Two‐stage estimation of limited dependent variable models with errors‐in‐variables

Summary This paper deals with censored or truncated regression models where the explanatory variables are measured with additive errors. We propose a two‐stage estimation procedure that combines the instrumental variable method and the minimum distance estimation. This approach produces consistent and asymptotically normally distributed estimators for model parameters. When the predictor and instrumental variables are normally distributed, we also propose a maximum likelihood based estimator and a two‐stage moment estimator. Simulation studies show that all proposed estimators perform satisfactorily for relatively small samples and relatively high degree of censoring. In addition, the maximum likelihood based estimators are fairly robust against non‐normal and /or heteroskedastic random errors in our simulations. The method can be generalized to panel data models.     

On the efficiency of a semi‐parametric GARCH model

Summary Financial time series exhibit time‐varying volatilities and non‐Gaussian distributions. There has been considerable research on the GARCH models for dealing with these issues related to financial data. Since in practice the true error distribution is unknown, various quasi maximum likelihood methods based on different assumptions on the error distribution have been studied in the literature. However, the specification of the distribution family or in particular the shape parameter of the density function is often incorrect. This leads to an efficiency loss quite common in such estimation procedures. To avoid the inaccuracy, semi‐parametric maximum likelihood approaches were introduced, where the estimators of the GARCH parameters are derived from the likelihood based on a non‐parametrically estimated density function. In general, the semi‐parametric likelihood function is trimmed for the extreme observations in order to derive a ‐consistent estimator. In this paper we consider the situation in which the untrimmed likelihood function is maximized to develop a new semi‐parametric estimator. The resulting estimator is consistent, asymptotically Gaussian with a vanishing bias term, and a limiting variance–covariance matrix that attains the information lower bound. This work also provides insight into the efficiencies (bias–variability trade‐off) of a general class of semi‐parametric estimators of GARCH models.     

On skewness and kurtosis of econometric estimators

Summary We derive the approximate results for two standardized measures of deviation from normality, namely, the skewness and excess kurtosis coefficients, for a class of econometric estimators. The results are built on a stochastic expansion of the moment condition used to identify the econometric estimator. The approximate results can be used not only to study the finite sample behaviour of a particular estimator, but also to compare the finite sample properties of two asymptotically equivalent estimators. We apply the approximate results to the spatial autoregressive model and find that our results approximate the non‐normal behaviours of the maximum likelihood estimator reasonably well. However, when the weights matrix becomes denser, the finite sample distribution of the maximum likelihood estimator departs more severely from normality and our results provide less accurate approximation.     

A new technique for simulating the likelihood of stochastic differential equations

This article presents a new simulation‐based technique for estimating the likelihood of stochastic differential equations. This technique is based on a result of Dacunha‐Castelle and Florens‐Zmirou. These authors proved that the transition densities of a nonlinear diffusion process with a constant diffusion coefficient can be written in a closed form involving a stochastic integral. We show that this stochastic integral can be easily estimated through simulations and we prove a convergence result. This simulator for the transition density is used to obtain the simulated maximum likelihood (SML) estimator. We show through some Monte Carlo experiments that our technique is highly computationally efficient and the SML estimator converges rapidly to the maximum likelihood estimator.     

Identification and estimation of local average derivatives in non‐separable models without monotonicity

Summary In many structural economic models there are no good arguments for additive separability of the error. Recently, this motivated intensive research on non‐separable structures. For instance, in Hoderlein and Mammen (2007) a non‐separable model in the single equation case was considered, and it was established that in the absence of the frequently employed monotonicity assumption local average structural derivatives (LASD) are still identified. In this paper, we introduce an estimator for the LASD. The estimator we propose is based on local polynomial fitting of conditional quantiles. We derive its large sample distribution through a Bahadur representation, and give some related results, e.g. about the asymptotic behaviour of the quantile process. Moreover, we generalize the concept of LASD to include endogeneity of regressors and discuss the case of a multivariate dependent variable. We also consider identification of structured non‐separable models, including single index and additive models. We discuss specification testing, as well as testing for endogeneity and for the impact of unobserved heterogeneity. We also show that fixed censoring can easily be addressed in this framework. Finally, we apply some of the concepts to demand analysis using British Consumer Data.     

