Emma M. Iglesias

HIGHER ORDER ASYMPTOTIC THEORY WHEN A PARAMETER IS ON A BOUNDARY WITH AN APPLICATION TO GARCH MODELS

Andrews (1999, Econometrica 67, 1341–1383) derived the first-order asymptotic theory for a very general class of estimators when a parameter is on a boundary. We derive the second-order asymptotic theory in this setting in some special cases. We focus on the behavior of the quasi maximum likelihood estimator (QMLE) in stationary and nonstationary generalized autoregressive conditionally heteroskedastic (GARCH) models when constraints are imposed in the maximization procedure. We show how in this case both a first- and a second-order bias appear in the estimator and how the bias can be quite large. We provide two types of bias correction mechanisms for the researcher to choose in practice: either to bias correct only for a first-order bias or for a first- and second-order bias. We show that when some constraints are imposed, it is advisable to bias correct not only for the first-order bias but also for the second-order bias.We thank Bruce Hansen and two referees for helpful comments. The first author gratefully acknowledges financial support from the MSU Intramural Research Grants Program. The second author gratefully acknowledges financial support from the ESRC.

BIVARIATE ARCH MODELS: FINITE-SAMPLE PROPERTIES OF QML ESTIMATORS AND AN APPLICATION TO AN LM-TYPE TEST

At the present time, there exists an important and growing econometric literature that deals with the application of multivariate-ARCH models to a variety of economic and financial data. However, the properties of the estimation procedures that are used have not yet been fully explored. This paper provides two main new results: the first concerns the large biases and variances that can arise when the ML estimation method is employed in a simple bivariate structure under the assumption of conditional heteroscedasticity; and the second examines how to use these analytical theoretical results to improve the size and the power of a test for multivariate ARCH effects. We analyse two models: one proposed in Wong and Li (1997) (where the disturbances are dependent but uncorrelated) and another proposed by Engle and Kroner (1995) and Liu and Polasek (1999, 2000) (where conditional correlation is allowed through a diagonal representation). We prove theoretically that a relatively large difference between the intercepts in the two conditional variance equations produces, in the first model, very large variances in some of the ML estimators and, in the second, very severe biases in some of the ML estimators of the parameters. Later we use our bias expressions to propose an LM type test of multivariate ARCH effects, showing that the size and the power of the test improve when we allow for bias correction in the estimators, and that the best recommendation in practical applications is always to use the expected hessian version of the LM. We address as well some constraints that should be included in the estimation of the models but which have so far been ignored. Finally, we present a SUR (seemingly unrelated) specification in both models, that provides an alternative way to retrieve the information matrix. We also extend Lumsdaine (1995) results in multivariate framework.

Finite sample theory of QMLE in ARCH models with dynamics in the mean equation

We provide simulation and theoretical results concerning the finite-sample theory of quasi-maximum-likelihood estimators in autoregressive conditional heteroskedastic (ARCH) models when we include dynamics in the mean equation. In the setting of the AR(q)-ARCH(p), we find that in some cases bias correction is necessary even for sample sizes of 100, especially when the ARCH order increases. We warn about the existence of important biases and potentially low power of the t-tests in these cases. We also propose ways to deal with them. We also find simulation evidence that when conditional heteroskedasticity increases, the mean-squared error of the maximum-likelihood estimator of the AR(1) parameter in the mean equation of an AR(1)-ARCH(1) model is reduced. Finally, we generalize the Lumsdaine [J. Bus. Econ. Stat. 13 (1995) pp. 1-10] invariance properties for the biases in these situations. Copyright 2008 The Authors. Journal compilation 2008 Blackwell Publishing Ltd

The Limiting Properties of the QMLE in a General Class of Asymmetric Volatility Models

In this paper we analyze the limiting properties of the estimated parameters in a general class of asymmetric volatility models which are closely related to the traditional exponential GARCH model. The new representation has three main advantages over the traditional EGARCH: (1) It allows a much more flexible representation of the conditional variance function. (2) It is possible to provide a complete characterization of the asymptotic distribution of the QML estimator based on the new class of nonlinear volatility models, something which has proven very difficult even for the traditional EGARCH. (3) It can produce asymmetric news impact curves where, contrary to the traditional EGARCH, the resulting variances do not excessively exceed the ones associated with the standard GARCH model, irrespectively of the sign of an impact of moderate size. Furthermore, the new class of models considered can create a wide array of news impact curves which provide the researcher with a richer choice set relative to the traditional. We also show in a Monte Carlo experiment the good finite sample performance of our asymptotic theoretical results and we compare them with those obtained from a parametric and the residual based bootstrap. Finally, we provide an empirical illustration.

Analysing one-month Euro-market interest rates by fractionally integrated models

This article considers the modelling of short-term interest rates with the ARFIMA model in six European countries based on daily data in the 1990s using the Modified Profile Likelihood estimation method. This allows one to study the different convergence processes that have been followed in each case. Empirical evidence shows that, even with this estimation method, the standard AIC tends to select models that in some cases are in accordance with traditional inference but in other cases may not be so. Analysing these results, the series for Switzerland appears to be an I(1) series, which conflicts with the findings in previous literature.

