Karim M Abadir | Imperial College London (original) (raw)
Papers by Karim M Abadir
Discussion Papers, 1995
Hypergeometric functions are a generalization of exponential functions. They are explicit, comput... more Hypergeometric functions are a generalization of exponential functions. They are explicit, computable functions that can also be manipulated analytically. The functions and series we use in quantitative economics are all special cases of them. In this paper, a unified approach to hypergeometric functions is given. As a result, some potentially useful general applications emerge in a number of areas such as in econometrics and economic theory. The greatest benefit from using these functions stems from the fact that they provide parsimonious explicit (and interpretable) solutions to a wide range of general problems.
8 The Unconventional Dynamics of Economic and Financial Aggregates Karim M. Abadir and Gabriel Ta... more 8 The Unconventional Dynamics of Economic and Financial Aggregates Karim M. Abadir and Gabriel Talmain CONTENTS 8.1 Introduction 205 8.2 The Economic Origins of the Nonlinear Long-Memory 206 8.3 Modeling Co-Movements for Series with Nonlinear Long-Memory ...
ABSTRACT The well-known transformation theorem (or change-of-variables theorem) is difficult to p... more ABSTRACT The well-known transformation theorem (or change-of-variables theorem) is difficult to prove, requiring knowledge of theorems in advanced analysis. For this reason, it is stated without proof in all the statistics textbooks we know of. Here, we provide a new and much simpler proof, by exploiting the fact that it is a density function which is being transformed, and the idea of conditioning.
Let { } follow a discrete Gaussian Vector Auto-Regression with deterministic components. We de... more Let { } follow a discrete Gaussian Vector Auto-Regression with deterministic components. We derive the exact finite-sample joint Moment Generating Function (MGF) of the quadratic forms that form the basis for the sufficient statistic. The formula is then specialized to the limiting MGF of functionals involving multivariate and univariate Ornstein-Uhlenbeck processes, drifts and time trends. Such processes arise asymptotically from more general non-Gaussian processes as well as the Gaussian { }, and have also been used in areas other than time series, such as the 'goodness of fit' literature.
Discussion Papers, 1994
Two variables are said to be cointegrated when they move closely together over time, after proper... more Two variables are said to be cointegrated when they move closely together over time, after proper scaling. Cointegration was taken to be the statistical expression of the notion of equilibrium in economics. But is it still possible to talk of cointegration when 'disequilibrium' economics prevail? We argue that it is, and that the duality is strongest between cointegration theory and economic theories of non-clearing markets. By setting up a simple generic non-parametric model, it is shown that Clower's dual decision hypothesis is a more direct and natural expression of the notion of cointegration than long-run equilibrium is. With sticky prices, quantities (e.g. consumption and income) move together more closely than they would otherwise. As a by-product, the model gives rise to (and justifies from an economics standpoint) a recent statistical approach to modelling economic time series. An observational equivalence between two econometric models is also presented.
Macroeconomic and aggregate financial series share an unconventional type of nonlinear dynamics. ... more Macroeconomic and aggregate financial series share an unconventional type of nonlinear dynamics. Existing techniques (like co-integration) model these dynamics incompletely, hence generating seemingly paradoxical results. To avoid this, we provide a methodology to disentangle the long-run relation between variables from their own dynamics, and illustrate with two applications.
J Optimiz Theor Appl, 2003
A unified framework to derive the distribution of conventional statistics under a unit root is pr... more A unified framework to derive the distribution of conventional statistics under a unit root is presented. It is based on formulae which can generate (analytically as well as numerically) the densities and distributions of statistics such as the t ratio, the normalized autocorrelation coefficient ...
