``Summary = summarize(Mdl)` returns distribution summary statistics `Summary`.`

## Examples

collapse all

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 a prior distribution $\pi \left({\left[{\Phi }_{1},...,{\Phi }_{4},\mathit{c}\right]}^{\prime },\Sigma \right)$ governs the behavior of the parameters. Consider using Minnesota regularization to obtain a parsimonious representation of the coefficient posterior distribution.

For each supported prior assumption, create the corresponding Bayesian VAR(4) model object for the three response variables by using `bayesvarm`. For each model that supports the option, specify all the following.

• The response variable names.

• Prior self-lag coefficients have variance 100. This large-variance setting allows the data to influence the posterior more than the prior.

• Prior cross-lag coefficients have variance 1. This small-variance setting tightens the cross-lag coefficients to zero during estimation.

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

• For the normal conjugate prior model, assume that the innovations covariance is the 3-D identity matrix.

```seriesnames = ["INFL" "UNRATE" "FEDFUNDS"]; numseries = numel(seriesnames); numlags = 4; DiffusePriorMdl = bayesvarm(numseries,numlags,'SeriesNames',seriesnames); ConjugatePriorMdl = bayesvarm(numseries,numlags,'ModelType','conjugate',... 'SeriesNames',seriesnames,'Center',0.75,'SelfLag',100,'Decay',2); SemiConjugatePriorMdl = bayesvarm(numseries,numlags,'ModelType','semiconjugate',... 'SeriesNames',seriesnames,'Center',0.75,'SelfLag',100,'CrossLag',1,'Decay',2); NormalPriorMdl = bayesvarm(numseries,numlags,'ModelType','normal',... 'SeriesNames',seriesnames,'Center',0.75,'SelfLag',100,'CrossLag',1,'Decay',2,... 'Sigma',eye(numseries));```

For each model, display summary of the prior distribution.

`summarize(DiffusePriorMdl)`
``` | Mean Std ------------------------- Constant(1) | 0 Inf Constant(2) | 0 Inf Constant(3) | 0 Inf AR{1}(1,1) | 0 Inf AR{1}(2,1) | 0 Inf AR{1}(3,1) | 0 Inf AR{1}(1,2) | 0 Inf AR{1}(2,2) | 0 Inf AR{1}(3,2) | 0 Inf AR{1}(1,3) | 0 Inf AR{1}(2,3) | 0 Inf AR{1}(3,3) | 0 Inf AR{2}(1,1) | 0 Inf AR{2}(2,1) | 0 Inf AR{2}(3,1) | 0 Inf AR{2}(1,2) | 0 Inf AR{2}(2,2) | 0 Inf AR{2}(3,2) | 0 Inf AR{2}(1,3) | 0 Inf AR{2}(2,3) | 0 Inf AR{2}(3,3) | 0 Inf AR{3}(1,1) | 0 Inf AR{3}(2,1) | 0 Inf AR{3}(3,1) | 0 Inf AR{3}(1,2) | 0 Inf AR{3}(2,2) | 0 Inf AR{3}(3,2) | 0 Inf AR{3}(1,3) | 0 Inf AR{3}(2,3) | 0 Inf AR{3}(3,3) | 0 Inf AR{4}(1,1) | 0 Inf AR{4}(2,1) | 0 Inf AR{4}(3,1) | 0 Inf AR{4}(1,2) | 0 Inf AR{4}(2,2) | 0 Inf AR{4}(3,2) | 0 Inf AR{4}(1,3) | 0 Inf AR{4}(2,3) | 0 Inf AR{4}(3,3) | 0 Inf Innovations Covariance Matrix | INFL UNRATE FEDFUNDS ------------------------------------ INFL | NaN NaN NaN | (NaN) (NaN) (NaN) UNRATE | NaN NaN NaN | (NaN) (NaN) (NaN) FEDFUNDS | NaN NaN NaN | (NaN) (NaN) (NaN) ```

Diffuse prior models put equal weight on all model coefficients. This specification allows the data to determine the posterior distribution.

