Conditional Error Distribution Research Articles

A semiparametric approach to the detection of change-points in volatility dynamics of financial data

Volatility is one of the most important features of financial time series data. There are often structural changes in volatility over time, and an accurate estimation of the volatility of financial time series requires careful identification of change-points. A common approach to modeling the volatility of time series data is based on the well-known GARCH model. Although the problem of change-point estimation of volatility dynamics derived from the GARCH model has been considered in the literature, these approaches rely on parametric assumptions of the conditional error distribution, which are often violated in financial time series. This may lead to inaccuracies in change-point detection, resulting in unreliable GARCH volatility estimates. This article introduces a novel change-point detection algorithm based on a semiparametric GARCH model. The proposed method retains the structural advantages of the GARCH process while incorporating the flexibility of nonparametric conditional error distribution. The approach utilizes a penalized likelihood derived from a semiparametric GARCH model and an efficient binary segmentation algorithm. The results show that in terms of change-point estimation and detection accuracy, the semiparametric method outperforms the commonly used Quasi-MLE (QMLE) and other variations of the GARCH models in wide-ranging scenarios.

Modeling stock market volatility in Croatia

Purpose: In this paper, the volatility of the Croatian stock market index CROBEX is investigated using the GARCH(1,1) model. Methodology: The novelty provided by this paper is the estimation of the GARCH(1,1) model by using three conditional error distributions (normal (Gaussian) distribution, Student’s-distribution with fixed degrees of freedom and generalized error distribution (GED) with fixed parameters). Results: The findings obtained in the research are in the line with previous research in this field (Erjavec & Cota, 2007; Sajter & Ćorić, 2009). The volatility of CROBEX returns is positively correlated with the volume of trade on the Zagreb Stock Exchange and movements on the main European and American stock markets. The movement of S&P 500 stock market index returns is transmitted from the previous day, providing signals for the direction of change of CROBEX index returns in the present. Conclusion: Therefore, this paper provides evidence that investors in Croatia strongly rely on the past information received from the American S&P500 stock market index. Furthermore, there seems to exist the co-movement between CROBEX and main European indexes on the same trading day.

Bayesian estimation and inference for log-ACD models

This paper adapts Bayesian Markov chain Monte Carlo methods for application to some auto-regressive conditional duration models. Subsequently, the properties of these estimators are examined and assessed across a range of possible conditional error distributions and dynamic specifications, including under error mis-specification. A novel model error distribution, employing a truncated skewed Student-t distribution is proposed and the Bayesian estimator assessed for it. The results of an extensive simulation study reveal that favourable estimation properties are achieved under a range of possible error distributions, but that the generalised gamma distribution assumption is most robust and best preserves these properties, including when it is incorrectly specified. The results indicate that the powerful numerical methods underlying the Bayesian estimator allow more efficiency than the (quasi-) maximum likelihood estimator for the cases considered.

Power arch modelling of the volatility of emerging equity markets

In developed equity markets the APARCH model of Ding, Granger and Engle [Ding, Z., Granger, C. and Engle, R., 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance 1, 83–106] has proven to be useful in modelling the leverage and asymmetry effects; power transformations and long memory; and non-normal conditional error distributions that characterise the data. Extending the analysis of Jayasuriya, Shambora and Rossiter [Jayasuriya, S., Shambora, W. and Rossiter, R., 2005. Asymmetric volatility in mature and emerging markets, Working Paper, Ohio University.] to a wider set of emerging markets this paper explores the applicability of the model to emerging markets. The key findings are as follows. First, unlike developed markets where a power term of unity and a conditional standard deviation model appears to be appropriate, emerging markets demonstrate a considerably greater range of power values. Second, unlike developed markets where non-normal conditional error distributions appear to fit the data well, there are a set of emerging markets for which estimation problems arise with a conditional t distribution, and a conditional normal distribution appears to be the preferred option. Third, the degree of volatility asymmetry appears to vary across the set of emerging markets, with the Middle Eastern and African markets having very different volatility asymmetry characteristics to those of the Latin American markets.

