Evaluating monthly volatility forecasts using proxies at different frequencies

This paper analyses the forecastability of stock returns monthly volatility. The forecast obtained from GARCH and AGARCH models with Normal and Student's t errors are evaluated with respect to proxies for the unobserved volatility obtained through sampling at different frequencies. It is found that aggregation of daily multi-step ahead GARCH-type forecasts provide rather accurate predictions of monthly volatility.

Inteligência artificial aplicada na previsão de mercados financeiros

Neste documento, são investigados vários métodos usados na inteligência artificial, com o objetivo de obter previsões precisas da evolução dos mercados financeiros. O uso de ferramentas lineares como os modelos AR, MA, ARMA e GARCH têm muitas limitações, pois torna-se muito difícil adaptá-los às não linearidades dos fenómenos que ocorrem nos mercados. Pelas razões anteriormente referidas, os algoritmos como as redes neuronais dinâmicas (TDNN, NARX e ESN), mostram uma maior capacidade de adaptação a estas não linearidades, pois não fazem qualquer pressuposto sobre as distribuições de probabilidade que caracterizam estes mercados. O facto destas redes neuronais serem dinâmicas, faz com que estas exibam um desempenho superior em relação às redes neuronais estáticas, ou outros algoritmos que não possuem qualquer tipo de memória. Apesar das vantagens reveladas pelas redes neuronais, estas são um sistema do tipo black box, o que torna muito difícil extrair informação dos pesos da rede. Isto significa que estes algoritmos devem ser usados com precaução, pois podem tornar-se instáveis.

Modeling and forecasting value-at-risk for the Portuguese stock market

The aim of this work project is to find a model that is able to accurately forecast the daily Value-at-Risk for PSI-20 Index, independently of the market conditions, in order to expand empirical literature for the Portuguese stock market. Hence, two subsamples, representing more and less volatile periods, were modeled through unconditional and conditional volatility models (because it is what drives returns). All models were evaluated through Kupiec’s and Christoffersen’s tests, by comparing forecasts with actual results. Using an out-of-sample of 204 observations, it was found that a GARCH(1,1) is an accurate model for our purposes.

Improving accuracy of inflation forecasts using macroeconomic information

This thesis examines the effects of macroeconomic factors on inflation level and volatility in the Euro Area to improve the accuracy of inflation forecasts with econometric modelling. Inflation aggregates for the EU as well as inflation levels of selected countries are analysed, and the difference between these inflation estimates and forecasts are documented. The research proposes alternative models depending on the focus and the scope of inflation forecasts. I find that models with a Generalized AutoRegressive Conditional Heteroskedasticity (GARCH) in mean process have better explanatory power for inflation variance compared to the regular GARCH models. The significant coefficients are different in EU countries in comparison to the aggregate EU-wide forecast of inflation. The presence of more pronounced GARCH components in certain countries with more stressed economies indicates that inflation volatility in these countries are likely to occur as a result of the stressed economy. In addition, other economies in the Euro Area are found to exhibit a relatively stable variance of inflation over time. Therefore, when analysing EU inflation one have to take into consideration the large differences on country level and focus on those one by one.

The information content of Canadian implied volatility indexes

For predicting future volatility, empirical studies find mixed results regarding two issues: (1) whether model free implied volatility has more information content than Black-Scholes model-based implied volatility; (2) whether implied volatility outperforms historical volatilities. In this thesis, we address these two issues using the Canadian financial data. First, we examine the information content and forecasting power between VIXC - a model free implied volatility, and MVX - a model-based implied volatility. The GARCH in-sample test indicates that VIXC subsumes all information that is reflected in MVX. The out-of-sample examination indicates that VIXC is superior to MVX for predicting the next 1-, 5-, 10-, and 22-trading days' realized volatility. Second, we investigate the predictive power between VIXC and alternative volatility forecasts derived from historical index prices. We find that for time horizons lesser than 10-trading days, VIXC provides more accurate forecasts. However, for longer time horizons, the historical volatilities, particularly the random walk, provide better forecasts. We conclude that VIXC cannot incorporate all information contained in historical index prices for predicting future volatility.

Simulation-Based Finite-Sample Tests for Heteroskedasticity and ARCH Effects.