`summarize(ConjugatePriorMdl)`
``` | Mean Std ------------------------------- Constant(1) | 0 33.3333 Constant(2) | 0 33.3333 Constant(3) | 0 33.3333 AR{1}(1,1) | 0.7500 3.3333 AR{1}(2,1) | 0 3.3333 AR{1}(3,1) | 0 3.3333 AR{1}(1,2) | 0 3.3333 AR{1}(2,2) | 0.7500 3.3333 AR{1}(3,2) | 0 3.3333 AR{1}(1,3) | 0 3.3333 AR{1}(2,3) | 0 3.3333 AR{1}(3,3) | 0.7500 3.3333 AR{2}(1,1) | 0 1.6667 AR{2}(2,1) | 0 1.6667 AR{2}(3,1) | 0 1.6667 AR{2}(1,2) | 0 1.6667 AR{2}(2,2) | 0 1.6667 AR{2}(3,2) | 0 1.6667 AR{2}(1,3) | 0 1.6667 AR{2}(2,3) | 0 1.6667 AR{2}(3,3) | 0 1.6667 AR{3}(1,1) | 0 1.1111 AR{3}(2,1) | 0 1.1111 AR{3}(3,1) | 0 1.1111 AR{3}(1,2) | 0 1.1111 AR{3}(2,2) | 0 1.1111 AR{3}(3,2) | 0 1.1111 AR{3}(1,3) | 0 1.1111 AR{3}(2,3) | 0 1.1111 AR{3}(3,3) | 0 1.1111 AR{4}(1,1) | 0 0.8333 AR{4}(2,1) | 0 0.8333 AR{4}(3,1) | 0 0.8333 AR{4}(1,2) | 0 0.8333 AR{4}(2,2) | 0 0.8333 AR{4}(3,2) | 0 0.8333 AR{4}(1,3) | 0 0.8333 AR{4}(2,3) | 0 0.8333 AR{4}(3,3) | 0 0.8333 Innovations Covariance Matrix | INFL UNRATE FEDFUNDS ----------------------------------------- INFL | 0.1111 0 0 | (0.0594) (0.0398) (0.0398) UNRATE | 0 0.1111 0 | (0.0398) (0.0594) (0.0398) FEDFUNDS | 0 0 0.1111 | (0.0398) (0.0398) (0.0594) ```

With a tighter prior variance around 0 for larger lags, the posterior of the conjugate model is likely to be more sparse that the posterior of the diffuse model.

`summarize(SemiConjugatePriorMdl)`
``` | Mean Std ------------------------------ Constant(1) | 0 100 Constant(2) | 0 100 Constant(3) | 0 100 AR{1}(1,1) | 0.7500 10 AR{1}(2,1) | 0 1 AR{1}(3,1) | 0 1 AR{1}(1,2) | 0 1 AR{1}(2,2) | 0.7500 10 AR{1}(3,2) | 0 1 AR{1}(1,3) | 0 1 AR{1}(2,3) | 0 1 AR{1}(3,3) | 0.7500 10 AR{2}(1,1) | 0 5 AR{2}(2,1) | 0 0.5000 AR{2}(3,1) | 0 0.5000 AR{2}(1,2) | 0 0.5000 AR{2}(2,2) | 0 5 AR{2}(3,2) | 0 0.5000 AR{2}(1,3) | 0 0.5000 AR{2}(2,3) | 0 0.5000 AR{2}(3,3) | 0 5 AR{3}(1,1) | 0 3.3333 AR{3}(2,1) | 0 0.3333 AR{3}(3,1) | 0 0.3333 AR{3}(1,2) | 0 0.3333 AR{3}(2,2) | 0 3.3333 AR{3}(3,2) | 