Regression models with time series errors attempt to explain the mean behavior of
a response series (*y _{t}*,

These models are particularly useful for econometric data. Use these models to:

Analyze the effects of a new policy on a market indicator (an intervention model).

Forecast population size adjusting for predictor effects, such as expected prevalence of a disease.

Study the behavior of a process adjusting for calendar effects. For example, you can analyze traffic volume by adjusting for the effects of major holidays. For details, see [3].

Estimate the trend by including time (

*t*) in the model.Forecast total energy consumption accounting for current and past prices of oil and electricity (distributed lag model).

Use these tools in Econometrics Toolbox™ to:

Specify a regression model with ARIMA errors (see

`regARIMA`

).Estimate parameters using a specified model, and response and predictor data (see

`estimate`

).Simulate responses using a model and predictor data (see

`simulate`

).Forecast responses using a model and future predictor data (see

`forecast`

).Infer residuals and estimated unconditional disturbances from a model using the model and predictor data (see

`infer`

).`filter`

innovations through a model using the model and predictor dataGenerate impulse responses (see

`impulse`

).Compare a regression model with ARIMA errors to an ARIMAX model (see

`arima`

).

A regression model with time series errors has the following form (in lag operator notation):

$$\begin{array}{c}{y}_{t}=c+{X}_{t}\beta +{u}_{t}\\ a\left(L\right)A\left(L\right){\left(1-L\right)}^{D}\left(1-{L}^{s}\right){u}_{t}=b\left(L\right)B\left(L\right){\epsilon}_{t},\end{array}$$ | (1) |

*t*= 1,...,*T*.*y*is the response series._{t}*X*is row_{t}*t*of*X*, which is the matrix of concatenated predictor data vectors. That is,*X*is observation_{t}*t*of each predictor series.*c*is the regression model intercept.*β*is the regression coefficient.*u*is the disturbance series._{t}*ε*is the innovations series._{t}$${L}^{j}{y}_{t}={y}_{t-j}.$$

$$a\left(L\right)=\left(1-{a}_{1}L-\mathrm{...}-{a}_{p}{L}^{p}\right),$$ 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*, seasonal autoregressive polynomial._{s}$${\left(1-L\right)}^{D},$$ 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*, seasonal moving average polynomial._{s}

Following Box and Jenkins methodology,
*u _{t}* is a stationary or unit root
nonstationary, regular, linear time series. However, if

Another deviation from the Box and Jenkins methodology is that
*u _{t}* does not have a constant term
(conditional mean), and therefore its unconditional mean is 0. However, the
regression model contains an intercept term,

If the unconditional disturbance process is nonstationary (i.e., the
nonseasonal or seasonal integration degree is greater than 0), then the
regression intercept, *c*, is not identifiable. For
details, see Intercept Identifiability in Regression Models with ARIMA Errors.

The software enforces stability and invertibility of the ARMA process. That is,

$$\psi (L)=\frac{b(L)B(L)}{a(L)A(L)}=1+{\psi}_{1}L+{\psi}_{2}{L}^{2}+\mathrm{...},$$

where the series {*ψ _{t}*}
must be absolutely summable. The conditions for
{

*a*(*L*) and*A*(*L*) are*stable*(i.e., the eigenvalues of*a*(*L*) = 0 and*A*(*L*) = 0 lie inside the unit circle).*b*(*L*) and*B*(*L*) are*invertible*(i.e., their eigenvalues lie of*b*(*L*) = 0 and*B*(*L*) = 0 inside the unit circle).

The software uses maximum likelihood for parameter estimation. You can choose
either a Gaussian or Student’s *t* distribution for the
innovations, *ε _{t}*.

The software treats predictors as nonstochastic variables for estimation and inference.

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

[2] Hyndman, R. J. (2010, October). “The ARIMAX Model
Muddle.” *Rob J. Hyndman*. Retrieved May 4, 2017 from
`https://robjhyndman.com/hyndsight/arimax/`

.

[3] Ruey, T. S. “Regression Models with Time Series
Errors.” *Journal of the American Statistical
Association.* Vol. 79, Number 385, March 1984, pp.
118–124.

`arima`

| `estimate`

| `filter`

| `forecast`

| `impulse`

| `infer`

| `regARIMA`

| `simulate`

- Alternative ARIMA Model Representations
- Intercept Identifiability in Regression Models with ARIMA Errors