Regression with ARMA errors for explanatory purpose

I have web log analysis data (AWStats) from a university library website. I'm looking at the number of visits per month divided by the number of faculty plus student enrollment (visits per headcount). This shows a downward trend, along with strong seasonality. Also, the undergrad enrollment has gone up steadily the last few years, while the graduate enrollment has stayed flat.

Therefore, I am fitting a regression with ARMA errors model, with the ratio of grad student headcount to undergrad headcount as an explanatory variable (since graduate students use the library more than undergrads). My interest is in explaining the downward trend, not forecasting. The time series plots for the response and explanatory variables look very similar, seasonal with spikes in the summer and a downward trend.

I have taken regular and seasonal differences for both variables and fit a model with ARMA errors. The estimate for the ratio is significant.

My question is, how can I estimate how much of the downward trend the does the regression variable explain? I don't think it explains all of it.

The AIC without the regression term is -20.02, and with the regression term is -35.16. The estimate of the slope is 3.71.

I'm more familiar with SAS, and I am using proc arima, but I could use R as well.

I want to emphasize that the question is not prediction. As there are more options to gather information online, it is natural that there might be less library usage per person. The question, at our particular institution, can part of the per-person usage be explained by the fact that the proportion of grad students in the entire student enrollment is less, as undergrad enrollment rises? We know, from survey data, that grad students use the library more. Then how much of the trend is attributable to that? 10%? 20%? 50%?

I will try posting some graphs, output, etc. when I get a chance.

• If you have access to SAS, also try using Proc UCM, it is very flexible in handing explanatory variables. In addition it could also handle level shifts/outliers in your data. Nov 14 '13 at 14:40
• I haven't read up on Proc UCM, I wasn't sure what it was doing. It's not using arima, right? I did want to know whether there was in fact a trend downward. I used regression with month dummy variables. Nov 14 '13 at 17:24
• I would recommend Proc UCM over ARIMA because, it is much more flexible and more interpretable especially when you want your time series data to be decomposed into trend and seasonal component. you can go to SAS website for more help. I can provide you with an eexample below. If you can post your data, I can see if I can model the data in proc ucm. support.sas.com/documentation/cdl/en/etsug/60372/HTML/default/… Nov 14 '13 at 20:01