Improved Breitung and Roling estimator for mixed-frequency models with application to forecasting inflation rates

Abstract Instead of applying the commonly used parametric Almon or Beta lag distribution of MIDAS, Breitung and Roling (J Forecast 34:588–603, 2015) suggested a nonparametric smoothed least-squares shrinkage estimator (henceforth $${SLS}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>SLS</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:math> ) for estimating mixed-frequency models. This $${SLS}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>SLS</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:math> approach ensures a flexible smooth trending lag distribution. However, even if the biasing parameter in $${SLS}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>SLS</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:math> solves the overparameterization problem, the cost is a decreased goodness-of-fit. Therefore, we suggest a modification of this shrinkage regression into a two-parameter smoothed least-squares estimator ( $${SLS}_{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>SLS</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> ). This estimator solves the overparameterization problem, and it has superior properties since it ensures that the orthogonality assumption between residuals and the predicted dependent variable holds, which leads to an increased goodness-of-fit. Our theoretical comparisons, supported by simulations, demonstrate that the increase in goodness-of-fit of the proposed two-parameter estimator also leads to a decrease in the mean square error of $${SLS}_{2},$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mrow> <mml:mi>SLS</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>,</mml:mo> </mml:mrow> </mml:math> compared to that of $${SLS}_{1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>SLS</mml:mi> </mml:mrow> <mml:mn>1</mml:mn> </mml:msub> </mml:math> . Empirical results, where the inflation rate is forecasted based on the oil returns, demonstrate that our proposed $${SLS}_{2}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mrow> <mml:mi>SLS</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msub> </mml:math> estimator for mixed-frequency models provides better estimates in terms of decreased MSE and improved R 2 , which in turn leads to better forecasts.