A wide range of tests for heteroskedasticity have been proposed in the econometric and statistics literature. Although a few exact homoskedasticity tests are available, the commonly employed procedures are quite generally based on asymptotic approximations which may not provide good size control in finite samples. There has been a number of recent studies that seek to improve the reliability of common heteroskedasticity tests using Edgeworth, Bartlett, jackknife and bootstrap methods. Yet the latter remain approximate. In this paper, we describe a solution to the problem of controlling the size of homoskedasticity tests in linear regression contexts. We study procedures based on the standard test statistics [e.g., the Goldfeld-Quandt, Glejser, Bartlett, Cochran, Hartley, Breusch-Pagan-Godfrey, White and Szroeter criteria] as well as tests for autoregressive conditional heteroskedasticity (ARCH-type models). We also suggest several extensions of the existing procedures (sup-type of combined test statistics) to allow for unknown breakpoints in the error variance. We exploit the technique of Monte Carlo tests to obtain provably exact p-values, for both the standard and the new tests suggested. We show that the MC test procedure conveniently solves the intractable null distribution problem, in particular those raised by the sup-type and combined test statistics as well as (when relevant) unidentified nuisance parameter problems under the null hypothesis. The method proposed works in exactly the same way with both Gaussian and non-Gaussian disturbance distributions [such as heavy-tailed or stable distributions]. The performance of the procedures is examined by simulation. The Monte Carlo experiments conducted focus on : (1) ARCH, GARCH, and ARCH-in-mean alternatives; (2) the case where the variance increases monotonically with : (i) one exogenous variable, and (ii) the mean of the dependent variable; (3) grouped heteroskedasticity; (4) breaks in variance at unknown points. We find that the proposed tests achieve perfect size control and have good power.

A Theoretical Comparison Between Integrated and Realized Volatilies

In this paper, we provide both qualitative and quantitative measures of the cost of measuring the integrated volatility by the realized volatility when the frequency of observation is fixed. We start by characterizing for a general diffusion the difference between the realized and the integrated volatilities for a given frequency of observations. Then, we compute the mean and variance of this noise and the correlation between the noise and the integrated volatility in the Eigenfunction Stochastic Volatility model of Meddahi (2001a). This model has, as special examples, log-normal, affine, and GARCH diffusion models. Using some previous empirical works, we show that the standard deviation of the noise is not negligible with respect to the mean and the standard deviation of the integrated volatility, even if one considers returns at five minutes. We also propose a simple approach to capture the information about the integrated volatility contained in the returns through the leverage effect.

Testing Normality : A GMM Approach

In this paper, we consider testing marginal normal distributional assumptions. More precisely, we propose tests based on moment conditions implied by normality. These moment conditions are known as the Stein (1972) equations. They coincide with the first class of moment conditions derived by Hansen and Scheinkman (1995) when the random variable of interest is a scalar diffusion. Among other examples, Stein equation implies that the mean of Hermite polynomials is zero. The GMM approach we adopted is well suited for two reasons. It allows us to study in detail the parameter uncertainty problem, i.e., when the tests depend on unknown parameters that have to be estimated. In particular, we characterize the moment conditions that are robust against parameter uncertainty and show that Hermite polynomials are special examples. This is the main contribution of the paper. The second reason for using GMM is that our tests are also valid for time series. In this case, we adopt a Heteroskedastic-Autocorrelation-Consistent approach to estimate the weighting matrix when the dependence of the data is unspecified. We also make a theoretical comparison of our tests with Jarque and Bera (1980) and OPG regression tests of Davidson and MacKinnon (1993). Finite sample properties of our tests are derived through a comprehensive Monte Carlo study. Finally, three applications to GARCH and realized volatility models are presented.

Testing Mean-Variance Efficiency in CAPM with Possibly Non-Gaussian Errors : An Exact Simulation-Based Approach

