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I was wondering if someone could provide some guidance to me. I would like to
Combine various financial data of mixed frequencies (some daily, weekly, some quarterly) to a composite index. I have been reading about MIDAS, but that seems to be high frequency to low frequency. Maybe I am missing something there.
Predict monthly forecasts of a time series using mixed frequency data (similar to the above).
MIDAS is useful when you have a low frequency series and you want to include high frequency data in the regression. So for instance, if you want to forecast quarterly GDP data and want to include daily S&P 500 data as a regressor instead of just using the quarter end value of S&P 500.
Usually we assume that the causality runs from S&P 500 to GDP. It becomes trickier when you start thinking about the impact of lagged GDP on S&P 500. You can usually get away with some kind of ad hoc adjustment like filling in the GDP growth rate for each daily period as equal to the quarterly growth rate, but it won't necessarily capture the relationship properly.
State space methods, such as Kalman Filters, are another common approach to handling mixed data series. The Philly Fed's ADS business conditions index and the literature on Nowcasting use this. It's a large, complicated literature that is probably beyond a small post to summarize. To get a general sense of the approach, they assume that the data is missing and then use Kalman Filters to estimate parameters given the missing data.
Another approach is to use a combination of forecasting models. In the above S&P 500 and GDP example, you could use a daily forecasting model that includes GDP and a quarterly forecasting model that includes it. You can use the daily model to forecast iteratively out for the remainder of the quarter and then use the quarterly model thereafter. This approach could be made significantly richer with monthly models, etc.
With respect to your second question, the hard part is estimating the model. Once you have the model set up and estimate the parameters of it, forecasting is usually straightforward.
To address the second question: I've done a couple of different things. When I did not care much about the result, I just took a straight line interpolation or fit a curve to make, for example quarterly into monthly. When I did care, and it was a lot of work, I found a monthly or weekly series similar to the lower frequency series, and used the changes in the higher frequency data to generate the lower. For example, although it's not what I did, but just for an example you could generate a weekly unemployment rate out of the monthly number by using the changes in the weekly claims numbers.
MIDAS is quite general framework and for conditional mean prediction involving mixed frequencies, MIDAS models are easier to estimate than state-space models.
You can certainly predict low frequency target variable using high frequency predictor(s). If you have many, you can do forecast combinations. Matlab and R packages for MIDAS have many functionalities, including forecast combinations.
You can also predict high frequency target using low frequency data using Reverse MIDAS model. See for example https://www.sciencedirect.com/science/article/pii/S0169207018300967
Also, you have VAR mixed frequency models that allow for two way causality relations, see https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2465448 (also published at joe).
