📈 economics

Estimation of MIDAS Regressions with Errors-in-the-Variables

This paper proposes a consistent estimation method for Mixed Data Sampling (MIDAS) models using a corrected score approach to account for measurement errors in both low- and high-frequency variables.

Original authors: Sukhbir Kaur, Sukhbir Singh, Kanchan Jain, Pooja Soni

Published 2026-04-28

📖 4 min read☕ Coffee break read

CC BY 4.0

Original authors: Sukhbir Kaur, Sukhbir Singh, Kanchan Jain, Pooja Soni

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict how much a bakery will sell tomorrow (the low-frequency variable) by looking at how many people walk past the shop every hour (the high-frequency variable).

This paper tackles a very specific, messy problem in economics: What happens when your data is "blurry" and your timing is "off"?

Here is the breakdown of the paper using everyday concepts.

1. The Problem: The "Blurry Lens" (Measurement Error)

In a perfect world, if 100 people walk past a shop, you record "100." But in the real world, your data is often "noisy." Maybe your sensor miscounts, or a person was partially hidden by a car. This is Measurement Error.

The researchers point out that economists often deal with two types of "blurry" data:

The Slow Data (Low Frequency): Like monthly unemployment rates. These are big-picture numbers that change slowly.
The Fast Data (High Frequency): Like daily stock prices or hourly foot traffic. These change rapidly.

The Catch: If you use standard math formulas to predict the "Slow Data" using "Fast Data" while both are blurry, the math breaks. It’s like trying to perform surgery while wearing thick, foggy goggles. You’ll end up making a "consistent" mistake—meaning no matter how much more data you collect, your answer will always be slightly wrong.

2. The Tool: The "MIDAS" Model (The Multi-Speed Gearbox)

To handle data moving at different speeds (monthly vs. daily), scientists use a tool called MIDAS (Mixed Data Sampling).

Think of MIDAS like a multi-speed gearbox in a car. Instead of forcing the fast-moving daily data to slow down to a monthly pace (which loses all the interesting details), MIDAS allows the model to "shift gears" and look at the fine-grained, daily details to predict the big monthly picture.

3. The Discovery: The "Broken Compass"

The paper looks at a popular way of using this "gearbox" (the Ghysels and Qian method). The authors proved mathematically that if your data has measurement errors, this popular method acts like a broken compass. Even if you walk for a thousand miles (collect more data), the compass will keep pointing you 10 degrees to the left of where you actually are. It is "inconsistent."

4. The Solution: The "Corrected Lens" (The New Estimator)

The authors propose a new mathematical way to fix this. They use a technique called the "Corrected Score Approach."

Imagine you know your camera lens has a specific smudge on it. Instead of just taking the photo and hoping for the best, you use a mathematical formula to "subtract" the smudge from the image.

The researchers say: "If we know roughly how much 'blurriness' (error) is in our sensors, we can add a correction factor to our equations to cancel out the mistake."

5. The Proof: The "Stress Test" (Simulations)

To prove their new "lens" works, they ran thousands of computer simulations (Monte Carlo studies). They threw "dirty" data at two different systems:

The Old Way: The "Broken Compass."
The New Way: Their "Corrected Lens."

The Results:

The Old Way: As they added more data, the errors stayed high. It was like trying to clear fog by just staring harder; it didn't work.
The New Way: As they added more data, the errors shrank toward zero. The "vision" became crystal clear.

Summary in a Nutshell

If you are trying to predict the future using data that is both moving at different speeds and slightly inaccurate, don't use standard tools—they will lead you astray. Instead, use the authors' "Corrected MIDAS" method, which mathematically "wipes the smudge off the lens," allowing you to see the true economic patterns clearly.

Technical Summary: Estimation of MIDAS Regressions with Errors-in-the-Variables

1. Problem Statement

The paper addresses a critical limitation in Mixed Data Sampling (MIDAS) regression models. While MIDAS models are highly effective for analyzing relationships between variables sampled at different frequencies (e.g., monthly inflation vs. quarterly GDP), they typically assume that the observed data is an accurate reflection of the underlying latent processes.

