Conditional Mean Models

Autoregressive (AR), moving average (MA), ARMA, ARIMA, ARIMAX, and seasonal 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

expand all

Create Model

`arima`	Create univariate autoregressive integrated moving average (ARIMA) model
`LagOp`	Create lag operator polynomial
`arma2ar`	Convert ARMA model to AR model
`arma2ma`	Convert ARMA model to MA model

Fit Model to Data

`estimate`	Fit univariate ARIMA or ARIMAX model to data
`infer`	Infer univariate ARIMA or ARIMAX model residuals or conditional variances
`summarize`	Display univariate ARIMA or ARIMAX model estimation results

Generate Simulations or Impulse Responses

`simulate`	Monte Carlo simulation of univariate ARIMA or ARIMAX models
`filter`	Filter disturbances using univariate ARIMA or ARIMAX model
`impulse`	Generate univariate ARIMA model impulse response function (IRF)
`armairf`	Generate or plot ARMA model impulse responses

Generate Minimum Mean Square Error Forecasts

forecast Forecast univariate ARIMA or ARIMAX model responses or conditional variances

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.
Volatility Modeling for Soft Commodities
This example demonstrates a diverse set of statistical methods, machine learning techniques, and time-series models that you can apply broadly in the field of volatility modeling.

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.

Related Information

Creating ARIMA Models Using Econometric Modeler App

Featured Examples

Model and Simulate Electricity Spot Prices Using the Skew-Normal Distribution

Simulate the future behavior of electricity spot prices from a time series model fitted to historical data. Because electricity spot prices can exhibit large deviations, the example models the innovations using a skew-normal distribution. The data set in this example contains simulated, daily electricity spot prices from January 1, 2010 through November 11, 2013.

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