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.     

Semi-nonparametric estimation and misspecification testing of diffusion models

Novel transition-based misspecification tests of semiparametric and fully parametric univariate diffusion models based on the estimators developed in [Kristensen, D., 2010. Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models. Journal of Econometrics 156, 239-259] are proposed. It is demonstrated that transition-based tests in general lack power in detecting certain departures from the null since they integrate out local features of the drift and volatility. As a solution to this, tests that directly compare drift and volatility estimators under the relevant null and alternative are also developed which exhibit better power against local alternatives.     

Estimation and testing of Euler equation models with time-varying reduced-form coefficients

This paper studies the estimation and testing of Euler equation models in the framework of the classical two-step minimum-distance method. The time-varying reduced-form model in the first step reflects the adaptation of private agents’ beliefs to the changing economic environment. The presumed ability of Euler conditions to deliver stable parameters indexing tastes and technology is interpreted as a time-invariant second-step model. This paper shows that, complementary to and independent of one another, both standard specification test and stability test are required for the evaluation of an Euler equation. As an empirical application, a widely used investment Euler equation is submitted to examination. The empirical outcomes appear to suggest that the standard investment model has not been a success for aggregate investment.     

Panel data models with multiple time-varying individual effects

This paper considers a panel data model with time-varying individual effects. The data are assumed to contain a large number of cross-sectional units repeatedly observed over a fixed number of time periods. The model has a feature of the fixed-effects model in that the effects are assumed to be correlated with the regressors. The unobservable individual effects are assumed to have a factor structure. For consistent estimation of the model, it is important to estimate the true number of individual effects. We propose a generalized methods of moments procedure by which both the number of individual effects and the regression coefficients can be consistently estimated. Some important identification issues are also discussed. Our simulation results indicate that the proposed methods produce reliable estimates.     

Estimation of semiparametric locally stationary diffusion models

This paper proposes a class of locally stationary diffusion processes. The model has a time varying but locally linear drift and a volatility coefficient that is allowed to vary over time and space. The model is semiparametric because we allow these functions to be unknown and the innovation process is parametrically specified, indeed completely known. We propose estimators of all the unknown quantities based on long span data. Our estimation method makes use of the property of local stationarity. We establish asymptotic theory for the proposed estimators as the time span increases, so we do not rely on infill asymptotics. We apply this method to interest rate data to illustrate the validity of our model. Finally, we present a simulation study to provide the finite-sample performance of the proposed estimators.     

The limit distribution of the estimates in cointegrated regression models with multiple structural changes

We study estimation and inference in cointegrated regression models with multiple structural changes allowing both stationary and integrated regressors. Both pure and partial structural change models are analyzed. We derive the consistency, rate of convergence and the limit distribution of the estimated break fractions. Our technical conditions are considerably less restrictive than those in Bai et al. [Bai, J., Lumsdaine, R.L., Stock, J.H., 1998. Testing for and dating breaks in multivariate time series. Review of Economic Studies 65, 395–432] who considered the single break case in a multi-equations system, and permit a wide class of practically relevant models. Our analysis is, however, restricted to a single equation framework. We show that if the coefficients of the integrated regressors are allowed to change, the estimated break fractions are asymptotically dependent so that confidence intervals need to be constructed jointly. If, however, only the intercept and/or the coefficients of the stationary regressors are allowed to change, the estimates of the break dates are asymptotically independent as in the stationary case analyzed by Bai and Perron [Bai, J., Perron, P., 1998. Estimating and testing linear models with multiple structural changes. Econometrica 66, 47–78]. We also show that our results remain valid, under very weak conditions, when the potential endogeneity of the non-stationary regressors is accounted for via an increasing sequence of leads and lags of their first-differences as additional regressors. Simulation evidence is presented to assess the adequacy of the asymptotic approximations in finite samples.     

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi–Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.     

Regression Models with Data‐based Indicator Variables*

Ordinary least squares estimation of an impulse‐indicator coefficient is inconsistent, but its variance can be consistently estimated. Although the ratio of the inconsistent estimator to its standard error has a t‐distribution, that test is inconsistent: one solution is to form an index of indicators. We provide Monte Carlo evidence that including a plethora of indicators need not distort model selection, permitting the use of many dummies in a general‐to‐specific framework. Although White's (1980) heteroskedasticity test is incorrectly sized in that context, we suggest an easy alteration. Finally, a possible modification to impulse ‘intercept corrections’ is considered.     

