Documentation

# summarize

Class: regARIMA

Display estimation results of regression model with ARIMA errors

## Syntax

``summarize(Mdl)``
``results = summarize(Mdl)``

## Description

example

````summarize(Mdl)` displays a summary of the regression model with ARIMA errors `Mdl`. If `Mdl` is an estimated model returned by `estimate`, then `summarize` prints estimation results to the MATLAB® Command Window. The display includes an estimation summary and a table of parameter estimates with corresponding standard errors, t statistics, and p-values. The estimation summary includes fit statistics, such as the Akaike Information Criterion (AIC), and the estimated innovations variance.If `Mdl` is an unestimated model returned by `regARIMA`, then `summarize` prints the standard object display (the same display that `regARIMA` prints during model creation). ```

example

````results = summarize(Mdl)` returns one of the following variables and does not print to the Command Window. If `Mdl` is an estimated model, then `results` is a structure containing estimation results.If `Mdl` is an unestimated model, then `results` is a `regARIMA` model object that is equal to `Mdl`. ```

## Input Arguments

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Regression model with ARIMA errors, specified as a `regARIMA` model object returned by `estimate` or `regARIMA`.

## Output Arguments

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Model summary, returned as a structure array or a `regARIMA` model object.

• If `Mdl` is an estimated model, then `results` is a structure array containing the fields in this table.

FieldDescription
`Description`Model summary description (string)
`SampleSize`Effective sample size (numeric scalar)
`NumEstimatedParameters`Number of estimated parameters (numeric scalar)
`LogLikelihood`Optimized loglikelihood value (numeric scalar)
`AIC`Akaike Information Criterion (numeric scalar)
`BIC`Bayesian Information Criterion (numeric scalar)
`Table`Maximum likelihood estimates of the model parameters with corresponding standard errors, t statistics (estimate divided by standard error), and p-values (assuming normality); a table with rows corresponding to model parameters

• If `Mdl` is an unestimated model, then `results` is a `regARIMA` model object that is equal to `Mdl`.

## Examples

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Regress the US gross domestic product (GDP) onto the US consumer price index (CPI) using a regression model with ARMA(1,1) errors, and summarize the results.

Load the US Macroeconomic data set and preprocess the data.

```load Data_USEconModel; logGDP = log(DataTable.GDP); dlogGDP = diff(logGDP); dCPI = diff(DataTable.CPIAUCSL);```

Fit the model to the data.

```ToEstMdl = regARIMA('ARLags',1,'MALags',1); EstMdl = estimate(ToEstMdl,dlogGDP,'X',dCPI,'Display','off');```

Display the estimates.

`summarize(EstMdl)`
``` ARMA(1,1) Error Model (Gaussian Distribution) Effective Sample Size: 248 Number of Estimated Parameters: 5 LogLikelihood: 798.406 AIC: -1586.81 BIC: -1569.24 Value StandardError TStatistic PValue __________ _____________ __________ __________ Intercept 0.014776 0.0014627 10.102 5.4243e-24 AR{1} 0.60527 0.08929 6.7787 1.2123e-11 MA{1} -0.16165 0.10956 -1.4755 0.14009 Beta(1) 0.002044 0.00070616 2.8946 0.0037969 Variance 9.3578e-05 6.0314e-06 15.515 2.7338e-54 ```

Estimate several models by passing the data to `estimate`. Vary the autoregressive and moving average degrees p and q, respectively. Estimation results contain the AIC, which you can extract and then compare among the models.

Simulate response and predictor data for the regression model with ARMA errors:

`$\begin{array}{l}{y}_{t}=2+{X}_{t}\left[\begin{array}{c}-2\\ 1.5\end{array}\right]+{u}_{t}\\ {u}_{t}=0.75{u}_{t-1}-0.5{u}_{t-2}+{\epsilon }_{t}+0.7{\epsilon }_{t-1},\end{array}$`

where ${\epsilon }_{t}$ is Gaussian with mean 0 and variance 1.

```Mdl = regARIMA('Intercept',2,'Beta',[-2; 1.5],... 'AR',{0.75, -0.5},'MA',0.7,'Variance',1); rng(2); % For reproducibility X = randn(1000,2); % Predictors y = simulate(Mdl,1000,'X',X);```

To determine the number of AR and MA lags, create and estimate multiple regression models with ARMA(p, q) errors. Vary p = 1,..,3 and q = 1,...,3 among the models. Suppress all estimation displays. Extract the AIC from the estimation results structure. The field `AIC` stores the AIC.

```pMax = 3; qMax = 3; AIC = zeros(pMax,qMax); % Preallocation for p = 1:pMax for q = 1:qMax ToEstMdl = regARIMA(p,0,q); EstMdl = estimate(ToEstMdl,y,'X',X,'Display','off'); results = summarize(EstMdl); AIC(p,q) = results.AIC; end end```

Compare the AIC values among the models.

`minAIC = min(min(AIC))`
```minAIC = 2.9280e+03 ```
`[bestP,bestQ] = find(AIC == minAIC)`
```bestP = 2 ```
```bestQ = 1 ```

The best fitting model is the regression model with AR(2,1) errors because its corresponding AIC is the lowest. This model also has the structure of the model used to simulate the data.