In this paper we propose exact likelihood-based mean-variance efficiency tests of the market portfolio in the context of Capital Asset Pricing Model (CAPM), allowing for a wide class of error distributions which include normality as a special case. These tests are developed in the frame-work of multivariate linear regressions (MLR). It is well known however that despite their simple statistical structure, standard asymptotically justified MLR-based tests are unreliable. In financial econometrics, exact tests have been proposed for a few specific hypotheses [Jobson and Korkie (Journal of Financial Economics, 1982), MacKinlay (Journal of Financial Economics, 1987), Gib-bons, Ross and Shanken (Econometrica, 1989), Zhou (Journal of Finance 1993)], most of which depend on normality. For the gaussian model, our tests correspond to Gibbons, Ross and Shanken’s mean-variance efficiency tests. In non-gaussian contexts, we reconsider mean-variance efficiency tests allowing for multivariate Student-t and gaussian mixture errors. Our framework allows to cast more evidence on whether the normality assumption is too restrictive when testing the CAPM. We also propose exact multivariate diagnostic checks (including tests for multivariate GARCH and mul-tivariate generalization of the well known variance ratio tests) and goodness of fit tests as well as a set estimate for the intervening nuisance parameters. Our results [over five-year subperiods] show the following: (i) multivariate normality is rejected in most subperiods, (ii) residual checks reveal no significant departures from the multivariate i.i.d. assumption, and (iii) mean-variance efficiency tests of the market portfolio is not rejected as frequently once it is allowed for the possibility of non-normal errors.

ARMA Representation of Integrated and Realized Variances

This paper derives the ARMA representation of integrated and realized variances when the spot variance depends linearly on two autoregressive factors, i.e., SR SARV(2) models. This class of processes includes affine, GARCH diffusion, CEV models, as well as the eigenfunction stochastic volatility and the positive Ornstein-Uhlenbeck models. We also study the leverage effect case, the relationship between weak GARCH representation of returns and the ARMA representation of realized variances. Finally, various empirical implications of these ARMA representations are considered. We find that it is possible that some parameters of the ARMA representation are negative. Hence, the positiveness of the expected values of integrated or realized variances is not guaranteed. We also find that for some frequencies of observations, the continuous time model parameters may be weakly or not identified through the ARMA representation of realized variances.

Finite-Sample Diagnostics for Multivariate Regressions with Applications to Linear Asset Pricing Models

In this paper, we propose several finite-sample specification tests for multivariate linear regressions (MLR) with applications to asset pricing models. We focus on departures from the assumption of i.i.d. errors assumption, at univariate and multivariate levels, with Gaussian and non-Gaussian (including Student t) errors. The univariate tests studied extend existing exact procedures by allowing for unspecified parameters in the error distributions (e.g., the degrees of freedom in the case of the Student t distribution). The multivariate tests are based on properly standardized multivariate residuals to ensure invariance to MLR coefficients and error covariances. We consider tests for serial correlation, tests for multivariate GARCH and sign-type tests against general dependencies and asymmetries. The procedures proposed provide exact versions of those applied in Shanken (1990) which consist in combining univariate specification tests. Specifically, we combine tests across equations using the MC test procedure to avoid Bonferroni-type bounds. Since non-Gaussian based tests are not pivotal, we apply the “maximized MC” (MMC) test method [Dufour (2002)], where the MC p-value for the tested hypothesis (which depends on nuisance parameters) is maximized (with respect to these nuisance parameters) to control the test’s significance level. The tests proposed are applied to an asset pricing model with observable risk-free rates, using monthly returns on New York Stock Exchange (NYSE) portfolios over five-year subperiods from 1926-1995. Our empirical results reveal the following. Whereas univariate exact tests indicate significant serial correlation, asymmetries and GARCH in some equations, such effects are much less prevalent once error cross-equation covariances are accounted for. In addition, significant departures from the i.i.d. hypothesis are less evident once we allow for non-Gaussian errors.

Affine and generalized affine models : Theory and applications

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Les modèles vectoriels et multiplicatifs avec erreurs non-négatives de séries chronologiques.

