## Abstract

Financial time series, such as nominal exchange rates and short-term interest rates, are often available with high frequency (say, daily), while macroeconomic time series, such as inflation rates, can be typically observed only with low frequency (say, monthly). In this mixed-frequency situation, one way to implement econometric/statistical inference based on two time series is to use them only with low frequency, but this necessarily ignores some of high frequency observations. In this paper, we propose a new nonparametric estimation method to make use of all high and low frequency observations. We in particular consider the case where a process {Yt} of interest is defined as the sum of two processes {Xt} and {Zt}: the former is observed with high frequency and the latter is so with low frequency, i.e., Yt=Xt+Zt, while the interested process {Yt} is, by the definition, observed with the same (low) frequency as {Zt}. Based on a convolution technique, which exploits the additive structure of the interested process, we propose new nonparametric estimators of probability density and conditional mean functions. We show that the new estimators are asymptotically normally distributed, whose limit variance is smaller than that of a standard estimator which only uses low frequency observations.