Regression models with ARIMA errors have the following form (in lag operator notation):
$$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ a\left(L\right)A\left(L\right){\left(1L\right)}^{D}\left(1{L}^{s}\right){u}_{t}=b\left(L\right)B\left(L\right){\epsilon}_{t},\end{array}$$
where
t = 1,...,T.
y_{t} is the response series.
X_{t} is row t of X, which is the matrix of concatenated predictor data vectors. That is, X_{t} is observation t of each predictor series.
c is the regression model intercept.
β is the regression coefficient.
u_{t} is the disturbance series.
ε_{t} is the innovations series.
$${L}^{j}{y}_{t}={y}_{tj}.$$
$$a\left(L\right)=\left(1{a}_{1}L\mathrm{...}{a}_{p}{L}^{p}\right),$$ which is the degree p, nonseasonal autoregressive polynomial.
$$A\left(L\right)=\left(1{A}_{1}L\mathrm{...}{A}_{{p}_{s}}{L}^{{p}_{s}}\right),$$ which is the degree p_{s}, seasonal autoregressive polynomial.
$${\left(1L\right)}^{D},$$ which is the degree D, nonseasonal integration polynomial.
$$\left(1{L}^{s}\right),$$ which is the degree s, seasonal integration polynomial.
$$b\left(L\right)=\left(1+{b}_{1}L+\mathrm{...}+{b}_{q}{L}^{q}\right),$$ which is the degree q, nonseasonal moving average polynomial.
$$B\left(L\right)=\left(1+{B}_{1}L+\mathrm{...}+{B}_{{q}_{s}}{L}^{{q}_{s}}\right),$$ which is the degree q_{s}, seasonal moving average polynomial.
For simplicity, use the shorthand notation Mdl = regARIMA(p,D,q)
to specify a regression model with ARIMA(p,D,q) errors, where p
, D
, and q
are nonnegative integers. Mdl
has the following default properties.
Property Name  Property Data Type 

AR  Length p cell vector of NaN s 
Beta  Empty vector [] of regression coefficients, corresponding to the predictor series 
D  Nonnegative scalar, corresponding to D 
Distribution  "Gaussian" , corresponding to the distribution of ε_{t} 
Intercept  NaN , corresponding to c 
MA  Length q cell vector of NaN s 
P  Number of AR terms plus degree of integration, p + D 
Q  Number of MA terms, q 
SAR  Empty cell vector 
SMA  Empty cell vector 
Variance  NaN , corresponding to the variance of ε_{t} 
Seasonality  0 , corresponding to s 
If you specify nonseasonal ARIMA errors, then
The properties D
and Q
are the inputs D
and q
, respectively.
Property P
= p
+ D
, which is the degree of the compound, nonseasonal autoregressive polynomial. In other words, P
is the degree of the product of the nonseasonal autoregressive polynomial, a(L) and the nonseasonal integration polynomial, (1 – L)^{D}.
The values of properties P
and Q
indicate how many presample observations the software requires to initialize the time series.
You can modify the properties of Mdl
using dot notation. For example, Mdl.Variance = 0.5
sets the innovation variance to 0.5.
For maximum flexibility in specifying a regression model with ARIMA errors, use namevalue pair arguments to, for example, set each of the autoregressive parameters to a value, or specify multiplicative seasonal terms. For example, Mdl = regARIMA('AR',{0.2 0.1})
defines a regression model with AR(2) errors, and the coefficients are a_{1} = 0.2 and a_{2} = 0.1.
You can only specify the nonseasonal autoregressive and moving average polynomial degrees, and nonseasonal integration degree using the shorthand notation regARIMA(p,D,q)
. Some tasks, such as forecasting and simulation, require you to specify values for parameters. You cannot specify parameter values using shorthand notation. For maximum flexibility, use namevalue pair arguments to specify regression models with ARIMA errors.
The nonseasonal ARIMA error model might contain the following polynomials:
The degree p autoregressive polynomial a(L) = 1 – a_{1}L – a_{2}L_{2} –...– a_{p}L^{p}. The eigenvalues of a(L) must lie within the unit circle (i.e., a(L) must be a stable polynomial).
The degree q moving average polynomial b(L) = 1 + b_{1}L + b_{2}L_{2} +...+ b_{q}L^{q}. The eigenvalues of b(L) must lie within the unit circle (i.e., b(L) must be an invertible polynomial).
The degree D nonseasonal integration polynomial is (1 – L)^{D}.
The following table contains the namevalue pair arguments that you use to specify the ARIMA error model (i.e., a regression model with ARIMA errors, but without a regression component and intercept):
$$\begin{array}{c}{y}_{t}={u}_{t}\\ a(L){(}^{1}=b(L){\epsilon}_{t}.\end{array}$$  (1) 
NameValue Pair Arguments for Nonseasonal ARIMA Error Models
Name  Corresponding Model Term(s) in Equation 1  When to Specify 

