Markov-switching models with endogenous explanatory variables II: A two-step MLE procedure

This paper proposes a two-step maximum likelihood estimation (MLE) procedure to deal with the problem of endogeneity in Markov-switching regression models. A joint estimation procedure provides us with an asymptotically most efficient estimator, but it is not always feasible, due to the ‘curse of dimensionality’ in the matrix of transition probabilities. A two-step estimation procedure, which ignores potential correlation between the latent state variables, suffers less from the ‘curse of dimensionality’, and it provides a reasonable alternative to the joint estimation procedure. In addition, our Monte Carlo experiments show that the two-step estimation procedure can be more efficient than the joint estimation procedure in finite samples, when there is zero or low correlation between the latent state variables.     

Estimating the structural credit risk model when equity prices are contaminated by trading noises

The transformed-data maximum likelihood estimation (MLE) method for structural credit risk models developed by Duan [Duan, J.-C., 1994. Maximum likelihood estimation using price data of the derivative contract. Mathematical Finance 4, 155–167] is extended to account for the fact that observed equity prices may have been contaminated by trading noises. With the presence of trading noises, the likelihood function based on the observed equity prices can only be evaluated via some nonlinear filtering scheme. We devise a particle filtering algorithm that is practical for conducting the MLE estimation of the structural credit risk model of Merton [Merton, R.C., 1974. On the pricing of corporate debt: The risk structure of interest rates. Journal of Finance 29, 449–470]. We implement the method on the Dow Jones 30 firms and on 100 randomly selected firms, and find that ignoring trading noises can lead to significantly over-estimating the firm’s asset volatility. The estimated magnitude of trading noise is in line with the direction that a firm’s liquidity will predict based on three common liquidity proxies. A simulation study is then conducted to ascertain the performance of the estimation method.     

Likelihood Inference for Gibbs Processes in the Analysis of Spatial Point Patterns*

Several authors have proposed stochastic and non‐stochastic approximations to the maximum likelihood estimate (MLE) for Gibbs point processes in modelling spatial point patterns with pairwise interactions. The approximations are necessary because of the difficulty of evaluating the normalizing constant. In this paper, we first provide a review of methods which yield crude approximations to the MLE. We also review methods based on Markov chain Monte Carlo techniques for which exact MLE has become feasible. We then present a comparative simulation study of the performance of such methods of estimation based on two simulation techniques, the Gibbs sampler and the Metropolis‐Hastings algorithm, carried out for the Strauss model.     

Prediction from ARFIMA models: Comparisons between MLE and semiparametric estimation procedures

This paper considers the effects on multi-step prediction of using semiparametric local Whittle estimators rather than MLE for long memory ARFIMA models. We consider various representations of the minimum MSE predictor with known parameters. We then conduct a detailed simulation study for when the true parameters are replaced with estimates. The predictor based on MLE is found to be superior, in the MSE sense, to the predictor based on the two-step local Whittle estimation. The “optimal” bandwidth local Whittle estimator produces worse predictions than the local Whittle using an agnostic bandwidth of the square root of the sample size.     

带未知异方差广义空间模型的有效估计

空间单元大小以及其它的经济特征上的差异,常常会导致空间异方差问题。本文给出了广义空间模型异方差问题的三种不同估计方法。第一种方法是将异方差形式参数化,来克服自由度的不足,使用ML估计进行实现。而针对异方差形式未知时,分别采用了基于2SLS的迭代GMM估计和更加直接的MCMC抽样方法加以解决,特别是MCMC方法表现得更加优美。蒙特卡罗模拟表明,给定异方差形式条件下, ML估计通过异方差参数化的方法依然可以获得较好的估计效果。而异方差形式未知的情况下,另外两种方法随着样本数的增大时也可以与ML的估计结果趋于一致。     

Likelihood-based estimation in a panel setting: Robustness,redundancy and validity of copulas

This paper considers the estimation of likelihood-based models in a panel setting. That is, we have panel data, and for each time period separately we have a correctly specified model that could be estimated by MLE. We want to allow non-independence over time. This paper shows how to improve on the QMLE. It then considers MLE based on joint distributions constructed using copulas. It discusses the efficiency gain from using the true copula, and shows that knowledge of the true copula is redundant only if the variance matrix of the relevant set of moment conditions is singular. It also discusses the question of robustness against misspecification of the copula, and proposes a test of the validity of the copula. GMM methods are argued to be useful analytically, and also for reasons of efficiency if the copula is robust but not correct.     

Closed-skew normality in stochastic frontiers with individual effects and long/short-run efficiency

This paper considers the estimation of Kumbhakar et al. (J Prod Anal. doi:10.1007/s11123-012-0303-1, 2012) (KLH) four random components stochastic frontier (SF) model using MLE techniques. We derive the log-likelihood function of the model using results from the closed-skew normal distribution. Our Monte Carlo analysis shows that MLE is more efficient and less biased than the multi-step KLH estimator. Moreover, we obtain closed-form expressions for the posterior expected values of the random effects, used to estimate short-run and long-run (in)efficiency as well as random-firm effects. The model is general enough to nest most of the currently used panel SF models; hence, its appropriateness can be tested. This is exemplified by analyzing empirical results from three different applications.     

