# Is it possible to combine two time series with different frequencies for forecasting?

I have two macroeconomic time series with annual GDP growth rates for a given country. One series has annual frequency (year-on-year growth) and historical data from mid 1990s until 2012 (about 20 data points). The other series starts in 2013 and has quarterly frequency (quarter-on-quarter growth; about 18 data points).

The objective is to conduct a forecasting exercise to verify the predictability of GDP. One possibility is to only use one of the series (either annual frequency series or quarterly frequency series), but this means there are only very few data points available - probably too few observations to get meaningful results.

I had the idea to somehow combine both series in order to increase the number of observations available for model training and testing, but I have not come across any references (libraries, papers etc.) in this respect.

Are there are any strategies to deal with this situation and, if so, what are the options?

If you are ok with yearly forecast, convert quarterly growth into yearly. like 10%, 10%, 10%, 10% quarterly growth would become 46.41 % yearly growth.

24 data points are less but this is what we have. Start with Moving average and then build on as GDP growth would have less variation.

Here is my 2 cents, though I am by no means an expert on time-series. But you might be able to treat this as a missing data problem. So you have yearly GDP growth, but you would like quarterly estimates for the 1990-2012. So basically you have the quarterly estimates for 1990-2012 as missing values.

You could create where $$X$$ are predictors for the growth rate and then $$Y$$ is the quarterly growth rate. You would break the 1990-2012 series into quarterly values and perhaps set the last quarter in the year equal to the actual value you got, while setting quarters 1-3 as missing. You would also have the quarterly data for 2013 forward in the same table, but now your quarterly data is not missing.

         X1  X2  X3 ...  Y
1990Q1                 NA
1990Q2                 NA
1990Q3                 NA
1990Q4                 Yearly Value 1990
.
.
.
2013Q1                 Quarterly value
2013Q2                 Quarterly value
.
.
.


So then you could use something like the EM algorithm (Expectation-Maximization) algorithm to predict the missing values for the quarter growth rates $$Y$$ from the covariates $$X$$. R has lots of packages for the EM algorithm, as well as Python etc.

I am not sure how much variation you would get in the predicted quarterly rates versus the average rate for the year, but you might be able to detect some interesting variations.

Of course, some of this depends upon the methods that are accepted within your discipline and stuff like that. And if anyone has a better idea--or correction to what I said--it would be interesting to hear. I have not done missing data problems in a few years.

Alert: Not an expert, but have had some experience with seasonal data of different frequencies.

Like other econometric timeseries, GDP data can be broken into Trend, Seasonality and Noise.

• To predict the trend, annual timeseries (constructed from summing quarters) might be enough to predict next Calendar Year's (CY) GDP

• To predict, next quarter's GDP, we need to know what % of Next CY's GDP comes from next quarter. This you can infer from 18 quarterly points, e.g the annual GDP is decomposed into: Q1:20%, Q2:30%, Q3:10%, Q4:40%

• To estimate noise in quarterly forecast, you can probably use last 18 quarter's stdev.

Other possibilities:

Option 1: You can convert all your quarterly numbers into annual numbers and get a longer time series Advantage: Simplicity Disadvantage: We're wasting information stored in quarterly frequency e.g. cyclical variations

Option 2: Convert all annual numbers into quarterly numbers by either expectation maximization through likelihood or multiple imputation methods Advantage: Information of frequency is retained Disadvantage: Since data is NMAR, quarterly values imputed for earlier periods might just not have enough value and may bias your results since it's a longer time interval