## Create Autoregressive Moving Average Models

These examples show how to create various autoregressive moving average
(ARMA) models by using the `arima`

function.

### Default ARMA Model

This example shows how to use the shorthand `arima(p,D,q)`

syntax to specify the default ARMA(*p*, *q*) model,

$${y}_{t}=6+0.2{y}_{t-1}-0.3{y}_{t-2}+3{x}_{t}+{\epsilon}_{t}+0.1{\epsilon}_{t-1}$$

By default, all parameters in the created model object have unknown values, and the innovation distribution is Gaussian with constant variance.

Specify the default ARMA(1,1) model:

Mdl = arima(1,0,1)

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

The output shows that the created model object, Mdl, has `NaN`

values for all model parameters: the constant term, the AR and MA coefficients, and the variance. You can modify the created model object using dot notation, or input it (along with data) to `estimate`

.

### ARMA Model with No Constant Term

This example shows how to specify an ARMA(*p*, *q*) model with constant term equal to zero. Use name-value syntax to specify a model that differs from the default model.

Specify an ARMA(2,1) model with no constant term,

$${y}_{t}={\varphi}_{1}{y}_{t-1}+{\varphi}_{2}{y}_{t-2}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1},$$

where the innovation distribution is Gaussian with constant variance.

Mdl = arima('ARLags',1:2,'MALags',1,'Constant',0)

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

The `ArLags`

and `MaLags`

name-value pair arguments specify the lags corresponding to nonzero AR and MA coefficients, respectively. The property `Constant`

in the created model object is equal to `0`

, as specified. The model has default values for all other properties, including `NaN`

values as placeholders for the unknown parameters: the AR and MA coefficients, and scalar variance.

You can modify the created model using dot notation, or input it (along with data) to `estimate`

.

### ARMA Model with Known Parameter Values

This example shows how to specify an ARMA(*p*, *q*) model with known parameter values. You can use such a fully specified model as an input to `simulate`

or `forecast`

.

Specify the ARMA(1,1) model

$${y}_{t}=0.3+0.7\varphi {y}_{t-1}+{\epsilon}_{t}+0.4{\epsilon}_{t-1},$$

where the innovation distribution is Student's *t* with 8 degrees of freedom, and constant variance 0.15.

tdist = struct('Name','t','DoF',8); Mdl = arima('Constant',0.3,'AR',0.7,'MA',0.4,... 'Distribution',tdist,'Variance',0.15)

Mdl = arima with properties: Description: "ARIMA(1,0,1) Model (t Distribution)" Distribution: Name = "t", DoF = 8 P: 1 D: 0 Q: 1 Constant: 0.3 AR: {0.7} at lag [1] SAR: {} MA: {0.4} at lag [1] SMA: {} Seasonality: 0 Beta: [1×0] Variance: 0.15

All parameter values are specified, that is, no object property is `NaN`

-valued.

### Specify ARMA Model Using Econometric Modeler App

In the **Econometric
Modeler** app, you can specify the lag structure, presence of a constant,
and innovation distribution of an ARMA(*p*,*q*)
model by following these steps. All specified coefficients are unknown but estimable parameters.

At the command line, open the

**Econometric Modeler**app.econometricModeler

Alternatively, open the app from the apps gallery (see

**Econometric Modeler**).In the

**Time Series**pane, select the response time series to which the model will be fit.On the

**Econometric Modeler**tab, in the**Models**section, click**ARMA**.The

**ARMA Model Parameters**dialog box appears.Specify the lag structure. To specify an ARMA(

*p*,*q*) model that includes all AR lags from 1 through*p*and all MA lags from 1 through*q*, use the**Lag Order**tab. For the flexibility to specify the inclusion of particular lags, use the**Lag Vector**tab. For more details, see Specifying Univariate Lag Operator Polynomials Interactively. Regardless of the tab you use, you can verify the model form by inspecting the equation in the**Model Equation**section.

For example:

To specify an ARMA(2,1) model that includes a constant, includes all AR and MA lags from 1 through their respective orders, and has a Gaussian innovation distribution:

Set

**Autoregressive Order**to`2`

.Set

**Moving Average Order**to`1`

.

To specify an ARMA(2,1) model that includes all AR and MA lags from 1 through their respective orders, has a Gaussian distribution, but does not include a constant:

Set

**Autoregressive Order**to`2`

.Set

**Moving Average Order**to`1`

.Clear the

**Include Constant Term**check box.

To specify an ARMA(2,1) model that includes all AR and MA lags from 1 through their respective orders, includes a constant term, and has

*t*-distributed innovations:Set

**Autoregressive Order**to`2`

.Set

**Moving Average Order**to`1`

.Click the

**Innovation Distribution**button, then select`t`

.

