Quantile

Abstract: SUMMARY This paper employs response surface regressions based on simulation experiments to calculate distribution functions for some well-known unit root and cointegration test statistics. The principal contributions of the paper are a set of data files that contain estimated response surface coefficients and a computer program for utilizing them. This program, which is freely available via the Internet, can easily be used to calculate both asymptotic and finite-sample critical values and P-values for any of the tests. Graphs of some of the tabulated distribution functions are provided. An empirical example deals with interest rates and inflation rates in Canada. Tests of the null hypothesis that a time-series process has a unit root have been widely used in recent years, as have tests of the null hypothesis that two or more integrated series are not cointegrated. The most commonly used unit root tests are based on the work of Dickey and Fuller (1979) and Said and Dickey (1984). These are known as Dickey-Fuller (DF) tests and Augmented Dickey-Fuller (ADF) tests, respectively. These tests have non-standard distributions, even asymptotically. The cointegration tests developed by Engle and Granger (1987) are closely related to DF and ADF tests, but they have different, non-standard distributions, which depend on the number of possibly cointegrated variables. Although the asymptotic theory of these unit root and cointegration tests is well developed, it is not at all easy for applied workers to calculate the marginal significance level, or P-value, associated with a given test statistic. Until a few years ago (MacKinnon, 1991), accurate critical values for cointegration tests were not available at all. In a recent paper (MacKinnon, 1994), I used simulation methods to estimate the asymptotic distributions of a large number of unit root and cointegration tests. I then obtained reasonably simple approximating equations that may be used to obtain approximate asymptotic P-values. In the present paper, I extend the results to allow for up to 12 variables, instead of six, and I correct two deficiencies of the earlier work. The first deficiency is that the approximating equations are considerably less accurate than the underlying estimated asymptotic distributions. The second deficiency is that, even though the simulation experiments provided information about the finite-sample distributions of the test statistics, the approximating equations were obtained only for the asymptotic case. The key to overcoming these two deficiencies is to use tables of response surface coefficients, from which estimated quantiles for any sample size may be calculated, instead of equations to

Abstract: The recent literature on maximum likelihood cointegration theory studies Gaussian vector autoregression (VAR) models allowing for some deterministic components in the form of polynomials in time. An examination is presented of such models for variables integrated at most of order one, when tests for cointegration involve statistics with non-standard asymptotic distributions. The asymptotic distributions of these test statistics are known to be functions of the distribution of certain matrices of stochastic variables involving integrals of Brownian motions. In fact, conditional on which restrictions on the coefficients of the polynomial in time are valid, different asymptotic distributions are obtained. The cases examined exhaust the hypotheses relevant to the cointegration rank analysis of I(1) variables in models involving up to linear trends and possibly seasonal dummies. The examination solves the numerical problem in making most of the interesting quantiles of these asymptotic distributions available to the applied econometrician.

Abstract: This paper employs response surface regressions based on simulation experiments to calculate asymptotic distribution functions for the Johansen-type likelihood ratio tests for cointegration. These are carried out in the context of the models recently proposed by Pesaran, Shin, and Smith (1997) that allow for the possibility of exogenous variables integrated of order one. The paper calculates critical values that are very much more accurate than those available previously. The principal contributions of the paper are a set of data files that contain estimated asymptotic quantiles obtained from response surface estimation and a computer program for utilizing them. This program, which is freely available via the Internet, can be used to calculate both asymptotic critical values and P-values. Copyright © 1999 John Wiley & Sons, Ltd.

Abstract: Value at Risk (VaR) has become the standard measure of market risk employed by financial institutions for both internal and regulatory purposes. VaR is defined as the value that a portfolio will lose with a given probability, over a certain time horizon (usually one or ten days). Despite its conceptual simplicity, its measurement is a very challenging statistical problem and none of the methodologies developed so far give satisfactory solutions. Interpreting the VaR as the quantile of future portfolio values conditional on current information, we propose a new approach to quantile estimation which does not require any of the extreme assumptions invoked by existing methodologies (such as normality or i.i.d. returns). The Conditional Autoregressive Value-at-Risk or CAViaR model moves the focus of attention from the distribution of returns directly to the behavior of the quantile. We specify the evolution of the quantile over time using a special type of autoregressive process and use the regression quantile framework introduced by Koenker and Bassett to determine the unknown parameters. Since the objective function is not differentiable, we use a differential evolutionary genetic algorithm for the numerical optimization. Utilizing the criterion that each period the probability of exceeding the VaR must be independent of all the past information, we introduce a new test of model adequacy, the Dynamic Quantile test. Applications to simulated and real data provide empirical support to this methodology and illustrate the ability of these algorithms to adapt to new risk environments.