Estimation for multivariate stable distributions with generalized empirical likelihood

This paper considers the generalized empirical likelihood (GEL) method for estimating the parameters of the multivariate stable distribution. The GEL method is considered to be an extension of the generalized method of moments (GMM). The multivariate stable distributions are widely applicable as they can accommodate both skewness and heavy tails. We treat the spectral measure, which summarizes scale and asymmetry, by discretization. In order to estimate all the model parameters simultaneously, we apply the estimating function constructed by equating empirical and theoretical characteristic functions. The efficacy of the proposed GEL method is demonstrated in Monte Carlo studies. An illustrative example involving daily returns of market indexes is also included.     

A Bayesian approach to bandwidth selection for multivariate kernel regression with an application to state-price density estimation

This paper presents a Bayesian approach to bandwidth selection for multivariate kernel regression. A Monte Carlo study shows that under the average squared error criterion, the Bayesian bandwidth selector is comparable to the cross-validation method and clearly outperforms the bootstrapping and rule-of-thumb bandwidth selectors. The Bayesian bandwidth selector is applied to a multivariate kernel regression model that is often used to estimate the state-price density of Arrow–Debreu securities with the S&P 500 index options data and the DAX index options data. The proposed Bayesian bandwidth selector represents a data-driven solution to the problem of choosing bandwidths for the multivariate kernel regression involved in the nonparametric estimation of the state-price density pioneered by Aït-Sahalia and Lo [Aït-Sahalia, Y., Lo, A.W., 1998. Nonparametric estimation of state-price densities implicit in financial asset prices. The Journal of Finance, 53, 499, 547.]     

Moving average stochastic volatility models with application to inflation forecast

We introduce a new class of models that has both stochastic volatility and moving average errors, where the conditional mean has a state space representation. Having a moving average component, however, means that the errors in the measurement equation are no longer serially independent, and estimation becomes more difficult. We develop a posterior simulator that builds upon recent advances in precision-based algorithms for estimating these new models. In an empirical application involving US inflation we find that these moving average stochastic volatility models provide better in-sample fitness and out-of-sample forecast performance than the standard variants with only stochastic volatility.     

Bayesian Model Selection with an Uninformative Prior*

Bayesian model selection with posterior probabilities and no subjective prior information is generally not possible because of the Bayes factors being ill‐defined. Using careful consideration of the parameter of interest in cointegration analysis and a re‐specification of the triangular model of Phillips (Econometrica, Vol. 59, pp. 283–306, 1991), this paper presents an approach that allows for Bayesian comparison of models of cointegration with ‘ignorance’ priors. Using the concept of Stiefel and Grassman manifolds, diffuse priors are specified on the dimension and direction of the cointegrating space. The approach is illustrated using a simple term structure of the interest rates model.     

On the statistical identification of DSGE models

Dynamic Stochastic General Equilibrium (DSGE) models are now considered attractive by the profession not only from the theoretical perspective but also from an empirical standpoint. As a consequence of this development, methods for diagnosing the fit of these models are being proposed and implemented. In this article we illustrate how the concept of statistical identification, that was introduced and used by Spanos [Spanos, Aris, 1990. The simultaneous-equations model revisited: Statistical adequacy and identification. Journal of Econometrics 44, 87–105] to criticize traditional evaluation methods of Cowles Commission models, could be relevant for DSGE models. We conclude that the recently proposed model evaluation method, based on the DSGE–VAR(λ)$\left(\lambda \right)$, might not satisfy the condition for statistical identification. However, our application also shows that the adoption of a FAVAR as a statistically identified benchmark leaves unaltered the support of the data for the DSGE model and that a DSGE–FAVAR can be an optimal forecasting model.     

A quasi-maximum likelihood method for estimating the parameters of multivariate diffusions

A quasi-maximum likelihood procedure for estimating the parameters of multi-dimensional diffusions is developed in which the transitional density is a multivariate Gaussian density with first and second moments approximating the true moments of the unknown density. For affine drift and diffusion functions, the moments are exactly those of the true transitional density and for nonlinear drift and diffusion functions the approximation is extremely good and is as effective as alternative methods based on likelihood approximations. The estimation procedure generalises to models with latent factors. A conditioning procedure is developed that allows parameter estimation in the absence of proxies.     

Information measures for generalized gamma family

The objective of this paper is to integrate the generalized gamma (GG)$(\mathit{GG})$ distribution into the information theoretic literature. We study information properties of the GG$\mathit{GG}$ distribution and provide an assortment of information measures for the GG$\mathit{GG}$ family, which includes the exponential, gamma, Weibull, and generalized normal distributions as its subfamilies. The measures include entropy representations of the log-likelihood ratio, AIC, and BIC, discriminating information between GG$\mathit{GG}$ and its subfamilies, a minimum discriminating information function, power transformation information, and a maximum entropy index of fit to histogram. We provide the full parametric Bayesian inference for the discrimination information measures. We also provide Bayesian inference for the fit of GG$\mathit{GG}$ model to histogram, using a semi-parametric Bayesian procedure, referred to as the maximum entropy Dirichlet (MED). The GG$\mathit{GG}$ information measures are computed for duration of unemployment and duration of CEO tenure.     

