A proposal for a new specification for a conditionally heteroskedastic variance model: the Quadratic Moving-Average Conditional Heteroskedasticity and an application to the D. Mark-U.S. dollar Exchange Rate

Ever since the appearance of the ARCH model [Engle(1982a)], an impressive array of variance specifications belonging to the same class of models has emerged [i.e. Bollerslev's (1986) GARCH; Nelson's (1990) EGARCH]. This recent domain has achieved very successful developments. Nevertheless, several empirical studies seem to show that the performance of such models is not always appropriate [Boulier(1992)]. In this paper we propose a new specification: the Quadratic Moving Average Conditional heteroskedasticity model. Its statistical properties, such as the kurtosis and the symmetry, as well as two estimators (Method of Moments and Maximum Likelihood) are studied. Two statistical tests are presented, the first one tests for homoskedasticity and the second one, discriminates between ARCH and QMACH specification. A Monte Carlo study is presented in order to illustrate some of the theoretical results. An empirical study is undertaken for the DM-US exchange rate.

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.

Bootstrapping Autoregressions with Conditional Heteroskedasticity of Unknown Form

Conditional heteroskedasticity is an important feature of many macroeconomic and financial time series. Standard residual-based bootstrap procedures for dynamic regression models treat the regression error as i.i.d. These procedures are invalid in the presence of conditional heteroskedasticity. We establish the asymptotic validity of three easy-to-implement alternative bootstrap proposals for stationary autoregressive processes with m.d.s. errors subject to possible conditional heteroskedasticity of unknown form. These proposals are the fixed-design wild bootstrap, the recursive-design wild bootstrap and the pairwise bootstrap. In a simulation study all three procedures tend to be more accurate in small samples than the conventional large-sample approximation based on robust standard errors. In contrast, standard residual-based bootstrap methods for models with i.i.d. errors may be very inaccurate if the i.i.d. assumption is violated. We conclude that in many empirical applications the proposed robust bootstrap procedures should routinely replace conventional bootstrap procedures for autoregressions based on the i.i.d. error assumption.

A class of improved heteroskedasticity-consistent covariance matrix estimators

The heteroskedasticity-consistent covariance matrix estimator proposed by White (1980), also known as HC0, is commonly used in practical applications and is implemented into a number of statistical software. Cribari–Neto, Ferrari & Cordeiro (2000) have developed a bias-adjustment scheme that delivers bias-corrected White estimators. There are several variants of the original White estimator that also commonly used by practitioners. These include the HC1, HC2 and HC3 estimators, which have proven to have superior small-sample behavior relative to White’s estimator. This paper defines a general bias-correction mechamism that can be applied not only to White’s estimator, but to variants of this estimator as well, such as HC1, HC2 and HC3. Numerical evidence on the usefulness of the proposed corrections is also presented. Overall, the results favor the sequence of improved HC2 estimators.

Inference in differences-in-differences with few treated groups and heteroskedasticity

Differences-in-Differences (DID) is one of the most widely used identification strategies in applied economics. However, how to draw inferences in DID models when there are few treated groups remains an open question. We show that the usual inference methods used in DID models might not perform well when there are few treated groups and errors are heteroskedastic. In particular, we show that when there is variation in the number of observations per group, inference methods designed to work when there are few treated groups tend to (under-) over-reject the null hypothesis when the treated groups are (large) small relative to the control groups. This happens because larger groups tend to have lower variance, generating heteroskedasticity in the group x time aggregate DID model. We provide evidence from Monte Carlo simulations and from placebo DID regressions with the American Community Survey (ACS) and the Current Population Survey (CPS) datasets to show that this problem is relevant even in datasets with large numbers of observations per group. We then derive an alternative inference method that provides accurate hypothesis testing in situations where there are few treated groups (or even just one) and many control groups in the presence of heteroskedasticity. Our method assumes that we can model the heteroskedasticity of a linear combination of the errors. We show that this assumption can be satisfied without imposing strong assumptions on the errors in common DID applications. With many pre-treatment periods, we show that this assumption can be relaxed. Instead, we provide an alternative inference method that relies on strict stationarity and ergodicity of the time series. Finally, we consider two recent alternatives to DID when there are many pre-treatment periods. We extend our inference methods to linear factor models when there are few treated groups. We also derive conditions under which a permutation test for the synthetic control estimator proposed by Abadie et al. (2010) is robust to heteroskedasticity and propose a modification on the test statistic that provided a better heteroskedasticity correction in our simulations.

