Interval estimation of value-at-risk based on GARCH models with heavy-tailed innovations

ARCH and GARCH models are widely used to model financial market volatilities in risk management applications. Considering a GARCH model with heavy-tailed innovations, we characterize the limiting distribution of an estimator of the conditional value-at-risk (VaR), which corresponds to the extremal quantile of the conditional distribution of the GARCH process. We propose two methods, the normal approximation method and the data tilting method, for constructing confidence intervals for the conditional VaR estimator and assess their accuracies by simulation studies. Finally, we apply the proposed approach to an energy market data set.     

Generalized R-estimators under conditional heteroscedasticity

In this paper, we extend the classical idea of Rank estimation of parameters from homoscedastic problems to heteroscedastic problems. In particular, we define a class of rank estimators of the parameters associated with the conditional mean function of an autoregressive model through a three-steps procedure and then derive their asymptotic distributions. The class of models considered includes Engel's ARCH model and the threshold heteroscedastic model. The class of estimators includes an extension of Wilcoxon-type rank estimator. The derivation of the asymptotic distributions depends on the uniform approximation of a randomly weighted empirical process by a perturbed empirical process through a very general weight-dependent partitioning argument.     

Nonparametric stochastic frontiers: A local maximum likelihood approach

This paper proposes a new approach to handle nonparametric stochastic frontier (SF) models. It is based on local maximum likelihood techniques. The model is presented as encompassing some anchorage parametric model in a nonparametric way. First, we derive asymptotic properties of the estimator for the general case (local linear approximations). Then the results are tailored to a SF model where the convoluted error term (efficiency plus noise) is the sum of a half normal and a normal random variable. The parametric anchorage model is a linear production function with a homoscedastic error term. The local approximation is linear for both the production function and the parameters of the error terms. The performance of our estimator is then established in finite samples using simulated data sets as well as with a cross-sectional data on US commercial banks. The methods appear to be robust, numerically stable and particularly useful for investigating a production process and the derived efficiency scores.     

Optimal convergence rates,Bahadur representation,and asymptotic normality of partitioning estimators

This paper studies the asymptotic properties of partitioning estimators of the conditional expectation function and its derivatives. Mean-square and uniform convergence rates are established and shown to be optimal under simple and intuitive conditions. The uniform rate explicitly accounts for the effect of moment assumptions, which is useful in semiparametric inference. A general asymptotic integrated mean-square error approximation is obtained and used to derive an optimal plug-in tuning parameter selector. A uniform Bahadur representation is developed for linear functionals of the estimator. Using this representation, asymptotic normality is established, along with consistency of a standard-error estimator. The finite-sample performance of the partitioning estimator is examined and compared to other nonparametric techniques in an extensive simulation study.     

A smoothed least squares estimator for threshold regression models

We propose a smoothed least squares estimator of the parameters of a threshold regression model. Our model generalizes that considered in Hansen [2000. Sample splitting and threshold estimation. Econometrica 68, 575–603] to allow the thresholding to depend on a linear index of observed regressors, thus allowing discrete variables to enter. We also do not assume that the threshold effect is vanishingly small. Our estimator is shown to be consistent and asymptotically normal thus facilitating standard inference techniques based on estimated standard errors or standard bootstrap for the slope and threshold parameters.     

Testing for ARCH in the presence of nonlinearity of unknown form in the conditional mean

Tests of ARCH are a routine diagnostic in empirical econometric and financial analysis. However, it is well known that misspecification of the conditional mean may lead to spurious rejection of the null hypothesis of no ARCH. Nonlinearity is a prime example of this phenomenon. There is little work on the extent of the effect of neglected nonlinearity on the properties of ARCH tests. We investigate this using new ARCH testing procedures that are robust to the presence of neglected nonlinearity. Monte Carlo evidence shows that the problem is serious and that the new methods alleviate this problem to a very large extent. We apply the new tests to exchange rate data and find substantial evidence of spurious rejection of the null hypothesis of no ARCH.     

GMM Estimation with Non‐causal Instruments under Rational Expectations*

Lanne and Saikkonen [Oxford Bulletin of Economics and Statistics (2011a) Vol. 73, pp. 581–592], show that the generalized method of moments (GMM) estimator is inconsistent, when the instruments are lags of variables that admit a non‐causal autoregressive representation. This article argues that this inconsistency depends on distributional assumptions, that do not always hold. In particular under rational expectations, the GMM estimator is found to be consistent. This result is derived in a linear context and illustrated by simulation of a nonlinear asset pricing model.     