Ssrn Electronic Journal, 1997
We derive formulae for the asymptotic density and distribution functions of the t-statistic for a... more We derive formulae for the asymptotic density and distribution functions of the t-statistic for autoregressive unit roots based on M-estimators. These formulae depend upon a nuisance parameter. Consequently, new critical values for this test would have to be generated for each different M-estimator that is used. We therefore provide a methodology for tractable and accurate analytic approximations of quantiles based on asymptotic expansions of functions. We use it to derive simple yet accurate approximations for the asymptotic distribution of these unit root M-tests. Using these asymptotic approximations, critical values of the tests can easily be obtained without even resorting to a computer. The approximation requires no new tabulation, and the resulting cumulative distribution function (cdf) has a maximum absolute error of 0.002 for typical quantiles (i.e. 1-10% quantiles).
The limiting marginal density of efficient estimators of bivariate cointegration vectors is deriv... more The limiting marginal density of efficient estimators of bivariate cointegration vectors is derived in closed form. The formula is exact, and it consists of highly efficient convergent expansion. It is used to plot the density. Furthermore, it is manipulated analytically to reveal features that could not be uncovered by Monte Carlo. For example, it is demonstrated that moments of any integer order exist, and the derived unconditional (marginal) density is compared to the conditional one which is normal.
ABSTRACT Ways of improving the efficiency of Monte-Carlo (MC) techniques are studied for dynamic ... more ABSTRACT Ways of improving the efficiency of Monte-Carlo (MC) techniques are studied for dynamic models. Such models cause the conventinal Antithetic Variate (AV) technique to fail, and will be proved to reduce the benefit from using Control Variates with nearly non-stationary series. This paper suggest modifications of the two conventional variance reduction techniques to enhance their efficiency. New classes of AV's are also proposed. Methods of reordering residuals are found to do less well than others wich rely on changing signs in the spirit of the traditional AV. Then, unconventional applications of these techniques of MC work for nearly nonstationary series. It generates econometric estimators at one and a half times the speed of conventional MC.
Summary This paper proposes a standard for notation in econometrics. It presents a fully integrat... more Summary This paper proposes a standard for notation in econometrics. It presents a fully integrated and internally consistent framework for notation and abbreviations, which is as close as possible to existing common practice. The symbols used are instantly recognizable and ...
We show how investors' liquidity patterns can provide a common framework to explain autocorrelati... more We show how investors' liquidity patterns can provide a common framework to explain autocorrelation of returns and volumes, and some calendar anomalies. The method helps us find new anomalies, and contribute to the explanation of older ones. We uncover a "festivities effect" that is composed of a pre-festivity period of negative returns and relatively low trading activity, and a post-festivity period of positive returns and increased trading activity. We demonstrate this effect for ten countries in the Middleand Far-East where the main festivities occur every year at a different time of the Western Gregorian calendar. In particular, we consider the Muslim Ramadan and Chinese New Year festivities in these countries.
This paper deals with models allowing for trending processes and cyclical component with error pr... more This paper deals with models allowing for trending processes and cyclical component with error processes that are possibly nonstationary, nonlinear, and non-Gaussian. Asymptotic confidence intervals for the trend, cyclical component, and memory parameters are obtained. The confidence intervals are applicable for a wide class of processes, exhibit good coverage accuracy, and are easy to implement. JEL Classification: C22.
Page 1. Econometric Theory, 11, 1995, 775-793. Printed in the United States of America. THE LIMIT... more Page 1. Econometric Theory, 11, 1995, 775-793. Printed in the United States of America. THE LIMITING DISTRIBUTION OF THE t RATIO UNDER A UNIT ROOT KARIM M. ABADIR University of Exeter An encompassing formula ...
We provide a methodology to calculate the expectation of a variate x in terms of the moments of a... more We provide a methodology to calculate the expectation of a variate x in terms of the moments of a transformation of x. Apart from the intrinsic interest in such a fundamental relation that relates the moments of a variate and its nonlinear transformations, our results can be used in practice to approximate E(x) by the low-order moments of a transformation
We extend the classical local Whittle estimation procedure to fractionally integrated I(d) proces... more We extend the classical local Whittle estimation procedure to fractionally integrated I(d) processes, where d>-3/2, thus covering stationary and non-stationary regions. Asymptotic properties of the bias of the estimator are investigated. It is shown that in a wider region than previously considered in the literature, the estimator exhibits the same asymptotic properties as in the stationary case. When the generating
Discussion Papers, 1995
Hypergeometric functions are a generalization of exponential functions. They are explicit, comput... more Hypergeometric functions are a generalization of exponential functions. They are explicit, computable functions that can also be manipulated analytically. The functions and series we use in quantitative economics are all special cases of them. In this paper, a unified approach to hypergeometric functions is given. As a result, some potentially useful general applications emerge in a number of areas such as in econometrics and economic theory. The greatest benefit from using these functions stems from the fact that they provide parsimonious explicit (and interpretable) solutions to a wide range of general problems.
