MCMC maximum likelihood for latent state models

This paper develops a pure simulation-based approach for computing maximum likelihood estimates in latent state variable models using Markov Chain Monte Carlo methods (MCMC). Our MCMC algorithm simultaneously evaluates and optimizes the likelihood function without resorting to gradient methods. The approach relies on data augmentation, with insights similar to simulated annealing and evolutionary Monte Carlo algorithms. We prove a limit theorem in the degree of data augmentation and use this to provide standard errors and convergence diagnostics. The resulting estimator inherits the sampling asymptotic properties of maximum likelihood. We demonstrate the approach on two latent state models central to financial econometrics: a stochastic volatility and a multivariate jump-diffusion models. We find that convergence to the MLE is fast, requiring only a small degree of augmentation.     

Efficient estimation of probit models with correlated errors

Maximum Likelihood (ML) estimation of probit models with correlated errors typically requires high-dimensional truncated integration. Prominent examples of such models are multinomial probit models and binomial panel probit models with serially correlated errors. In this paper we propose to use a generic procedure known as Efficient Importance Sampling (EIS) for the evaluation of likelihood functions for probit models with correlated errors. Our proposed EIS algorithm covers the standard GHK probability simulator as a special case. We perform a set of Monte Carlo experiments in order to illustrate the relative performance of both procedures for the estimation of a multinomial multiperiod probit model. Our results indicate substantial numerical efficiency gains for ML estimates based on the GHK–EIS procedure relative to those obtained by using the GHK procedure.     

Simulation-based estimation of models with lagged latent variables

We extend here our earlier work (Laroque-Salanié, 1989) and propose a dynamic simulated pseudo-maximum likelihood method to deal with a very general class of dynamic non-linear models, including models with lagged latent variables. We test this method on Monte Carlo-generated data for a canonical disequilibrium model. It appears to provide very satisfactory estimates at little computational cost. However, accurate estimation of the standard errors of the estimates may require some care in non-differentiable models.     

Determinants and Dynamics of Current Account Reversals: An Empirical Analysis

We use panel probit models with unobserved heterogeneity, state dependence and serially correlated errors in order to analyse the determinants and the dynamics of current account reversals for a panel of developing and emerging countries. The likelihood‐based inference of these models requires high‐dimensional integration for which we use efficient importance sampling. Our results suggest that current account balance, terms of trades, foreign reserves and concessional debt are important determinants of current account reversal. Furthermore, we find strong evidence for serial dependence in the occurrence of reversals. While the likelihood criterion suggest that state dependence and serially correlated errors are essentially observationally equivalent, measures of predictive performance provide support for the hypothesis that the serial dependence is mainly due to serially correlated country‐specific shocks related to local political or macroeconomic events.     

Full maximum likelihood estimation of second- order autoregressive error models

This paper develops an exact maximum likelihood technique for estimating linear models with second-order autoregressive errors, which utilizes the full set of observations, and explicitly constrains the estimates of the error process to satisfy a priori stationarity conditions. A non- linear solution technique which is new to econometrics and works very efficiently is put forward as part of the estimating procedure. Empirical results are presented which emphasize the importance of utilizing the full set of observations and the associated stationarity restrictions.     

Alternative estimators for factor garch models—A monte carlo comparison

This paper proposes four estimators for factor GARCH models: two-stage univariate GARCH (2SUE), two-stage quasi-maximum likelihood (2SML), quasi-maximum likelihood with known factor weights (RMLE), quasi-maximum likelihood with unknown factor weights (MLE). A Monte-Carlo study is designed for bivariate one-factor GARCH models to examine the finite sample properties. Results are presented for biases, ratios of standard errors to standard deviations, ratios of variances, coverage of confidence intervals, effects of misspecified factor weights, and finite sample properties of the 2SUE for factor GARCH-in-mean models.     

Some remarks on measurement errors and the identification of panel data models*

We demonstrate that despite the common worry about the possible correlations between the unobserved individual effects and the explanatory variables in panel data models the likelihood approach can provide a unified framework towards the study of the identification of a panel data model subject to measurement errors. In fact, it can also serve as a basis for deriving efficient estimation methods.     

