Asymptotic distribution of the cointegrating vector estimator in error correction models with conditional heteroskedasticity

This paper explores the asymptotic distribution of the cointegrating vector estimator in error correction models with conditionally heteroskedastic errors. Asymptotic properties of the maximum likelihood estimator (MLE) of the cointegrating vector, which estimates the cointegrating vector and the multivariate GARCH process jointly, are provided. The MLE of the cointegrating vector follows mixture normal, and its asymptotic distribution depends on the conditional heteroskedasticity and the kurtosis of standardized innovations. The reduced rank regression (RRR) estimator and the regression-based cointegrating vector estimators do not consider conditional heteroskedasticity, and thus the efficiency gain of the MLE emerges as the magnitude of conditional heteroskedasticity increases. The simulation results indicate that the relative power of the t-statistics based on the MLE improves significantly as the GARCH effect increases.     

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.     

Some properties of the LIML estimator in a dynamic panel structural equation

We investigate the finite sample and asymptotic properties of the within-groups (WG), the random-effects quasi-maximum likelihood (RQML), the generalized method of moment (GMM) and the limited information maximum likelihood (LIML) estimators for a panel autoregressive structural equation model with random effects when both T (time-dimension) and N (cross-section dimension) are large. When we use the forward-filtering due to Alvarez and Arellano (2003), the WG, the RQML and GMM estimators are significantly biased when both T and N are large while T/N is different from zero. The LIML estimator gives desirable asymptotic properties when T/N converges to a constant.     

A note on multinomial maximum likelihood estimation under ordered restrictions and the EM algorithm

A frequently occurring problem is to find the maximum likelihood estimation (MLE) of p subject to p∈C (C⊂ P the probability vectors in R ^k ). The problem has been discussed by many authors and they mainly focused when p is restricted by linear constraints or log-linear constraints. In this paper, we construct the relationship between the the maximum likelihood estimation of p restricted by p∈C and EM algorithm and demonstrate that the maximum likelihood estimator can be computed through the EM algorithm (Dempster et al. in J R Stat Soc Ser B 39:1–38, 1997). Several examples are analyzed by the proposed method.     

Fixed effects estimation of structural parameters and marginal effects in panel probit models

Fixed effects estimators of nonlinear panel models can be severely biased due to the incidental parameters problem. In this paper, I characterize the leading term of a large-T expansion of the bias of the MLE and estimators of average marginal effects in parametric fixed effects panel binary choice models. For probit index coefficients, the former term is proportional to the true value of the coefficients being estimated. This result allows me to derive a lower bound for the bias of the MLE. I then show that the resulting fixed effects estimates of ratios of coefficients and average marginal effects exhibit no bias in the absence of heterogeneity and negligible bias for a wide variety of distributions of regressors and individual effects in the presence of heterogeneity. I subsequently propose new bias-corrected estimators of index coefficients and marginal effects with improved finite sample properties for linear and nonlinear models with predetermined regressors.     

A comparison of the Box-Cox maximum likelihood estimator and the non-linear two-stage least squares estimator

This paper investigates the limiting behaviour of the ‘maximum likelihood estimator’(MLE) based on normality, as well as the nonlinear two-stage least squares estimator (NL2S), for the i.i.d. and regression models in which the Box-Cox transformation is applied to the dependent variable. Since the transformed variable cannot in general be normally distributed, the untransformed variable is assumed to have a two-parameter gamma distribution. Tables of probability limits and asymptotic variance demonstrate that, in this case, the inconsistency of the ‘normal MLE’ is often quite pronounced, while the NL2S is consistent and typically well behaved.     

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.     

Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes

This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter _d0$d_0$ are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood.     

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 the asymptotic optimality of the LIML estimator with possibly many instruments

We consider the estimation of the coefficients of a linear structural equation in a simultaneous equation system when there are many instrumental variables. We derive some asymptotic properties of the limited information maximum likelihood (LIML) estimator when the number of instruments is large; some of these results are new as well as old, and we relate them to results in some recent studies. We have found that the variance of the limiting distribution of the LIML estimator and its modifications often attain the asymptotic lower bound when the number of instruments is large and the disturbance terms are not necessarily normally distributed, that is, for the micro-econometric models of some cases recently called many instruments and many weak instruments.     