Nonparametric regression with nearly integrated regressors under long‐run dependence

We study the nonparametric estimation of a regression function with nonstationary (integrated or nearly integrated) covariates and the error series of the regressor process following a fractional integrated autoregressive moving average model. A local linear estimation method is developed to estimate the unknown regression function. The asymptotic results of the resulting estimator at both interior points and boundaries are obtained. The asymptotic distribution is mixed normal, associated with the local time of an Ornstein–Uhlenbeck fractional Brownian motion. Furthermore, we study the Nadaraya–Watson estimator and we examine its asymptotic results. As a result, it shares exactly the same asymptotic results as those for the local linear estimator for the zero energy situation. However, for the non‐zero energy case, the local linear estimator is superior to the Nadaraya–Watson estimator in terms of optimal convergence rate. We also present a comparison of our results with the conventional results for stationary covariates. Finally, we conduct a Monte Carlo simulation to illustrate the finite sample performance of the proposed estimator.     

Lessons learned about steered molecular dynamics simulations and free energy calculations

The calculation of free energy profiles is central in understanding differential enzymatic activity, for instance, involving chemical reactions that require QM‐MM tools, ligand migration, and conformational rearrangements that can be modeled using classical potentials. The use of steered molecular dynamics (sMD) together with the Jarzynski equality is a popular approach in calculating free energy profiles. Here, we first briefly review the application of the Jarzynski equality to sMD simulations, then revisit the so‐called stiff‐spring approximation and the consequent expectation of Gaussian work distributions and, finally, reiterate the practical utility of the second‐order cumulant expansion, as it coincides with the parametric maximum‐likelihood estimator in this scenario. We illustrate this procedure using simulations of CO, both in aqueous solution and in a carbon nanotube as a model system for biologically relevant nanoheterogeneous environments. We conclude the use of the second‐order cumulant expansion permits the use of faster pulling velocities in sMD simulations, without introducing bias due to large dispersion in the non‐equilibrium work distribution.     

The distribution of the maximum likelihood estimator in up-and-down experiments for quantal dose-response data.

Standard maximum likelihood logistic or probit regression has been used in biopharmaceutical practice for inference about tolerance threshold distributions in situations where subjects (patients) have been allocated doses according to an up-and-down design. For example, a steeper dose-response curve than expected was reported in one such study. This article demonstrates that the maximum likelihood estimator systematically and considerably exaggerates the regression parameter with moderately large sample sizes. Thus a probable explanation for finding a steeper curve than expected is the method used to analyze the experiment, that is, the bias in the maximum likelihood estimator. An additional consequence of this bias is that the mean/median/ED50 are estimated with a misleading precision. In particular, confidence intervals are much too narrow. As a conclusion, we warn against conventional logistic or probit regression in combination with up-and-down designs.     

Confidence Interval Estimation for Sensitivity at a Fixed Level of Specificity for Combined Biomarkers

Standard maximum likelihood logistic or probit regression has been used in biopharmaceutical practice for inference about tolerance threshold distributions in situations where subjects (patients) have been allocated doses according to an up-and-down design. For example, a steeper dose-response curve than expected was reported in one such study. This article demonstrates that the maximum likelihood estimator systematically and considerably exaggerates the regression parameter with moderately large sample sizes. Thus a probable explanation for finding a steeper curve than expected is the method used to analyze the experiment, that is, the bias in the maximum likelihood estimator. An additional consequence of this bias is that the mean/median/ED₅₀ are estimated with a misleading precision. In particular, confidence intervals are much too narrow. As a conclusion, we warn against conventional logistic or probit regression in combination with up-and-down designs.     

Quasi‐maximum likelihood estimation of discretely observed diffusions

Summary This paper introduces a quasi‐maximum likelihood estimator for discretely observed diffusions when a closed‐form transition density is unavailable. Higher‐order Wagner–Platen strong approximation is used to derive the first two conditional moments and a normal density function is used in estimation. Simulation study shows that the proposed estimator has high numerical precision and good numerical robustness. This method is applicable to a large class of diffusions.     

Estimating GARCH models: when to use what?

