## 14.9 Multicollinearity

For the discussion of multicollinearity and how to diagnose it in the regression model, we refer the reader to Section 15.3 of the Svetunkov (2022a) textbook. When it comes to dynamic models, the situation might differ, so we will focus on several aspects that might not be relevant to regression.

First, in the conventional ARIMA model (Chapter 8), multicollinearity is inevitable by construction because of the autocorrelations between actual values. This is why sometimes heteroskedasticity- and autocorrelation-consistent (HAC) estimators of the covariance matrix (see Section 15.4 of Hanck et al., 2020) of parameters are used instead of the standard ones. They are designed to fix the issue and produce standard errors of parameters that are close to those without the problem.

Second, in the case of state space models, and specifically in ETS, multicollinearity might not cause as severe issues as in the case of regression. For example, it is possible to use all the values of a categorical variable (Section 10.5), avoiding the trap of dummy variables. The values of a categorical variable, in this case, are considered as changes relative to the baseline. The classic example of this is the seasonal model, for example, ETS(A,A,A), where the seasonal components can be considered as a set of parameters for dummy variables, expanded from the seasonal categorical variable (e.g. months of year variable). If we set $$\gamma=0$$, thus making the seasonality deterministic, the ETS can still be estimated even though all variable values are used. This becomes apparent with the conventional ETS model, for example, from the forecast package for R:

etsModel <- forecast::ets(AirPassengers, "AAA")
# Calculate determination coefficients for seasonal states
determ(etsModel\$states[,-c(1:2)])
##        s1        s2        s3        s4        s5        s6        s7        s8
## 0.9999992 0.9999992 0.9999991 0.9999991 0.9999992 0.9999992 0.9999992 0.9999991
##        s9       s10       s11       s12
## 0.9999991 0.9999991 0.9999992 0.9999992

As we see, the states of the model are almost perfectly correlated, but still, the model works and does not have the issue that the classical linear regression would have. This is because the state-space models are constructed and estimated differently than the conventional regression (see Section 10).

### References

• Hanck, C., Arnold, M., Gerber, A., Schmelzer, M., 2020. Introduction to Econometrics with R. https://www.econometrics-with-r.org/index.html (version: 2020-08-12)
• Svetunkov, I., 2022a. Statistics for business analytics. https://openforecast.org/sba/ (version: 31.10.2022)