Estimation for spatial dynamic panel data with fixed effects: The case of spatial cointegration

Yu et al. (2008) establish asymptotic properties of quasi-maximum likelihood estimators for a stable spatial dynamic panel model with fixed effects when both the number of individuals n and the number of time periods T are large. This paper investigates unstable cases where there are unit roots generated by temporal and spatial correlations. We focus on the spatial cointegration model where some eigenvalues of the data generating process are equal to 1 and the outcomes of spatial units are cointegrated as in a vector autoregressive system. The asymptotics of the QML estimators are developed by reparameterization, and bias correction for the estimators is proposed. We also consider the 2SLS and GMM estimations when T could be small.     

Model selection criteria in multivariate models with multiple structural changes

This paper considers the issue of selecting the number of regressors and the number of structural breaks in multivariate regression models in the possible presence of multiple structural changes. We develop a modified Akaike information criterion (AIC), a modified Mallows’ C_p criterion and a modified Bayesian information criterion (BIC). The penalty terms in these criteria are shown to be different from the usual terms. We prove that the modified BIC consistently selects the regressors and the number of breaks whereas the modified AIC and the modified C_p criterion tend to overfit with positive probability. The finite sample performance of these criteria is investigated through Monte Carlo simulations and it turns out that our modification is successful in comparison to the classical model selection criteria and the sequential testing procedure robust to heteroskedasticity and autocorrelation.     

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.     

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.     

A family of empirical likelihood functions and estimators for the binary response model

The minimum discrimination information principle is used to identify an appropriate parametric family of probability distributions and the corresponding maximum likelihood estimators for binary response models. Estimators in the family subsume the conventional logit model and form the basis for a set of parametric estimation alternatives with the usual asymptotic properties. Sampling experiments are used to assess finite sample performance.     

Maximum likelihood estimation and Lagrange multiplier tests for panel seemingly unrelated regressions with spatial lag and spatial errors: An application to hedonic housing prices in Paris

This paper proposes maximum likelihood estimators for panel seemingly unrelated regressions with both spatial lag and spatial error components. We study the general case where spatial effects are incorporated via spatial errors terms and via a spatial lag dependent variable and where the heterogeneity in the panel is incorporated via an error component specification. We generalize the approach of Wang and Kockelman (2007) and propose joint and conditional Lagrange multiplier tests for spatial autocorrelation and random effects for this spatial SUR panel model. The small sample performance of the proposed estimators and tests are examined using Monte Carlo experiments. An empirical application to hedonic housing prices in Paris illustrate these methods. The proposed specification uses a system of three SUR equations corresponding to three types of flats within 80 districts of Paris over the period 1990-2003. We test for spatial effects and heterogeneity and find reasonable estimates of the shadow prices for housing characteristics.     

Estimation of possibly misspecified semiparametric conditional moment restriction models with different conditioning variables

Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578] and Ai and Chen [2003. Efficient estimation of conditional moment restrictions models containing unknown functions. Econometrica 71, 1795–1843] propose sieve minimum distance (SMD) estimation of both finite dimensional parameter (θ)$(\theta )$ and infinite dimensional parameter (h) that are identified through a conditional moment restriction model, in which h could depend on endogenous variables. This paper modifies their SMD procedure to allow for different conditioning variables to be used in different equations, and derives the asymptotic properties when the model may be misspecified  . Under low-level sufficient conditions, we show that: (i) the modified SMD estimators of both θ$\theta$ and h   converge to some pseudo-true values in probability; (ii) the SMD estimators of smooth functionals, including the θ$\theta$ estimator and the average derivative estimator, are asymptotically normally distributed; and (iii) the estimators for the asymptotic covariances of the SMD estimators of smooth functionals are consistent and easy to compute. These results allow for asymptotically valid tests of various hypotheses on the smooth functionals regardless of whether the semiparametric model is correctly specified or not.     

Performance of conditional Wald tests in IV regression with weak instruments

We compare the powers of five tests of the coefficient on a single endogenous regressor in instrumental variables regression. Following Moreira [2003, A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048], all tests are implemented using critical values that depend on a statistic which is sufficient under the null hypothesis for the (unknown) concentration parameter, so these conditional tests are asymptotically valid under weak instrument asymptotics. Four of the tests are based on k-class Wald statistics (two-stage least squares, LIML, Fuller's [Some properties of a modification of the limited information estimator. Econometrica 45, 939–953], and bias-adjusted TSLS); the fifth is Moreira's (2003) conditional likelihood ratio (CLR) test. The heretofore unstudied conditional Wald (CW) tests are found to perform poorly, compared to the CLR test: in many cases, the CW tests have almost no power against a wide range of alternatives. Our analysis is facilitated by a new algorithm, presented here, for the computation of the asymptotic conditional p-value of the CLR test.