8 The Unconventional Dynamics of Economic and Financial Aggregates Karim M. Abadir and Gabriel Ta... more 8 The Unconventional Dynamics of Economic and Financial Aggregates Karim M. Abadir and Gabriel Talmain CONTENTS 8.1 Introduction 205 8.2 The Economic Origins of the Nonlinear Long-Memory 206 8.3 Modeling Co-Movements for Series with Nonlinear Long-Memory ...
ABSTRACT The well-known transformation theorem (or change-of-variables theorem) is difficult to p... more ABSTRACT The well-known transformation theorem (or change-of-variables theorem) is difficult to prove, requiring knowledge of theorems in advanced analysis. For this reason, it is stated without proof in all the statistics textbooks we know of. Here, we provide a new and much simpler proof, by exploiting the fact that it is a density function which is being transformed, and the idea of conditioning.
Let { } follow a discrete Gaussian Vector Auto-Regression with deterministic components. We de... more Let { } follow a discrete Gaussian Vector Auto-Regression with deterministic components. We derive the exact finite-sample joint Moment Generating Function (MGF) of the quadratic forms that form the basis for the sufficient statistic. The formula is then specialized to the limiting MGF of functionals involving multivariate and univariate Ornstein-Uhlenbeck processes, drifts and time trends. Such processes arise asymptotically from more general non-Gaussian processes as well as the Gaussian { }, and have also been used in areas other than time series, such as the 'goodness of fit' literature.
Discussion Papers, 1994
Two variables are said to be cointegrated when they move closely together over time, after proper... more Two variables are said to be cointegrated when they move closely together over time, after proper scaling. Cointegration was taken to be the statistical expression of the notion of equilibrium in economics. But is it still possible to talk of cointegration when 'disequilibrium' economics prevail? We argue that it is, and that the duality is strongest between cointegration theory and economic theories of non-clearing markets. By setting up a simple generic non-parametric model, it is shown that Clower's dual decision hypothesis is a more direct and natural expression of the notion of cointegration than long-run equilibrium is. With sticky prices, quantities (e.g. consumption and income) move together more closely than they would otherwise. As a by-product, the model gives rise to (and justifies from an economics standpoint) a recent statistical approach to modelling economic time series. An observational equivalence between two econometric models is also presented.
Macroeconomic and aggregate financial series share an unconventional type of nonlinear dynamics. ... more Macroeconomic and aggregate financial series share an unconventional type of nonlinear dynamics. Existing techniques (like co-integration) model these dynamics incompletely, hence generating seemingly paradoxical results. To avoid this, we provide a methodology to disentangle the long-run relation between variables from their own dynamics, and illustrate with two applications.
J Optimiz Theor Appl, 2003
A unified framework to derive the distribution of conventional statistics under a unit root is pr... more A unified framework to derive the distribution of conventional statistics under a unit root is presented. It is based on formulae which can generate (analytically as well as numerically) the densities and distributions of statistics such as the t ratio, the normalized autocorrelation coefficient ...
Ssrn Electronic Journal, 1997
We derive formulae for the asymptotic density and distribution functions of the t-statistic for a... more We derive formulae for the asymptotic density and distribution functions of the t-statistic for autoregressive unit roots based on M-estimators. These formulae depend upon a nuisance parameter. Consequently, new critical values for this test would have to be generated for each different M-estimator that is used. We therefore provide a methodology for tractable and accurate analytic approximations of quantiles based on asymptotic expansions of functions. We use it to derive simple yet accurate approximations for the asymptotic distribution of these unit root M-tests. Using these asymptotic approximations, critical values of the tests can easily be obtained without even resorting to a computer. The approximation requires no new tabulation, and the resulting cumulative distribution function (cdf) has a maximum absolute error of 0.002 for typical quantiles (i.e. 1-10% quantiles).
