MULTIVARIATE ARCH MODELS: FINITE SAMPLE PROPERTIES OF ML 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.Multivariate GARCH, Bias evaluation.

The evolving SARS-CoV-2 epidemic in Africa: Insights from rapidly expanding genomic surveillance

INTRODUCTION Investment in Africa over the past year with regard to severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) sequencing has led to a massive increase in the number of sequences, which, to date, exceeds 100,000 sequences generated to track the pandemic on the continent. These sequences have profoundly affected how public health officials in Africa have navigated the COVID-19 pandemic. RATIONALE We demonstrate how the first 100,000 SARS-CoV-2 sequences from Africa have helped monitor the epidemic on the continent, how genomic surveillance expanded over the course of the pandemic, and how we adapted our sequencing methods to deal with an evolving virus. Finally, we also examine how viral lineages have spread across the continent in a phylogeographic framework to gain insights into the underlying temporal and spatial transmission dynamics for several variants of concern (VOCs). RESULTS Our results indicate that the number of countries in Africa that can sequence the virus within their own borders is growing and that this is coupled with a shorter turnaround time from the time of sampling to sequence submission. Ongoing evolution necessitated the continual updating of primer sets, and, as a result, eight primer sets were designed in tandem with viral evolution and used to ensure effective sequencing of the virus. The pandemic unfolded through multiple waves of infection that were each driven by distinct genetic lineages, with B.1-like ancestral strains associated with the first pandemic wave of infections in 2020. Successive waves on the continent were fueled by different VOCs, with Alpha and Beta cocirculating in distinct spatial patterns during the second wave and Delta and Omicron affecting the whole continent during the third and fourth waves, respectively. Phylogeographic reconstruction points toward distinct differences in viral importation and exportation patterns associated with the Alpha, Beta, Delta, and Omicron variants and subvariants, when considering both Africa versus the rest of the world and viral dissemination within the continent. Our epidemiological and phylogenetic inferences therefore underscore the heterogeneous nature of the pandemic on the continent and highlight key insights and challenges, for instance, recognizing the limitations of low testing proportions. We also highlight the early warning capacity that genomic surveillance in Africa has had for the rest of the world with the detection of new lineages and variants, the most recent being the characterization of various Omicron subvariants. CONCLUSION Sustained investment for diagnostics and genomic surveillance in Africa is needed as the virus continues to evolve. This is important not only to help combat SARS-CoV-2 on the continent but also because it can be used as a platform to help address the many emerging and reemerging infectious disease threats in Africa. In particular, capacity building for local sequencing within countries or within the continent should be prioritized because this is generally associated with shorter turnaround times, providing the most benefit to local public health authorities tasked with pandemic response and mitigation and allowing for the fastest reaction to localized outbreaks. These investments are crucial for pandemic preparedness and response and will serve the health of the continent well into the 21st century

some recent developments in econometric test methodology

abstract (introduction): a crucial element in the development of econometric methodology over the past decade or so has been the concern with testing, as opposed to estimating, econometric models. this has been brought about partly as the result of the subject reaching a new level of maturity and partly because the lack of success of econometric modelling in the seventies, particularly at the macro level, made clear the need for a deeper concern with model evaluation. modern econometric practice advocates that a given specification should be subject to a rigourous testing procedure and it is now becoming routine to test for misspecifications such as omitted variables, serially correlated disturbances and structural change and, in addition,to test for heteroscedasticity and incorrect functional form. this kind of intensive misspecification testing leads to problems of distortions in the inference procedures but leading econometricians believe that the importance of carrying out such tests overrides these problems. ... hendry advocated that this should be done notwithstanding the difficulties involved in calculating and controlling type 1 and type 2 errors. while it is important to test econometric models rigourously, it is also important to seek to structure the testing procedure in such a way that problems of data mining are minimised. in particular we seek test procedures to test for the presence of, possibly, several misspecifications simultaneously in such a way that: (a) the overall type 1 error probability is controlled within acceptable limits, and (b) the test procedure while having good power properties provides some opportunity for detecting individual types of misspecification. in this paper i consider some recent advances in test methodology which contribute to the development of such procedures. there are two general approaches to conducting tests for misspecification in econometric models. in the first approach, we obtain some sample statistic whose distribution is known under the null hypothesis, i.e. when the model is assumed correct, and if the statistic assumes a significant value this is taken as evidence that the model is misspecified in some unknown way. the d.w. statistic is sometimes used for such a test because it is relatively sensitive to various departures from the null hypothesis. in this case we do not necessarily regard a significant test result as implying that the disturbances are first order serially correlated and proceed with a cochrane orcutt type estimation procedure. the significant test result is simply taken as evidence that something is wrong. tests of this type are known as pure significance tests and do not require the specification of a particular alternative model.