Multimodality in GARCH regression models

It is shown empirically that mixed autoregressive moving average regression models with generalized autoregressive conditional heteroskedasticity (Reg-ARMA-GARCH models) can have multimodality in the likelihood that is caused by a dummy variable in the conditional mean. Maximum likelihood estimates at the local and global modes are investigated and turn out to be qualitatively different, leading to different model-based forecast intervals. In the simpler GARCH(p,q) regression model, we derive analytical conditions for bimodality of the corresponding likelihood. In that case, the likelihood is symmetrical around a local minimum. We propose a solution to avoid this bimodality.     

Joint tests for regularity and autocorrelation in allocation systems

In the context of allocation models with vector autoregressive errors we propose a convenient procedure, based on the Lagrange multiplier principle, for testing any possible combination of absence of serial correlation, homogeneity, and symmetry against any possible alternative which specifies autocorrelation of an arbitrary given order. We also derive generic expressions for the maximum likelihood estimation of the models under six possible combinations of constraints. The methodology is illustrated with the Rotterdam model and the differential AIDS model, both estimated from the same quarterly British data.     

Robust prediction of domain compositions from uncertain data using isometric logratio transformations in a penalized multivariate Fay–Herriot model

Assessing regional population compositions is an important task in many research fields. Small area estimation with generalized linear mixed models marks a powerful tool for this purpose. However, the method has limitations in practice. When the data are subject to measurement errors, small area models produce inefficient or biased results since they cannot account for data uncertainty. This is particularly problematic for composition prediction, since generalized linear mixed models often rely on approximate likelihood inference. Obtained predictions are not reliable. We propose a robust multivariate Fay–Herriot model to solve these issues. It combines compositional data analysis with robust optimization theory. The nonlinear estimation of compositions is restated as a linear problem through isometric logratio transformations. Robust model parameter estimation is performed via penalized maximum likelihood. A robust best predictor is derived. Simulations are conducted to demonstrate the effectiveness of the approach. An application to alcohol consumption in Germany is provided.     

Maximum likelihood estimation of monotone and concave production frontiers

In this paper we bring together the previously separate parametric and nonparametric approaches to production frontier estimation by developing composed error models for maximum likelihood estimation from nonparametrically specified classes of frontiers. This approach avoids the untestable restrictions of parametric functional forms and also provides a statistical foundation for nonparametric frontier estimation. We first examine the single output setting and then extend our formulation to the multiple output setting. The key step in developing the estimation problems is to identify operational constraint sets to ensure estimation from the desired class of frontiers. We also suggest algorithms for solving the resulting constrained likelihood function optimization problems.The refereeing process of this paper was handled through R. Robert Russell. Helpful comments from Bob Russell and two anonymous referees are gratefully acknowedged. We are, of course, solely responsible for any remaining errors or omissions.     

Sequential estimation of shape parameters in multivariate dynamic models

Sequential maximum likelihood and GMM estimators of distributional parameters obtained from the standardised innovations of multivariate conditionally heteroskedastic dynamic regression models evaluated at Gaussian PML estimators preserve the consistency of mean and variance parameters while allowing for realistic distributions. We assess their efficiency, and obtain moment conditions leading to sequential estimators as efficient as their joint ML counterparts. We also obtain standard errors for VaR and CoVaR, and analyse the effects on these measures of distributional misspecification. Finally, we illustrate the small sample performance of these procedures through simulations and apply them to analyse the risk of large eurozone banks.     

Identification and estimation of Gaussian affine term structure models

This paper develops new results for identification and estimation of Gaussian affine term structure models. We establish that three popular canonical representations are unidentified, and demonstrate how unidentified regions can complicate numerical optimization. A separate contribution of the paper is the proposal of minimum-chi-square estimation as an alternative to MLE. We show that, although it is asymptotically equivalent to MLE, it can be much easier to compute. In some cases, MCSE allows researchers to recognize with certainty whether a given estimate represents a global maximum of the likelihood function and makes feasible the computation of small-sample standard errors.     

On Maximum Likelihood Estimation of Dynamic Panel Data Models

We analyse the finite sample properties of maximum likelihood estimators for dynamic panel data models. In particular, we consider transformed maximum likelihood (TML) and random effects maximum likelihood (RML) estimation. We show that TML and RML estimators are solutions to a cubic first‐order condition in the autoregressive parameter. Furthermore, in finite samples both likelihood estimators might lead to a negative estimate of the variance of the individual‐specific effects. We consider different approaches taking into account the non‐negativity restriction for the variance. We show that these approaches may lead to a solution different from the unique global unconstrained maximum. In an extensive Monte Carlo study we find that this issue is non‐negligible for small values of T and that different approaches might lead to different finite sample properties. Furthermore, we find that the Likelihood Ratio statistic provides size control in small samples, albeit with low power due to the flatness of the log‐likelihood function. We illustrate these issues modelling US state level unemployment dynamics.     

