Economic statistics can tell us about the past and present, but if we are to plan for the future ahead of time, say, to control inventory, we're going to need forecasts of some sort. Forecasts can, of course, be qualitative or quantitative in nature, but here I will focus on producing point and interval forecasts using the univariate Box-Jenkins method. It's worth noting that properly constructed ARIMA models produce optimal univariate forecasts; the mean-squared forecast error is smaller than for any other univariate forecast (produced by univariate, linear, fixed coefficient models).
If we're going to construct a model we better be able to write it out, so let's have a quick review of some notation used for expressing seasonal non-seasonal multiplicative models. There are also seasonal non-seasonal additive models, but let's leave them aside for the purposes of this blog post.
The general form of seasonal non-seasonal multiplicative models can be written compactly as: \[ \phi_{p}(B) \Phi_{p} (B^{s}) \nabla^{d} \nabla_{s}^{D} \tilde{z} = \Theta_{Q}(B^{s})\theta_{q}(B)a_{t} \] where $\phi_{p}(B)$ is the nonseasonal AR operator, $\theta_{q}(B)$ is the nonseasonal MA operator, $\Phi_{p} (B^{s})$ is the seasonal AR operator, $\Theta_{Q}(B^{s})$ is the seasonal MA operator, $\nabla^{d}$ is the nonseasonal differencing operator, $\nabla_{s}^{D}$ is the seasonal differencing operator, and $\tilde{z} = z_{t} - \mu$, is the data expressed in deviations from the mean.
This sort of process is often referred to as an ARIMA(p,d,q)(P,D,Q)$_{s}$ process. Lower case letters p, d, and q denote the nonseasonal orders while the upper case letter P, D, Q denote the seasonal orders. The letter s denotes the type of periodicity; 4 for quarterly; 12 for monthly.
Using the Box-Jenkins method, I identified and estimated the following tentative model for the aforementioned CSO data: an ARIMA(1,1,0)(0,1,0)$_{12}$ model. The model can be expressed in the following form: \[(1 - \phi_{1}B)(1 - B)(1 - B^{12}) \tilde{z} = a_{t}\] or equivalently as: \[ \phi_{1}(B) \nabla \nabla_{12} \tilde{z} = a_{t} \] It's easy to see from either of these equations that the model consists of one non-seasonal autoregressive term, it is in first differences, and it has been seasonally differenced once with monthly periodicity. Furthermore, there are no non-seasonal moving average terms, nor are there any seasonal autoregressive terms or seasonal moving average terms.
I should say that this compact notation may look a bit daunting at first, but in actual fact it's far clearer than the common algebraic form normally used to express difference equations. For argument's sake, let's write the above model in common algebraic form. Using a bit of algebra, we get the following series of equations: \begin{aligned} (1 - \phi_{1}B) (1 - B) z_{t} - z_{t-12} &= a_{t}\\ (1 - \phi_{1}B) z_{t} - z_{t-12} - z_{t-1} + z_{t-13} &= a_{t}\\ z_{t} - z_{t-12} - z_{t-1} + z_{t-13} - \phi_{1}z_{t-1} + \phi_{1}z_{t-13} + \phi_{1}z_{t-2} - \phi_{1}z_{t-14} &= a_{t} \end{aligned} \begin{aligned} (z_{t} - z_{t-12}) - (z_{t-1} - z_{t-13}) &= \phi_{1}z_{t-1} - \phi_{1}z_{t-13} - \phi_{1}z_{t-2} + \phi_{1}z_{t-14} + a_{t}\\ (z_{t} - z_{t-12}) - (z_{t-1} - z_{t-13}) &= \phi_{1}(z_{t-1} - z_{t-13} - z_{t-2} + z_{t-14}) + a_{t} \end{aligned} Expressed in common algebraic form, this model is less easy to identify hence the preference for using the compact notation (it's not only about saving paper!).
After performing diagnostic checks on the ARIMA(1,1,0)(0,1,0)$_{12}$, I decided that, for the purposes of this blog post, the model could be used to produce forecasts. I have plotted a fit of the ARIMA(1,1,0)(0,1,0)$_{12}$ model below and the figure also shows both point and interval out-of-sample forecasts. Since I omitted a legend from the plot, the dashed line is the model fit, the solid black line is the actual data, and you should be able to make sense of the line (point forecasts) and shading (interval forecasts) in the out-of-sample period.
Rather than go through all of the steps of the analysis here, I've instead written an R script to show how I carried out the analysis using R. The script also contains the code that I used to colour in between the forecast intervals in gray. This wasn't so easy, but I managed to do it without using any packages from the library. In fact, to carry out the analysis I only resorted to using my own R function (presented in a previous post). The plotting scheme that I used to present the fit of the model and the out-of-sample forecasts above is one that I created using the base package graphics only.
Note of caution: admittedly, I identified other possible ARIMA(p,d,q)(P,D,Q)$_{s}$ models that I did not subsequently estimate or diagnostically check. If this were a thorough analysis of the data, these alternative models would also have to be entertained. In any event, this post was just meant to be a quick taster. As a larger exercise, I might compare this model to some of the other tentative models in a later post. In other words, the model presented here is only a candidate for the final model.
Note of caution: admittedly, I identified other possible ARIMA(p,d,q)(P,D,Q)$_{s}$ models that I did not subsequently estimate or diagnostically check. If this were a thorough analysis of the data, these alternative models would also have to be entertained. In any event, this post was just meant to be a quick taster. As a larger exercise, I might compare this model to some of the other tentative models in a later post. In other words, the model presented here is only a candidate for the final model.
