# How do I forecast quarterly public expenses based on annual budgets and potentially other variables?

I have some time series data from 2008 and forward (see below) on quarterly public expenses and annual public budgets. I would like to forecast the last two quarters of 2018 as precisely as possible, but I am unsure about what methods I should use.

I have been considering a vector autoregressive model, but there is an issue with different frequencies on the data. I guess I could split the budget into 4 equal parts.

I have also considered computing the deviation between the two series and then use a univariate model instead, but that would rule out adding additional explaining variables such as demography.

If possible I would like some suggestions on what type of models I should look into. I am not an expert in times series analysis and especially not the multivariate ones, but I hope it is possible.

I hope you can help. Thank you!

+------------+-----------------+
|    Date    |    Expenses     |
+------------+-----------------+
| 2008-01-01 | 66.386.086.765  |
| ...        | ...             |
| 2016-01-01 | 78.200.910.570  |
| 2016-04-01 | 167.604.150.482 |
| 2016-07-01 | 251.017.792.801 |
| 2016-10-01 | 337.595.471.006 |
| 2017-01-01 | 77.580.707.994  |
| 2017-04-01 | 166.599.846.864 |
| 2017-07-01 | 251.331.721.551 |
| 2017-10-01 | 337.247.375.085 |
| 2018-01-01 | 76.467.286.000  |
| 2018-04-01 | 166.634.900.000 |
| 2018-07-01 | ?               |
| 2018-10-01 | ?               |
+------------+-----------------+
+------+-----------------+
| Date |     Budget      |
+------+-----------------+
| 2008 | 295.130.222.937 |
| ...  | ...             |
| 2016 | 348.210.251.795 |
| 2017 | 346.244.633.739 |
| 2018 | 346.127.455.000 |
+------+-----------------+

• Those are cumulative quarterly expenses, not actual quarterly expenses, right? – whuber Aug 16 at 13:28
• Yes, you are right. I have change it to actual quarterly expenses in my model now – Frederik Aug 21 at 10:31

There are several ways to approach this issue ranging from simple heuristics to rigorous econometric models.

Among the heuristics is to simply interpolate the values for higher temporal levels down to a more disaggregate periodicity. Many software packages offer automatic methods for doing this.

A more rigorous approach for mixed temporality is Ghysels' MIDAS regression. He and some colleagues have developed an R module for this, Mixed Frequency Data Sampling Regression Models: The R Package midasr which is referenced here... https://www.jstatsoft.org/article/view/v072i04/v72i04.pdf

• Thanks for the comments. I've mostly been looking at a SARMAX model so far as suggested by IrishStat, but also shortly at a MIDAS model. It seems to me like MIDAS models are a solution when the regressors have higher frequency the independent variable and not the other way around. Is that correctly understood? Because if that is the case, then a MIDAS model is not feasible solution – Frederik Aug 21 at 10:27
• My understanding is that it's bi-directional as a function of how the data is operationalized. – user332577 Aug 21 at 23:21

Aside of the issue of converting data to a higher level of frequency, your approach should be to build a SARMAX model incorporating predictors and the history of predictors, the history of Y itself while incorporating pulses, level/step shifts , seasonal pulses and local time trends. Follow https://autobox.com/pdfs/A.pdf to get a useful https://autobox.com/pdfs/SARMAX.pdf . Stay way clear of simple ols regression as suggested here https://autobox.com/pdfs/regvsbox-old.pdf unless suggested (i.e. not proven to be inadeqaute ) by the analytics.