Editing Mixed-data sampling

Econometric models involving data sampled at different frequencies are of general interest. '''Mixed-data sampling (MIDAS)''' is an [[econometric]] regression developed by [[Eric Ghysels]] with several co-authors. There is now a substantial literature on MIDAS regressions and their applications, including Ghysels, Santa-Clara and Valkanov (2006),<ref>Ghysels, Eric, Pedro Santa-Clara and Rossen Valkanov (2006) ''Predicting Volatility: How to Get Most Out of Returns Data Sampled at Different Frequencies'', Journal of Econometrics, 131, 59-95</ref> Ghysels, Sinko and Valkanov,<ref>Ghysels, Eric and Arthur Sinko and Rossen Valkanov (2006) ''MIDAS Regressions: Further Results and New Directions'', Econometric Reviews, 26, 53-90.</ref> Andreou, Ghysels and Kourtellos (2010)<ref>Andreou, Elena & Eric Ghysels & Andros Kourtellos (2010) "Regression Models with Mixed Sampling Frequencies", Journal of Econometrics, 158, 246-261.</ref> and Andreou, Ghysels and Kourtellos (2013).<ref>Andreou, Elena & Eric Ghysels & Andros Kourtellos (2013) "Should macroeconomic forecasters use daily financial data and how?",  Journal of Business and Economic Statistics 31, 240-251.</ref>

==MIDAS Regressions==

A MIDAS regression is a direct forecasting tool which can relate future low-frequency data with current and lagged high-frequency indicators, and yield different forecasting models for each forecast horizon. It can flexibly deal with data sampled at different frequencies and provide a direct forecast of the low-frequency variable. It incorporates each individual high-frequency data in the regression, which solves the problems of losing potentially useful information and including mis-specification.

A simple regression example has the [[independent variable]] appearing at a higher frequency than the [[dependent variable]]:

:<math>y_t = \beta_0 + \beta_1 B(L^{1/m};\theta)x_t^{(m)} + \varepsilon_t^{(m)},</math>

where ''y'' is the dependent variable, ''x'' is the regressor, ''m'' denotes the frequency &ndash; for instance if ''y'' is yearly <math>x_t^{(4)}</math> is quarterly &ndash; <math>\varepsilon</math> is the disturbance and <math>B(L^{1/m};\theta)</math> is a lag distribution, for instance the [[Beta function]] or the [[Distributed lag#Finite distributed lags|Almon Lag]].  For example <math>B(L^{1/m};\theta) = \sum_{k=0}^K B(k; \theta) L^{k/m}</math>.

The regression models can be viewed in some cases as substitutes for the [[Kalman filter]] when applied in the context of mixed frequency data. Bai, Ghysels and Wright (2013)<ref>Bai, Jennie and Eric Ghysels and Jonathan Wright (2013) ''State Space Models and MIDAS Regressions'', Econometric Reviews, 32, 779–813.</ref> examine the relationship between MIDAS regressions and Kalman filter state space models applied to mixed frequency data. In general, the latter involves a system of equations, whereas, in contrast, MIDAS
regressions involve a (reduced form) single equation. As a consequence, MIDAS regressions might be less efficient, but also less prone to specification errors. In cases where the MIDAS regression is only an approximation, the approximation errors tend to be small.

==Machine Learning MIDAS Regressions==

The MIDAS can also be used for [[machine learning]] time series and panel data [[nowcasting (economics)|nowcasting]].<ref>{{Cite journal |last=Babii |first=Andrii |last2=Ghysels |first2=Eric |last3=Striaukas |first3=Jonas |date=2022-07-03 |title=Machine Learning Time Series Regressions With an Application to Nowcasting |url=https://www.tandfonline.com/doi/full/10.1080/07350015.2021.1899933 |journal=Journal of Business & Economic Statistics |language=en |volume=40 |issue=3 |pages=1094–1106 |doi=10.1080/07350015.2021.1899933 |issn=0735-0015|arxiv=2005.14057 }}</ref><ref>{{Cite journal |last=Babii |first=Andrii |last2=Ball |first2=Ryan T. |last3=Ghysels |first3=Eric |last4=Striaukas |first4=Jonas |date=2022-07-26 |title=Machine learning panel data regressions with heavy-tailed dependent data: Theory and application |url=https://www.sciencedirect.com/science/article/pii/S0304407622001282 |journal=Journal of Econometrics |pages=105315 |doi=10.1016/j.jeconom.2022.07.001 |issn=0304-4076|arxiv=2008.03600 }}</ref> The machine learning MIDAS regressions involve [[Legendre polynomials]]. High-dimensional mixed frequency time series regressions involve certain data structures that once taken into account should improve the performance of unrestricted estimators in small samples. These structures are represented by groups covering lagged dependent variables and groups of lags for a single (high-frequency) covariate. To that end, the machine learning MIDAS approach exploits the sparse-group [[Lasso (statistics)|LASSO]] (sg-LASSO) regularization that accommodates conveniently such structures.<ref>Simon, N., J. Friedman, T. Hastie, and R. Tibshirani (2013): ''A sparse-group LASSO'',
Journal of Computational and Graphical Statistics, 22(2), 231-245.</ref> The attractive feature of the sg-LASSO estimator is that it allows us to combine effectively the approximately sparse and dense signals.

==Software packages==

Several software packages feature MIDAS regressions and related econometric methods. These include:

* MIDAS Matlab Toolbox<ref>{{cite web|url=https://www.mathworks.com/matlabcentral/fileexchange/45150-midas-matlab-toolbox|title=MIDAS Matlab Toolbox maintained by Hang Qian}}</ref>
* midasr, R package<ref>{{cite web|url=https://cran.r-project.org/web/packages/midasr/index.html|title=midasr: Mixed Data Sampling Regression maintained by Virmantas Kvedaras and Vaidotas Zemlys-Balevicius|date=23 February 2021 }}</ref>
* midasml, R package for High-Dimensional Mixed Frequency Time Series Data<ref>{{cite web|url=https://cran.r-project.org/web/packages/midasml/index.html|title=midasml: Estimation and Prediction Methods for High-Dimensional Mixed Frequency Time Series Data maintained by Jonas Striaukas|date=29 April 2022 }}</ref>
* EViews<ref>{{cite web|url=https://www.eviews.com/EViews9/ev95midas.html|title=EViews 9.5 MIDAS Forecasting Demonstration}}</ref>
* Python<ref>{{cite web|url=https://github.com/sapphire921/midas_pro|title=MIDAS Python code|website=[[GitHub]] }}</ref>
* Julia<ref>{{cite web|url=https://github.com/mikemull/Midas.jl|title=MIDAS Julia|website=[[GitHub]] }}</ref>
* Stata，midasreg

== Alternatives ==
In some situations it might be possible to alternatively use [[temporal disaggregation]] methods (for [[upsampling]] time series data from e.g. monthly to daily).<ref>F. T. Denton. Adjustment of monthly or quarterly series to annual totals: An approach based on quadratic minimization. Journal of the American Statistical Association, Mar. 1971</ref>

==References==
{{Reflist}}

==See also==
* [[Distributed lag]]
* [[ARMAX]]

[[Category:Econometric modeling]]
[[Category:Time series models]]
[[Category:Statistical forecasting]]


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