# Documentation

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# Conditional Mean Models

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

## Classes

 `arima` Create ARIMA or ARIMAX time series model `LagOp` Create lag operator polynomial (LagOp) object

## Functions

 `arima` Create ARIMA or ARIMAX time series model `LagOp` Create lag operator polynomial (LagOp) object `arma2ar` Convert ARMA model to AR model `arma2ma` Convert ARMA model to MA model
 `estimate` Estimate ARIMA or ARIMAX model parameters `infer` Infer ARIMA or ARIMAX model residuals or conditional variances `print` Display parameter estimation results for ARIMA or ARIMAX models
 `simulate` Monte Carlo simulation of ARIMA or ARIMAX models `filter` Filter disturbances using ARIMA or ARIMAX model `impulse` Impulse response function `armairf` Generate ARMA model impulse responses
 `forecast` Forecast ARIMA or ARIMAX process

## Examples and How To

### Create Model

Specify Conditional Mean Models Using arima

Create various ARIMA models.

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.

AR Model Specifications

Create various stationary ARMA models.

MA Model Specifications

Create various, invertible moving average models.

ARMA Model Specifications

Create various stationary autoregressive, moving average models.

ARIMA Model Specifications

Create various autoregressive integrated moving average models.

ARIMAX Model Specifications

Create various ARIMAX models.

Multiplicative ARIMA Model Specifications

Create various multiplicative ARIMA models.

Specify Multiplicative ARIMA Model

Create a seasonal ARIMA model.

Specify Conditional Mean and Variance Models

Create a composite conditional mean and variance model.

### Fit Model to Data

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

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 Conditional Mean and Variance Models

Estimate a composite conditional mean and variance model.

Infer Residuals for Diagnostic Checking

Infer residuals from a fitted ARIMA model.

### 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

Plot the impulse response function for various models.

### Generate Minimum Mean Square Error Forecasts

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 Using ARIMAX Model

Forecast an ARIMAX model by computing MMSE forecasts or using Monte Carlo simulation.

## Concepts

Conditional Mean Models

Learn about the characteristics and forms of conditional mean models.

Autoregressive Model

Learn about autoregressive models.

Moving Average Model

Learn about moving average models.

Autoregressive Moving Average Model

Learn about autoregressive, moving average models.

ARIMA Model

Learn about autoregressive integrated moving average models.

Multiplicative ARIMA Model

Learn about addressing seasonality and potential seasonal unit roots using multiplcative ARIMA models.

ARIMA Model Including Exogenous Covariates

Learn about ARIMA models that include a linear term for exogenous variables.

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.

Presample Data for Conditional Mean Model Estimation

Specify presample data to initialize the model.

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.

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.

Monte Carlo Forecasting of Conditional Mean Models

Learn about Monte Carlo forecasting.

Impulse Response Function

Learn about impulse response functions.

MMSE Forecasting of Conditional Mean Models

Learn about MMSE forecasting.

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