Summary The class of generalized autoregressive conditional heteroscedastic (GARCH) models has proved particularly valuable in modelling time series with time varying volatility. These include financial data, which can be particularly heavy tailed. It is well understood now that the tail heaviness of the innovation distribution plays an important role in determining the relative performance of the two competing estimation methods, namely the maximum quasi‐likelihood estimator based on a Gaussian likelihood (GMLE) and the log‐transform‐based least absolutely deviations estimator (LADE) (see Peng and Yao 2003 Biometrika,90, 967–75). A practically relevant question is when to use what. We provide in this paper a solution to this question. By interpreting the LADE as a version of the maximum quasilikelihood estimator under the likelihood derived from assuming hypothetically that the log‐squared innovations obey a Laplace distribution, we outline a selection procedure based on some goodness‐of‐fit type statistics. The methods are illustrated with both simulated and real data sets. Although we deal with the estimation for GARCH models only, the basic idea may be applied to address the estimation procedure selection problem in a general regression setting.     

Asymptotic approximations in the near‐integrated model with a non‐zero initial condition

This paper considers various asymptotic approximations in the near‐integrated first‐order autoregressive model with a non‐zero initial condition. We first extend the work of Knight and Satchell (1993), who considered the random walk case with a zero initial condition, to derive the expansion of the relevant joint moment generating function in this more general framework. We also consider, as alternative approximations, the stochastic expansion of Phillips (1987c) and the continuous‐time approximation of Perron (1991a). We assess, via a Monte Carlo simulation study, the extent to which these alternative methods provide adequate approximations to the finite sample distribution of the least‐squares estimator in a first‐order autoregressive model. The results show that, when the initial condition is non‐zero, Perron's (1991a) continuous‐time approximation performs very well while the others only offer improvements when the initial condition is zero.     

Non‐parametric regression for binary dependent variables

Summary Finite‐sample properties of non‐parametric regression for binary dependent variables are analyzed. Non parametric regression is generally considered as highly variable in small samples when the number of regressors is large. In binary choice models, however, it may be more reliable since its variance is bounded. The precision in estimating conditional means as well as marginal effects is investigated in settings with many explanatory variables (14 regressors) and small sample sizes (250 or 500 observations). The Klein–Spady estimator, Nadaraya–Watson regression and local linear regression often perform poorly in the simulations. Local likelihood logit regression, on the other hand, is 25 to 55% more precise than parametric regression in the Monte Carlo simulations. In an application to female labour supply, local logit finds heterogeneity in the effects of children on employment that is not detected by parametric or semiparametric estimation. (The semiparametric estimator actually leads to rather similar results as the parametric estimator.)     

Fully modified narrow‐band least squares estimation of weak fractional cointegration

Summary We consider estimation of the cointegrating relation in the weak fractional cointegration model, where the strength of the cointegrating relation (difference in memory parameters) is less than one‐half. A special case is the stationary fractional cointegration model, which has found important applications recently, especially in financial economics. Previous research on this model has considered a semi‐parametric narrow‐band least squares (NBLS) estimator in the frequency domain, but in the stationary case its asymptotic distribution has been derived only under a condition of non‐coherence between regressors and errors at the zero frequency. We show that in the absence of this condition, the NBLS estimator is asymptotically biased, and also that the bias can be consistently estimated. Consequently, we introduce a fully modified NBLS estimator which eliminates the bias, and indeed enjoys a faster rate of convergence than NBLS in general. We also show that local Whittle estimation of the integration order of the errors can be conducted consistently based on NBLS residuals, but the estimator has the same asymptotic distribution as if the errors were observed only under the condition of non‐coherence. Furthermore, compared to much previous research, the development of the asymptotic distribution theory is based on a different spectral density representation, which is relevant for multivariate fractionally integrated processes, and the use of this representation is shown to result in lower asymptotic bias and variance of the narrow‐band estimators. We present simulation evidence and a series of empirical illustrations to demonstrate the feasibility and empirical relevance of our methodology.     

Non‐standard rates of convergence of criterion‐function‐based set estimators for binary response models

This paper establishes consistency and non‐standard rates of convergence for set estimators based on contour sets of criterion functions for a semi‐parametric binary response model under a conditional median restriction. The model can be partially identified due to potentially limited‐support regressors and an unknown distribution of errors. A set estimator analogous to the maximum score estimator is essentially cube‐root consistent for the identified set when a continuous but possibly bounded regressor is present. Arbitrarily fast convergence occurs when all regressors are discrete. We also establish the validity of a subsampling procedure for constructing confidence sets for the identified set. As a technical contribution, we provide more convenient sufficient conditions on the underlying empirical processes for cube‐root convergence and a sufficient condition for arbitrarily fast convergence, both of which can be applied to other models. Finally, we carry out a series of Monte Carlo experiments, which verify our theoretical findings and shed light on the finite‐sample performance of the proposed procedures.     

