# Conditional Mean Models

In time series econometrics, the dynamic behavior of a variable over time
is often of interest. A dynamic conditional mean model specifies the
expected value of a response process
*y _{t}* as a function of historical
information.

To model the dynamic behavior of a univariate linear conditional mean
model, use the Econometrics Toolbox™
`arima`

function at the command line
or you can create models interactively with the **Econometric
Modeler** app. By using `arima`

, you can create a
wide variety of autoregressive integrated moving average (ARIMA) models,
including optionally specifying seasonal components for a SARIMA model,
linearly adjusting for exogenous predictors for an ARIMAX model, or
specifying a GARCH variance model, for example, to create a composite
conditional mean and variance model. For more details on programmatic and
interactive ARIMA model creation, see Creating Univariate Conditional Mean Models.

For multivariate conditional mean models, see Vector Autoregression Models, and, for linear regression models that assume an ARIMA error process, see Autocorrelated and Heteroscedastic Disturbances.

## Apps

Econometric Modeler | Analyze and model econometric time series |

## Functions

## Topics

### Interactive Workflows

**Analyze Time Series Data Using Econometric Modeler**

Interactively visualize and analyze univariate or multivariate time series data.**Implement Box-Jenkins Model Selection and Estimation Using Econometric Modeler App**

Interactively implement the Box-Jenkins methodology to select the appropriate number of lags for a univariate conditional mean model. Then, fit the model to data and export the estimated model to the command line to generate forecasts.**Perform ARIMA Model Residual Diagnostics Using Econometric Modeler App**

Interactively evaluate model assumptions after fitting data to an ARIMA model by performing residual diagnostics.**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.

### Create Model

**Select ARIMA Model for Time Series Using Box-Jenkins Methodology**

Apply Box-Jenkins methodology to select an ARIMA model for the quarterly Australian consumer price index.**Creating Univariate Conditional Mean Models**

Create univariate conditional mean models using`arima`

or the Econometric Modeler app.**Specifying Univariate Lag Operator Polynomials Interactively**

Specify univariate lag operator polynomial terms for time series model estimation using Econometric Modeler.**Modify Properties of Conditional Mean Model Objects**

Change modifiable model properties using dot notation.**Specify Conditional Mean Model Innovation Distribution**

Specify Gaussian or t distributed innovations process, or a conditional variance model for the variance process.**Specify t Innovation Distribution Using Econometric Modeler App**

Interactively specify a*t*innovation distribution for an ARIMA model.**Create Autoregressive Models**

Create stationary autoregressive models using`arima`

or the Econometric Modeler app.**Create Moving Average Models**

Create invertible moving average models using`arima`

or the Econometric Modeler app.**Create Autoregressive Moving Average Models**

Create stationary and invertible autoregressive moving average models using`arima`

or the Econometric Modeler app.**Create Autoregressive Integrated Moving Average Models**

Create autoregressive integrated moving average models using`arima`

or the Econometric Modeler app.**Create ARIMA Models That Include Exogenous Covariates**

Create ARIMAX models using`arima`

or the Econometric Modeler app.**Create Multiplicative ARIMA Models**

Create multiplicative ARIMA models using`arima`

or the Econometric Modeler app.**Create Multiplicative Seasonal ARIMA Model for Time Series Data**

Create a seasonal ARIMA model.**Specify Conditional Mean and Variance Models**

Create a composite conditional mean and variance model.

### Fit Model to Data

**Presample Data for Conditional Mean Model Estimation**

Specify presample data to initialize the model.**Time Base Partitions for ARIMA Model Estimation**

When you fit a time series model to data, lagged terms in the model require initialization, usually with observations at the beginning of the sample.**Box-Jenkins Differencing vs. ARIMA Estimation**

Compare Box-Jenkins and ARIMA estimation.**Choose ARMA Lags Using BIC**

Select ARMA model using information criteria.**Estimate Multiplicative ARIMA Model Using Econometric Modeler App**

Interactively estimate a multiplicative seasonal ARIMA model.**Estimate Multiplicative ARIMA Model**

Estimate a multiplicative seasonal ARIMA model.**Model Seasonal Lag Effects Using Indicator Variables**

Estimate a seasonal ARIMA model by specifying a multiplicative model or using seasonal dummies.**Estimate ARIMAX Model Using Econometric Modeler App**

Interactively specify and estimate an ARIMAX model.**Estimate Conditional Mean and Variance Model**

Estimate a composite conditional mean and variance model.**Infer Residuals for Diagnostic Checking**

Infer residuals from a fitted ARIMA model.**Maximum Likelihood Estimation for Conditional Mean Models**

Learn how maximum likelihood is carried out for conditional mean models.**Conditional Mean Model Estimation with Equality Constraints**

Constrain the model during estimation using known parameter values.**Initial Values for Conditional Mean Model Estimation**

Specify initial parameter values for estimation.**Optimization Settings for Conditional Mean Model Estimation**

Troubleshoot estimation issues by specifying alternative optimization options.

### Generate Simulations or Impulse Responses

**Simulate Stationary Processes**

Simulate stationary autoregressive models and moving average models.**Simulate Trend-Stationary and Difference-Stationary Processes**

Illustrate the distinction between trend-stationary and difference-stationary processes by simulation.**Simulate Multiplicative ARIMA Models**

Simulate sample paths from a multiplicative seasonal ARIMA model.**Simulate Conditional Mean and Variance Models**

Simulate responses and conditional variances from a composite conditional mean and variance model.**Plot the Impulse Response Function of Conditional Mean Model**

Plot the impulse response function of univariate autoregressive moving average models.**Monte Carlo Simulation of Conditional Mean Models**

Learn about Monte Carlo simulation.**Presample Data for Conditional Mean Model Simulation**

Learn about presample requirements for simulation.**Transient Effects in Conditional Mean Model Simulations**

Learn how to minimize transient effects.

### Generate Minimum Mean Square Error Forecasts

**Compare Predictive Performance After Creating Models Using Econometric Modeler**

Interactively choose lags for an ARIMA model by comparing the AIC values of estimated models. Then, export several models to the command line to compare their predictive performance.**Forecast Multiplicative ARIMA Model**

Forecast a multiplicative seasonal ARIMA model.**Convergence of AR Forecasts**

Evaluate the asymptotic convergence of forecasts from an AR model, and compare forecasts made with and without using presample data.**Forecast Conditional Mean and Variance Model**

Forecast responses and conditional variances from a composite conditional mean and variance model.**Forecast IGD Rate from ARX Model**

Forecast an ARIMAX model by computing MMSE forecasts or using Monte Carlo simulation.**Specify Presample and Forecast Period Data to Forecast ARIMAX Model**

This example shows how to partition a timeline into presample, estimation, and forecast periods, and it shows how to supply the appropriate number of observations to initialize a dynamic model for estimation and forecasting.**Monte Carlo Forecasting of Conditional Mean Models**

Learn about Monte Carlo forecasting.**MMSE Forecasting of Conditional Mean Models**

Learn about MMSE forecasting.