# Basic - Modelling Two Series, one is an index

I'm trying to model two time series. One is a seasonally adjusted # of new jobs number against and index of business development. Its been a long time since I took econometrics, so I'm hoping someone can jump start my brain...

Here is a plot of the two series, but I've divided the blue line (jobs number) by 10 just so it looks somewhat similar to the orange number. I tried an OLS, but it didn't make much sense. FYI - I'm using MATLAB.

I hope this makes some sense. Should I be doing an polynomial fit? Or is there some time series thing that makes more sense? Or simply divide things until they look more similar?

Thanks a lot!

• When you say "model two time series" -- you should be more explicit. Are you trying to predict one from the other (which from which?) or do something else? Jan 3 '15 at 0:44
• Sorry - yeah I'm trying to predict the Jobs one from the index. So orange line to predict blue. Jan 3 '15 at 3:16

Two classes of time-series models, which are somewhat similar to each other, spring to mind as potentially being useful in this context; ARMAX models and ARDL models. Let me set the scene by jogging your memory and reminding you what these models look like (in brief, so please jog!).

ARMAX

An ARMAX model would provide you with a framework to explain the number of new jobs not only by the index of business development (the explanatory variable), but also by its own lagged values (AR terms) and disturbances (MA terms). This kind of model, known as an ARMAX(p,q) model, can take the following form $$y_{t} = \beta x_{t} + \phi_{1}y_{t-1} + \cdots + \phi_{p}y_{t-p} + \epsilon_{t} + \theta_{1}\epsilon_{t-1} \cdots + \theta_{q}\epsilon_{t-q}$$ where the number of new jobs and the index of business development are denoted by $y_{t}$ and $x_{t}$, respectively. My favourite online reference for these models is this Rob Hyndman blog post and I recommend checking it out for more details (subtleties).

ARDL

An ARDL model would provide you with a framework to explain the number of new jobs by its own lags (AR terms) and by both contemporaneous and lagged values of the index of business development. This kind of model, known as an ARDL(p,q) model, can be written in the form $$y_{t} = \delta + \sum_{i=1}^{p} \alpha_{i}y_{t-i} + \sum_{j=0}^{q} \beta_{j}x_{t-j} + u_{t}$$ where $u_{t}$ is an error term. As you should see, it's similar to ARMAX - constant terms can be included or not.

For purposes of demonstration, let's consider an ARDL(1,1) model to see the potential usefulness of this particular class of models in the present context. The ARDL(1,1) model can be expressed as follows $$y_{t} = \delta + \alpha_{1}y_{t-1} + \beta_{0}x_{t} + \beta_{1}x_{t-1} + u_{t}.$$

Nested within this model are some nice special cases with interesting interpretations! I'll mention just a few which could be useful to you and I'll leave a reference in case you want to track down the others. Note that the special cases can be extended into the more general ARDL(p,q) model.

1. Static Regression ($\alpha_{1} = \beta_{1} = 0$): $$y_{t} = \delta + \beta_{0}x_{t} + u_{t}$$
2. Leading-Indicator Model ($\alpha_{1} = \beta_{0} = 0$): $$y_{t} = \delta + \beta_{1}x_{t-1} + u_{t}$$
3. Partial Adjustment Model ($\beta_{1} = 0$): $$y_{t} = \delta + \alpha_{1}y_{t-1} + \beta_{0}x_{t} + u_{t}$$
4. Error Correction Model ($\beta_{0} + \beta_{1} = 1 - \alpha_{1}$): $$\Delta y_{t} = \delta + \beta_{0}\Delta x_{t} + (\alpha_{1}-1)(y_{t-1}-x_{t-1}) + u_{t}$$

OK, so there we have a bunch of models that could be useful, but in order to build the empirical model some investigation is required. In other words, to select a model you'll need to employ some strategy.

Note that the nested nature of the ARDL model provides a modelling environment conducive to the general-to-specific approach of time-series econometric modelling, which is associated with David F. Hendry, so there's one place to look.

Hopefully that has jump started your brain enough to proceed, but let me end with some suggestions (listed in no particular order).

1. The cross correlation function (CCF) can be used to see if one variable leads another. Uncovering leading behaviour would help justify a leading indicator model, for example, but the data will have to speak.
2. Perform (non-)stationarity tests; both formal and informal. Looks as though you'll need to induce stationarity (although, this depends; see next point.).
3. Test for cointegration if necessary.
4. Use the following time-domain tools to help identify structure in each time-series; the autocorrelation function (ACF), partial autocorrelation function (PACF), and residual autocorrelation function. These tools will help specify the final model and in the latter case also help perform diagnostic checks. Also, eyeballing the data in levels will not show you many of the hidden dynamics that these tools may give light to.
5. Don't rule out the ARIMA class of models, but also consider other explanatory variables if you have reason to.
6. If you really want to explain, consider using theory as a guide to build a proper econometric model; are new jobs a function of firm birth / death rates, tourist visits, something else, etc.?
7. If you want to revert to a univariate framework, there are automated procedures that can be used in R to build both ARIMA and ETS models.

Based on the information provided, that's all I can say. Fundamentally, some modelling will have to be done on your behalf. I hope I've helped.

• Thanks for the writeup! Its helped, but thinking I should have paid a bit more attention in econometrics! I'm trying to work on it now, been too long since I've done this! Jan 8 '15 at 21:34