Direct semi‐parametric estimation of fixed effects panel data varying coefficient models

In this paper, we present a new technique to estimate varying coefficient models of unknown form in a panel data framework where individual effects are arbitrarily correlated with the explanatory variables in an unknown way. The estimator is based on first differences and then a local linear regression is applied to estimate the unknown coefficients. To avoid a non‐negligible asymptotic bias, we need to introduce a higher‐dimensional kernel weight. This enables us to remove the bias at the price of enlarging the variance term and, hence, achieving a slower rate of convergence. To overcome this problem, we propose a one‐step backfitting algorithm that enables the resulting estimator to achieve optimal rates of convergence for this type of problem. It also exhibits the so‐called oracle efficiency property. We also obtain the asymptotic distribution. Because the estimation procedure depends on the choice of a bandwidth matrix, we also provide a method to compute this matrix empirically. The Monte Carlo results indicate the good performance of the estimator in finite samples.     

Asymptotic confidence intervals for impulse responses of near‐integrated processes

Summary Many economic time series are characterized by high persistence which typically requires nonstandard limit theory for inference. This paper proposes a new method for constructing confidence intervals for impulse response functions and half‐lives of nearly non‐stationary processes. It is based on inverting the acceptance region of the likelihood ratio statistic under a sequence of null hypotheses of possible values for the impulse response or the half‐life. This paper shows the consistency of the restricted estimator of the localizing constant which ensures the validity of the asymptotic inference. The proposed method is used to study the persistence of shocks to real exchange rates.     

Identification and estimation of semi‐parametric censored dynamic panel data models of short time periods

In this paper, we present a semi‐parametric identification and estimation method for censored dynamic panel data models of short time periods and their average partial effects with only two periods of data. The proposed method transforms the semi‐parametric specification of censored dynamic panel data models into a parametric family of distribution functions of observables without specifying the distribution of the initial condition. Then the censored dynamic panel data models are globally identified under a standard maximum likelihood estimation framework. The identifying assumptions are related to the completeness of the families of known parametric distribution functions corresponding to censored dynamic panel data models. Dynamic tobit models and two‐part dynamic regression models satisfy the key assumptions. We propose a sieve maximum likelihood estimator and we investigate the finite sample properties of these sieve‐based estimators using Monte Carlo analysis. Our empirical application using the Medical Expenditure Panel Survey shows that individuals consume more health care when their incomes increase, after controlling for past health expenditures.     

Non‐parametric identification of the mixed proportional hazards model with interval‐censored durations

Summary This note presents identification results for the mixed proportional hazards model when duration data are interval‐censored. Earlier positive results on identification under interval‐censoring require both parametric specification on how covariates enter the hazard functions and assumptions of unbounded support for covariates. New results provided here show how one can dispense with both of these assumptions. The mixed proportional hazards model is non‐parametrically identified with interval‐censored duration data, provided covariates have support on an open set and the hazard function is a non‐constant continuous function of covariates.     

Estimation of fixed effects panel data partially linear additive regression models

In this paper, we investigate the estimation problem of fixed effects panel data partially linear additive regression models. Semi‐parametric fixed effects panel data regression models are tools that are well suited to econometric analysis and the analysis of cDNA micro‐arrays. By applying a polynomial spline series approximation and a profile least‐squares procedure, we propose a semi‐parametric least‐squares dummy variables estimator (SLSDVE) for the parametric component and a series estimator for the non‐parametric component. Under very weak conditions, we show that the SLSDVE is asymptotically normal and that the series estimator achieves the optimal convergence rate of the non‐parametric regression. In addition, we propose a two‐stage local polynomial estimation for the non‐parametric component by applying the additive structure and the series estimator. The resultant estimator is asymptotically normal and the asymptotic distribution of each additive component is the same as it would be if the other components were known with certainty. We conduct simulation studies to demonstrate the finite sample performance of the proposed procedures and we also present an illustrative empirical application.