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

Abstract: In this paper, a Mixed Data Sampling (MIDAS) model is studied when both low and high frequency variables are contaminated with measurement error. It is shown that the profile likelihood estimator becomes inconsistent in the presence of measurement error. Using the corrected score approach along with profile likelihood approach, a consistent estimator for parameters of MIDAS Measurement Error model is proposed. Small and large sample properties of the estimator are examined by performing a monte carlo simulation study and considering the effect of sample size, number of lags and profiling parameter.

• The paper demonstrates that conventional profile likelihood estimators yield inconsistent MIDAS regression parameters when measurement error is present.
• It introduces a corrected estimator using a score-based approach that accounts for measurement error variances to achieve consistency and asymptotic normality.
• Empirical results via Monte Carlo simulations highlight that low-frequency errors chiefly bias regression coefficients while high-frequency errors impact lag parameter estimates.

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

Introduction

This paper addresses the fundamental issue of estimating Mixed Data Sampling (MIDAS) regressions when both endogenous and exogenous variables are contaminated by measurement error. MIDAS models allow the fusion of high-frequency and low-frequency time series, a critical framework for econometric applications where observations are sampled at different intervals (e.g., monthly inflation rates vs. quarterly GDP). While prior methodologies, particularly the profile likelihood estimator [Ghysels and Qian, 2019], are computationally appealing for MIDAS regression, this paper formally demonstrates that they fail to yield consistent parameter estimates when measurement error is present in the observed series.

Problem Formulation

The classical ADL-MIDAS regression considers a low-frequency dependent variable alongside lagged high-frequency exogenous variables, with lags parameterized by polynomials (e.g., Beta or Almon). Profile likelihood estimation expedites computation by reducing the maximization to a lower-dimensional parameter space. However, realistic economic and financial datasets often contain non-negligible measurement error in both low- and high-frequency variables due to, e.g., preliminary releases and subsequent data revisions.

The paper formalizes the ADL-MIDAS regression model under measurement error, showing analytically that the naïve application of the profile likelihood estimator to contaminated data fundamentally breaks the consistency guarantee. The authors rigorously derive how the probability limit of the estimator deviates from the true parameter, with the limit depending on measurement error variances and the lag polynomial structure.

Methodology

A corrected estimation procedure is proposed utilizing the corrected score approach [Nakamura, 1990], combined with the profile likelihood framework. The crucial assumption is that the variances of measurement error in observed series are known or reliably estimated from auxiliary data. The corrected likelihood incorporates explicit terms accounting for the variance inflation induced by measurement error, resulting in a closed-form estimator for the regression coefficients and lag polynomial parameters.

Large sample properties are derived: the corrected estimators are proved to be consistent and asymptotically normal, with variance expressions that reflect the additional noise from measurement error. The methodology respects the parsimonious parameterization of lag coefficients, ensuring computational tractability even as the number of lags grows.

Empirical Evaluation

Monte Carlo simulations systematically evaluate both naïve and corrected estimators across a range of sample sizes, lag lengths ($j^{max}$), lag polynomial hyperparameters ($\theta$), and measurement error variances. The results indicate:

• Existing profile estimator is inconsistent in the presence of measurement error: Bias and variability stabilize above zero as sample size increases, failing to converge.
• Corrected estimator is statistically consistent: Bias and variability diminish toward zero with increasing sample size.
• Bias amplification is dominated by low-frequency measurement error for regression coefficients, while high-frequency measurement error more strongly affects lag polynomial and noise variance estimates.
• Increasing the number of lags exacerbates bias and variability, emphasizing the necessity of parsimonious lag selection.
• The distribution of lag weights (parameterized by $\theta$) significantly influences bias properties, with distant-past-weighting lag structures being more susceptible to noise amplification.

Strong numerical results underscore that measurement error effects are non-negligible and require explicit correction; the corrected estimator maintains consistency and desirable asymptotic properties even under substantial measurement contamination.

Implications and Future Directions

The theoretical insight that naïve profile likelihood estimation is inconsistent under measurement error mandates reconsideration of empirical strategies in econometrics and financial forecasting with mixed frequency data. The corrected profile estimator’s reliance on known error variance suggests future research agendas in robust estimation of measurement error variance, potentially leveraging replication or external validation datasets.

Practically, this methodology enables more reliable estimation and forecasting in settings where data revision and measurement noise are endemic, e.g., real-time macroeconomic indicators. Theoretically, the results reinforce the necessity of integrating measurement error correction into model estimation for mixed-frequency systems.

Open avenues include joint estimation of measurement error variances, extension to more general nonlinear MIDAS specifications, and the integration of state-space or Bayesian correction strategies to mitigate identification and computational complexity.

Conclusion

The paper establishes that profile likelihood estimation in MIDAS regressions fails to maintain consistency when applied to measurement error contaminated data. A corrected profile likelihood estimator, leveraging prior knowledge of error variances, achieves both consistency and asymptotic normality. Empirical evaluation confirms the estimator’s superiority in bias and variance reduction across typical mixed-frequency econometric scenarios. The findings imply that measurement error correction must be a priority in practical MIDAS regression applications, and the corrected approach lays a foundation for ongoing advances in robust mixed-frequency econometric modeling.

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