Extended Neyman smooth goodness-of-fit tests,applied to competing heavy-tailed distributions

A simplified version of the Neyman (1937) “Smooth” goodness-of-fit test is extended to account for the presence of estimated model parameters, thereby removing overfitting bias. Using a Lagrange Multiplier approach rather than the Likelihood Ratio statistic proposed by Neyman greatly simplifies the calculations. Polynomials, splines, and the step function of Pearson’s test are compared as alternative perturbations to the theoretical uniform distribution. The extended tests have negligible size distortion and more power than standard tests. The tests are applied to competing symmetric leptokurtic distributions with US stock return data. These are generally rejected, primarily because of the presence of skewness.     

An Introduction to Best Empirical Models when the Parameter Space is Infinite Dimensional*

Ploberger and Phillips (Econometrica, Vol. 71, pp. 627–673, 2003) proved a result that provides a bound on how close a fitted empirical model can get to the true model when the model is represented by a parameterized probability measure on a finite dimensional parameter space. The present note extends that result to cases where the parameter space is infinite dimensional. The results have implications for model choice in infinite dimensional problems and highlight some of the difficulties, including technical difficulties, presented by models of infinite dimension. Some implications for forecasting are considered and some applications are given, including the empirically relevant case of vector autoregression (VAR) models of infinite order.     

Econometric analysis of jump-driven stochastic volatility models

This paper introduces and studies the econometric properties of a general new class of models, which I refer to as jump-driven stochastic volatility models, in which the volatility is a moving average of past jumps. I focus attention on two particular semiparametric classes of jump-driven stochastic volatility models. In the first, the price has a continuous component with time-varying volatility and time-homogeneous jumps. The second jump-driven stochastic volatility model analyzed here has only jumps in the price, which have time-varying size. In the empirical application I model the memory of the stochastic variance with a CARMA(2,1) kernel and set the jumps in the variance to be proportional to the squared price jumps. The estimation, which is based on matching moments of certain realized power variation statistics calculated from high-frequency foreign exchange data, shows that the jump-driven stochastic volatility model containing continuous component in the price performs best. It outperforms a standard two-factor affine jump–diffusion model, but also the pure-jump jump-driven stochastic volatility model for the particular jump specification.     

Limited information Bayesian analysis of a simultaneous equation with an autocorrelated error term and its application to the U.S. gasoline market

Using Markov Chain Monte Carlo algorithms within the limited information Bayesian framework, we estimate the parameters of the structural equation of interest and test weak exogeneity in a simultaneous equation model with white noise as well as autocorrelated error terms. A numerical example and an estimation of the supply and demand equations of the U.S. gasoline market show that if we ignore autocorrelation we obtain unreasonable posterior distributions of the parameters of interest. Also we find that the hypothesis of the asymmetric effect of the changes in oil price on the changes in gasoline price is rejected. Oil inventory has a significant negative effect on the gasoline price.     

Model averaging by jackknife criterion in models with dependent data

The past decade witnessed a literature on model averaging by frequentist methods. For the most part, the asymptotic optimality of various existing frequentist model averaging estimators has been established under i.i.d. errors. Recently, Hansen and Racine [Hansen, B.E., Racine, J., 2012. Jackknife model averaging. Journal of Econometrics 167, 38–46] developed a jackknife model averaging (JMA) estimator, which has an important advantage over its competitors in that it achieves the lowest possible asymptotic squared error under heteroscedastic errors. In this paper, we broaden Hansen and Racine’s scope of analysis to encompass models with (i) a non-diagonal error covariance structure, and (ii) lagged dependent variables, thus allowing for dependent data. We show that under these set-ups, the JMA estimator is asymptotically optimal by a criterion equivalent to that used by Hansen and Racine. A Monte Carlo study demonstrates the finite sample performance of the JMA estimator in a variety of model settings.     

Inference with non-Gaussian Ornstein–Uhlenbeck processes for stochastic volatility

Continuous-time stochastic volatility models are becoming an increasingly popular way to describe moderate and high-frequency financial data. Barndorff-Nielsen and Shephard (2001a) proposed a class of models where the volatility behaves according to an Ornstein–Uhlenbeck (OU) process, driven by a positive Lévy process without Gaussian component. These models introduce discontinuities, or jumps, into the volatility process. They also consider superpositions of such processes and we extend that to the inclusion of a jump component in the returns. In addition, we allow for leverage effects and we introduce separate risk pricing for the volatility components. We design and implement practically relevant inference methods for such models, within the Bayesian paradigm. The algorithm is based on Markov chain Monte Carlo (MCMC) methods and we use a series representation of Lévy processes. MCMC methods for such models are complicated by the fact that parameter changes will often induce a change in the distribution of the representation of the process and the associated problem of overconditioning. We avoid this problem by dependent thinning methods. An application to stock price data shows the models perform very well, even in the face of data with rapid changes, especially if a superposition of processes with different risk premiums and a leverage effect is used.     