The method of elimination and substitution in the GMM estimation of mixed regressive,spatial autoregressive models

This paper proposes a computationally simple GMM for the estimation of mixed regressive spatial autoregressive models. The proposed method explores the advantage of the method of elimination and substitution in linear algebra. The modified GMM approach reduces the joint (nonlinear) estimation of a complete vector of parameters into estimation of separate components. For the mixed regressive spatial autoregressive model, the nonlinear estimation is reduced to the estimation of the (single) spatial effect parameter. We identify situations under which the resulting estimator can be efficient relative to the joint GMM estimator where all the parameters are jointly estimated.     

Learning in a multilateral bargaining experiment

This paper analyzes data from an investigation of a majoritarian bargaining experiment. A learning model is proposed to account for the evolution of play in this experiment. It is also suggested that an adjustment must be made to account for the panel structure of the data. Such adjustments have been used in other fields and are known to be important as unadjusted standard errors may be severely biased downward. These results indicate that this adjustment also has an important effect in this application. Furthermore, an efficient estimator that takes into account heterogeneity across players is proposed. A unique learning model to account for the paths of play under two different amendment rules cannot be rejected with the standard estimator with adjusted standard errors, however it can be rejected using the efficient estimator. The data and the estimated learning model suggest that after proposing “fair” divisions, subjects adapt and their proposals change rapidly in the treatment where uneven proposals are almost always accepted. Their beliefs in the estimated learning model are influenced by more than just the most recent outcomes.     

Sparse estimators and the oracle property,or the return of Hodges’ estimator

We point out some pitfalls related to the concept of an oracle property as used in Fan and Li [2001. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360; 2002. Variable selection for Cox's proportional hazards model and frailty model. Annals of Statistics 30, 74–99; 2004. New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. Journal of the American Statistical Association 99, 710–723] which are reminiscent of the well-known pitfalls related to Hodges’ estimator. The oracle property is often a consequence of sparsity of an estimator. We show that any estimator satisfying a sparsity property has maximal risk that converges to the supremum of the loss function; in particular, the maximal risk diverges to infinity whenever the loss function is unbounded. For ease of presentation the result is set in the framework of a linear regression model, but generalizes far beyond that setting. In a Monte Carlo study we also assess the extent of the problem in finite samples for the smoothly clipped absolute deviation (SCAD) estimator introduced in Fan and Li [2001. Variable selection via nonconcave penalized likelihood and its oracle properties. Journal of the American Statistical Association 96, 1348–1360]. We find that this estimator can perform rather poorly in finite samples and that its worst-case performance relative to maximum likelihood deteriorates with increasing sample size when the estimator is tuned to sparsity.     

Jackknife estimation of stationary autoregressive models

This paper explores the properties of jackknife methods of estimation in stationary autoregressive models. Some general results concerning the correct weights for bias reduction under various sampling schemes are provided and the asymptotic properties of a jackknife estimator based on non-overlapping sub-samples are derived for the case of a stationary autoregression of order p$p$ when the number of sub-samples is either fixed or increases with the sample size at an appropriate rate. The results of a detailed investigation into the finite sample properties of various jackknife and alternative estimators are reported and it is found that the jackknife can deliver substantial reductions in bias in autoregressive models. This finding is robust to departures from normality, ARCH effects and misspecification. The median-unbiasedness and mean squared error properties are also investigated and compared with alternative methods as are the coverage rates of jackknife-based confidence intervals.     

Least squares model averaging by Mallows criterion

This paper is in response to a recent paper by Hansen (2007) who proposed an optimal model average estimator with weights selected by minimizing a Mallows criterion. The main contribution of Hansen’s paper is a demonstration that the Mallows criterion is asymptotically equivalent to the squared error, so the model average estimator that minimizes the Mallows criterion also minimizes the squared error in large samples. We are concerned with two assumptions that accompany Hansen’s approach. The first is the assumption that the approximating models are strictly nested in a way that depends on the ordering of regressors. Often there is no clear basis for the ordering and the approach does not permit non-nested models which are more realistic from a practical viewpoint. Second, for the optimality result to hold the model weights are required to lie within a special discrete set. In fact, Hansen noted both difficulties and called for extensions of the proof techniques. We provide an alternative proof which shows that the result on the optimality of the Mallows criterion in fact holds for continuous model weights and under a non-nested set-up that allows any linear combination of regressors in the approximating models that make up the model average estimator. These results provide a stronger theoretical basis for the use of the Mallows criterion in model averaging by strengthening existing findings.     