SEMIPARAMETRIC ESTIMATION OF SEPARABLE MODELS WITH POSSIBLY LIMITED DEPENDENT VARIABLES

In this paper we introduce a general method for estimating semiparametrically the different components in separable models. The family of separable models is quite popular in economic research because this structure offers clear interpretation, has straightforward economic consequences, and is often justified by theory. This family is also of statistical interest because it allows us to estimate high-dimensional complexity semiparametrically without running into the curse of dimensionality. We consider even the case when multiple indices appear in the objective function; thus we can estimate models that are typical in economic analysis, such as those that contain limited dependent variables. The idea of the new method is mainly based on a generalized profile likelihood approach. Although this requires some hypotheses on the conditional error distribution, it yields a quite general usable method with low computational costs but high accuracy even for small samples. We give estimation procedures and provide some asymptotic theory. Implementation is discussed; simulations and an application demonstrate its feasibility and good finite-sample behavior.This research was financially supported by Dirección General de Investigación del Ministerio de Ciencia y Tecnología under research grants BEC2001-1121 and BEC2001-1270; Dirección General de Enseñanza Superior del Ministerio de Educación y Ciencia under Subprograma de estancias de investigadores españoles en centros de investigación españoles y extranjeros, ref. PR2000-0096; and by the Danish Social Science Research fund. We also thank M. Delgado, O. Linton, two anonymous referees, and all participants of the working group STAPH on functional statistics in Toulouse, the activities of which are available on line at http://www.lsp.upstlse.fr/Fp/Ferraty/staph.html.

Mean/Variance Relation and the Conditional Distribution

This paper examines the relation between the expected return and the conditional variance using three conditional error distributions 1) the conditional normal error distribution, 2) the Generalized Error Distribution, and 3) the skewed student's t-distribution. Using a GARCH-M model modified by allowing skewness in mean, we find support for a significant and positive mean/variance relation when the skewed student's t-distribution is used. Our results show that the time variations in conditional skewness influence the dynamics of the conditional mean and conditional variance, as reflected by the reduced volatility persistence and a significant mean/variance relationship. This further stresses the point that there is an intimate relation between return, volatility and skewness; within the GARCH-M framework, conditional skewness plays a role analogous to heteroskedasticity by smoothing out the conditional mean and conditional variance process.

Sieve Empirical Likelihood and Extensions of the Generalized Least Squares

The empirical likelihood cannot be used directly sometimes when an infinite dimensional parameter of interest is involved. To overcome this difficulty, the sieve empirical likelihoods are introduced in this paper. Based on the sieve empirical likelihoods, a unified procedure is developed for estimation of constrained parametric or non‐parametric regression models with unspecified error distributions. It shows some interesting connections with certain extensions of the generalized least squares approach. A general asymptotic theory is provided. In the parametric regression setting it is shown that under certain regularity conditions the proposed estimators are asymptotically efficient even if the restriction functions are discontinuous. In the non‐parametric regression setting the convergence rate of the maximum estimator based on the sieve empirical likelihood is given. In both settings, it is shown that the estimator is adaptive for the inhomogeneity of conditional error distributions with respect to predictor, especially for heteroscedasticity.

An Assessment of Empirical Model Performance When Financial Market Transactions are Observed at Different Data Frequencies: An Application to East Asian Exchange Rates

This paper compares the performance of alternative models of east Asian exchange rates at different data frequencies. Selected models employ different specifications of the conditional variance and the conditional error distribution. Conditional variance specifications include: homoscedasticity, GARCH, LGARCH, and EGARCH. Conditional error distribution specifications include normal and Student t. The best exchange rate model specification is clearly conditional on data frequency. Higher frequency (daily, weekly) data commonly exhibit characteristics that demand more sophisticated estimation methods than analysts commonly employ. These characteristics generally vanish at lower (monthly, quarterly) frequencies. Overall we find significant benefit from accommodating heteroscedasticity and leptokurtic properties of the conditional distribution as data frequency increases. Using a likelihood ratio test we compare the relative gain from addressing heteroscedasticity (through use of GARCH models) versus accommodation of leptokurtosis. This comparison suggests that the gains from correct specification of the conditional distribution dominate those obtained from addressing problems of heteroscedasticity.