AR  Nonseasonal AR coefficients: a_{1}, a_{2},...,a_{p} 

ARLags  Lags corresponding to nonzero, nonseasonal AR coefficients 

D  Degree of nonseasonal differencing, D 

Distribution  Distribution of the innovation process, ε_{t} 

MA  Nonseasonal MA coefficients: b_{1}, b_{2},...,b_{q} 

MALags  Lags corresponding to nonzero, nonseasonal MA coefficients 

Variance  Scalar variance, σ^{2}, of the innovation process, ε_{t}  To set equality constraints for σ^{2}. For example, for an ARIMA error model with known innovation variance 0.1, specify 
Use the namevalue pair arguments in the following table in conjunction with those in NameValue Pair Arguments for Nonseasonal ARIMA Error Models to specify the regression components of the regression model with ARIMA errors:
$$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ a(L){(}^{1}=b(L){\epsilon}_{t}.\end{array}$$  (2) 
NameValue Pair Arguments for the Regression Component of the regARIMA Model
Name  Corresponding Model Term(s) in Equation 2  When to Specify 

Beta  Regression coefficient values corresponding to the predictor series, β 

Intercept  Intercept term for the regression model, c 

If the time series has seasonality s, then
The degree p_{s} seasonal autoregressive polynomial is A(L) = 1 – A _{1}L – A_{2}L^{2} –...– A_{ps}L^{ps}.
The degree q_{s} seasonal moving average polynomial is B(L) 1 + B _{1}L + B_{2}L^{2} +...+ B_{qs}L^{qs}.
The degree s seasonal integration polynomial is (1 – L^{s}).
Use the namevalue pair arguments in the following table in conjunction with those in tables NameValue Pair Arguments for Nonseasonal ARIMA Error Models and NameValue Pair Arguments for the Regression Component of the regARIMA Model to specify the regression model with multiplicative seasonal ARIMA errors:
$$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ a(L){(}^{1}A(L)(1{L}^{s}){u}_{t}=b(L)B(L){\epsilon}_{t}.\end{array}$$  (3) 
NameValue Pair Arguments for Seasonal ARIMA Models
Argument  Corresponding Model Term(s) in Equation 3  When to Specify 

SAR  Seasonal AR coefficients: A_{1}, A_{2},...,A_{ps} 

SARLags  Lags corresponding to nonzero seasonal AR coefficients, in the periodicity of the responses 

SMA  Seasonal MA coefficients: B_{1}, B_{2},...,B_{qs} 

SMALags  Lags corresponding to the nonzero seasonal MA coefficients, in the periodicity of the responses 

Seasonality  Seasonal periodicity, s 

You cannot assign values to the properties P
and Q
. For multiplicative ARIMA error models,
regARIMA
sets P
equal to p + D + p_{s} + s.
regARIMA
sets Q
equal to q + q_{s}
You can specify the predictor variables in the regression component, and the error model lag structure and innovation distribution, using the Econometric Modeler app. The app treats all coefficients as unknown and estimable.
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 Regression Models section contains supported regression models. To specify a multiple linear regression (MLR) model, select MLR
. To specify regression models with ARMA errors, select RegARMA
.
After you select a model, the app displays the Type
Model Parameters dialog box, where Type
is the model type. This figure shows the RegARMA Model Parameters dialog box.
Adjustable parameters depend on the model Type
. In general, adjustable parameters include:
Predictor variables for the linear regression component, listed in the Predictors section.
For regression models with ARMA errors, you must include at least one predictor in the model. To include a predictor, select the corresponding check box in the Include? column.
For MLR models, you can clear all check boxes in the Include? column. In this case, you can specify a constant mean model (interceptonly model) by selecting the Include Intercept check box. Or, you can specify an erroronly model by clearing the Include Intercept check box.
The innovation distribution and nonseasonal lags for the error model, for regression models with ARMA errors.
As you adjust parameter values, the equation in the Model Equation section changes to match your specifications. Adjustable parameters correspond to input and namevalue pair arguments described in the previous sections and in the regARIMA
reference page.
For more details on specifying models using the app, see Fitting Models to Data and Specifying Lag Operator Polynomials Interactively.