0 0.3333 AR{3}(1,3) | 0 0.3333 AR{3}(2,3) | 0 0.3333 AR{3}(3,3) | 0 3.3333 AR{4}(1,1) | 0 2.5000 AR{4}(2,1) | 0 0.2500 AR{4}(3,1) | 0 0.2500 AR{4}(1,2) | 0 0.2500 AR{4}(2,2) | 0 2.5000 AR{4}(3,2) | 0 0.2500 AR{4}(1,3) | 0 0.2500 AR{4}(2,3) | 0 0.2500 AR{4}(3,3) | 0 2.5000 Innovations Covariance Matrix | INFL UNRATE FEDFUNDS ----------------------------------------- INFL | 0.1111 0 0 | (0.0594) (0.0398) (0.0398) UNRATE | 0 0.1111 0 | (0.0398) (0.0594) (0.0398) FEDFUNDS | 0 0 0.1111 | (0.0398) (0.0398) (0.0594) ```
`summarize(NormalPriorMdl)`
``` | Mean Std ------------------------------ Constant(1) | 0 100 Constant(2) | 0 100 Constant(3) | 0 100 AR{1}(1,1) | 0.7500 10 AR{1}(2,1) | 0 1 AR{1}(3,1) | 0 1 AR{1}(1,2) | 0 1 AR{1}(2,2) | 0.7500 10 AR{1}(3,2) | 0 1 AR{1}(1,3) | 0 1 AR{1}(2,3) | 0 1 AR{1}(3,3) | 0.7500 10 AR{2}(1,1) | 0 5 AR{2}(2,1) | 0 0.5000 AR{2}(3,1) | 0 0.5000 AR{2}(1,2) | 0 0.5000 AR{2}(2,2) | 0 5 AR{2}(3,2) | 0 0.5000 AR{2}(1,3) | 0 0.5000 AR{2}(2,3) | 0 0.5000 AR{2}(3,3) | 0 5 AR{3}(1,1) | 0 3.3333 AR{3}(2,1) | 0 0.3333 AR{3}(3,1) | 0 0.3333 AR{3}(1,2) | 0 0.3333 AR{3}(2,2) | 0 3.3333 AR{3}(3,2) | 0 0.3333 AR{3}(1,3) | 0 0.3333 AR{3}(2,3) | 0 0.3333 AR{3}(3,3) | 0 3.3333 AR{4}(1,1) | 0 2.5000 AR{4}(2,1) | 0 0.2500 AR{4}(3,1) | 0 0.2500 AR{4}(1,2) | 0 0.2500 AR{4}(2,2) | 0 2.5000 AR{4}(3,2) | 0 0.2500 AR{4}(1,3) | 0 0.2500 AR{4}(2,3) | 0 0.2500 AR{4}(3,3) | 0 2.5000 Innovations Covariance Matrix | INFL UNRATE FEDFUNDS ----------------------------------- INFL | 1 0 0 | (0) (0) (0) UNRATE | 0 1 0 | (0) (0) (0) FEDFUNDS | 0 0 1 | (0) (0) (0) ```

Semiconjugate and normal conjugate prior models yield a richer prior specification than the conjugate and diffuse models.

Consider the 3-D VAR(4) model of Inspect Minnesota Prior Assumptions Among Models. Assume that the prior distribution is diffuse.

Load the US macroeconomic data set. Compute the inflation rate, stabilize the unemployment and federal funds rates, and remove missing values.

