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We propose a solution to the measurement error problem that plagues the estimation of the relation between the expected return of the stock market and its conditional variance due to the latency of these conditional moments. We use intra-period returns to construct a nonparametric proxy for the latent conditional variance in the first step which is subsequently used as an input in the second step to estimate the parameters characterizing the risk-return tradeoff via a GMM approach. We propose a bias-correction to the standard GMM estimator derived under a double asymptotic framework, wherein the number of intra-period returns, N, as well as the number of low frequency time periods, T , simultaneously go to infinity. Simulation exercises show that the bias-correction is particularly relevant for small values of N which is the case in empirically realistic scenarios. The methodology lends itself to additional applications, such as the empirical evaluation of factor models, wherein the factor betas may be estimated using intra-period returns and the unexplained returns or alphas subsequently recovered at lower frequencies.

We consider a Kronecker product structure for large covariance matrices, which has the feature that the number of free parameters increases logarithmically with the dimensions of the matrix. We propose an estimation method of the free parameters based on the log linear property of this structure, and also a Quasi-Likelihood method. We establish the rate of convergence of the estimated parameters when the size of the matrix diverges. We also establish a CLT for our method. We apply the method to portfolio choice for S&P500 daily returns and compare with sample covariance based methods and with the recent Fan et al. (2013) method.

We consider a class of nonparametric time series regression models in which the regressor takes values in a sequence space and the data are stationary and weakly dependent. We propose an infinite dimensional Nadaraya-Watson type estimator with a bandwidth sequence that shrinks the effects of long lags. We investigate its asymptotic properties in detail under both static and dynamic regressions contexts. First we show pointwise consistency of the estimator under a set of mild regularity conditions. We establish a CLT for the estimator at a point under stronger conditions as well as for a feasibly studentized version of the estimator, thereby allowing pointwise inference to be conducted. We establish the uniform consistency over a compact set of logarithmically increasing dimension. We specify the explicit rates of convergence in terms of the Lambert W function, and show that the optimal rate that balances the squared bias and variance is of logarithmic order, the precise rate depending on the smoothness of the regression function and the dependence of the data in a non-trivial way.

We propose a Kronecker product structure for large covariance or correlation matrices. One feature of this model is that it scales logarithmically with dimension in the sense that the number of free parameters increases logarithmically with the dimension of the matrix. We propose an estimation method of the parameters based on a log-linear property of the structure, and also a quasi-maximum likelihood estimation (QMLE) method. We establish the rate of convergence of the estimated parameters when the size of the matrix diverges. We also establish a central limit theorem (CLT) for our method. We derive the asymptotic distributions of the estimators of the parameters of the spectral distribution of the Kronecker product correlation matrix, of the extreme logarithmic eigenvalues of this matrix, and of the variance of the minimum variance portfolio formed using this matrix. We also develop tools of inference including a test for over-identification. We apply our methods to portfolio choice for S&P500 daily returns and compare with sample covariance-based methods and with the recent Fan, Liao, and Mincheva (2013) method.

Estimation of heteroskedasticity and autocorrelation consistent covariance matrices (HACs) is a well established problem in time series. Results have been established under a variety of weak conditions on temporal dependence and heterogeneity that allow one to conduct inference on a variety of statistics, see Newey and West (1987), Hansen (1992), de Jong and Davidson (2000), and Robinson (2004). Indeed there is an extensive literature on automating these procedures starting with Andrews (1991). Alternative methods for conducting inference include the bootstrap for which there is also now a very active research program in time series especially, see Lahiri (2003) for an overview. One convenient method for time series is the subsampling approach of Politis, Romano, andWolf (1999). This method was used by Linton, Maasoumi, andWhang (2003) (henceforth LMW) in the context of testing for stochastic dominance. This paper is concerned with the practical problem of conducting inference in a vector time series setting when the data is unbalanced or incomplete. In this case, one can work only with the common sample, to which a standard HAC/bootstrap theory applies, but at the expense of throwing away data and perhaps losing effciency. An alternative is to use some sort of imputation method, but this requires additional modelling assumptions, which we would rather avoid.1 We show how the sampling theory changes and how to modify the resampling algorithms to accommodate the problem of missing data. We also discuss effciency and power. Unbalanced data of the type we consider are quite common in financial panel data, see for example Connor and Korajczyk (1993). These data also occur in cross-country studies.

We develop inference tools in a semiparametric partially linear regression model with missing response data. A class of estimators is defined that includes as special cases: a semiparametric regression imputation estimator, a marginal average estimator and a (marginal) propensity score weighted estimator. We show that any of our class of estimators is asymptotically normal. The three special estimators have the same asymptotic variance. They achieve the semiparametric efficiency bound in the homoskedastic Gaussian case. We show that the Jackknife method can be used to consistently estimate the asymptotic variance. Our model and estimators are defined with a view to avoid the curse of dimensionality, that severely limits the applicability of existing methods. The empirical likelihood method is developed. It is shown that when missing responses are imputed using the semiparametric regression method the empirical log-likelihood is asymptotically a scaled chi-square variable. An adjusted empirical log-likelihood ratio, which is asymptotically standard chi-square, is obtained. Also, a bootstrap empirical log-likelihood ratio is derived and its distribution is used to approximate that of the imputed empirical log-likelihood ratio. A simulation study is conducted to compare the adjusted and bootstrap empirical likelihood with the normal approximation based method in terms of coverage accuracies and average lengths of confidence intervals. Based on biases and standard errors, a comparison is also made by simulation between the proposed estimators and the related estimators.

For vectors x and w, let r(x,w) be a function that can be nonparametrically estimated consistently and asymptotically normally. We provide consistent, asymptotically normal estimators for the functions g and h, where r(x,w) = h[g(x),w], g is linearly homogeneous and h is monotonic in g. This framework encompasses homothetic and homothetically separable functions. Such models reduce the curse of dimensionality, provide a natural generalization of linear index models, and are widely used in utility, production, and cost function applications. Extensions to related functional forms include a generalized partly linear model with unknown link function. We provide simulation evidence on the small sample performance of our estimator, and we apply our method to a Chinese production dataset.

We propose a procedure for estimating the critical values of the extended Kolmogorov- Smirnov tests of First and Second Order Stochastic Dominance in the general K-prospect case. We allow for the observations to be serially dependent and, for the first time, we can accommodate general dependence amongst the prospects which are to be ranked. Also, the prospects may be the residuals from certain conditional models, opening the way for conditional ranking. We also propose a test of Prospect Stochastic Dominance. Our method is based on subsampling and we show that the resulting tests are consistent and powerful against some N-1=2 local alternatives. We also propose some heuristic methods for selecting subsample size and demonstrate in simulations that they perform reasonably.

We propose a new estimator for nonparametric regression based on local likelihood estimation using an estimated error score function obtained from the residuals of a preliminary nonparametric regression. We show that our estimator is asymptotically equivalent to the infeasible local maximum likelihood estimator [Staniswalis (1989)], and hence improves on standard kernel estimators when the error distribution is not normal. We investigate the finite sample performance of our procedure on simulated data.

We introduce a nonparametric smoothing procedure for nonparametric factor analaysis of multivariate time series. The asymptotic properties of the proposed procedures are derived. We present an application based on the residuals from the Fair macromodel.