The limiting marginal density of efficient estimators of bivariate cointegration vectors is deriv... more The limiting marginal density of efficient estimators of bivariate cointegration vectors is derived in closed form. The formula is exact, and it consists of highly efficient convergent expansion. It is used to plot the density. Furthermore, it is manipulated analytically to reveal features that could not be uncovered by Monte Carlo. For example, it is demonstrated that moments of any integer order exist, and the derived unconditional (marginal) density is compared to the conditional one which is normal.
ABSTRACT Ways of improving the efficiency of Monte-Carlo (MC) techniques are studied for dynamic ... more ABSTRACT Ways of improving the efficiency of Monte-Carlo (MC) techniques are studied for dynamic models. Such models cause the conventinal Antithetic Variate (AV) technique to fail, and will be proved to reduce the benefit from using Control Variates with nearly non-stationary series. This paper suggest modifications of the two conventional variance reduction techniques to enhance their efficiency. New classes of AV's are also proposed. Methods of reordering residuals are found to do less well than others wich rely on changing signs in the spirit of the traditional AV. Then, unconventional applications of these techniques of MC work for nearly nonstationary series. It generates econometric estimators at one and a half times the speed of conventional MC.
Summary This paper proposes a standard for notation in econometrics. It presents a fully integrat... more Summary This paper proposes a standard for notation in econometrics. It presents a fully integrated and internally consistent framework for notation and abbreviations, which is as close as possible to existing common practice. The symbols used are instantly recognizable and ...
We show how investors' liquidity patterns can provide a common framework to explain autocorrelati... more We show how investors' liquidity patterns can provide a common framework to explain autocorrelation of returns and volumes, and some calendar anomalies. The method helps us find new anomalies, and contribute to the explanation of older ones. We uncover a "festivities effect" that is composed of a pre-festivity period of negative returns and relatively low trading activity, and a post-festivity period of positive returns and increased trading activity. We demonstrate this effect for ten countries in the Middleand Far-East where the main festivities occur every year at a different time of the Western Gregorian calendar. In particular, we consider the Muslim Ramadan and Chinese New Year festivities in these countries.
This paper deals with models allowing for trending processes and cyclical component with error pr... more This paper deals with models allowing for trending processes and cyclical component with error processes that are possibly nonstationary, nonlinear, and non-Gaussian. Asymptotic confidence intervals for the trend, cyclical component, and memory parameters are obtained. The confidence intervals are applicable for a wide class of processes, exhibit good coverage accuracy, and are easy to implement. JEL Classification: C22.
Page 1. Econometric Theory, 11, 1995, 775-793. Printed in the United States of America. THE LIMIT... more Page 1. Econometric Theory, 11, 1995, 775-793. Printed in the United States of America. THE LIMITING DISTRIBUTION OF THE t RATIO UNDER A UNIT ROOT KARIM M. ABADIR University of Exeter An encompassing formula ...
We provide a methodology to calculate the expectation of a variate x in terms of the moments of a... more We provide a methodology to calculate the expectation of a variate x in terms of the moments of a transformation of x. Apart from the intrinsic interest in such a fundamental relation that relates the moments of a variate and its nonlinear transformations, our results can be used in practice to approximate E(x) by the low-order moments of a transformation
We extend the classical local Whittle estimation procedure to fractionally integrated I(d) proces... more We extend the classical local Whittle estimation procedure to fractionally integrated I(d) processes, where d>-3/2, thus covering stationary and non-stationary regions. Asymptotic properties of the bias of the estimator are investigated. It is shown that in a wider region than previously considered in the literature, the estimator exhibits the same asymptotic properties as in the stationary case. When the generating