The Existence and Asymptotic Properties of a Backfitting Projection Algorithm Under Weak Conditions

We derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function. The procedure employs a recent projection interpretation of popular kernel estimators provided by Mammen et al. (1997), and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in Bickel et al. (1995). Our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components, and in this sense improves on the method analyzed in Opsomer and Ruppert (1997). We provide 'high level' conditions independent of the sampling scheme. We then verify that these conditions are satisfied in a time series autoregression under weak conditions.Additive models, alternating projections, backfitting, kernel smoothing, local polynomials, nonparametric regression

The existence and asymptotic properties of a backfitting projection algorithm under weak conditions.

We derive the asymptotic distribution of a new backfitting procedure for estimating the closest additive approximation to a nonparametric regression function. The procedure employs a recent projection interpretation of popular kernel estimators provided by Mammen, Marron, Turlach and Wand and the asymptotic theory of our estimators is derived using the theory of additive projections reviewed in Bickel, Klaassen, Ritov and Wellner. Our procedure achieves the same bias and variance as the oracle estimator based on knowing the other components, and in this sense improves on the method analyzed in Opsomer and Ruppert. We provide ‘‘high level’’ conditions independent of the sampling scheme. We then verify that these conditions are satisfied in a regression and a time series autoregression under weak conditions.

A comparison of in-sample forecasting methods

In-sample forecasting is a recent continuous modification of well-known forecasting methods based on aggregated data. These aggregated methods are known as age-cohort methods in demography, economics, epidemiology and sociology and as chain ladder in non-life insurance. Data is organized in a two-way table with age and cohort as indices, but without measures of exposure. It has recently been established that such structured forecasting methods based on aggregated data can be interpreted as structured histogram estimators. Continuous in-sample forecasting transfers these classical forecasting models into a modern statistical world including smoothing methodology that is more efficient than smoothing via histograms. All in-sample forecasting estimators are collected and their performance is compared via a finite sample simulation study. All methods are extended via multiplicative bias correction. Asymptotic theory is being developed for the histogram-type method of sieves and for the multiplicatively corrected estimators. The multiplicative bias corrected estimators improve all other known in-sample forecasters in the simulation study. The density projection approach seems to have the best performance with forecasting based on survival densities being the runner-up

Estimating Yield Curves by Kernel Smoothing Methods

We introduce a new method for the estimation of discount functions, yield curves and forward curves for coupon bonds. Our approach is nonparametric and does not assume a particular functional form for the discount function although we do show how to impose various important restrictions in the estimation. Our method is based on kernel smoothing and is defined as the minimum of some localized population moment condition. The solution to the sample problem is not explicit and our estimation procedure is iterative, rather like the backfitting method of estimating additive nonparametric models. We establish the asymptotic normality of our methods using the asymptotic representation of our estimator as an infinite series with declining coefficients. The rate of convergence is standard for one dimensional nonparametric regression.Coupon bonds; forward curve; Hilbert space; local linear; nonparametric regression; yield curve

More Efficient Kernel Estimation in Nonparametric Regression with Autocorrelated Errors

We propose a modification of kernel time series regression estimators that improves efficiency when the innovation process is autocorrelated. The procedure is based on a pre-whitening transformation of the dependent variable that has to be estimated from the data. We establish the asymptotic distribution of our estimator under weak dependence conditions. It is shown that the proposed estimation procedure is more efficient than the conventional kernel method. We also provide simulation evidence to suggest that gains can be achieved in moderate sized samples.Backfitting, efficiency, kernel estimation, time series.

Generalised additive dependency inflated models including aggregated covariates

Let us assume that X, Y and U are observed and that the conditional mean of U given X and Y can be expressed via an additive dependency of X, λ(X)Y and X + Y for some unspecified function . This structured regression model can be transferred to a hazard model or a density model when applied on some appropriate grid, and has important forecasting applications via structured marker dependent hazards models or structured density models including age-period-cohort relationships. The structured regression model is also important when the severity of the dependent variable has a complicated dependency on waiting times X, Y and the total waiting time X+Y . In case the conditional mean of U approximates a density, the regression model can be used to analyse the age-period-cohort model, also when exposure data are not available. In case the conditional mean of U approximates a marker dependent hazard, the regression model introduces new relevant age-period-cohort time scale interdependencies in understanding longevity. A direct use of the regression relationship introduced in this paper is the estimation of the severity of outstanding liabilities in non-life insurance companies. The technical approach taken is to use B-splines to capture the underlying one-dimensional unspecified functions. It is shown via finite sample simulation studies and an application for forecasting future asbestos related deaths in the UK that the B-spline approach works well in practice. Special consideration has been given to ensure identifiability of all models considered

Yield Curve Estimation by Kernel Smoothing Methods

We introduce a new method for the estimation of discount functions, yield curves and forward curves from government issued coupon bonds. Our approach is nonparametric and does not assume a particular functional form for the discount function although we do show how to impose various restrictions in the estimation. Our method is based on kernel smoothing and is defined as the minimum of some localized population moment condition. The solution to the sample problem is not explicit and our estimation procedure is iterative, rather like the backfitting method of estimating additive nonparametric models. We establish the asymptotic normality of our methods using the asymptotic representation of our estimator as an infinite series with declining coefficients. The rate of convergence is standard for one dimensional nonparametric regression. We investigate the finite sample performance of our method, in comparison with other well-established methods, in a small simulation experiment.

Generalized linear time series regression

We consider a cross-section model that contains an individual component, a deterministic time trend and an unobserved latent common time series component. We show the following oracle property: the parameters of the latent time series and the parameters of the deterministic time trend can be estimated with the same asymptotic accuracy as if the parameters of the individual component were known. We consider this model in two settings: least squares fits of linear specifications of the individual component and the parameters of the deterministic time trend and, more generally, quasilikelihood estimation in a generalized linear time series model

Nonparametric instrumental variable methods for dynamic treatment evaluation

We develop a nonparametric instrumental variable approach for the estimation of average treatment effects on hazard rates and conditional survival probabilities, without model structure.We derive constructive identification proofs for average treatment effects under noncompliance and dynamic selection, exploiting instrumental variation taking place during ongoing spells. We derive asymptotic distributions of the corresponding estimators. This includes a detailed examination of noncompliance in a dynamic context. In an empirical application, we evaluate the French labor market policy reform PARE which abolished the dependence of unemployment insurance benefits on the elapsed unemployment duration and simultaneously introduced additional active labor market policy measures. The estimated effect of the reform on the survival function of the duration of unemployment duration is positive and significant. Neglecting selectivity leads to an underestimation of the effects in absolute terms

In-Sample Forecasting Applied to Reserving and Mesothelioma Mortality

This paper shows that recent published mortality projections with unobserved exposure can be understood as structured density estimation. The structured density is only observed on a sub-sample corresponding to historical calendar time. The mortality forecast is obtained by extrapolating the structured density to future calendar times using that the components of the density are identified within sample. The new method is illustrated on the important practical problem of forecasting mesothelioma for the UK population. Full asymptotic theory is provided. The theory is given in such generality that it also introduces mathematical statistical theory for the recent continuous chain ladder model. This allows a modern approach to classical reserving techniques used every day in any non-life insurance company around the globe. Applications to mortality data and non-life insurance data are provided along with relevant small sample simulation studies