Pseudo-maximum likelihood estimation in two classes of semiparametric diffusion models

Two classes of semiparametric diffusion models are considered, where either the drift or the diffusion term is parameterized, while the other term is left unspecified. We propose a pseudo-maximum likelihood estimator (PMLE) of the parametric component that maximizes the likelihood with a preliminary estimator of the unspecified term plugged in. It is demonstrated how models and estimators can be used in a two-step specification testing strategy of semiparametric and fully parametric models, and shown that approximate/simulated versions of the PMLE inherit the properties of the actual but infeasible estimator. A simulation study investigates the finite sample performance of the PMLE.     

Delay times of sequential procedures for multiple time series regression models

We consider a multiple regression model in which the explanatory variables are specified by time series. To sequentially test for the stability of the regression parameters in time, we introduce a detector which is based on the first excess time of a CUSUM-type statistic over a suitably constructed threshold function. The aim of this paper is to study the delay time associated with this detector. As our main result, we derive the limit distribution of the delay time and provide thereby a theory that extends the benchmark average run-length concept utilized in most of the sequential monitoring literature. To highlight the applicability of the limit results in finite samples, we present a Monte Carlo simulation study and an application to macroeconomic data.     

Unpredictability in economic analysis,econometric modeling and forecasting

Unpredictability arises from intrinsic stochastic variation, unexpected instances of outliers, and unanticipated extrinsic shifts of distributions. We analyze their properties, relationships, and different effects on the three arenas in the title, which suggests considering three associated information sets. The implications of unanticipated shifts for forecasting, economic analyses of efficient markets, conditional expectations, and inter-temporal derivations are described. The potential success of general-to-specific model selection in tackling location shifts by impulse-indicator saturation is contrasted with the major difficulties confronting forecasting.     

Quasi-maximum likelihood estimation of volatility with high frequency data

This paper investigates the properties of the well-known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise, by extending the classic asymptotic results of quasi-maximum likelihood estimation. When trying to estimate the integrated volatility and the variance of noise, this parametric approach remains consistent, efficient and robust as a quasi-estimator under misspecified assumptions. Moreover, it shares the model-free feature with nonparametric alternatives, for instance realized kernels, while being advantageous over them in terms of finite sample performance. In light of quadratic representation, this estimator behaves like an iterative exponential realized kernel asymptotically. Comparisons with a variety of implementations of the Tukey–Hanning₂ kernel are provided using Monte Carlo simulations, and an empirical study with the Euro/US Dollar future illustrates its application in practice.     

GMM Estimation with Non‐causal Instruments*

This note provides a warning against careless use of the generalized method of moments (GMM) with time series data. We show that if time series follow non‐causal autoregressive processes, their lags are not valid instruments, and the GMM estimator is inconsistent. Moreover, endogeneity of the instruments may not be revealed by the J‐test of overidentifying restrictions that may be inconsistent and has, in general, low finite‐sample power. Our explicit results pertain to a simple linear regression, but they can easily be generalized. Our empirical results indicate that non‐causality is quite common among economic variables, making these problems highly relevant.     

Contemporaneous threshold autoregressive models: Estimation,testing and forecasting

This paper proposes a contemporaneous smooth transition threshold autoregressive model (C-STAR) as a modification of the smooth transition threshold autoregressive model surveyed in Teräsvirta [1998. Modelling economic relationships with smooth transition regressions. In: Ullah, A., Giles, D.E.A. (Eds.), Handbook of Applied Economic Statistics. Marcel Dekker, New York, pp. 507–552.], in which the regime weights depend on the ex ante probability that a latent regime-specific variable will exceed a threshold value. We argue that the contemporaneous model is well suited to rational expectations applications (and pricing exercises), in that it does not require the initial regimes to be predetermined. We investigate the properties of the model and evaluate its finite-sample maximum likelihood performance. We also propose a method to determine the number of regimes based on a modified Hansen [1992. The likelihood ratio test under nonstandard conditions: testing the Markov switching model of GNP. Journal of Applied Econometrics 7, S61–S82.] procedure. Furthermore, we construct multiple-step ahead forecasts and evaluate the forecasting performance of the model. Finally, an empirical application of the short term interest rate yield is presented and discussed.     