```load Data_USEconModel seriesnames = ["INFL" "UNRATE" "FEDFUNDS"]; DataTimeTable.INFL = 100*[NaN; price2ret(DataTimeTable.CPIAUCSL)]; DataTimeTable.DUNRATE = [NaN; diff(DataTimeTable.UNRATE)]; DataTimeTable.DFEDFUNDS = [NaN; diff(DataTimeTable.FEDFUNDS)]; seriesnames(2:3) = "D" + seriesnames(2:3); rmDataTimeTable = rmmissing(DataTimeTable);```

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

```numseries = numel(seriesnames); numlags = 4; PriorMdl = bayesvarm(numseries,numlags,'SeriesNames',seriesnames);```

Estimate the posterior distribution.

`PosteriorMdl = estimate(PriorMdl,rmDataTimeTable{:,seriesnames});`
```Bayesian VAR under diffuse priors Effective Sample Size: 197 Number of equations: 3 Number of estimated Parameters: 39 | Mean Std ------------------------------- Constant(1) | 0.1007 0.0832 Constant(2) | -0.0499 0.0450 Constant(3) | -0.4221 0.1781 AR{1}(1,1) | 0.1241 0.0762 AR{1}(2,1) | -0.0219 0.0413 AR{1}(3,1) | -0.1586 0.1632 AR{1}(1,2) | -0.4809 0.1536 AR{1}(2,2) | 0.4716 0.0831 AR{1}(3,2) | -1.4368 0.3287 AR{1}(1,3) | 0.1005 0.0390 AR{1}(2,3) | 0.0391 0.0211 AR{1}(3,3) | -0.2905 0.0835 AR{2}(1,1) | 0.3236 0.0868 AR{2}(2,1) | 0.0913 0.0469 AR{2}(3,1) | 0.3403 0.1857 AR{2}(1,2) | -0.0503 0.1647 AR{2}(2,2) | 0.2414 0.0891 AR{2}(3,2) | -0.2968 0.3526 AR{2}(1,3) | 0.0450 0.0413 AR{2}(2,3) | 0.0536 0.0223 AR{2}(3,3) | -0.3117 0.0883 AR{3}(1,1) | 0.4272 0.0860 AR{3}(2,1) | -0.0389 0.0465 AR{3}(3,1) | 0.2848 0.1841 AR{3}(1,2) | 0.2738 0.1620 AR{3}(2,2) | 0.0552 0.0876 AR{3}(3,2) | -0.7401 0.3466 AR{3}(1,3) | 0.0523 0.0428 AR{3}(2,3) | 0.0008 0.0232 AR{3}(3,3) | 0.0028 0.0917 AR{4}(1,1) | 0.0167 0.0901 AR{4}(2,1) | 0.0285 0.0488 AR{4}(3,1) | -0.0690 0.1928 AR{4}(1,2) | -0.1830 0.1520 AR{4}(2,2) | -0.1795 0.0822 AR{4}(3,2) | 0.1494 0.3253 AR{4}(1,3) | 0.0067 0.0395 AR{4}(2,3) | 0.0088 0.0214 AR{4}(3,3) | -0.1372 0.0845 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) ```

Summarize the posterior distribution; compare each estimation display type.

`summarize(PosteriorMdl); % The default is 'table'.`
``` | Mean Std ------------------------------- Constant(1) | 0.1007 0.0832 Constant(2) | -0.0499 0.0450 Constant(3) | -0.4221 0.1781 AR{1}(1,1) | 0.1241 0.0762 AR{1}(2,1) | -0.0219 0.0413 AR{1}(3,1) | -0.1586 0.1632 AR{1}(1,2) | -0.4809 0.1536 AR{1}(2,2) | 0.4716 0.0831 AR{1}(3,2) | -1.4368 0.3287 AR{1}(1,3) | 0.1005 0.0390 AR{1}(2,3) | 0.0391 0.0211 AR{1}(3,3) | -0.2905 0.0835 AR{2}(1,1) | 0.3236 0.0868 AR{2}(2,1) | 0.0913 0.0469 AR{2}(3,1) | 0.3403 0.1857 AR{2}(1,2) | -0.0503 0.1647 AR{2}(2,2) | 0.2414 0.0891 AR{2}(3,2) | -0.2968 0.3526 AR{2}(1,3) | 0.0450 0.0413 AR{2}(2,3) | 0.0536 0.0223 AR{2}(3,3) | -0.3117 0.0883 AR{3}(1,1) | 0.4272 0.0860 AR{3}(2,1) | -0.0389 0.0465 AR{3}(3,1) | 0.2848 0.1841 AR{3}(1,2) | 0.2738 0.1620 AR{3}(2,2) | 0.0552 0.0876 AR{3}(3,2) | -0.7401 0.3466 AR{3}(1,3) | 0.0523 0.0428 AR{3}(2,3) | 0.0008 0.0232 AR{3}(3,3) | 0.0028 0.0917 AR{4}(1,1) | 0.0167 0.0901 AR{4}(2,1) | 0.0285 0.0488 AR{4}(3,1) | -0.0690 0.1928 AR{4}(1,2) | -0.1830 0.1520 AR{4}(2,2) | -0.1795 0.0822 AR{4}(3,2) | 0.1494 0.3253 AR{4}(1,3) | 0.0067 0.0395 AR{4}(2,3) | 0.0088 0.0214 AR{4}(3,3) | -0.1372 0.0845 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) ```

The default is the same default tabular display that `estimate` prints.

`summarize(PosteriorMdl,'equation');`
``` 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) ```

