Can someone please tell me what they think of my explanation of the Box-Jenkins approach to modeling time series? Do you have anything to add (in particular to my explanation as to the intuition behind stationarity and the spurious regression problem)?
"I use the Box-Jenkins approach to develop a model for $Y_t$. This approach involves the following steps:
Calculate ACF of raw $Y_t$. If $Y_t$ is non-stationary, you should difference the data or take logarithms.
Observe the ACF structure of the transformed variable. Compare the observed structure of the ACF to the theoretical properties of different ARMA processes. Use this information to propose a model(s). If the ACF displays seasonal patterns, this should be modelled, e.g., through seasonal differencing.
Estimate your model and check the residuals for autocorrelation. If the model is correctly specified, the residuals should not be autocorrelated, i.e., they should be white noise. If the residuals are autocorrelated, then you should go back to step 2.
You may be asking, why don’t we model $Y_t$ as it is? Well, for two reasons:
Firstly, ARMA models are not going to be useful if the series is non-stationary. To understand this, you should understand what non-stationarity is. Intuitively, it means the behaviour of the series changes over time. More technically, it means the mean, variance, and/or covariance structure change with time. For example, if the mean of the series is increasing with time, then the mean value during the sample is probably not going to be a very good predictor of the future value of the series. Of course, this depends how quickly it is changing and how far in the future you want to forecast. Therefore, it should be clear, intuitively, that modelling non-stationary data, although entirely possible, is not the best idea.
Secondly, we don’t want to fall into any spurious regression traps, i.e., we don’t want to go around claiming $Y_t$ is causally related to $X_t$ simply because a regression of $Y_t$ on $X_t$ results in a statistically significant coefficient estimate. Because time series variables are (often) trending, you should expect a statistically significant slope coefficient even if there is no causal relationship. Therefore, it is not a great idea to model these variables as they are. In other words, you are probably less likely to avoid this spurious regression problem if you model changes in $Y_t$ , or percentage changes, than the series themselves. There is mathematics behind this too, but the intuition should suffice."