L'objectif du présent mémoire vise à présenter des modèles de séries chronologiques multivariés impliquant des vecteurs aléatoires dont chaque composante est non-négative. Nous considérons les modèles vMEM (modèles vectoriels et multiplicatifs avec erreurs non-négatives) présentés par Cipollini, Engle et Gallo (2006) et Cipollini et Gallo (2010). Ces modèles représentent une généralisation au cas multivarié des modèles MEM introduits par Engle (2002). Ces modèles trouvent notamment des applications avec les séries chronologiques financières. Les modèles vMEM permettent de modéliser des séries chronologiques impliquant des volumes d'actif, des durées, des variances conditionnelles, pour ne citer que ces applications. Il est également possible de faire une modélisation conjointe et d'étudier les dynamiques présentes entre les séries chronologiques formant le système étudié. Afin de modéliser des séries chronologiques multivariées à composantes non-négatives, plusieurs spécifications du terme d'erreur vectoriel ont été proposées dans la littérature. Une première approche consiste à considérer l'utilisation de vecteurs aléatoires dont la distribution du terme d'erreur est telle que chaque composante est non-négative. Cependant, trouver une distribution multivariée suffisamment souple définie sur le support positif est plutôt difficile, au moins avec les applications citées précédemment. Comme indiqué par Cipollini, Engle et Gallo (2006), un candidat possible est une distribution gamma multivariée, qui impose cependant des restrictions sévères sur les corrélations contemporaines entre les variables. Compte tenu que les possibilités sont limitées, une approche possible est d'utiliser la théorie des copules. Ainsi, selon cette approche, des distributions marginales (ou marges) peuvent être spécifiées, dont les distributions en cause ont des supports non-négatifs, et une fonction de copule permet de tenir compte de la dépendance entre les composantes. Une technique d'estimation possible est la méthode du maximum de vraisemblance. Une approche alternative est la méthode des moments généralisés (GMM). Cette dernière méthode présente l'avantage d'être semi-paramétrique dans le sens que contrairement à l'approche imposant une loi multivariée, il n'est pas nécessaire de spécifier une distribution multivariée pour le terme d'erreur. De manière générale, l'estimation des modèles vMEM est compliquée. Les algorithmes existants doivent tenir compte du grand nombre de paramètres et de la nature élaborée de la fonction de vraisemblance. Dans le cas de l'estimation par la méthode GMM, le système à résoudre nécessite également l'utilisation de solveurs pour systèmes non-linéaires. Dans ce mémoire, beaucoup d'énergies ont été consacrées à l'élaboration de code informatique (dans le langage R) pour estimer les différents paramètres du modèle. Dans le premier chapitre, nous définissons les processus stationnaires, les processus autorégressifs, les processus autorégressifs conditionnellement hétéroscédastiques (ARCH) et les processus ARCH généralisés (GARCH). Nous présentons aussi les modèles de durées ACD et les modèles MEM. Dans le deuxième chapitre, nous présentons la théorie des copules nécessaire pour notre travail, dans le cadre des modèles vectoriels et multiplicatifs avec erreurs non-négatives vMEM. Nous discutons également des méthodes possibles d'estimation. Dans le troisième chapitre, nous discutons les résultats des simulations pour plusieurs méthodes d'estimation. Dans le dernier chapitre, des applications sur des séries financières sont présentées. Le code R est fourni dans une annexe. Une conclusion complète ce mémoire.

Gamma stochastic volatility models

This paper presents gamma stochastic volatility models and investigates its distributional and time series properties. The parameter estimators obtained by the method of moments are shown analytically to be consistent and asymptotically normal. The simulation results indicate that the estimators behave well. The insample analysis shows that return models with gamma autoregressive stochastic volatility processes capture the leptokurtic nature of return distributions and the slowly decaying autocorrelation functions of squared stock index returns for the USA and UK. In comparison with GARCH and EGARCH models, the gamma autoregressive model picks up the persistence in volatility for the US and UK index returns but not the volatility persistence for the Canadian and Japanese index returns. The out-of-sample analysis indicates that the gamma autoregressive model has a superior volatility forecasting performance compared to GARCH and EGARCH models.

Expectativas y volatilidad condicionada: los tipos de interés en el mercado interbancario

En este trabajo examinamos si la teoría de expectativas con primas de liquidez constantes puede explicar la estructura temporal de los tipos de interés de pequeños vencimientos en el mercado interbancario de depósitos español, para datos mensuales desde 1977 hasta 1995. Utilizamos el contraste de Campbell y Shiller (1987) basado en un modelo VAR cointegrado. A partir de las estimaciones consistentes de dicho modelo obtenemos la magnitud y persistencia de los shocks a través de la simulación de la respuesta al impulso, y estimaciones eficientes de los parámetros modelizando la varianza condicional que es variable en el tiempo. En este sentido, se proponen varios esquemas de volatilidad que permiten plantear distintas aproximaciones de la incertidumbre en un entorno multiecuacional GARCH y que están basadas en el modelo de expectativas propuesto. La evidencia empírica muestra que se incumple la teoría de las expectativas, que existe una dinámica conjunta a corto plazo para los tipos de interés y el diferencial que está definida por un modelo VAR(4)-GARCH( 1,1)-BEKK (que está próximo a la integrabilidad en varianza), y que existen distintos factores de riesgo que afectan a las primas en los plazos estudiados