The compactification of generalized linear models

The main purpose of this paper is to unify and extend the existing theory of 'estimated zeroes' in log-linear and logit models. To this end it is shown that every generalized linear model (GLM) can be embedded in a larger model with a compact parameter space and a continuous likelihood (a 'CGLM'). Clearly in a CGLM the maximum likelihood estimate (MLE) always exists, easing a major data analysis problem. In the mean-value parametrization, the construction of the CGLM is remarkably simple; except in a rather pathological and rare case, the estimated expected values are always finite., In the β-parametrization however, the compactification is more complex; the MLE need not correspond with a finite β, as is well known for estimated zeros in log-linear models. The boundary distributions of CGLMs are classified in four categories: 'Inadmissible', 'degenerate', 'Chentsov', and 'constrained'. For a large class of GLMs, including all GLMs with canonical link functions and probit models, the MLE in the corresponding CGLM exists and is unique. Even stronger, the likelihood has no other local maxima. We give equivalent algebraic and geometric conditions (in the vein of Haberman (1974, 1977) and Albert and Anderson (1984) respectively), necessary for the existence of the MLE in the GLM corresponding to a finite β. For a large class of GLMs these conditions are also sufficient. Even for log-linear models this seams to be a new result.     

Least squares, preliminary test and Stein-type estimation in general vector AR(p) models

In the general vector autoregressive process AR ( p ), multivariate least square estimation (LSE)/maximum likelihood estimation (MLE) of a subset of the parameters is considered when the complementary subset is suspected to be redundant. This may be viewed as a special case of linear constraints of autoregressive parameters. We incorporate this nonsample information in the estimation process and propose preliminary test and Stein-type estimators for the target subset of parameters. Under local alternatives their asymptotic properties are investigated and compared with those of unrestricted and restricted LSE. The dominance picture of the estimators is presented.     

Estimation of nonlinear models with mismeasured regressors using marginal information

We consider the estimation of nonlinear models with mismeasured explanatory variables, when information on the marginal distribution of the true values of these variables is available. We derive a semi‐parametric MLE that is shown to be $\sqrt{n}$ consistent and asymptotically normally distributed. In a simulation experiment we find that the finite sample distribution of the estimator is close to the asymptotic approximation. The semi‐parametric MLE is applied to a duration model for AFDC welfare spells with misreported welfare benefits. The marginal distribution of the correctly measured welfare benefits is obtained from an administrative source. Copyright © 2010 John Wiley & Sons, Ltd.     

GMM estimation of spatial autoregressive models with moving average disturbances

In this paper, we introduce the one-step generalized method of moments (GMM) estimation methods considered in Lee (2007a) and Liu, Lee, and Bollinger (2010) to spatial models that impose a spatial moving average process for the disturbance term. First, we determine the set of best linear and quadratic moment functions for GMM estimation. Second, we show that the optimal GMM estimator (GMME) formulated from this set is the most efficient estimator within the class of GMMEs formulated from the set of linear and quadratic moment functions. Our analytical results show that the one-step GMME can be more efficient than the quasi maximum likelihood (QMLE), when the disturbance term is simply i.i.d. With an extensive Monte Carlo study, we compare its finite sample properties against the MLE, the QMLE and the estimators suggested in Fingleton (2008a).     

Bayesian Inference in Spatial Sample Selection Models

In this study, we consider Bayesian methods for the estimation of a sample selection model with spatially correlated disturbance terms. We design a set of Markov chain Monte Carlo algorithms based on the method of data augmentation. The natural parameterization for the covariance structure of our model involves an unidentified parameter that complicates posterior analysis. The unidentified parameter – the variance of the disturbance term in the selection equation – is handled in different ways in these algorithms to achieve identification for other parameters. The Bayesian estimator based on these algorithms can account for the selection bias and the full covariance structure implied by the spatial correlation. We illustrate the implementation of these algorithms through a simulation study and an empirical application.     

Almost unbiased estimation in multiplicative models

In the lognormal linear models the estimation of constant term presents problems. In this paper we use weighted jackknife procedure (suggested by Hinkley 1977) for reducing the bias of the maximum likelihood estimator. The resulting estimator is unbiased upto order (1/T),T being the number of observations, and has the same MSE as that of the MLE to the same order of approximation; moreover, being the jackknife estimator it enjoys all the desirable large sample properties like any other jackknife estimator. The research of this author is partially supported through a research grant from NSERC of Canada.     