Further Results on Pseudo-Maximun Likelihood Estimation and Testing the Constant Elasticity of Variance Continuous Time Model

[Abstract]: Constant elasticity volatility processes have been shown to be useful, for example, to encompass a number of existing models that have closed-form likelihood functions. In this article, we extend the existing literature in two directions: first we find explicit closed form solutions of the pseudo maximum likelihood estimators (MLEs) by discretizing the diffusion function and we provide their asymptotic theory in the context of the constant elasticity of variance (CEV) model characterized by a general CEV parameter ������� ≥ 0. Second we obtain bias expansions for those pseudo MLEs also in terms of ������� ≥ 0. We provide a general framework since only the cases with ������� = 0 and ������� = 0.5 have been considered in the literature so far. When the time series is not positive almost surely, we need to impose the restriction that ������� is a non-negative integer.We wish to thank the Editor and two referees for very helpful comments. The first author is very grateful for the financial support from the Spanish Ministry of Science and Innovation, projects ECO2015-63845-P and PGC2018-101327-B-I00

The Robustness of the Higher-Order 2SLS and General k-Class Bias Approximations to Non-Normal Disturbances

In a seminal paper Nagar (1959) obtained first and second moment approximations for the k-class of estimators in a general static simultaneous equation model under the assumption that the structural disturbances were i.i.d and normally distributed. Later Mikhail (1972) obtained a higher-order bias approximation for 2SLS under the same assumptions as Nagar while Iglesias and Phillips (2010) obtained the higher order approximation for the general k-class of estimators. These approximations show that the higher order biases can be important especially in highly overidentified cases. In this paper we show that Mikhail.s higher order bias approximation for 2SLS continues to be valid under symmetric, but not necessarily normal, disturbances with an arbitrary degree of kurtosis but not when the disturbances are asymmetric. A modified approximation for the 2SLS bias is then obtained which includes the case of asymmetric disturbances. The results are then extended to the general k-class of estimators.

The bias to order T-Â 2 for the general k-class estimator in a simultaneous equation model

Kinal (1980) showed that k-class estimators for which k Bias k-class estimators Simultaneous equations

Moment Approximation for Least Squares Estimators in Dynamic Regression Models with a Unit Root

This discussion paper led to a publication in 'The Econometrics Journal' .Asymptotic expansions are employed in a dynamic regression model with a unit root inorder to find approximations for the bias, the variance and for the mean squared error of theleast-squares estimator of all coefficients. It is found that in this particular context suchexpansions exist only when the autoregressive model contains at least one non-redundant exogenousexplanatory variable and that local to zero asymptotic approaches are here without avail.Surprisingly the large sample and small disturbance asymptotic techniques give closely relatedresults, which is not the case in stable dynamic regression models. The expressions for momentapproximations are specialized to the random walk with (trend in) drift model and their accuracyis examined in Monte Carlo experiments.

Alternative Bias Approximations in Regressions with a Lagged-Dependent Variable

The small sample bias of the least-squares coefficient estimator is examined in the dynamic multiple linear regression model with normally distributed whitenoise disturbances and an arbitrary number of regressors which are all exogenous except for the one-period lagged-dependent variable. We employ large sample ( T → ∞) and small disturbance (σ → 0) asymptotic theory and derive and compare expressions to O (T −1 ) and to O (σ 2), respectively, for the bias in the least-squares coefficient vector. In some simulations and for an empirical example, we examine the mean (squared) error of these expressions and of corrected estimation procedures that yield estimates that are unbiased to O (T −l ) and to O (σ 2), respectively. The large sample approach proves to be superior, easily applicable, and capable of generating more efficient and less biased estimators.