Pseudo conditional maximum likelihood estimation of the dynamic logit model for binary panel data

We show how the dynamic logit model for binary panel data may be approximated by a quadratic exponential model. Under the approximating model, simple sufficient statistics exist for the subject-specific parameters introduced to capture the unobserved heterogeneity between subjects. The latter must be distinguished from the state dependence which is accounted for by including the lagged response variable among the regressors. By conditioning on the sufficient statistics, we derive a pseudo conditional likelihood estimator of the structural parameters of the dynamic logit model, which is simple to compute. Asymptotic properties of this estimator are studied in detail. Simulation results show that the estimator is competitive in terms of efficiency with estimators recently proposed in the econometric literature.     

Endogeneity in quantile regression models: A control function approach

This paper considers a linear triangular simultaneous equations model with conditional quantile restrictions. The paper adjusts for endogeneity by adopting a control function approach and presents a simple two-step estimator that exploits the partially linear structure of the model. The first step consists of estimation of the residuals of the reduced-form equation for the endogenous explanatory variable. The second step is series estimation of the primary equation with the reduced-form residual included nonparametrically as an additional explanatory variable. This paper imposes no functional form restrictions on the stochastic relationship between the reduced-form residual and the disturbance term in the primary equation conditional on observable explanatory variables. The paper presents regularity conditions for consistency and asymptotic normality of the two-step estimator. In addition, the paper provides some discussions on related estimation methods in the literature.     

Dynamic quantile models

This paper introduces the Dynamic Additive Quantile (DAQ) model that ensures the monotonicity of conditional quantile estimates. The DAQ model is easily estimable and can be used for computation and updating of the Value-at-Risk. An asymptotically efficient estimator of the DAQ is obtained by maximizing an objective function based on the inverse KLIC measure. An alternative estimator proposed in the paper is the Method of L-Moments estimator (MLM). The MLM estimator is consistent, but generally not fully efficient. Goodness-of-fit tests and diagnostic tools for the assessment of the model are also provided. For illustration, the DAQ model is estimated from a series of returns on the Toronto Stock Exchange (TSX) market index.     

Local structural quantile effects in a model with a nonseparable control variable

I consider a semiparametric version of the nonseparable triangular model of Chesher [Chesher, A., 2003. Identification in nonseparable models. Econometrica 71, 1405–1441]. The proposed model is linear in coefficients, where the coefficients are unknown functions of unobserved latent variables. Using a control variable idea and quantile regression methods, I propose a simple two-step estimator for the coefficients evaluated at particular values of the latent variables. Under the condition that the instruments are locally relevant (i.e. they affect a particular conditional quantile of interest of the endogenous variable) I establish consistency and asymptotic normality. Simulation experiments confirm the theoretical results.     

Smoothly mixing regressions

This paper extends the conventional Bayesian mixture of normals model by permitting state probabilities to depend on observed covariates. The dependence is captured by a simple multinomial probit model. A conventional and rapidly mixing MCMC algorithm provides access to the posterior distribution at modest computational cost. This model is competitive with existing econometric models, as documented in the paper's illustrations. The first illustration studies quantiles of the distribution of earnings of men conditional on age and education, and shows that smoothly mixing regressions are an attractive alternative to nonBayesian quantile regression. The second illustration models serial dependence in the S&P 500 return, and shows that the model compares favorably with ARCH models using out of sample likelihood criteria.     

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.     

On efficient estimation of the ordered response model

In this paper, we derive efficiency bounds for the ordered response model when the distribution of the errors is unknown. Furthermore, we develop an estimator that is efficient under suitable conditions. Interestingly, neither the bounds nor the estimator are trivial extensions of what has been proposed in the literature for the binary response model. The estimator is composed of quadratic B-splines, and estimation is performed by the method of sieves. In addition, the estimator of the distribution function is restricted to be a proper distribution function. An empirical example on the effect of fees on attendance rates at universities and community colleges is also included; we get substantively different results by relaxing the assumption that the distribution of the errors is normal.     

Modelling optimal strategies for the allocation of wealth in multicurrency investments

This paper analyzes rates of return on financial assets denominated in five major currencies and provides a framework for the determination of optimal strategies for the allocation of wealth in multicurrency investments. Three models are estimated: a univariate autoregressive conditional heteroskedasticity (ARCH) model, an extended ARCH model using the random coefficient (RC) procedure, and a pure RC model. A comparison of the forecasts of these models with those generated by a random walk model demonstrates that forecasts based on the RC/extended ARCH procedure are superior to those based on the random walk model and those based on direct ARCH estimation. These results could be useful for both international investors for the allocation of their wealth among fixed-income investment securities and central banks for the management of their external reserve assets.