Interaction between monetary policy and stock prices: a comparison between the Caribbean and the US

We analyse the interaction between monetary policy and stock prices in Barbados, Jamaica and Trinidad and Tobago (T&T), both individually and jointly as the Caribbean countries using structural VARs, as proposed in Bjornland and Leitemo (2009). Annual and monthly frequencies are used for Barbados while, due to data availability constraints, only annual data is employed for Jamaica and T&T. First, our results show that in Barbados, with monthly (and annual) data, a monetary policy shock that increases the Treasury bill rate by 100 basis points causes stock prices to increase by 0.038% (and fall by 0.06%), while a stock price shock that increases stock prices by 1% results in an increase in the Treasury bill rate of 30 (and 190) basis points, respectively. For Jamaica, a monetary policy shock causes stock prices to fall by 0.3%, while a stock price shock that increases stock prices by 1% results in an increase in the Treasury bill rate of 400 basis points. Likewise for T&T, a shock to monetary policy causes stock prices to fall by 0.1% and a shock leading to a 1% increase in real stock prices causes the Treasury bill to increase by 330 basis points. When we analyse the three Caribbean countries jointly, a positive 1% stock price shock causes the Treasure bill rate to increase by 700 basis points and a positive monetary policy shock cause stock price to fall by 0.027%. Therefore, our results in relation to the signs of the relationships with annual data are similar to those of the US in Bjornland and Leitemo (2009), however the magnitudes are substantially different. The effect of a monetary policy shock is greater in the US, while the effect of a stock price shock is smaller in the US than in our Caribbean economy. We argue that this reflects clear differences between the US and Caribbean economies. Caribbean countries have slower information channels, for example, by targeting the 30-day Certificate of Deposit (COD) rate instead of the overnight Treasury bill rate as in the US. This supports our results that only with annual data we find similar relationships as in the US with monthly data. Moreover, the higher economic instability in the Caribbean is clearly observed in the larger effect that a stock price increase has on interest rates versus the USA.

Almost Unbiased Estimation in Simultaneous Equation Models With Strong and/or Weak Instruments

We propose two simple bias-reduction procedures that apply to estimators in a general static simultaneous equation model and that are valid under relatively weak distributional assumptions for the errors. Standard jackknife estimators, as applied to 2SLS, may not reduce the bias of the exogenous variable coefficient estimators since the estimator biases are not monotonically nonincreasing with sample size (a necessary condition for successful bias reduction) and they have moments only up to the order of overidentification. Our proposed approaches do not have either of these drawbacks. (1) In the first procedure, both endogenous and exogenous variable parameter estimators are unbiased to order T − 2 and when implemented for k-class estimators for which k < 1, the higher-order moments will exist. (2) An alternative second approach is based on taking linear combinations of k-class estimators for k < 1. In general, this yields estimators that are unbiased to order T − 1 and that possess higher moments. We also prove theoretically how the combined k-class estimator produces a smaller mean squared error than 2SLS when the degree of overidentification of the system is 0, 1, or at least 8. The performance of the two procedures is compared with 2SLS in a number of Monte Carlo experiments using a simple two-equation model. Finally, an application shows the usefulness of our new estimator in practice versus competitor estimators.

An analysis of extreme movements of exchange rates of the main currencies traded in the Foreign Exchange market

This article analyses the extreme movements of exchange rates of the seven main currencies traded in the Foreign Exchange market against the US dollar: Euro, British pound, Canadian dollar, Japanese Yen, Swiss franc, Australian dollar and New Zealand dollar by using tail index indicators. Payaslioğlu (2009) considers the case of the Turkish exchange rate using the traditional Hill (1975) estimator as a tool. In this article, we employ also an alternative estimator proposed in Iglesias and Linton (2009) that is shown to have, in some cases, improved finite sample properties and it provides substantially different results versus the Hill estimator. We find that for the Euro, Japanese Yen, Swiss franc, Canadian, Australian and New Zealand dollars, the Hill estimator provides a better measure to analyse the extreme behaviour; while for the British pound, the Iglesias and Linton alternative estimator is superior by using Hausman-type tests of misspecification. Measures of value at risk are also provided for the seven markets. We also find that the largest estimated value at risk by far is for the Japanese Yen, followed by the Swiss franc, the Canadian dollar, the Euro, the New Zealand dollar and the Australian dollar. The UK pound has the smallest value at risk when extreme movements occur.

Another look about the evolution of the risk premium: a VAR-GARCH-M model

Abstract In this paper we model the Spanish interest rate in the period 1979–1998, estimating the time-varying risk premium for France, Germany and Spain and allowing for relationships among them. For this purpose, we select a VAR(1)-GARCH(1,1)-M(1) from among competing models. Results clearly support the existence of a time-varying risk premium for Spain that depends on the German volatility.

Small Sample Estimation Bias in GARCH Models with Any Number of Exogenous Variables in the Mean Equation

In this article we show how bias approximations for the quasi maximum likelihood estimators of the parameters in Generalized Autoregressive Conditional Heteroskedastic (GARCH)(p, q) models change when any number of exogenous variables are included in the mean equation. The approximate biases are shown to vary in an additive and proportional way in relation to the number of exogenous variables, and they do not depend on the moments of the regressors under the correct specification of the model. This suggests a rule of thumb in testing for misspecification in GARCH models. We also extend the theoretical bias approximations given in Linton (1997) for the GARCH(1, 1). Because the expressions are not in closed form, we concentrate in detail, and for simplicity of interpretation, on the ARCH(1) model. At each stage, we check our theoretical results by simulation and generally, we find that the approximations are quite accurate for sample sizes of at least 50. We find that the biases are not trivial in some circumstances and we discuss how the bias approximations may be used, in practice, to reduce the bias. We also carry out simulations for the GARCH(1,1) model and show that the biases change as predicted by the approximations when the mean equation is augmented. Finally, we illustrate the usefulness of our approach for U.S. monthly inflation rates.