Finite sample multivariate structural change tests with application to energy demand models

This paper considers finite sample motivated structural change tests in the multivariate linear regression model with application to energy demand models, in which case commonly used structural change tests remain asymptotic. As in Dufour and Kiviet [1996. Exact tests for structural change in first-order dynamic models. Journal of Econometrics 70, 39–68], we account for intervening nuisance parameters through a two-stage maximized Monte Carlo test procedure. Our contributions can be classified into five categories: (i) we extend tests for which a finite-sample theory has been supplied for Gaussian distributions to the non-Gaussian context; (ii) we show that Bai et al. [1998. Testing and dating common breaks in multi-variate time series. The Review of Economic Studies 65 (3), 395–432] test severely over-rejects and propose exact variants of this test; (iii) we consider predictive break test approaches which generalize tests in Dufour [1980. Dummy variables and predictive tests for structural change. Economics Letters 6, 241–247] and Dufour and Kiviet [1996. Exact tests for structural change in first-order dynamic models. Journal of Econometrics 70, 39–68]; (iv) we propose exact (non-Bonferonni based) extensions of the multivariate outliers test from Wilks [1963. Multivariate statistical outliers. Sankhya Series A 25, 407–426] to models with covariates; (v) we apply these tests to the energy demand system analyzed by Arsenault et al. [1995. A total energy demand model of Québec: forecasting properties. Energy Economics 17 (2), 163–171]. For two out of the six industrial sectors analyzed over the 1962–2000 period, break and further goodness-of-fit and diagnostic tests allow to identify (and correct) specification problems arising from historical regulatory changes or (possibly random) industry-specific effects. The procedures we propose have potential useful applications in statistics, econometrics and finance (e.g. event studies).     

A class of stochastic unit-root bilinear processes: Mixing properties and unit-root test

A class of stochastic unit-root bilinear processes, allowing for GARCH-type effects with asymmetries, is studied. Necessary and sufficient conditions for the strict and second-order stationarity of the error process are given. The strictly stationary solution is shown to be strongly mixing under mild additional assumptions. It follows that, in this model, the standard (non-stochastic) unit-root tests of Phillips–Perron and Dickey–Fuller are asymptotically valid to detect the presence of a (stochastic) unit-root. The finite sample properties of these tests are studied via Monte-Carlo experiments.     

Estimation of stable distributions by indirect inference

This article deals with the estimation of the parameters of an α-stable distribution with indirect inference, using the skewed-t distribution as an auxiliary model. The latter distribution appears as a good candidate since it has the same number of parameters as the α-stable distribution, with each parameter playing a similar role. To improve the properties of the estimator in finite sample, we use constrained indirect inference. In a Monte Carlo study we show that this method delivers estimators with good properties in finite sample. We provide an empirical application to the distribution of jumps in the S&P 500 index returns.     

Econometric analysis of linearized singular dynamic stochastic general equilibrium models

In this paper I propose an alternative to calibration of linearized singular dynamic stochastic general equilibrium models. Given an a-theoretical econometric model as a representative of the data generating process, I will construct an information measure which compares the conditional distribution of the econometric model variables with the corresponding singular conditional distribution of the theoretical model variables. The singularity problem will be solved by using convolutions of both distributions with a non-singular distribution. This information measure will then be maximized to the deep parameters of the theoretical model, which links these parameters to the parameters of the econometric model and provides an alternative to calibration. This approach will be illustrated by an application to a linearized version of the stochastic growth model of King, Plosser and Rebelo.     

Estimating distributions of potential outcomes using local instrumental variables with an application to changes in college enrollment and wage inequality

This paper extends the method of local instrumental variables developed by Heckman and Vytlacil [Heckman, J., Vytlacil E., 2005. Structural equations, treatment, effects and econometric policy evaluation. Econometrica 73(3), 669–738] to the estimation of not only means, but also distributions of potential outcomes. The newly developed method is illustrated by applying it to changes in college enrollment and wage inequality using data from the National Longitudinal Survey of Youth of 1979. Increases in college enrollment cause changes in the distribution of ability among college and high school graduates. This paper estimates a semiparametric selection model of schooling and wages to show that, for fixed skill prices, a 14% increase in college participation (analogous to the increase observed in the 1980s), reduces the college premium by 12% and increases the 90–10 percentile ratio among college graduates by 2%.