Just a general question that I couldn't find too much on.
What would be some good approaches to one step ahead forecasting of financial time series with mixed frequencies?
Often a lot of the available data influencing the price of say a stock or commodity is published at different frequencies, some daily, some weekly, some monthly etc. which in my head makes it tricky to use normal models for anything but the longest time interval.
Edit: found this paper which goes some way to cover the topic
You can use MIDAS regressions to predict low-frequency target variable with some set of high-frequency (hf) covariates. If you have a larger set of hf covariates, the simplest way is to use forecast combinations, see [1] of such an application of MIDAS regression model to nowcast GDP growth rate using large set of hf covariates. Also, see Matlab toolbox for MIDAS regressions in many different contexts and actual data examples.
Briefly about MIDAS regression:
Formally, let $t\in [T]$ be the low-frequency (lf) time period, say quarterly and $y_t$ be the lf target variable (so you observe it each quarter). Further, let $x_{t-j/m}$, $j \in \{0, \dots, m-1\}$, be hf covariate which we observe within period $(t-1, t]$, i.e. we observe $x$ $m$ times per $t$ time period, and the notation $j/m$ means that $x$ is lagged by a fraction of time period $t$. Say we have daily hf. Assuming 66 days within the quarter (reasonable assumption e.g when taking trading days but it can be generalized to any number, and it can even vary quarter...), this means that we have 66 $x$ observations for each $y$ observation. To use information efficiently, we may wish to use all 66 observations in our regression model, so we can write the model as
$ \qquad \qquad y_t = \alpha + \sum_{j=0}^{m-1} \beta_{j} x_{t-1-j/m} + \epsilon_t$
where we subtract additional lag in $x$ to make the regression predictive. This is what is called U-MIDAS (unrestricted MIDAS) regression model. When $m$ is large, as in the quarterly/daily example, the model suffers from parameter proliferation problem (a lot of parameters and typically small samples to estimate them). In MIDAS regression we parameterize $\sum_{j=0}^{m-1} \beta_{j}$ lag polynomial such that it depends on a few, say one or two, parameters. In this case, the model is
$ \qquad \qquad y_t = \alpha + \beta \omega (\theta) X_{t-1} + \epsilon_t$
where $\beta$ is the usual regression slope coefficient, $\omega (\theta)$ is some weight function typically taken to be exponential Almon or beta density function, $\theta$ is a low-dimensional parameter of $\omega(.)$ that determines the shape of the weight function and $X_{t-1} \in \mathbf{R}^m$ is a vector of hf lags. Slope coefficient in fact is needed only if you want to know the aggregate effect of $X_{t-1}$ on $y_t$, while if your goal is the prediction you actually do not need it. In the former case, $\omega$ function has to scaled to sum to one for $\beta$ to be identified.
To estimate U-MIDAS you use OLS, while for MIDAS you typically need non-linear LS estimator. There is a very elegant way to avoid NLS estimation (sometimes NLS is problematic) by profiling out $\theta$ parameter, see [2]. In this case, you only need OLS to estimate MIDAS regression.
You can also check wiki on MIDAS models: https://en.wikipedia.org/wiki/Mixed-data_sampling
Hope this helps!
Matlab toolbox: https://nl.mathworks.com/matlabcentral/fileexchange/45150-midas-matlab-toolbox
R package: https://cran.r-project.org/web/packages/midasr/midasr.pdf
Refs:
[1] Andreou, E., Ghysels, E., & Kourtellos, A. (2013). Should macroeconomic forecasters use daily financial data and how?. Journal of Business & Economic Statistics, 31(2), 240-251.
[2] Ghysels, E., & Qian, H. (2019). Estimating MIDAS regressions via OLS with polynomial parameter profiling. Econometrics and statistics, 9, 1-16.