Inference in differences-in-differences with few treated groups and heteroskedasticity

Differences-in-Differences (DID) is one of the most widely used identification strategies in applied economics. However, how to draw inferences in DID models when there are few treated groups remains an open question. We show that the usual inference methods used in DID models might not perform well when there are few treated groups and errors are heteroskedastic. In particular, we show that when there is variation in the number of observations per group, inference methods designed to work when there are few treated groups tend to (under-) over-reject the null hypothesis when the treated groups are (large) small relative to the control groups. This happens because larger groups tend to have lower variance, generating heteroskedasticity in the group x time aggregate DID model. We provide evidence from Monte Carlo simulations and from placebo DID regressions with the American Community Survey (ACS) and the Current Population Survey (CPS) datasets to show that this problem is relevant even in datasets with large numbers of observations per group. We then derive an alternative inference method that provides accurate hypothesis testing in situations where there are few treated groups (or even just one) and many control groups in the presence of heteroskedasticity. Our method assumes that we know how the heteroskedasticity is generated, which is the case when it is generated by variation in the number of observations per group. With many pre-treatment periods, we show that this assumption can be relaxed. Instead, we provide an alternative application of our method that relies on assumptions about stationarity and convergence of the moments of the time series. Finally, we consider two recent alternatives to DID when there are many pre-treatment groups. We extend our inference method to linear factor models when there are few treated groups. We also propose a permutation test for the synthetic control estimator that provided a better heteroskedasticity correction in our simulations than the test suggested by Abadie et al. (2010).

Bayesian estimation in generalized autoregressive conditional heteroskedasticity models

Esta tesis doctoral nace con el propósito de entender, analizar y sobre todo modelizar el comportamiento estadístico de las series financieras. En este sentido, se puede afirmar que los modelos que mejor recogen las especiales características de estas series son los modelos de heterocedasticidad condicionada en tiempo discreto,si los intervalos de tiempo en los que se recogen los datos lo permiten, y en tiempo continuo si tenemos datos diarios o datos intradía. Con esta finalidad, en esta tesis se proponen distintos estimadores bayesianos para la estimación de los parámetros de los modelos GARCH en tiempo discreto (Bollerslev (1986)) y COGARCH en tiempo continuo (Kluppelberg et al. (2004)). En el capítulo 1 se introducen las características de las series financieras y se presentan los modelos ARCH, GARCH y COGARCH, así como sus principales propiedades. Mandelbrot (1963) destacó que las series financieras no presentan estacionariedad y que sus incrementos no presentan autocorrelación, aunque sus cuadrados sí están correlacionados. Señaló también que la volatilidad que presentan no es constante y que aparecen clusters de volatilidad. Observó la falta de normalidad de las series financieras, debida principalmente a su comportamiento leptocúrtico, y también destacó los efectos estacionales que presentan las series, analizando como se ven afectadas por la época del año o el día de la semana. Posteriormente Black (1976) completó la lista de características especiales incluyendo los denominados leverage effects relacionados con como las fluctuaciones positivas y negativas de los precios de los activos afectan a la volatilidad de las series de forma distinta.

Volatility regimes for the VIX index

This article presents a Markov chain framework to characterize the behavior of the CBOE Volatility Index (VIX index). Two possible regimes are considered: high volatility and low volatility. The specification accounts for deviations from normality and the existence of persistence in the evolution of the VIX index. Since the time evolution of the VIX index seems to indicate that its conditional variance is not constant over time, I consider two different versions of the model. In the first one, the variance of the index is a function of the volatility regime, whereas the second version includes an autoregressive conditional heteroskedasticity (ARCH) specification for the conditional variance of the index.

Cluster de volatilidade no mercado de taxas de câmbio USD/GBP

Mestrado em Controlo de Gestão e dos Negócios

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.

Specification and testing of multiplicative time-varying GARCH models with applications

In this article, we develop a specification technique for building multiplicative time-varying GARCH models of Amado and Teräsvirta (2008, 2013). The variance is decomposed into an unconditional and a conditional component such that the unconditional variance component is allowed to evolve smoothly over time. This nonstationary component is defined as a linear combination of logistic transition functions with time as the transition variable. The appropriate number of transition functions is determined by a sequence of specification tests. For that purpose, a coherent modelling strategy based on statistical inference is presented. It is heavily dependent on Lagrange multiplier type misspecification tests. The tests are easily implemented as they are entirely based on auxiliary regressions. Finite-sample properties of the strategy and tests are examined by simulation. The modelling strategy is illustrated in practice with two real examples: an empirical application to daily exchange rate returns and another one to daily coffee futures returns.

Growth and Survival Determinants of Chinese Private Firms: Fieldwork evidence and econometric estimates

This paper reports on one of the first empirical attempts to investigate small firm growth and survival, and their determinants, in the Peoples’ Republic of China. The work is based on field work evidence gathered from a sample of 83 Chinese private firms (mainly SMEs) collected initially by face-to-face interviews, and subsequently by follow-up telephone interviews a year later. We extend the models of Gibrat (1931) and Jovanovic (1982), which traditionally focus on size and age alone (e.g. Brock and Evans, 1986), to a ‘comprehensive’ growth model with two types of additional explanatory variables: firm-specific (e.g. business planning); and environmental (e.g. choice of location). We estimate two econometric models: a ‘basic’ age-size-growth model; and a ‘comprehensive’ growth model, using Heckman’s two-step regression procedure. Estimation is by log-linear regression on cross-section data, with corrections for sample selection bias and heteroskedasticity. Our results refute a pure Gibrat model (but support a more general variant) and support the learning model, as regards the consequences of size and age for growth; and our extension to a comprehensive model highlights the importance of location choice and customer orientation for the growth of Chinese private firms. In the latter model, growth is explained by variables like planning, R&D orientation, market competition, elasticity of demand etc. as well as by control variables. Our work on small firm growth achieves two things. First, it upholds the validity of ‘basic’ size-age-growth models, and successfully applies them to the Chinese economy. Second, it extends the compass of such models to a ‘comprehensive’ growth model incorporating firm-specific and environmental variables.

