Beta-Bernstein Smoothing for Regression Curves with Compact Support

ABSTRACT. The problem of boundary bias is associated with kernel estimation for regression curves with compact support. This paper proposes a simple and uni(r)ed approach for remedying boundary bias in non-parametric regression, without dividing the compact support into interior and boundary areas and without applying explicitly different smoothing treatments separately. The approach uses the beta family of density functions as kernels. The shapes of the kernels vary according to the position where the curve estimate is made. Theyare symmetric at the middle of the support interval, and become more and more asymmetric nearer the boundary points. The kernels never put any weight outside the data support interval, and thus avoid boundary bias. The method is a generalization of classical Bernstein polynomials, one of the earliest methods of statistical smoothing. The proposed estimator has optimal mean integrated squared error at an order of magnitude n ^−4/5, equivalent to that of standard kernel estimators when the curve has an unbounded support.     

Forecasting on British election night 1997

An account is given of methods used to predict the outcome of the 1997 general election from early declared results, for use by the British Broadcasting Corporation (BBC) in its election night television and radio coverage. Particular features of the 1997 election include extensive changes to constituency boundaries, simultaneous local elections in many districts and strong tactical voting. A new technique is developed, designed to eliminate systematic sources of bias such as differential refusal, for incorporating prior information from the BBC's exit poll. The sequence of forecasts generated on election night is displayed, with commentary.     

Search for optimal designs in a three stagenested random model

In this paper we define a class of unbalanced designs, denoted by C_k,s,t, for estimating the components of variance in a k-stage nested random effects linear model. This class contains many of the designs proposed in the literature for nested components of variance models. We focus on the three-state model and discuss the determination of locally optimal designs within this class using a systematic computer search. For large sample sizes we show that approximate optimal designs may be obtained using a limit argument combined with numerical optimization. A comparison of our designs with previously published designs suggests that, in many cases, our designs result in substantial gains in efficiency.     

Understanding exponential smoothing via kernel regression

Exponential smoothing is the most common model-free means of forecasting a future realization of a time series. It requires the specification of a smoothing factor which is usually chosen from the data to minimize the average squared residual of previous one-step-ahead forecasts. In this paper we show that exponential smoothing can be put into a nonparametric regression framework and gain some interesting insights into its performance through this interpretation. We also use theoretical developments from the kernel regression field to derive, for the first time, asymptotic properties of exponential smoothing forecasters.     

Multivariate bandwidth selection for local linear regression

The existence and properties of optimal bandwidths for multivariate local linear regression are established, using either a scalar bandwidth for all regressors or a diagonal bandwidth vector that has a different bandwidth for each regressor. Both involve functionals of the derivatives of the unknown multivariate regression function. Estimating these functionals is difficult primarily because they contain multivariate derivatives. In this paper, an estimator of the multivariate second derivative is obtained via local cubic regression with most cross-terms left out. This estimator has the optimal rate of convergence but is simpler and uses much less computing time than the full local estimator. Using this as a pilot estimator, we obtain plug-in formulae for the optimal bandwidth, both scalar and diagonal, for multivariate local linear regression. As a simpler alternative, we also provide rule-of-thumb bandwidth selectors. All these bandwidths have satisfactory performance in our simulation study.     

多元线性回归分析及其实际应用

本文主要介绍了多元线性回归分析的数学模型.同时结合实例演示了应用R软件实现多元线性回归的过程.     

Discrete–Time Approximations of the Holmström–Milgrom Brownian–Motion Model of Intertemporal Incentive Provision

This paper studies the relation between discrete–time and continuous–time principal–agent models. We derive the continuous–time model as a limit of discrete–time models with ever shorter periods and show that optimal incentive schemes in the discrete–time models approximate the optimal incentive scheme in the continuous model, which is linear in accounts. Under the additional assumption that the principal observes only cumulative total profits at the end and the agent can destroy profits unnoticed, an incentive scheme that is linear in total profits is shown to be approximately optimal in the discrete–time model when the length of the period is small.     

Is There a Curse of Dimensionality for Contraction Fixed Points in the Worst Case?

This paper analyzes the complexity of the contraction fixed point problem: compute an ε‐approximation to the fixed point V*Γ(V*) of a contraction mapping Γ that maps a Banach space B_d of continuous functions of d variables into itself. We focus on quasi linear contractions where Γ is a nonlinear functional of a finite number of conditional expectation operators. This class includes contractive Fredholm integral equations that arise in asset pricing applications and the contractive Bellman equation from dynamic programming. In the absence of further restrictions on the domain of Γ, the quasi linear fixed point problem is subject to the curse of dimensionality, i.e., in the worst case the minimal number of function evaluations and arithmetic operations required to compute an ε‐approximation to a fixed point V*∈B_d increases exponentially in d. We show that the curse of dimensionality disappears if the domain of Γ has additional special structure. We identify a particular type of special structure for which the problem is strongly tractable even in the worst case, i.e., the number of function evaluations and arithmetic operations needed to compute an ε‐approximation of V* is bounded by Cε^−p where C and p are constants independent of d. We present examples of economic problems that have this type of special structure including a class of rational expectations asset pricing problems for which the optimal exponent p1 is nearly achieved.     

Computing Normal Form Perfect Equilibria for Extensive Two‐Person Games

This paper presents an algorithm for computing an equilibrium of an extensive two‐person game with perfect recall. The method is computationally efficient by virtue of using the sequence form, whose size is proportional to the size of the game tree. The equilibrium is traced on a piecewise linear path in the sequence form strategy space from an arbitrary starting vector. If the starting vector represents a pair of completely mixed strategies, then the equilibrium is normal form perfect. Computational experiments compare the sequence form and the reduced normal form, and show that only the sequence form is tractable for larger games.     

Pivotal Statistics for Testing Structural Parameters in Instrumental Variables Regression

We propose a novel statistic for conducting joint tests on all the structural parameters in instrumental variables regression. The statistic is straightforward to compute and equals a quadratic form of the score of the concentrated log–likelihood. It therefore attains its minimal value equal to zero at the maximum likelihood estimator. The statistic has a χ² limiting distribution with a degrees of freedom parameter equal to the number of structural parameters. The limiting distribution does not depend on nuisance parameters. The statistic overcomes the deficiencies of the Anderson–Rubin statistic, whose limiting distribution has a degrees of freedom parameter equal to the number of instruments, and the likelihood based, Wald, likelihood ratio, and Lagrange multiplier statistics, whose limiting distributions depend on nuisance parameters. Size and power comparisons reveal that the statistic is a (asymptotic) size–corrected likelihood ratio statistic. We apply the statistic to the Angrist–Krueger (1991) data and find similar results as in Staiger and Stock (1997).