In the '`equation'` display, rows correspond to response equations in the VAR system, and columns correspond to lagged response variables within equations. Elements in the table correspond to the posterior means of the corresponding coefficient; under each mean in parentheses is the standard deviation of the posterior.

`summarize(PosteriorMdl,'matrix');`
``` VAR Coefficient Matrix of Lag 1 | INFL(-1) DUNRATE(-1) DFEDFUNDS(-1) -------------------------------------------------- INFL | 0.1241 -0.4809 0.1005 | (0.0762) (0.1536) (0.0390) DUNRATE | -0.0219 0.4716 0.0391 | (0.0413) (0.0831) (0.0211) DFEDFUNDS | -0.1586 -1.4368 -0.2905 | (0.1632) (0.3287) (0.0835) VAR Coefficient Matrix of Lag 2 | INFL(-2) DUNRATE(-2) DFEDFUNDS(-2) -------------------------------------------------- INFL | 0.3236 -0.0503 0.0450 | (0.0868) (0.1647) (0.0413) DUNRATE | 0.0913 0.2414 0.0536 | (0.0469) (0.0891) (0.0223) DFEDFUNDS | 0.3403 -0.2968 -0.3117 | (0.1857) (0.3526) (0.0883) VAR Coefficient Matrix of Lag 3 | INFL(-3) DUNRATE(-3) DFEDFUNDS(-3) -------------------------------------------------- INFL | 0.4272 0.2738 0.0523 | (0.0860) (0.1620) (0.0428) DUNRATE | -0.0389 0.0552 0.0008 | (0.0465) (0.0876) (0.0232) DFEDFUNDS | 0.2848 -0.7401 0.0028 | (0.1841) (0.3466) (0.0917) VAR Coefficient Matrix of Lag 4 | INFL(-4) DUNRATE(-4) DFEDFUNDS(-4) -------------------------------------------------- INFL | 0.0167 -0.1830 0.0067 | (0.0901) (0.1520) (0.0395) DUNRATE | 0.0285 -0.1795 0.0088 | (0.0488) (0.0822) (0.0214) DFEDFUNDS | -0.0690 0.1494 -0.1372 | (0.1928) (0.3253) (0.0845) Constant Term INFL | 0.1007 | (0.0832) DUNRATE | -0.0499 | 0.0450 DFEDFUNDS | -0.4221 | 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) ```

In the `'matrix'` display, each table contains the posterior mean of the corresponding coefficient matrix. Under each mean in parentheses the posterior standard deviation.

Consider the 3-D VAR(4) model of Inspect Minnesota Prior Assumptions Among Models. Assume that the parameters follow a semiconjugate prior model.

Load the US macroeconomic data set. Compute the inflation rate, stabilize the unemployment aand federal funds rates, and remove missing values.

