Estimation methods for stochastic volatility models: a survey

Abstract.  Although stochastic volatility (SV) models have an intuitive appeal, their empirical application has been limited mainly due to difficulties involved in their estimation. The main problem is that the likelihood function is hard to evaluate. However, recently, several new estimation methods have been introduced and the literature on SV models has grown substantially. In this article, we review this literature. We describe the main estimators of the parameters and the underlying volatilities focusing on their advantages and limitations both from the theoretical and empirical point of view. We complete the survey with an application of the most important procedures to the S&P 500 stock price index.     

The British low-wage sector and the employment prospects of the unemployed

Are low wages a way for the unemployed to switch to higher-paying jobs? Using data from the British Household Panel Survey, the labour market dynamics of unemployed, low-paid and higher-paid employed men are analysed. Moreover, the respective (un)employment duration and occupational skill level are accounted for. Results show that in general low wages significantly reduce the risk of future unemployment and increase the chances of ascending the salary ladder, especially in the case of long-term unemployment (>360 days). Furthermore, the occupational skill level has a substantial influence on the upward mobility of low-paid jobs: individuals working in the initial period in a low-paid and higher-skilled occupation have on average an 11 percentage points higher probability of entering higher pay compared to when working in a low-paid and low-skilled occupation.     

Modelling of repairable systems with various degrees of repair

The availability of a stochastic repairable system depends on the failure behaviour and on repair strategies. In this paper, we deal with a general repair model for a system using auxiliary counting processes and corresponding intensities which include various degrees of repair (between minimal repair and perfect repair). For determining the model parameters we need estimators depending on failure times and repair times: maximum likelihood (ML) estimator and Bayes estimators are considered. Special results are obtained by the use of Weibull-type intensities and random observation times.     

ML– and semiparametric estimation in logistic models with incomplete covariate data

ML–estimation of regression parameters with incomplete covariate information usually requires a distributional assumption regarding the concerned covariates that implies a source of misspecification. Semiparametric procedures avoid such assumptions at the expense of efficiency. In this paper a simulation study with small sample size is carried out to get an idea of the performance of the ML–estimator under misspecification and to compare it with the semiparametric procedures when the former is based on a correct assumption. The results show that there is only a little gain by correct parametric assumptions, which does not justify the possibly large bias when the assumptions are not met. Additionally, a simple modification of the complete case estimator appears to be nearly semiparametric efficient.     

Modified efficient importance sampling for partially non‐Gaussian state space models

The construction of an importance density for partially non‐Gaussian state space models is crucial when simulation methods are used for likelihood evaluation, signal extraction, and forecasting. The method of efficient importance sampling is successful in this respect, but we show that it can be implemented in a computationally more efficient manner using standard Kalman filter and smoothing methods. Efficient importance sampling is generally applicable for a wide range of models, but it is typically a custom‐built procedure. For the class of partially non‐Gaussian state space models, we present a general method for efficient importance sampling. Our novel method makes the efficient importance sampling methodology more accessible because it does not require the computation of a (possibly) complicated density kernel that needs to be tracked for each time period. The new method is illustrated for a stochastic volatility model with a Student's t distribution.     

Numerical Maximisation of Likelihood: A Neglected Alternative to EM?

There is by now a long tradition of using the EM algorithm to find maximum‐likelihood estimates (MLEs) when the data are incomplete in any of a wide range of ways, even when the observed‐data likelihood can easily be evaluated and numerical maximisation of that likelihood is available as a conceptually simple route to the MLEs. It is rare in the literature to see numerical maximisation employed if EM is possible. But with excellent general‐purpose numerical optimisers now available free, there is no longer any reason, as a matter of course, to avoid direct numerical maximisation of likelihood. In this tutorial, I present seven examples of models in which numerical maximisation of likelihood appears to have some advantages over the use of EM as a route to MLEs. The mathematical and coding effort is minimal, as there is no need to derive and code the E and M steps, only a likelihood evaluator. In all the examples, the unconstrained optimiser nlm available in R is used, and transformations are used to impose constraints on parameters. I suggest therefore that the following question be asked of proposed new applications of EM: Can the MLEs be found more simply and directly by using a general‐purpose numerical optimiser?     

RIDGE ESTIMATORS FOR PROBIT REGRESSION: WITH AN APPLICATION TO LABOUR MARKET DATA

In this paper we propose ridge regression estimators for probit models since the commonly applied maximum likelihood (ML) method is sensitive to multicollinearity. An extensive Monte Carlo study is conducted where the performance of the ML method and the probit ridge regression (PRR) is investigated when the data are collinear. In the simulation study we evaluate a number of methods of estimating the ridge parameter k that have recently been developed for use in linear regression analysis. The results from the simulation study show that there is at least one group of the estimators of k that regularly has a lower mean squared error than the ML method for all different situations that have been evaluated. Finally, we show the benefit of the new method using the classical Dehejia and Wahba dataset which is based on a labour market experiment.     

A Tutorial on Palm Distributions for Spatial Point Processes

This tutorial provides an introduction to Palm distributions for spatial point processes. Initially, in the context of finite point processes, we give an explicit definition of Palm distributions in terms of their density functions. Then we review Palm distributions in the general case. Finally, we discuss some examples of Palm distributions for specific models and some applications.     

Celebrity endorsements: Investigating the interactive effects of internalization,identification and product type on consumers’ attitudes and intentions

In line with Kang and Herr's work (2006), this research questions the ELM's contention that celebrity credibility serves foremost as a peripheral element in a persuasive communication context. Nevertheless, in a different light to the Kang and Herr's ‘either-central-or-peripheral-role’ of a source characteristic, this research advances that celebrity credibility plays concomitantly central and peripheral roles in a persuasive message context depending on product involvement and brand-purchase motive. Particularly, this research uses the notions of ‘source internalization’ and ‘source identification’ (Kelman, 1961) to theorize that source credibility can have a concomitant dual role (peripheral and central) in a persuasion context. More precisely, this research investigates the interactive effects of source identification and internalization with product involvement as well as brand-purchase motives on consumers' attitudes and intentions. Source internalization is predicted to have persuasive effects in the contexts of high-involvement as well as informational products. Source identification is predicted to have persuasive effects in the context of low-involvement as well as transformational products. The findings of two experiments show that celebrity credibility acts through only a single route (i.e., only internalization has persuasive effects) uniformly across different product involvement levels and brand-purchase motive types. We interpret these results with the lens of the ‘match-up’ hypothesis (Kamins, 1990).     

Estimating the gravity model when zero trade flows are frequent and economically determined

ABSTRACT

Since the 1950s, we have known that the presence of zero-valued dependent variables can seriously bias econometric estimates whether the zeros are included or excluded. Yet the widely-used gravity model is frequently estimated on samples that include large fractions of zeros. An influential paper by Santos Silva and Tenreyro – based on simulations that include no economically-determined zeros – concludes that the bias problems resulting from zeros and those resulting from heteroscedasticity and nonlinearity can be solved using the Poisson Pseudo-Maximum-Likelihood (PPML) model including the zero values. This paper begins by adapting the Santos Silva and Tenreyro experimental design to include economically-determined zeros to see whether this conclusion continues to hold. With this design, it finds that alternative estimators have lower bias than PPML. Changing to a Monte Carlo design that replicates the much-higher real-world frequency of predicted values near zero restores the finding of lower bias with the PPML estimator. The results highlight the need for very careful design of Monte Carlo experiments when evaluating alternative estimators of the gravity model.