Efficient Estimation with Non-Standard Sampling or Missing Endogenous Variables, and Conditional Density Modelling with Unobserved Copula-Connected Shocks

In Chapter 1, it is shown how to use a smoothed empirical likelihood approach to conduct efficient semi-parametric inference in models characterised as conditional moment equalities when data are collected by variable probability sampling.
Results from a simulation experiment suggest that the smoothed-empirical-likelihood-based estimator can estimate the model parameters very well in small to moderately sized stratified samples.

In Chapter 2, a novel univariate conditional density model is proposed to decompose asset returns into a sum of copula-connected unobserved ‘good’ and ‘bad’ shocks. The novelty of this approach comes from two factors: correlation between unobserved shocks is modelled explicitly, and the presence of copula-connected discrete jumps is allowed for.
The proposed framework is very flexible and subsumes other models, such as ‘bad environments, good environments’. The proposed model shows certain hidden characteristics of returns, explains investors’ behaviour in greater detail, and yields better forecasts of risk measures. The in-sample and out-of-sample performance of the proposed model is better than that of 40 popular GARCH variants. A Monte Carlo simulation shows that the proposed model recovers the structural parameters of the unobserved dynamics. This model is estimated on S&P 500 data, and time-dependent non-negative covariance between ‘good’ and ‘bad’ shocks with a leverage-like effect is found to be an essential component of the total variance. Asymmetric reaction to shocks is present almost in all characteristics of returns. The conditional distribution of returns seems to be very time-dependent with skewness both in the centre and tails. Continuous shocks are more important than discrete jumps for return modelling, at least at the daily frequency.

In Chapter 3, the semi-parametric efficiency bound is derived for estimating finite-dimensional parameters identified via a system of conditional moment equalities when at least one of the endogenous variables (which can either be endogenous outcomes, or endogenous explanatory variables, or both) is missing for some individuals in the sample.
An interesting result is obtained that if there are no endogenous variables that are not missing, i.e. all the endogenous variables in the model are missing, then estimation using only the validation subsample (the sub-sample of observations for which the endogenous variables are non-missing) is asymptotically efficient.
An estimator based on the full sample is proposed, and it is shown that it achieves the semi-parametric efficiency bound.
A simulation study reveals that the proposed estimator can work well in medium-sized samples and that the resulting efficiency gains (measured as the ratio of the variance of an efficient estimator based on the validation sample and the variance of our estimator) are comparable with the maximum gain the simulation design can deliver.