```load Data_USEconModel seriesnames = ["INFL" "UNRATE" "FEDFUNDS"]; DataTimeTable.INFL = 100*[NaN; price2ret(DataTimeTable.CPIAUCSL)]; DataTimeTable.DUNRATE = [NaN; diff(DataTimeTable.UNRATE)]; DataTimeTable.DFEDFUNDS = [NaN; diff(DataTimeTable.FEDFUNDS)]; seriesnames(2:3) = "D" + seriesnames(2:3); rmDataTimeTable = rmmissing(DataTimeTable);```

Create a semiconjugate Bayesian VAR(4) prior model for the three response series. Specify the response variable names, and suppress the estimation display.

```numseries = numel(seriesnames); numlags = 4; PriorMdl = bayesvarm(numseries,numlags,'Model','semiconjugate',... 'SeriesNames',seriesnames);```

Estimate the posterior distribution. Suppress the estimation display.

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

Because the posterior of a semiconjugate model is analytically intractable, `PosteriorMdl` is an `empiricalbvarm` model object storing the draws from the Gibbs sampler.

Summarize the posterior distribution; return the estimation summary.

`Summary = summarize(PosteriorMdl);`
``` | Mean Std ------------------------------- Constant(1) | 0.1830 0.0718 Constant(2) | -0.0808 0.0413 Constant(3) | -0.0161 0.1309 AR{1}(1,1) | 0.2246 0.0650 AR{1}(2,1) | -0.0263 0.0340 AR{1}(3,1) | -0.0263 0.0775 AR{1}(1,2) | -0.0837 0.0824 AR{1}(2,2) | 0.3665 0.0740 AR{1}(3,2) | -0.1283 0.0948 AR{1}(1,3) | 0.1362 0.0323 AR{1}(2,3) | 0.0154 0.0198 AR{1}(3,3) | -0.0538 0.0685 AR{2}(1,1) | 0.2518 0.0700 AR{2}(2,1) | 0.0928 0.0352 AR{2}(3,1) | 0.0373 0.0628 AR{2}(1,2) | -0.0097 0.0632 AR{2}(2,2) | 0.1657 0.0709 AR{2}(3,2) | -0.0254 0.0688 AR{2}(1,3) | 0.0329 0.0308 AR{2}(2,3) | 0.0341 0.0199 AR{2}(3,3) | -0.1451 0.0637 AR{3}(1,1) | 0.2895 0.0665 AR{3}(2,1) | 0.0013 0.0332 AR{3}(3,1) | -0.0036 0.0530 AR{3}(1,2) | 0.0322 0.0538 AR{3}(2,2) | -0.0150 0.0667 AR{3}(3,2) | -0.0369 0.0568 AR{3}(1,3) | 0.0368 0.0298 AR{3}(2,3) | -0.0083 0.0194 AR{3}(3,3) | 0.1516 0.0603 AR{4}(1,1) | 0.0452 0.0644 AR{4}(2,1) | 0.0225 0.0325 AR{4}(3,1) | -0.0097 0.0470 AR{4}(1,2) | -0.0218 0.0468 AR{4}(2,2) | -0.1125 0.0611 AR{4}(3,2) | 0.0013 0.0491 AR{4}(1,3) | 0.0180 0.0273 AR{4}(2,3) | 0.0084 0.0179 AR{4}(3,3) | -0.0815 0.0594 Innovations Covariance Matrix | INFL DUNRATE DFEDFUNDS ------------------------------------------- INFL | 0.2983 -0.0219 0.1750 | (0.0307) (0.0121) (0.0500) DUNRATE | -0.0219 0.0890 -0.1495 | (0.0121) (0.0093) (0.0290) DFEDFUNDS | 0.1750 -0.1495 1.4730 | (0.0500) (0.0290) (0.1514) ```
`Summary`
```Summary = struct with fields: Description: "3-Dimensional VAR(4) Model" NumEstimatedParameters: 39 Table: [39x2 table] CoeffMap: [39x1 string] CoeffMean: [39x1 double] CoeffStd: [39x1 double] SigmaMean: [3x3 double] SigmaStd: [3x3 double] ```

Summary is a structure array of fields containing posterior estimation information. For example, the `CoeffMap` field contains a list of the coefficient names. The order of the names corresponds to the order the all coefficient vector inputs and outputs. Display `CoeffMap`.

`Summary.CoeffMap`
```ans = 39x1 string "AR{1}(1,1)" "AR{1}(1,2)" "AR{1}(1,3)" "AR{2}(1,1)" "AR{2}(1,2)" "AR{2}(1,3)" "AR{3}(1,1)" "AR{3}(1,2)" "AR{3}(1,3)" "AR{4}(1,1)" "AR{4}(1,2)" "AR{4}(1,3)" "Constant(1)" "AR{1}(2,1)" "AR{1}(2,2)" "AR{1}(2,3)" "AR{2}(2,1)" "AR{2}(2,2)" "AR{2}(2,3)" "AR{3}(2,1)" "AR{3}(2,2)" "AR{3}(2,3)" "AR{4}(2,1)" "AR{4}(2,2)" "AR{4}(2,3)" "Constant(2)" "AR{1}(3,1)" "AR{1}(3,2)" "AR{1}(3,3)" "AR{2}(3,1)" ⋮ ```

