Mixed-data sampling

Key takeaways

Mixed-data sampling belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Mixed-data sampling to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Mixed-data sampling from memory before moving on to harder problems.

Reference excerpt

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), Ghysels, Sinko and Valkanov, Andreou, Ghysels and Kourtellos (2010) and Andreou, Ghysels and Kourtellos (2013).

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:

y t = β 0 + β 1 B ( L 1 / m ; θ ) x t ( m ) + ε t ( m ) , {\displaystyle y_{t}=\beta _{0}+\beta _{1}B(L^{1/m};\theta )x_{t}^{(m)}+\varepsilon _{t}^{(m)},}

where y is the dependent variable, x is the regressor, m denotes the frequency – for instance if y is yearly x t ( 4 ) {\displaystyle x_{t}^{(4)}} is quarterly – ε {\displaystyle \varepsilon } is the disturbance and B ( L 1 / m ; θ ) {\displaystyle B(L^{1/m};\theta )} is a lag distribution, for instance the Beta function or the Almon Lag. For example B ( L 1 / m ; θ ) = ∑ k = 0 K B ( k ; θ ) L k / m {\displaystyle B(L^{1/m};\theta )=\sum _{k=0}^{K}B(k;\theta )L^{k/m}} . 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) 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. 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 (sg-LASSO) regularization that accommodates conveniently such structures. 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 midasr, R package midasml, R package for High-Dimensional Mixed Frequency Time Series Data EViews Python Julia 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).

References

Worked examples

Example 1 — a first encounter with Mixed-data sampling

Start with the simplest possible case. Write down what Mixed-data sampling claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Mixed-data sampling before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Mixed-data sampling ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Mixed-data sampling

In research: Mixed-data sampling appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Mixed-data sampling in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Mixed-data sampling is common in secondary-school and first-year university syllabi. It links to neighbouring topics Econometric modeling, Econometrics stubs, Statistical forecasting, so understanding it makes those chapters shorter.
In everyday life: Look for Mixed-data sampling outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Mixed-data sampling in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Mixed-data sampling means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Mixed-data sampling out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Mixed-data sampling in simple terms?

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.

Why does Mixed-data sampling matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Mixed-data sampling?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Mixed-data sampling.