Growth and Survival Determinants of Chinese Private Firms: Fieldwork evidence and econometric estimates

This paper reports on one of the first empirical attempts to investigate small firm growth and survival, and their determinants, in the Peoples’ Republic of China. The work is based on field work evidence gathered from a sample of 83 Chinese private firms (mainly SMEs) collected initially by face-to-face interviews, and subsequently by follow-up telephone interviews a year later. We extend the models of Gibrat (1931) and Jovanovic (1982), which traditionally focus on size and age alone (e.g. Brock and Evans, 1986), to a ‘comprehensive’ growth model with two types of additional explanatory variables: firm-specific (e.g. business planning); and environmental (e.g. choice of location). We estimate two econometric models: a ‘basic’ age-size-growth model; and a ‘comprehensive’ growth model, using Heckman’s two-step regression procedure. Estimation is by log-linear regression on cross-section data, with corrections for sample selection bias and heteroskedasticity. Our results refute a pure Gibrat model (but support a more general variant) and support the learning model, as regards the consequences of size and age for growth; and our extension to a comprehensive model highlights the importance of location choice and customer orientation for the growth of Chinese private firms. In the latter model, growth is explained by variables like planning, R&D orientation, market competition, elasticity of demand etc. as well as by control variables. Our work on small firm growth achieves two things. First, it upholds the validity of ‘basic’ size-age-growth models, and successfully applies them to the Chinese economy. Second, it extends the compass of such models to a ‘comprehensive’ growth model incorporating firm-specific and environmental variables.

Determinants of school efficiency : The case of primary schools in the State of Geneva, Switzerland

The public primary school system in the State of Geneva, Switzerland, is characterized by centrally evaluated pupil performance measured with the use of standardized tests. As a result, consistent data are collected among the system. The 2010-2011 dataset is used to develop a two-stage data envelopment analysis (DEA) of school efficiency. In the first stage, DEA is employed to calculate an individual efficiency score for each school. It shows that, on average, each school could reduce its inputs by 7% whilst maintaining the same quality of pupil performance. The cause of inefficiency lies in perfectible management. In the second stage, efficiency is regressed on school characteristics and environmental variables;external factors outside of the control of headteachers. The model is tested for multicollinearity, heteroskedasticity and endogeneity. Four variables are identified as statistically significant. School efficiency is negatively influenced by (1) the provision of special education, (2) the proportion of disadvantaged pupils enrolled at the school and (3) operations being held on multiple sites, but positively influenced by school size (captured by the number of pupils). The proportion of allophone pupils; schools located in urban areas and the provision of reception classes for immigrant pupils are not significant. Although the significant variables influencing school efficiency are outside of the control of headteachers, it is still possible to either boost the positive impact or curb the negative impact. Dans le canton de Genève (Suisse), les écoles publiques primaires sont caractérisées par un financement assuré par les collectivités publiques (canton et communes) et par une évaluation des élèves à l'aide d'épreuves standardisées à trois moments distincts de leur scolarité. Cela permet de réunir des informations statistiques consistantes. La base de données de l'année 2010-2011 est utilisée dans une analyse en deux étapes de l'efficience des écoles. Dans une première étape, la méthode d'analyse des données par enveloppement (DEA) est utilisée pour calculer un score d'efficience pour chaque école. Cette analyse démontre que l'efficience moyenne des écoles s'élève à 93%. Chaque école pourrait, en moyenne, réduire ses ressources de 7% tout en conservant constants les résultats des élèves aux épreuves standardisées. La source de l'inefficience réside dans un management des écoles perfectible. Dans une seconde étape, les scores d'efficience sont régressés sur les caractéristiques des écoles et sur des variables environnementales. Ces variables ne sont pas sous le contrôle (ou l'influence) des directeurs d'école. Le modèle est testé pour la multicolinéartié, l'hétéroscédasticité et l'endogénéité. Quatre variables sont statistiquement significatives. L'efficience des écoles est influencée négativement par (1) le fait d'offrir un enseignement spécialisé en classe séparée, (2) la proporition d'élèves défavorisés et (3) le fait d'opérer sur plusieurs sites différents. L'efficience des écoles est influencée positivement par la taille de l'école, mesurée par le nombre d'élèves. La proporition d'élèves allophones, le fait d'être situé dans une zone urbaine et d'offrir des classes d'accueil pour les élèves immigrants constituent autant de variables non significatives. Le fait que les variables qui influencent l'efficience des écoles ne soient pas sous le contrôle des directeurs ne signifie pas qu'il faille céder au fatalisme. Différentes pistes sont proposées pour permettre soit de réduire l'impact négatif soit de tirer parti de l'impact positif des variables significatives.