Unrestricted Mixed Data Sampling (MIDAS): MIDAS Regressions with Unrestricted Lag Polynomials

Abstract

Mixed data sampling (MIDAS) regressions allow us to estimate dynamic equations that explain a low frequency variable by high frequency variables and their lags. When the difference in sampling frequencies between the regressand and the regressors is large, distributed lag functions are typically employed to model dynamics avoiding parameter proliferation. In macroeconomic applications, however, differences in sampling frequencies are often small. In such a case, it might not be necessary to employ distributed lag functions. We discuss the pros and cons of unrestricted lag polynomials in MIDAS regressions. We derive unrestricted-MIDAS (U-MIDAS) regressions from linear high frequency models, discuss identification issues and show that their parameters can be estimated by ordinary least squares. In Monte Carlo experiments, we compare U-MIDAS with MIDAS with functional distributed lags estimated by non-linear least squares. We show that U-MIDAS performs better than MIDAS for small differences in sampling frequencies. However, with large differing sampling frequencies, distributed lag functions outperform unrestricted polynomials. The good performance of U-MIDAS for small differences in frequency is confirmed in empirical applications on nowcasting and short-term forecasting euro area and US gross domestic product growth by using monthly indicators.

Taming Volatile High Frequency Data with Long Lag Structure: An Optimal Filtering Approach for Forecasting

“…More recently MIDAS has also been used on macroeconomic data for example in Clements & Galvão (2008) and Clements & Galvão (2009) or Armesto et al (2010) and Andreou et al (2011). More recently Foroni et al (2012) have shown, that if differences in frequencies are small, for instance for a mixture of quarterly and monthly data, an unrestricted MIDAS setup (U-MIDAS) is equivalent or even superior compared with standard MIDAS setups. An unrestricted MIDAS setup requires less computational and modelling efforts compared with standard MIDAS setups.…”

“…The unrestricted MIDAS approach was promoted by Foroni et al (2012). Instead of approximating the parameters of each high-frequency observation of the high frequency variable X t , this approach estimates the weights as unrestricted parameters.…”

A daily indicator of economic growth for the euro area

“…The first and the simplest is bridge equations, i.e., linear regressions that link ('bridge') high frequency variables to low frequency ones. The second is the unrestricted MIDAS (U-MIDAS), linear variant to the mixed-data sampling (MIDAS) introduced by Ghysels et al (2004), where low and high frequency variables are linked through highly parsimonious distributed lag polynomials (see Foroni et al, 2013;Carriero et al, 2013, for a complete description of the U-MIDAS approach). As a target variable we adopt GDP growth, which, due to its broad coverage of the economy, is generally accepted as the key economic indicator of economic activity.…”

On inflation expectations in the NKPC model

“…These models are called MIDAS models, see Ghysels et al (2006Ghysels et al ( , 2007, Foroni et al (2015), and Breitung and Roling (2015) among many possible references.…”

“…The parameters are estimated unrestrictedly, thereby following the format recommended in Foroni et al (2015), which is called the unrestricted MIDAS model, or in short, UMIDAS. Table 1 gives a selection of the estimation results.…”