# How to treat different frequencies of a time series data?

I have annual GDP, annual CPI, monthly exchange rate, and monthly export data. I would like to go for estimation being monthly stock return as the dependent variable. What are the methods to treat the annual figures in to monthly and/or the monthly in to annual? Please, I need your recommendations or books to read?

• One thing: in the title, you should use “frequencies”, not “dimensions.” The current title is confusing. – Brian Romanchuk Jan 4 '19 at 3:08

CPI is a stock while GDP is a flow. Re-sampling of stocks to higher frequencies can be approximated with a number of choices, but probably the most common is linear interpolation. In some contexts filling forward and filling backward are common. But in principle you could fit any function you want through all the points you have and use the values from the function instead. Flows are a bit more complicated because annual flows are the sum of monthly flows. Dividing the annual flow into twelfths across the entire year is analogous to filling forward or filling backward (depending on what month of the year is the date of annual measurement). For example, if GDP is 144 and measured in December of 2018, assigning every month in 2018 a value of 12 is similar to filling backwards. But again, as long as you remember that the flows are supposed to add up across months to equal or approximate annual flows you can you a number of functional forms to approximate them.

This is a big topic. I recommend searching for papers related to "interpolated time series". Resampling and Subsampling for Financial Time Series by E Paparoditis, DN Politis might be a good, if advanced, place to start. Modeling and Forecasting Time Series Sampled at Different Frequencies by Casals, Jerez, and Sotoca is another useful reference. In comparing the effectiveness different methods, Interpolation in Time Series: An Introductive Overview of Existing Methods, Their Performance Criteria and Uncertainty Assessment by Lepot, Aubin, and Clemens provides a nice list of methods but doesn't focus on the stock versus flow distinction.

1. Deterministic methods (fill forward or back)
2. Nearest-neighbor
3. Polynomial interpolation
4. Distance weighting methods
5. Fourier transformation based methods
6. Regression models
7. Auto-regressive models
8. Machine learning based methods
9. Kernel methods
10. K-nearest neighbors
11. Box-Jenkins models
12. Kriging-based methods

As there are no other answers, I will offer an informal answer.

Converting from annual to monthly data is effectively inserting information into the series. As a result, the choice of conversion will affect the analysis. It is safer to convert everything to the lowest frequency (annual), as that has much less effect on results.

To convert monthly to annual data, two standard techniques are to either take the latest value (December) or the annual average. It would make sense to have the choice aligned with the methodology for the annual data.

Unfortunately, I do not have books to suggest that cover this topic.

• To convert monthly to annual for export data, surely the straightforward technique is addition? I'm assuming here that the data are value of exports for each month. – Adam Bailey Jan 4 '19 at 11:37
• Sum of all the months in the year? Sure, but that is just a scaling of the average (12 times the monthly average). – Brian Romanchuk Jan 5 '19 at 4:23