Robust inference with GMM estimators

The local robustness properties of generalized method of moments (GMM) estimators and of a broad class of GMM based tests are investigated in a unified framework. GMM statistics are shown to have bounded influence if and only if the function defining the orthogonality restrictions imposed on the underlying model is bounded. Since in many applications this function is unbounded, it is useful to have procedures that modify the starting orthogonality conditions in order to obtain a robust version of a GMM estimator or test. We show how this can be obtained when a reference model for the data distribution can be assumed. We develop a flexible algorithm for constructing a robust GMM (RGMM) estimator leading to stable GMM test statistics. The amount of robustness can be controlled by an appropriate tuning constant. We relate by an explicit formula the choice of this constant to the maximal admissible bias on the level or (and) the power of a GMM test and the amount of contamination that one can reasonably assume given some information on the data. Finally, we illustrate the RGMM methodology with some simulations of an application to RGMM testing for conditional heteroscedasticity in a simple linear autoregressive model. In this example we find a significant instability of the size and the power of a classical GMM testing procedure under a non-normal conditional error distribution. On the other side, the RGMM testing procedures can control the size and the power of the test under non-standard conditions while maintaining a satisfactory power under an approximatively normal conditional error distribution.

Semiparametric qualitative response model estimation with unknown heteroscedasticity or instrumental variables

This paper provides estimators of discrete choice models, including binary, ordered, and multinomial response (choice) models. The estimators closely resemble ordinary and two-stage least squares. The distribution of the model's latent variable error is unknown and may be related to the regressors, e.g., the model could have errors that are heteroscedastic or correlated with regressors. The estimator does not require numerical searches, even for multinomial choice. For ordered and binary choice models the estimator is root N consistent and asymptotically normal. A consistent estimator of the conditional error distribution is also provided.

Estimating EGARCH-M models: Science or art?

This paper shows that the EGARCH-M model should be estimated with caution. Regardless of the assumption made regarding the conditional error distribution, the EGARCH-M model is sensitive to the choice of starting values and the degree of computer precision. However, a simple forecasting example shows that these issues only become a concern when the EGARCH-M model is estimated on a high-precision computer, under the assumption that the conditional errors follow a generalized error distribution. The instability of the risk aversion parameter in the mean equation is the greatest influence on these results.

Nearest Neighbor Regression with Heavy-Tailed Errors

There has been an increasing interest in modelling regression with heavy-tailed conditional error distributions, mostly in the parametric setting. Nonparametric regression procedures have been studied almost exclusively for the cases where the conditional variance of the regressed variable is finite in the region of interest. We initiate a study of the infinite variance case. Some results in strong uniform consistency of the nearest neighbor estimator with rates are proven. The technique used provides new results and insights when higher conditional moments exist. Some asymptotic distribution theory has also been obtained when the conditional errors are in the domain of attraction of a stable law.

Structuring rainfall-landuse-runoff models for a large catchment in N.S.W.

To retain the advantages of conceptual modelling while addressing the difficulty of parameter estimation, a procedure is proposed in which model structure is developed progressively, with model performance and associated parameter uncertainty being evaluated at each stage, to determine whether and which further processes should be added. When developing a structure to model runoff from a large catchment, a combination of global statistical performance measures and information concerning conditional error distribution was required; the former to enable the optimal performance of each structural variant to be compared, the latter to assess conformity of the errors to standard statistical assumptions and hence to indicate how the current variant should be augmented.