# Is this explanation of the Box-Jenkins approach correct?

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:

1. Calculate ACF of raw $Y_t$. If $Y_t$ is non-stationary, you should difference the data or take logarithms.

2. 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.

3. 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."

no ! 1.Calculate ACF of raw Y t . If Y t is non-stationary, you should difference the data or take logarithms

The need for logarithms or any power transformation are justified when the errors from a reasonable model are computed and suggest non-constancy. The F test and T tst of coefficients is all about the error variance NOT the variance of the original series around it's mean

The 1960's cookbook suggested (incorrectly) suggested using a power transformation ( e.g. logarithms/reciprocals/square root etc. ) Unfortunately most textbooks and relevant software perpetuate this fallacy.

To see an example of this "bad approach" and a suggested "modern approach" please see http://www.autobox.com/pdfs/vegas_ibf_09a.pdf which discusses (among other things) the Airline Series.

I have helped write some of the code that is referenced in this citation

No !

3.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.

If the mean of the errors is not constant then we might examine the residuals to find Pulses, Level/Step Shifts, Seasonal Pulses and/or Local Time Trends. If any of these are present then the variance of the errors is inflated. Since the acf is the covariance divided by the variance, the result is a downwards bias of the acf . This is called the "Alice in Wonderland Effect". Because you didn't validate the assumption of the constant mean of th errors you conclude that there is no autocorrelation !

This then would take us back to step 2. OTHERWISE as follows

After removing any needed Interventions (p/ls/sp/time) then one needs to validate that the parameters of the model are invariant over time. If this hypothesis can't be rejected then one needs to confirm the variance of the errors is constant over time. Now go back to step 2

no !

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.

In order to receive consistent, reliable results, the non-stationary data needs to be transformed into stationary data . An ARIMA filter/equation can transform a non-stationary process to a stationary process perhaps including a power transform or a set of weights leading to GLS. If the resultant transformed/filtered data ( model errors) are non-statioany ( e.g. time varying process) then the model is of little use unless you can model the "model dynamics" via Threshold Autoregressive Models (TAR) suggest by Tong. BUT if the model errors can't be proven to be non-staionary the your model may be useful.

• I don't understand anything you said... – Christian Sep 14 '13 at 10:06
• @Christian Perhaps you can contact me directly and I will try and help you. See my contact info . – IrishStat Sep 14 '13 at 10:27
• I can't access your contact details. Please contact me via email, if you have time. – Christian Sep 16 '13 at 1:33
• autobox.com/cms/index.php/blog/entry/… might be of help. My email address is dave@autobox.com – IrishStat Sep 16 '13 at 9:43
• Thank you very much. I will have a read of it as soon as I can and get back to you. – Christian Sep 16 '13 at 14:35