# Fit VAR Model of CPI and Unemployment Rate

This example shows how to estimate the parameters of a VAR(4) model. The response series are quarterly measures of the consumer price index (CPI) and the unemployment rate.

Load the `Data_USEconModel` data set.

`load Data_USEconModel`

Plot the two series on separate plots.

```figure; plot(DataTable.Time,DataTable.CPIAUCSL); title('Consumer Price Index'); ylabel('Index'); xlabel('Date');``` ```figure; plot(DataTable.Time,DataTable.UNRATE); title('Unemployment rate'); ylabel('Percent'); xlabel('Date');``` The CPI appears to grow exponentially.

Stabilize the CPI by converting it to a series of growth rates. Synchronize the two series by removing the first observation from the unemployment rate series.

```rcpi = price2ret(DataTable.CPIAUCSL); unrate = DataTable.UNRATE(2:end);```

Create a default VAR(4) model using the shorthand syntax.

`Mdl = varm(2,4)`
```Mdl = varm with properties: Description: "2-Dimensional VAR(4) Model" SeriesNames: "Y1" "Y2" NumSeries: 2 P: 4 Constant: [2Ã-1 vector of NaNs] AR: {2Ã-2 matrices of NaNs} at lags [1 2 3 ... and 1 more] Trend: [2Ã-1 vector of zeros] Beta: [2Ã-0 matrix] Covariance: [2Ã-2 matrix of NaNs] ```

`Mdl` is a `varm` model object. It serves as a template for model estimation. MATLABï¿½ considers any `NaN` values as unknown parameter values to be estimated. For example, the `Constant` property is a 2-by-1 vector of `NaN` values. Therefore, model constants are model parameters to be estimated.

Fit the model to the data.

`EstMdl = estimate(Mdl,[rcpi unrate])`
```EstMdl = varm with properties: Description: "AR-Stationary 2-Dimensional VAR(4) Model" SeriesNames: "Y1" "Y2" NumSeries: 2 P: 4 Constant: [0.00171639 0.316255]' AR: {2Ã-2 matrices} at lags [1 2 3 ... and 1 more] Trend: [2Ã-1 vector of zeros] Beta: [2Ã-0 matrix] Covariance: [2Ã-2 matrix] ```

`EstMdl` is a `varm` model object. `EstMdl` is structurally the same as `Mdl`, but all parameters are known. To inspect the estimated parameters, you can display them using dot notation.

Display the coefficient of the first lag term.

`EstMdl.AR{1}`
```ans = 0.3090 -0.0032 -4.4834 1.3433 ```

Display an estimation summary including all parameters, standard errors, and p-values for testing the null hypothesis that the coefficient is 0.

`summarize(EstMdl)`
``` AR-Stationary 2-Dimensional VAR(4) Model Effective Sample Size: 241 Number of Estimated Parameters: 18 LogLikelihood: 811.361 AIC: -1586.72 BIC: -1524 Value StandardError TStatistic PValue ___________ _____________ __________ __________ Constant(1) 0.0017164 0.0015988 1.0735 0.28303 Constant(2) 0.31626 0.091961 3.439 0.0005838 AR{1}(1,1) 0.30899 0.063356 4.877 1.0772e-06 AR{1}(2,1) -4.4834 3.6441 -1.2303 0.21857 AR{1}(1,2) -0.0031796 0.0011306 -2.8122 0.004921 AR{1}(2,2) 1.3433 0.065032 20.656 8.546e-95 AR{2}(1,1) 0.22433 0.069631 3.2217 0.0012741 AR{2}(2,1) 7.1896 4.005 1.7951 0.072631 AR{2}(1,2) 0.0012375 0.0018631 0.6642 0.50656 AR{2}(2,2) -0.26817 0.10716 -2.5025 0.012331 AR{3}(1,1) 0.35333 0.068287 5.1742 2.2887e-07 AR{3}(2,1) 1.487 3.9277 0.37858 0.705 AR{3}(1,2) 0.0028594 0.0018621 1.5355 0.12465 AR{3}(2,2) -0.22709 0.1071 -2.1202 0.033986 AR{4}(1,1) -0.047563 0.069026 -0.68906 0.49079 AR{4}(2,1) 8.6379 3.9702 2.1757 0.029579 AR{4}(1,2) -0.00096323 0.0011142 -0.86448 0.38733 AR{4}(2,2) 0.076725 0.064088 1.1972 0.23123 Innovations Covariance Matrix: 0.0000 -0.0002 -0.0002 0.1167 Innovations Correlation Matrix: 1.0000 -0.0925 -0.0925 1.0000 ```