Documentation

Default ARIMA Model

The default ARIMA(p,D,q) model in Econometrics Toolbox™ is the nonseasonal model of the form

You can write this equation in condensed form using lag operator notation:

In either equation, the default innovation distribution is Gaussian with mean zero and constant variance.

At the command line, you can specify a model of this form using the shorthand syntax arima(p,D,q). For the input arguments p, D, and q, enter the number of nonseasonal AR terms (p), the order of nonseasonal integration (D), and the number of nonseasonal MA terms (q), respectively.

When you use this shorthand syntax, arima creates an arima model with these default property values.

To assign nondefault values to any properties, you can modify the created model object using dot notation.

Notice that the inputs D and q are the values arima assigns to properties D and Q. However, the input argument p is not necessarily the value arima assigns to the model property P. P stores the number of presample observations needed to initialize the AR component of the model. For nonseasonal models, the required number of presample observations is p + D.

To illustrate, consider specifying the ARIMA(2,1,1) model

where the innovation process is Gaussian with (unknown) constant variance.

Mdl = arima(2,1,1)

Mdl = 
  arima with properties:

     Description: "ARIMA(2,1,1) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 3
               D: 1
               Q: 1
        Constant: NaN
              AR: {NaN NaN} at lags [1 2]
             SAR: {}
              MA: {NaN} at lag [1]
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN

Notice that the model property P does not have value 2 (the AR degree). With the integration, a total of p + D (here, 2 + 1 = 3) presample observations are needed to initialize the AR component of the model.

The created model, Mdl, has NaNs for all parameters. A NaN value signals that a parameter needs to be estimated or otherwise specified by the user. All parameters must be specified to forecast or simulate the model.

To estimate parameters, input the model object (along with data) to estimate. This returns a new fitted arima model object. The fitted model object has parameter estimates for each input NaN value.

Calling arima without any input arguments returns an ARIMA(0,0,0) model specification with default property values:

DefaultMdl = arima

DefaultMdl = 
  arima with properties:

     Description: "ARIMA(0,0,0) Model (Gaussian Distribution)"
    Distribution: Name = "Gaussian"
               P: 0
               D: 0
               Q: 0
        Constant: NaN
              AR: {}
             SAR: {}
              MA: {}
             SMA: {}
     Seasonality: 0
            Beta: [1×0]
        Variance: NaN

Specify Nonseasonal Models Using Name-Value Pairs

The best way to specify models to arima is using name-value pair arguments. You do not need, nor are you able, to specify a value for every model object property. arima assigns default values to any properties you do not (or cannot) specify.

In condensed, lag operator notation, nonseasonal ARIMA(p,D,q) models are of the form

You can extend this model to an ARIMAX(p,D,q) model with the linear inclusion of exogenous variables. This model has the form

For the distribution of the innovations, ε_t, there are two choices:

The arima default for the innovations is an iid Gaussian process with constant (scalar) variance.

In order to estimate, forecast, or simulate a model, you must specify the parametric form of the model (e.g., which lags correspond to nonzero coefficients, the innovation distribution) and any known parameter values. You can set any unknown parameters equal to NaN, and then input the model to estimate (along with data) to get estimated parameter values.

arima (and estimate) returns a model corresponding to the model specification. You can modify models to change or update the specification. Input models (with no NaN values) to forecast or simulate for forecasting and simulation, respectively. Here are some example specifications using name-value arguments.

You can specify the following name-value arguments to create nonseasonal arima models.

Specify Multiplicative Models Using Name-Value Pairs

For a time series with periodicity s, define the degree p_s seasonal AR operator polynomial, $$\Phi (L)=(1-{\Phi }_1{L}^{p_1}-\dots -{\Phi }_{p_s}{L}^{p_s})$$, and the degree q_s seasonal MA operator polynomial, $$\Theta (L)=(1+{\Theta }_1{L}^{q_1}+\dots +{\Theta }_{q_s}{L}^{q_s})$$. Similarly, define the degree p nonseasonal AR operator polynomial, $$\varphi (L)=(1-{\varphi }_{1}L-\dots -{\varphi }_p{L}^p)$$, and the degree q nonseasonal MA operator polynomial,

A multiplicative ARIMA model with degree D nonseasonal integration and degree s seasonality is given by

In addition to the arguments for specifying nonseasonal models (described in Name-Value Arguments for Nonseasonal ARIMA Models), you can specify these name-value arguments to create a multiplicative arima model. You can extend an ARIMAX model similarly to include seasonal effects.

Specify Conditional Mean Model Using Econometric Modeler App

You can specify the lag structure and innovation distribution of seasonal and nonseasonal conditional mean models using the Econometric Modeler app. The app treats all coefficients as unknown and estimable, including the degrees of freedom parameter for a t innovation distribution.

At the command line, open the Econometric Modeler app.

econometricModeler

Alternatively, open the app from the apps gallery (see Econometric Modeler).

In the app, you can see all supported models by selecting a time series variable for the response in the Data Browser. Then, on the Econometric Modeler tab, in the Models section, click the arrow to display the models gallery.

The ARMA/ARIMA Models section contains supported conditional mean models.

For conditional mean model estimation, SARIMA and SARIMAX are the most flexible models. You can create any conditional mean model that excludes exogenous predictors by clicking SARIMA, or you can create any conditional mean model that includes at least one exogenous predictor by clicking SARIMAX.

After you select a model, the app displays the Type Model Parameters dialog box, where Type is the model type. This figure shows the SARIMAX Model Parameters dialog box.

Adjustable parameters in the dialog box depend on Type. In general, adjustable parameters include:

As you adjust parameter values, the equation in the Model Equation section changes to match your specifications. Adjustable parameters correspond to input and name-value pair arguments described in the previous sections and in the arima reference page.

For more details on specifying models using the app, see Fitting Models to Data and Specifying Lag Operator Polynomials Interactively.