The degrees of freedom parameter of the

*t*distribution is an unknown but estimable parameter.To specify an ARMA(8,4) model containing nonconsecutive lags

$${y}_{t}-{\varphi}_{1}{y}_{t-1}-{\varphi}_{4}{y}_{t-4}-{\varphi}_{8}{y}_{t-8}={\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+{\theta}_{4}{\epsilon}_{t-4},$$

where

*ε*is a series of IID Gaussian innovations:_{t}Click the

**Lag Vector**tab.Set

**Autoregressive Lags**to`1 4 8`

.Set

**Moving Average Lags**to`1 4`

.Clear the

**Include Constant Term**check box.

After you specify a model, click **Estimate** to
estimate all unknown parameters in the model.

### What Are Autoregressive Moving Average Models?

#### ARMA(*p*,*q*) Model

For some observed time series, a very high-order AR or MA model is needed to model the underlying process well. In this case, a combined autoregressive moving average (ARMA) model can sometimes be a more parsimonious choice.

An ARMA model expresses the conditional mean of
*y _{t}* as a function of both past
observations, $${y}_{t-1},\dots ,{y}_{t-p}$$, and past innovations, $${\epsilon}_{t-1},\dots ,{\epsilon}_{t-q}.$$The number of past observations that

*y*depends on,

_{t}*p*, is the AR degree. The number of past innovations that

*y*depends on,

_{t}*q*, is the MA degree. In general, these models are denoted by ARMA(

*p*,

*q*).

The form of the ARMA(*p*,*q*) model in
Econometrics Toolbox™ is

$${y}_{t}=c+{\varphi}_{1}{y}_{t-1}+\dots +{\varphi}_{p}{y}_{t-p}+{\epsilon}_{t}+{\theta}_{1}{\epsilon}_{t-1}+\dots +{\theta}_{q}{\epsilon}_{t-q},$$ | (1) |

In lag operator polynomial notation, $${L}^{i}{y}_{t}={y}_{t-i}$$. Define the degree *p* AR lag operator polynomial $$\varphi (L)=(1-{\varphi}_{1}L-\dots -{\varphi}_{p}{L}^{p})$$. Define the degree *q* MA lag operator polynomial $$\theta (L)=(1+{\theta}_{1}L+\dots +{\theta}_{q}{L}^{q})$$. You can write the ARMA(*p*,*q*)
model as

$$\varphi (L){y}_{t}=c+\theta (L){\epsilon}_{t}.$$ | (2) |

The signs of the coefficients in the AR lag operator polynomial, $$\varphi (L)$$, are opposite to the right side of Equation 1. When specifying and interpreting AR coefficients in Econometrics Toolbox, use the form in Equation 1.

#### Stationarity and Invertibility of the ARMA Model

Consider the ARMA(*p*,*q*) model in lag operator
notation,

$$\varphi (L){y}_{t}=c+\theta (L){\epsilon}_{t}.$$

From this expression, you can see that

$${y}_{t}=\mu +\frac{\theta (L)}{\varphi (L)}{\epsilon}_{t}=\mu +\psi (L){\epsilon}_{t},$$ | (3) |

$$\mu =\frac{c}{\left(1-{\varphi}_{1}-\dots -{\varphi}_{p}\right)}$$

is the unconditional mean of the process, and $$\psi (L)$$ is a rational, infinite-degree lag operator polynomial, $$(1+{\psi}_{1}L+{\psi}_{2}{L}^{2}+\dots )$$.

**Note**

The `Constant`

property of an `arima`

model
object corresponds to *c*, and not the unconditional mean
*μ*.

By Wold’s decomposition [2], Equation 3 corresponds to a stationary stochastic process
provided the coefficients $${\psi}_{i}$$ are absolutely summable. This is the case when the AR polynomial, $$\varphi (L)$$, is *stable*, meaning all its roots lie
outside the unit circle. Additionally, the process is *causal*
provided the MA polynomial is *invertible*, meaning all its
roots lie outside the unit circle.

Econometrics Toolbox enforces stability and invertibility of ARMA processes. When you
specify an ARMA model using `arima`

, you get an error if you enter
coefficients that do not correspond to a stable AR polynomial or invertible MA
polynomial. Similarly, `estimate`

imposes stationarity and
invertibility constraints during estimation.

## References

[1] Box, George E. P., Gwilym M. Jenkins, and Gregory C. Reinsel. *Time Series Analysis: Forecasting and Control*. 3rd ed. Englewood Cliffs, NJ: Prentice Hall, 1994.

[2] Wold, Herman. "A Study in the Analysis of Stationary Time
Series." *Journal of the Institute of Actuaries* 70 (March 1939): 113–115.
https://doi.org/10.1017/S0020268100011574.

## See Also

### Apps

### Objects

### Functions

## Related Topics

- Analyze Time Series Data Using Econometric Modeler
- Specifying Univariate Lag Operator Polynomials Interactively
- Creating Univariate Conditional Mean Models
- Modify Properties of Conditional Mean Model Objects
- Specify Conditional Mean Model Innovation Distribution
- Plot the Impulse Response Function of Conditional Mean Model