A control function approach for testing the usefulness of trending variables in forecast models and linear regression

Many predictors employed in forecasting macroeconomic and finance variables display a great deal of persistence. Tests for determining the usefulness of these predictors are typically oversized, overstating their importance. Similarly, hypothesis tests on cointegrating vectors will typically be oversized if there is not an exact unit root. This paper uses a control variable approach where adding stationary covariates with certain properties to the model can result in asymptotic normal inference for prediction regressions and cointegration vector estimates in the presence of possibly non-unit root trending covariates. The properties required for this result are derived and discussed.     

Consistent Model Selection by an Automatic Gets Approach*

We establish the consistency of the selection procedures embodied in PcGets, and compare their performance with other model selection criteria in linear regressions. The significance levels embedded in the PcGets Liberal and Conservative algorithms coincide in very large samples with those implicit in the Hannan–Quinn (HQ) and Schwarz information criteria (SIC), respectively. Thus, both PcGets rules are consistent under the same conditions as HQ and SIC. However, PcGets has a rather different finite‐sample behaviour. Pre‐selecting to remove many of the candidate variables is confirmed as enhancing the performance of SIC.     

Forecasting by factors,by variables,by both or neither?

We consider forecasting with factors, variables and both, modeling in-sample using Autometrics so all principal components and variables can be included jointly, while tackling multiple breaks by impulse-indicator saturation. A forecast-error taxonomy for factor models highlights the impacts of location shifts on forecast-error biases. Forecasting US GDP over 1-, 4- and 8-step horizons using the dataset from Stock and Watson (2009) updated to 2011:2 shows factor models are more useful for nowcasting or short-term forecasting, but their relative performance declines as the forecast horizon increases. Forecasts for GDP levels highlight the need for robust strategies, such as intercept corrections or differencing, when location shifts occur as in the recent financial crisis.     

Regression models with mixed sampling frequencies

We study regression models that involve data sampled at different frequencies. We derive the asymptotic properties of the NLS estimators of such regression models and compare them with the LS estimators of a traditional model that involves aggregating or equally weighting data to estimate a model at the same sampling frequency. In addition we propose new tests to examine the null hypothesis of equal weights in aggregating time series in a regression model. We explore the above theoretical aspects and verify them via an extensive Monte Carlo simulation study and an empirical application.     

A martingale approach for testing diffusion models based on infinitesimal operator

I develop an omnibus specification test for diffusion models based on the infinitesimal operator. The infinitesimal operator based identification of the diffusion process is equivalent to a “martingale hypothesis” for the processes obtained by a transformation of the original diffusion model. My test procedure is then constructed by checking the “martingale hypothesis” via a multivariate generalized spectral derivative based approach that delivers a N(0,1) asymptotical null distribution for the test statistic. The infinitesimal operator of the diffusion process is a closed-form function of drift and diffusion terms. Consequently, my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case. Moreover, different transformed martingale processes contain separate information about the drift and diffusion specifications. This motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected. Simulation studies show that the proposed tests have reasonable size and excellent power performance. An empirical application of my test procedure using Eurodollar interest rates finds that most popular short-rate models are rejected and the drift misspecification plays an important role in such rejections.     

An efficient nonparametric estimator for models with nonlinear dependence

We provide a convenient econometric framework for the analysis of nonlinear dependence in financial applications. We introduce models with constrained nonparametric dependence, which specify the conditional distribution or the copula in terms of a one-dimensional functional parameter. Our approach is intermediate between standard parametric specifications (which are in general too restrictive) and the fully unrestricted approach (which suffers from the curse of dimensionality). We introduce a nonparametric estimator defined by minimizing a chi-square distance between the constrained densities in the family and an unconstrained kernel estimator of the density. We derive the nonparametric efficiency bound for linear forms and show that the minimum chi-square estimator is nonparametrically efficient for linear forms.