How to include control variables in an Intervention analysis with ARIMA?

Question

I want to conduct an Intervention Analysis using SPSS. Thereby, I want to find out if a specific regulation introduced in 2013 has an effect on leveraged loan volume. Thus, the dependent variable is the time series of leveraged loan volume from 2009 to 2015. However, I would like to control for three exogenous variables which are significantly correlated with leveraged loan volume.

Leverage loan volume before and after introduction of the new regulation in May 2013:

Then I used Log transformation to make the variance constant and removed the outlier in August 2011 using Winsorization. Afterwards, I just observed the pre intervention phase and used the Box Jenkins methodology fit the Arima model.

My differenced and stationary series looks as follows for the pre intervention phase.

After Identification, Estimation and Diagnostic check I found that Arima (3,1,0) is the best fit.

Now Im ready for the Intervention analysis and as the new guidelines became effective in May 2013 I guess it is a abrupt permanent or abrupt gradual intervention. For the intervention I would use the step function and assign dummy variables 0 before intervention and 1 after intervention.

Answer 1

The three series were automatically analyzed using AUTOBOX. AUTOBOX was directed that the SAP series was stochastic and the LAW series was not. The initial identification requited pre-whitening the stochastic input and the outout series. . This lead to the following starting model . Estimating this model we obtained . Performing Intervention Detection suggested two trends , some season activity and a pulse. . Estimation provided . The residuals from this model (here ) were analyzed and an AR(1) augmentation was automatically suggested . The residuals from this model were analyzed to assess both the need for a power transform and a weighted least squares transform that might be needed to stabilize the variance of the. error process. It is fairly obvious that the error variance is not homogeneous over time and fairly obvious that there is no persistant coupling of the error process with the level of the output series , more like a permanent change in the error process. This is supported by the Tsay test . Finally we have the resultant model whose error seem to be uncorrelated and whose time plot suggests white noise . THe Actual/Fit and Forecast graph is here with the forecasts clearer here

Answer 2

It appears that you are using the de jure date of the intervention as one of your predictors. You can add the other control variables and any needed ARIMA structure and any needed pulse indicators to complete the model. If you wish to identify the de facto date of the intervention while incorporating other control variables you may have to look for other software like AUTOBOX, which I have helped develop.

I am not an expert in SPSS so I suggest that you contact their support desk and ask them how to automatically detect the nature and form of needed (empirically detected) intervention variables while also automatically identifying any needed ARIMA structure and also while detecting the appropriate ADL/PDL for your user-suggested control variables.

MODIFIED TO RESPOND TO OP'S QUESTION ABOUT TRANSFER FUNCTION MODEL IDENTIFICAION:

In the case where y or x is non-stationary the suggested procedure is to difference each series according to it’s characteristics i.e. the appropriate differencing operator required for each series. In this case the appropriate differencing operator for y is a seasonal differencing operator while no differencing is need for x. A typical flaw in reasoning is often found on the web suggesting using the same differencing operator for both series and more incorrectly using just first differences rather than seasonal differences OR BOTH. The reason for the bad advice is often that it is easier to explain rather than doing it correctly i.e. using series specific differencing factors. Early computer implementation took the easy path of using the same differencing operator. A generally correct except for the error regarding the utilization of utilizing a common differencing operators is here https://onlinecourses.science.psu.edu/stat510/node/75/

The original citation for exactly how to deal with differencing operators is found at Section 5.3 of the 1976 seminal text by Box and Jenkins (Holden-Day) : TIME SERIES ANALYSIS Forecasting and Control ISBN 0=8162-1104-3 which details the possibility of unique differencing operators predicated upon the ARIMA model for each series. The following advice from http://robjhyndman.com/uwafiles/fpp-notes.pdf has a cautionary but often ignored “if necessary” and a very important “Difference variables until all stationary” comment .

The following flow diagram may help:

In the case where y or x is non-stationary the suggested procedure is to difference each series according to it’s characteristics i.e. the appropriate differencing operator required for each series. In this case the appropriate differencing operator for y is a seasonal differencing operator while no differencing is need for x. A typical flaw in reasoning is often found on the web suggesting using the same differencing operator for both series and more incorrectly in this case using just first differences rather than seasonal differences. The reason for the bad advice is often that it is easier to explain rather than doing it correctly i.e. using series specific differencing factors. Early computer implementation took the easy path of using the same differencing operator. A generally correct except for the error regarding the utilization of utilizing a common differencing operators is here https://onlinecourses.science.psu.edu/stat510/node/75/

The original citation for exactly how to deal with differencing operators is found at Section 5.3 of the 1976 seminal text by Box and Jenkins (Holden-Day) : TIME SERIES ANALYSIS Forecasting and Control ISBN 0=8162-1104-3 which details the possibility of unique differencing operators predicated upon the ARIMA model for each series. The following advice from http://robjhyndman.com/uwafiles/fpp-notes.pdf has a cautionary but often ignored “if necessary” and a very important “Difference variables until all stationary” comment .With this data that mean differ y seasonally (order 12) and don’t difference x.

In the case where y or x is non-stationary the suggested procedure is to difference each series according to it’s characteristics i.e. the appropriate differencing operator required for each series. In this case the appropriate differencing operator for y is a seasonal differencing operator while no differencing is need for x. A typical flaw in reasoning is often found on the web suggesting using the same differencing operator for both series and more incorrectly in this case using just first differences rather than seasonal differences. The reason for the bad advice is often that it is easier to explain rather than doing it correctly i.e. using series specific differencing factors. Early computer implementation took the easy path of using the same differencing operator. A generally correct except for the error regarding the utilization of utilizing a common differencing operators is here https://onlinecourses.science.psu.edu/stat510/node/75/

The original citation for exactly how to deal with differencing operators is found at Section 5.3 of the 1976 seminal text by Box and Jenkins (Holden-Day) : TIME SERIES ANALYSIS Forecasting and Control ISBN 0=8162-1104-3 which details the possibility of unique differencing operators predicated upon the ARIMA model for each series. The following advice from http://robjhyndman.com/uwafiles/fpp-notes.pdf has a cautionary but often ignored “if necessary” and a very important “Difference variables until all stationary” comment .With this data that mean differ y seasonally (order 12) and don’t difference x.

A wise strategy is to prewhiten the user-specified stochastic (control) series in order to tentatively identify a possible transfer function the add the deteromistic control series . This is done in AUTOBOX simultaneously and may be done elsewhere. Then I would add the tentative ARIMA structure and then I would identify the deterministic series that you didn't suggest. This little trick is done via Intervention Detection procedures culminating after stepdown with a possibly useful model and an error sequence that is information free.

You appear to have a reasonable understanding of what to do. The degree of difficulty that may come into play is trying to get your current software to do these tasks.

http://www.autobox.com/stack/dpr-isf27.ppt slide 70 and on may provide some insight .