Mixed Causal–Noncausal Autoregressions: Bimodality Issues in Estimation and Unit Root Testing1

This paper stresses the bimodality of the likelihood function of the Mixed causal–noncausal AutoRegressions (MAR), and it is shown that the bimodality issue becomes more salient as the causal root approaches unity from below. The consequences are important as the roots of the local maxima are typically interchanged, attributing the noncausal component to the causal one and vice-versa. This severely changes the interpretation of the results, and the properties of unit root tests of the backward root are adversely affected. To circumvent the bimodality issue, this paper proposes an estimation strategy which (i) increases noticeably the probability of attaining the global MLE; and (ii) selects carefully the maximum used for the unit root test against a MAR stationary alternative.     

Mixed Causal–Noncausal Autoregressions: Bimodality Issues in Estimation and Unit Root Testing1

This paper stresses the bimodality of the likelihood function of the Mixed causal–noncausal AutoRegressions (MAR), and it is shown that the bimodality issue becomes more salient as the causal root approaches unity from below. The consequences are important as the roots of the local maxima are typically interchanged, attributing the noncausal component to the causal one and vice‐versa. This severely changes the interpretation of the results, and the properties of unit root tests of the backward root are adversely affected. To circumvent the bimodality issue, this paper proposes an estimation strategy which (i) increases noticeably the probability of attaining the global MLE; and (ii) selects carefully the maximum used for the unit root test against a MAR stationary alternative.     

有效价差的极大似然估计

有效价差是刻画金融资产交易成本的一种重要度量。本文基于Roll的价格模型,利用对数价格极差分布的近似正态特征,提出了一种有效价差的近似极大似然估计,并通过数值模拟比较了这一新的估计与以往文献中提出的Roll的协方差估计、贝叶斯估计以及High-Low估计在各种不同状况下的精度。模拟的结果表明,无论是在连续交易的理想状态还是交易不连续且价格不能被完全观测到的非理想状态下,极大似然估计和High-Low估计的精度均高于协方差和贝叶斯估计;当波动率相对较小的时候,极大似然估计的精度优于High-Low估计;另外,在非理想情形下,极大似然估计要比High-Low估计更加稳健。     

Parametric links for binary choice models: A Fisherian–Bayesian colloquy

The familiar logit and probit models provide convenient settings for many binary response applications, but a larger class of link functions may be occasionally desirable. Two parametric families of link functions are investigated: the Gosset link based on the Student t latent variable model with the degrees of freedom parameter controlling the tail behavior, and the Pregibon link based on the (generalized) Tukey λ family, with two shape parameters controlling skewness and tail behavior. Both Bayesian and maximum likelihood methods for estimation and inference are explored, compared and contrasted. In applications, like the propensity score matching problem discussed below, where it is critical to have accurate estimates of the conditional probabilities, we find that misspecification of the link function can create serious bias. Bayesian point estimation via MCMC performs quite competitively with MLE methods; however nominal coverage of Bayes credible regions is somewhat more problematic.     

Consistent likelihood-based estimation of a star-shaped distribution

We consider the classical problem of nonparametrically estimating a star-shaped distribution, i.e., a distribution function F on [0,∞) with the property that F(u)/u is nondecreasing on the set {u : F(u) < 1}. This problem is intriguing because of the fact that a well defined maximum likelihood estimator (MLE) exists, but this MLE is inconsistent. In this paper, we argue that the likelihood that is commonly used in this context is somewhat unnatural and propose another, so called ‘smoothed likelihood’. However, also the resulting MLE turns out to be inconsistent. We show that more serious smoothing of the likelihood yields consistent estimators in this model.     

A general asymptotic theory for time-series models

This paper develops a general asymptotic theory for the estimation of strictly stationary and ergodic time–series models. Under simple conditions that are straightforward to check, we establish the strong consistency, the rate of strong convergence and the asymptotic normality of a general class of estimators that includes LSE, MLE and some M-type estimators. As an application, we verify the assumptions for the long-memory fractional ARIMA model. Other examples include the GARCH(1,1) model, random coefficient AR(1) model and the threshold MA(1) model.     

Small‐scale Inference: Empirical Bayes and Confidence Methods for as Few as a Single Comparison

Empirical Bayes methods of estimating the local false discovery rate (LFDR) by maximum likelihood estimation (MLE), originally developed for large numbers of comparisons, are applied to a single comparison. Specifically, when assuming a lower bound on the mixing proportion of true null hypotheses, the LFDR MLE can yield reliable hypothesis tests and confidence intervals given as few as one comparison. Simulations indicate that constrained LFDR MLEs perform markedly better than conventional methods, both in testing and in confidence intervals, for high values of the mixing proportion, but not for low values. (A decision‐theoretic interpretation of the confidence distribution made those comparisons possible.) In conclusion, the constrained LFDR estimators and the resulting effect‐size interval estimates are not only effective multiple comparison procedures but also they might replace p‐values and confidence intervals more generally. The new methodology is illustrated with the analysis of proteomics data.