In real-world economic applications, both low-frequency ( $Z_t$ ) and high-frequency ( $\xi_t$ ) variables are often subject to measurement error (ME) due to data collection lags, revisions, or processing errors. The authors identify that the standard profile likelihood estimator—widely used for its computational efficiency—becomes inconsistent when measurement error is present in either the endogenous or exogenous variables. This inconsistency leads to biased parameter estimates, undermining the reliability of economic forecasts and structural analyses.

2. Methodology

The authors focus on the Autoregressive Distributed Lag - MIDAS (ADL-MIDAS) model, which incorporates an autoregressive component for the low-frequency variable and a functional lag polynomial (such as the Beta polynomial) for the high-frequency variable.

Key methodological steps include:

Modeling Measurement Error: The authors define a model where the observed variables $y_t$ and $x_t$ are the sum of the true latent variables ( $Z_t, \xi_t$ ) and additive, normally distributed measurement errors ( $u_t, v_t$ ).
Demonstrating Inconsistency: By applying a Taylor expansion to the gradient of the log-likelihood function under the presence of ME, the authors mathematically prove that the probability limit of the standard estimator does not equal the true parameter value ( $\text{plim } \hat{\gamma}_e \neq \gamma$ ).
Corrected Score Approach: To rectify this, the authors adopt the corrected score methodology (based on Nakamura, 1990). They modify the log-likelihood function by adding a correction term that accounts for the expected values of the measurement error variances ( $\sigma^2_u$ and $\sigma^2_v$ ).
Profile Likelihood Estimation: The authors use a profiling technique to concentrate the likelihood function. They first profile out the error variance ( $\sigma^2_\epsilon$ ) and then optimize the MIDAS polynomial parameters ( $\theta$ ) and the regression coefficients ( $\beta$ ) using the corrected log-likelihood.

3. Key Contributions

Theoretical Proof of Inconsistency: The paper provides the first formal demonstration that the standard ADL-MIDAS profile estimator fails to maintain consistency in the presence of measurement error in both variables.
Development of a Consistent Estimator: The primary contribution is the proposal of a new Corrected Profile Estimator for the ADL-MIDAS ME model.
Asymptotic Derivation: The authors derive the large-sample properties of the new estimator, proving that it is not only consistent but also asymptotically normally distributed.
Simulation Framework: They provide a robust Monte Carlo simulation study that evaluates the estimator's performance across various dimensions, including sample size ( $T$ ), number of lags ( $j_{max}$ ), and the magnitude of measurement error.

4. Results

The Monte Carlo simulations yielded several significant findings:

Consistency Validation: Unlike the naive estimator, whose bias and variability stabilize at non-zero levels as sample size increases, the proposed corrected estimator's bias and variability tend toward zero as $T \to \infty$ .
Impact of Lag Length ( $j_{max}$ ): Increasing the number of lags increases the complexity and the bias/variability of the estimator. The authors emphasize the need for a parsimonious model to balance information capture with estimation stability.
Error Amplification Effects:
- Measurement error in the low-frequency variable has a more severe impact on the regression coefficients ( $\beta$ ).
- Measurement error in the high-frequency variable has a more significant impact on the MIDAS hyperparameter ( $\theta$ ) and the equation error variance ( $\sigma^2_\epsilon$ ).
Parameter Sensitivity: The performance of the estimator is sensitive to the value of $\theta$ . Higher values of $\theta$ (which place more weight on recent high-frequency observations) tend to result in lower bias for the regression coefficients.

5. Significance

This research is highly significant for econometricians and policy analysts. By providing a mathematically sound and computationally feasible method to handle contaminated mixed-frequency data, it allows for more accurate modeling of economic dynamics. The study bridges a gap between MIDAS modeling and the "Errors-in-Variables" literature, offering a practical tool for researchers working with imperfectly measured macroeconomic indicators.