SummaryIn this paper we propose a new method to estimate nonparametrically a time varying parameter model when some qualitative information from outside data (e.g. seasonality) is available. In this framework we make two main contributions. First, the resulting estimator is shown to belong to the class of generalized ridge estimators and under some conditions its rate of convergence is optimal within its smoothness class. Furthermore, if the outside data information is full lled by the underlying model, the estimator shows e ciency gains in small sample sizes. Second, for the implementation process, since the estimation procedure envolves the computation of the inverse of a high order matrix we provide an algorithm that avoids this computation and, also, a data-driven method is derived to select the control parameters. The practical performance of the method is demonstrated in a simulation study and in an application to the demand of soft drinks in Canada.

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This paper provides a detailed analysis of the asymptotic properties of a kernel estimator for a seemingly unrelated regression equations model with time-varying coefficients (tv-SURE) under general conditions. Theoretical results together with a simulation study differentiate the cases for which the estimation of a tv-SURE outperforms the estimation of a single regression equations model with time-varying coefficients. The study shows that Zellner’s results cannot be straightforwardly extended to the time-varying case. The tv-SURE is applied to the Fama and French five-factor model using data from four different international markets. Finally, we provide the estimation under cross-restriction and discuss a testing procedure.

We propose a two-stage procedure to estimate conditional beta pricing models that allow for flexibility in the dynamics of assets' covariances with risk factors and market prices of risk (MPR).First, conditional covariances are estimated nonparametrically for each asset and period using the time-series of previous data. Then, time-varying MPR are estimated from the cross-section of returns and covariances using the entire sample. We prove the consistency and asymptotic normality of the estimators. Results from a Monte Carlo simulation for the three-factor model of Fama and French (1993) suggest that nonparametrically estimated betas outperform rolling betas under different specifications of beta dynamics. Using return data on the 25 size and book-tomarket sorted portfolios, we find that MPR associated with the three Fama-French factors exhibit substantial variation through time. Finally, the flexible version of the three-factor model beats alternative parametric specifications in terms of forecasting future returns.

Spain is being hit hard by the COVID-19 pandemic. During the first wave, from mid-March to early June 2020, the disease caused nearly 30,000 deaths in a population of 47 million. This article quantifies the unevenness in the distribution of epidemiological variables across the Spanish territory. The study is relevant because Spain is divided into regions that hold devolved authority for providing health care services to their citizens. Using inequality metrics, the study shows: i) By mid-April inequality in the epidemiological variables reached a stationary value that changed little with the incorporation of new cases and deaths. At the end of the outbreak, cumulative cases and deaths were fairly unevenly distributed across Spanish provinces; ii) Inequality shows a monotonic downward trend throughout the outbreak showing a decrease from the onset to the end ranging from 22% to 49% in cases and between 17% and 42% in deaths; iii) Over 90% of the inequality observed can be attributed to differences between regions, while less than 10% is due to the differences across provinces within regions. Awareness of the existence and nature of the inequality observed in the epidemiological variables is needed to develop successful policies to improve health services in Spain.