Common mistake 1 with Dynamic regression (ARIMAX)

The big one: forecasting with contemporaneous predictors. If x enters at time t, then to forecast y at t+1 you need x at t+1 - which you do not have. The app forecasts x for you and warns that the intervals are consequently too narrow. Lag the predictors instead, and the problem disappears entirely.

Common mistake 2 with Dynamic regression (ARIMAX)

Putting a predictor in that is caused BY y. That is simultaneity, and the coefficient is then meaningless.

Common mistake 3 with Dynamic regression (ARIMAX)

Over-fitting. Six predictors on eighty observations will fit the sample beautifully and forecast badly.

Common mistake 4 with Dynamic regression (ARIMAX)

Forgetting that a predictor which is itself hard to forecast adds its own error to yours.

EconStatsCraft › Dynamic regression (ARIMAX)

Dynamic regression (ARIMAX)

Plain ARIMA can only extrapolate a series from its own past. This forecasts one series using OTHER series as predictors - a regression with ARIMA errors. It checks the differencing order for you, tells you by AIC/BIC whether the predictors actually earn their place, and is explicit about what the forecast assumes.

What it needs

Series to forecast - The one variable you want a forecast for.
Predictor series - One or more other series you believe carry information about it.

The intuition

A plain ARIMA forecasts a series from its own past. Dynamic regression adds other series as drivers, so the forecast can respond to things ARIMA cannot see. The subtlety is what happens to the errors: regress two trending series on each other with independent errors and you get a spurious fit, so the ARIMA part is not an ornament on the regression, it is what keeps the standard errors honest.

When to use it

When you want to forecast ONE series and you believe other series carry information about it beyond its own history - sales from consumer confidence, inflation from the exchange rate, electricity demand from temperature. If you care equally about all the series and want to model them as a system, use a VAR instead.

Where the estimator comes from

The model is y_t=x_t'\beta+\eta_t where the DISTURBANCE, not the raw series, follows an ARIMA: \phi(L)(1-L)^d\eta_t=\theta(L)\varepsilon_t. Estimating \beta by OLS and then fitting ARIMA to the residuals is not equivalent and is not efficient - the two must be estimated jointly by maximum likelihood, which is what SARIMAX does. When d>0 the differencing applies to y and to x alike, so the model in differences is \Delta y_t=\Delta x_t'\beta+\dots, and \beta retains its interpretation as the effect of a change in x on a change in y. A drift term is needed whenever d+D=1, or the trend in y has nowhere to go and loads onto whichever regressor also trends.

The model

y_t=\beta'x_t+\eta_t,\qquad \phi(L)(1-L)^{d}\eta_t=\theta(L)\varepsilon_t,\qquad \varepsilon_t\sim\text{WN}(0,\sigma^2)

What it assumes

The predictors are exogenous - they are not themselves caused by y. If they are, you need a VAR or a simultaneous system.
No variable is I(2). The app checks and stops if one is.
After the ARIMA part absorbs the dynamics, the residuals are white noise. The Ljung–Box result below the table tells you whether they are.
The relationship is stable over the sample - no structural break in β.

Statistical properties

The coefficients are regression coefficients on the differenced data. They are NOT dynamic multipliers: the full effect of a permanent change in x accumulates through the ARIMA structure and must be computed from the impulse response.
Forecasting requires future values of the regressors. If they are contemporaneous, those futures must themselves be forecast, and the resulting intervals do not include that extra uncertainty unless explicitly widened.
Lagged regressors avoid the problem entirely: if x enters at lag 3, its next three values are already observed and the forecast needs no assumption about them. This is usually the better specification for forecasting.
Both series must be integrated of the same order, or the regression is unbalanced. If they are cointegrated, an error-correction model is the correct specification and retains the long-run relationship that differencing destroys.
A predictor on a very different numerical scale wrecks the conditioning of the optimiser and produces meaningless standard errors, so the predictors are standardised internally and the coefficients reported back on the original scale.
Ljung-Box on the residuals is the check that the ARIMA part has done its job. Autocorrelation left over means the standard errors on beta are still wrong.

How to read the output

The predictor coefficients are the regression; the ar and ma terms describe what is left over in the errors. Read them separately.
The "Do the predictors earn their place?" table is the honest test. If BIC RISES, the predictors are not helping and a plain ARIMA is the better model - report that, do not hide it.
A significant predictor coefficient does not mean the model forecasts better. Look at AIC/BIC and, better still, run Forecast backtest.
When d is chosen for you, both y and the predictors are differenced. The coefficients are then changes-on-changes, not levels-on-levels.

A worked reading

Retail sales on consumer confidence, the interest rate and unemployment, each lagged two months. The unit-root step makes it ARIMAX(2,2,1). Confidence lagged two months carries the model: 0.9460 (SE 0.1106, z = 8.55), while the interest rate (-1.0153, p = 0.43) and unemployment (0.6684, p = 0.67) add nothing individually. Against the same ARIMA with no predictors, BIC falls from 966.4 to 910.1, so as a group they earn their place. Two things in that output are worth more than the coefficients. The MA term sits at exactly -1.0000 with an absurd standard error, which is the signature of over-differencing - d = 2 where d = 1 would do, and the MA root cancelling the extra difference. And Ljung-Box rejects at p = 0.026, so structure is left in the residuals and the forecast intervals below are narrower than they deserve to be. A fit that reports both of those honestly is more useful than one that hides them.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Predictor significant but residuals autocorrelated	The ARIMA order is inadequate	The t-statistic is not trustworthy yet
Coefficient collapsing when differencing is applied	The level relationship was spurious	The differenced answer is the honest one
Forecast intervals implausibly narrow	Regressor forecast uncertainty excluded	Lag the predictors, or widen the interval
Series of different integration orders	Unbalanced regression	Difference the higher-order one, or test for cointegration
Predictors and y cointegrated	Differencing throws away the long run	Use an error-correction or ARDL specification
Huge z-statistics on the AR terms	Ill-conditioned from predictor scale	Standardise; this is silently catastrophic otherwise

Common mistakes

The big one: forecasting with contemporaneous predictors. If x enters at time t, then to forecast y at t+1 you need x at t+1 - which you do not have. The app forecasts x for you and warns that the intervals are consequently too narrow. Lag the predictors instead, and the problem disappears entirely.
Putting a predictor in that is caused BY y. That is simultaneity, and the coefficient is then meaningless.
Over-fitting. Six predictors on eighty observations will fit the sample beautifully and forecast badly.
Forgetting that a predictor which is itself hard to forecast adds its own error to yours.

Where to read this properly

Hyndman and Athanasopoulos, Forecasting: Principles and Practice (2021), ch. 10.
Pankratz, Forecasting with Dynamic Regression Models (1991).
Box, Jenkins, Reinsel and Ljung, Time Series Analysis (2015), ch. 10 and 11.

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Dynamic regression (ARIMAX)

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

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