I came across the below statement in this article:
"An appropriate model for analyzing this time series in a least squares framework is to take the logarithm of the data and regress it on its lagged values at lag 1 and 12. This corresponds to a multiplicative SARIMA(1, 0, 0)(1, 0, 0)12 model fitted by OLS. Another possibility would be to take first differences instead of logs which leads to very similar results, with slightly inferior fits."
Are there any general rules about when regression and ARIMA models are equivalent?
(I want to use the R package strucchange for monitoring, but I am not happy with having to fit all my models as regression models)
Thanks!
Best Answer
In theory, the multiplicative SARIMA(1,0,0)(1,0,0)12 is not the same as regressing the data on its lags of order 1 and 12. To see this, the multiplicative SARIMA model is defined as:
$$ (1 - \phi_1L)(1 - \phi_2L^{12}) y_t = \varepsilon_t \,, \quad \varepsilon \sim NID(0, \sigma^2) \,, $$
where $L$ is the lag operator such that $L^i y_t = y_{t-i}$.
Expanding the product of polynomials we get:
$$ 1 - \phi_1Ly_t - \phi_2L^{12}y_t + \phi_1\phi_2L^{13}y_t = \varepsilon_t \\ y_t = \phi_1 y_{t-1} + \phi_2y_{t-12} - \phi_1\phi_2y_{t-13} + \varepsilon_t \,. $$
Thus, we can see that the multiplicative SARIMA model includes a lag of order 13 whose coefficient is restricted to be the product of coefficients $\phi_1\phi_{12}$. (For illustration, this result was used here.)
In practice, the unrestricted regression model with lags 1 and 12 may give a similar fit to the actual multiplicative SARIMA model, it will depend on the data. However, I wouldn't recommend working with ARIMA models as regression models since there are software packages that provide features and utilities specific for ARIMA models.
The
strucchangepackage requires as input a model fitted bylm, i.e. the output of a linear regression model. My answer to this post gives an example about how to adapt the ideas implemented instrucchangeto the case of ARIMA models. However, be aware that there is literature specific for the detection of structural breaks in the parameters of time series models.Edit (added references)
As the title mentions the equivalence between ARIMA and regression models, you may also be interested in this post, which compares the interpretation of an AR model with exogenous regressors and a linear regression model.