Exact likelihood computation for nonlinear DSGE models with heteroskedastic innovations

Phenomena such as the Great Moderation have increased the attention of macroeconomists towards models where shock processes are not (log-)normal. This paper studies a class of discrete-time rational expectations models where the variance of exogenous innovations is subject to stochastic regime shifts. We first show that, up to a second-order approximation using perturbation methods, regime switching in the variances has an impact only on the intercept coefficients of the decision rules. We then demonstrate how to derive the exact model likelihood for the second-order approximation of the solution when there are as many shocks as observable variables. We illustrate the applicability of the proposed solution and estimation methods in the case of a small DSGE model.     

Linear and nonlinear regression with stable errors

In this paper we describe methods and evaluate programs for linear regression by maximum likelihood when the errors have a heavy tailed stable distribution. The asymptotic Fisher information matrix for both the regression coefficients and the error distribution parameters are derived, giving large sample confidence intervals for all parameters. Simulated examples are shown where the errors are stably distributed and also where the errors are heavy tailed but are not stable, as well as a real example using financial data. The results are then extended to nonlinear models and to non-homogeneous error terms.     

Estimating structural equation models with non-normal variables by using transformations

We discuss structural equation models for non-normal variables. In this situation the maximum likelihood and the generalized least-squares estimates of the model parameters can give incorrect estimates of the standard errors and the associated goodness-of-fit chi-squared statistics. If the sample size is not large, for instance smaller than about 1000, asymptotic distribution-free estimation methods are also not applicable. This paper assumes that the observed variables are transformed to normally distributed variables. The non-normally distributed variables are transformed with a Box–Cox function. Estimation of the model parameters and the transformation parameters is done by the maximum likelihood method. Furthermore, the test statistics (i.e. standard deviations) of these parameters are derived. This makes it possible to show the importance of the transformations. Finally, an empirical example is presented.     

Bayesian inference for nonlinear structural time series models

We consider efficient methods for likelihood inference applied to structural models. In particular, we introduce a particle filter method which concentrates upon disturbances in the Markov state of the approximating solution to the structural model. A particular feature of such models is that the conditional distribution of interest for the disturbances is often multimodal. We provide a fast and effective method for approximating such distributions. We estimate a neoclassical growth model using this approach. An asset pricing model with persistent habits is also considered. The methodology we employ allows many fewer particles to be used than alternative procedures for a given precision.     

Properties and estimation of asymmetric exponential power distribution

The new distribution class, Asymmetric Exponential Power Distribution (AEPD), proposed in this paper generalizes the class of Skewed Exponential Power Distributions (SEPD) in a way that in addition to skewness introduces different decay rates of density in the left and right tails. Our parametrization provides an interpretable role for each parameter. We derive moments and moment-based measures: skewness, kurtosis, expected shortfall. It is demonstrated that a maximum entropy property holds for the AEPD distributions. We establish consistency, asymptotic normality and efficiency of the maximum likelihood estimators over a large part of the parameter space by dealing with the problems created by non-smooth likelihood function and derive explicit analytical expressions of the asymptotic covariance matrix; where the results apply to the SEPD class they enlarge on the current literature. Also we give a convenient stochastic representation of the distribution; our Monte Carlo study illustrates the theoretical results. We also provide some empirical evidence for the usefulness of employing AEPD errors in GARCH type models for predicting downside market risk of financial assets.     

A Note on the Power of Money-Output Causality Tests

This study suggests that some empirical findings against money-output causality can be the consequence of ignoring autoregressive conditional heteroskedastic (ARCH) errors. Monte Carlo results confirm that ARCH effects drastically reduce the power of the standard causality test. The maximum likelihood approach allowing for ARCH effects, on the other hand, provides a good power performance. Using different specifications and sample period, Friedman and Kuttner (1993) and Thomas (1994) report limited evidence of money causing output. We detect significant ARCH effects in the models considered by these studies. Once ARCH effects are explicitly accounted for, we find that the monetary effect is significant though its magnitude is quite small.