## Input Arguments

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Prior or posterior Bayesian VAR model, specified as a model object in this table.

Model ObjectDescription
`conjugatebvarm`Dependent, matrix-normal-inverse-Wishart conjugate model returned by `bayesvarm`, `conjugatebvarm`, or `estimate`
`semiconjugatebvarm`Independent, normal-inverse-Wishart semiconjugate prior model returned by `bayesvarm` or `semiconjugatebvarm`
`diffusebvarm`Diffuse prior model returned by `bayesvarm` or `diffusebvarm`
`empiricalbvarm`Prior or posterior model characterized by random draws from respective distributions, returned by `empiricalbvarm` or `estimate`

Distribution summary display style, specified as a value in this table.

ValueDescription
`'off'``summarize` does not print to the command line.
`'table'`

`summarize` prints the following:

• Estimation information

• Tabular summary of coefficient posterior means and standard deviations; each row corresponds to a coefficient, and each column corresponds to an estimate type

• Posterior mean of the innovations covariance matrix with standard deviations in parentheses

`'equation'`

`summarize` prints the following:

• Estimation information

• Tabular summary of posterior means and standard deviations; each row corresponds to a response variable in the system, and each column corresponds to a coefficient in the equation (for example, the column labeled `Y1(-1)` contains the estimates of the lag 1 coefficient of the first response variable in each equation)

• Posterior mean of the innovations covariance matrix with standard deviations in parentheses.

`'matrix'`

`summarize` prints the following:

• Estimation information

• Separate tabular displays of posterior means and standard deviations (in parentheses) for each parameter in the model Φ1,…, Φp, c, δ, Β, and Σ

Data Types: `char` | `string`

## Output Arguments

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Distribution summary statistics, returned as a structure array containing these fields:

FieldDescriptionData type
`Description`Model descriptionstring scalar
`NumEstimatedParameters`Number of coefficientsnumeric scalar
`Table`Table of coefficient distribution means and standard deviations; each row corresponds to a coefficient and each column corresponds to a statistictable
`CoeffMap`Coefficient namesstring vector
`CoeffMean`Coefficient distribution means numeric vector, rows correspond to `CoeffMap`
`CoeffStd`Coefficient distribution standard deviationsnumeric vector, rows correspond to `CoeffMap`
`SigmaMean`Innovations covariance distribution mean matrixnumeric matrix, rows and columns correspond to response equations
`SigmaStd`Innovations covariance distribution standard deviation matrixnumeric matrix, rows and columns correspond to response equations

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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), 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.

## Version History

Introduced in R2020a