# Autocorrelated and Heteroscedastic Disturbances

To explicitly model for serial correlation in the disturbance series, create a regression model with ARIMA errors (`regARIMA`

model object). Alternatively, to acknowledge the presence of nonsphericality, you can estimate a heteroscedastic-and-autocorrelation-consistent (HAC) coefficient covariance matrix, or implement feasible generalized least squares (FGLS). For more details on HAC and FGLS estimators, see Time Series Regression X: Generalized Least Squares and HAC Estimators.

For conditional mean model tools that support ARIMA model creation and analysis, see Conditional Mean Models.

## Apps

Econometric Modeler | Analyze and model econometric time series |

## Functions

## Examples and How To

### Create Model

**Create Regression Models with ARIMA Errors**

Create regression models with autoregressive integrated moving average errors using`regARIMA`

or the Econometric Modeler app.**Specify the Default Regression Model with ARIMA Errors**

Create a default regression model with ARIMA errors using`regARIMA`

.**Create Regression Models with AR Errors**

Create regression models with AR errors using`regARIMA`

.**Create Regression Models with MA Errors**

Create regression models with MA errors using`regARIMA`

.**Create Regression Models with ARMA Errors**

Create regression models with ARMA errors using`regARIMA`

or the Econometric Modeler app.**Create Regression Models with ARIMA Errors**

Create regression models with ARIMA errors using`regARIMA`

.**Create Regression Models with SARIMA Errors**

Create regression models with SARIMA errors using`regARIMA`

.**Specify ARIMA Error Model Innovation Distribution**

Choose between Gaussian- or t-distributed innovations.**Specify Regression Model with SARIMA Errors**

Create a regression model with multiplicative seasonal ARIMA errors.**Modify regARIMA Model Properties**

Change aspects of an existing model.**Plot Impulse Response of Regression Model with ARIMA Errors**

Plot impulse response functions of various regression models with ARIMA errors.**Alternative ARIMA Model Representations**

Convert between ARMAX and regression models with ARMA errors.

### Fit Model to Data

**Estimate Regression Model with ARMA Errors Using Econometric Modeler App**

Interactively specify and estimate a regression model with ARMA errors.**Estimate a Regression Model with ARIMA Errors**

Estimate the sensitivity of the US Gross Domestic Product (GDP) to changes in the Consumer Price Index (CPI) using`estimate`

.**Estimate a Regression Model with Multiplicative ARIMA Errors**

Fit a regression model with multiplicative ARIMA errors to data using`estimate`

.**Alternative ARIMA Model Representations**

Convert between ARMAX and regression models with ARMA errors.**Choose Lags for ARMA Error Model**

To select the nonseasonal autoregressive and moving average lag polynomial degrees for a regression model with ARMA errors, use Akaike Information Criterion (AIC).**Plot a Confidence Band Using HAC Estimates**

Plot corrected confidence bands using Newey-West robust standard errors.**Change the Bandwidth of a HAC Estimator**

Change the bandwidth when estimating a HAC coefficient covariance, and compare estimates over varying bandwidths and kernels.**Compare Robust Regression Techniques**

Address influential outliers using regression models with ARIMA errors, bags of regression trees, and Bayesian linear regression.**Share Results of Econometric Modeler App Session**

Export variables to the MATLAB^{®}Workspace, generate plain text and live functions that return a model estimated in an app session, or generate a report recording your activities on time series and estimated models in an Econometric Modeler app session.

### Generate Simulations or Impulse Responses

**Simulate Regression Models with ARMA Errors**

Simulate observations from various regression models with ARMA errors.**Simulate Regression Models with Nonstationary Errors**

Simulate regression model with nonstationary and exponential errors.**Simulate Regression Models with Multiplicative Seasonal Errors**

Simulate regression model with stationary and difference stationary errors.**Forecast a Regression Model with ARIMA Errors**

Forecast a regression model with ARIMA(3,1,2) errors using`forecast`

and`simulate`

.

### Generate Minimum Mean Square Error Forecasts

**Forecast a Regression Model with ARIMA Errors**

Forecast a regression model with ARIMA(3,1,2) errors using`forecast`

and`simulate`

.**Forecast a Regression Model with Multiplicative Seasonal ARIMA Errors**

Forecast a multiplicative seasonal ARIMA model using`forecast`

.**Verify Predictive Ability Robustness of a regARIMA Model**

Forecast a regression model with ARIMA errors, and check the model predictability robustness.

## Concepts

**Econometric Modeler App Overview**The Econometric Modeler app is an interactive tool for visualizing and analyzing univariate or multivariate time series data.

**Specifying Univariate Lag Operator Polynomials Interactively**Specify univariate lag operator polynomial terms for time series model estimation using Econometric Modeler.

**Impulse Response of Regression Models with ARIMA Errors**Learn about impulse response functions of regression models with ARIMA errors.

**Nonspherical Models**Learn about innovations that exhibit autocorrelation and heteroscedasticity.

**Regression Models with Time Series Errors**Learn about regression models with ARIMA errors.

**Time Series Regression Models**Define different types of time series regression models.

**Initial Values for regARIMA Model Estimation**Learn how MATLAB uses initial parameter values during estimation.

**Intercept Identifiability in Regression Models with ARIMA Errors**Learn about intercept identifiability in regression model with ARIMA errors.

**Select Regression Model with ARIMA Errors**Learn how to select an appropriate regression model with ARIMA errors.

**Maximum Likelihood Estimation of regARIMA Models**Learn about maximum likelihood estimation for regression models with ARIMA errors.

**Optimization Settings for regARIMA Model Estimation**Learn about optimization settings for regression model with ARIMA errors estimation.

**Presample Values for regARIMA Model Estimation**Learn how MATLAB uses presample values during estimation.

**regARIMA Model Estimation Using Equality Constraints**Estimate regression model with ARIMA errors with equality constraints.

**Monte Carlo Simulation of Regression Models with ARIMA Errors**Learn about generating independent, random draws from a regression model with ARIMA errors.

**Presample Data for regARIMA Model Simulation**Learn about the presample data required to simulate a regression model with ARIMA errors.

**Transient Effects in regARIMA Model Simulations**Learn about how presample data affects a simulated path.

**Monte Carlo Forecasting of regARIMA Models**Learn about forecasting a regression model with ARIMA errors using many simulated paths.

**MMSE Forecasting Regression Models with ARIMA Errors**Learn about minimum mean square error forecasts.