Probabilistic approach to assessing macroeconomic uncertainties – PRAM

When using stable and tempered stable distributions, the practitioner faces the two following problems: 1) the densities do not have explicit expressions; 2) the existence of unknown dynamics in macroeconomic series. In the case of tempered stable distributions, the following point is added: 3) the difficulty of doing Monte Carlo simulations. If the dynamics mentioned in point 2) is neglected and if we limit ourself to a simple framework (in particular univariate) with a small number of unknown parameters, 1) is easily overcome by using the standard maximum likelihood estimator, using numerical evaluations of the stable densities. In practice, the dynamics can not be neglected, without risking erroneous statistical conclusions. We called « Quasi marginal maximum likelihood » the method which consists in estimating the marginal distribution as if the observations were independent, taking into account the dynamics to correct, in a non parametric way, for the asymptotic variance of the estimator.
The stable distributions being defined by their characteristic functions, a natural alternative consists in using an estimator based on the empirical characteristic function. To consider the point 2) one can also specify a parametric dynamics. We explored all these approaches. In particular, we proposed and studied a model of explosive bubbles.

In an article that has just been accepted for publication, we study the asymptotic properties of the quasi marginal maximum likelihood marginal estimator, which is the maximum likelihood criterion in the presence of a general dynamics. This estimator applies when the marginal distribution is well specified, but the dynamic is a nuisance parameter that is not specified. We give general results that apply in particular to stable marginal distributions. In a working paper, which is currently under review, we consider the estimator based on a comparison between the characteristic function of a stable law and the empirical characteristic function of the residuals of a GARCH model. Estimators of this type are usually based on the empirical characteristic function of the observations, while ours is based on residuals. We show that the estimator has a nonstandard asymptotic distribution (see the figure of the illustration). In several other studies, we investigate the numerical aspects, in particular we compare several methods for simulating tempered stable laws (see point 3) of the previous boxed text). To mimic the phenomenon of explosive bubbles, we use a non-causal AR model with Cauchy innovations. This model has natural interpretation and allows for a complete probabilistic and statistical study.

We are trying to apply the general results that we have obtained on the estimator of quasi marginal maximum marginal likelihood to the case where the marginal distribution is tempered stable. The theoretical difficulty is to specify a parameterization that would be the most convenient for the statistical analysis, as we did for the stable laws. The practical difficulty comes from the large number of parameters that have to be estimated. We also seek to obtain more precise information on the law of the estimator based on the empirical characteristic function of the residuals. Finally, we should extend our findings in a multivariate framework.

The paper on the quasi marginal maximum likelihood estimator has been accepted for publication in « Journal of Business and Economic Statistics » (JBES). This work has been presented in several conferences. The paper on the empirical characteristic function is under review for « Econometric Theory » (ET). This will be presented in the conference that will be held in Minsk in September. The article on the bubble is submitted. An article related to modeling macroeconomic series has just been published in « Applied Economics Letters ». Other studies have given rise to papers, many of which are submitted.

The general aim of the project, which is the joint initiative between the UK and French researchers, is to develop and apply new methods which will be used for the evaluation of uncertainties associated with forecasting of main macroeconomic, mostly monetary, indicators. By uncertainties we understand the probabilities that the macroeconomic characteristics like inflation, interest and exchange rates will reach extreme values, indicating deflation of very high inflation, drastic devaluation or revaluation and radical changes in interest rates. The methodological stimulation of the project has been provided by recent development of a new class of probability distributions, the so-called tempered stable distributions. We propose that the macroeconomic uncertainties should be modelled by these distributions which fit better to data than the traditionally used normal (or related) distributions. Such novel approach enables improving on the degree of accuracy in calculation probabilities of realisation of events like drastic devaluations, occurrences of high inflation or deflation and hitting monetary targets. More specifically, this approach will be applied for constructing forecasts, prediction of turning points and assessing risks related to monetary policies. Regarding forecasting, the emphasis is not on the extrapolation of actual tendencies but rather on deviations from such tendencies. The objectives also shares roots with some financial risk management theories used in finance, especially for modelling of option pricing. However, the methods and techniques will be developed in different direction, by concentrating on changes in parameters over time, mutual dependencies and mixing different types of distributions.Specific objectives of the project concern the development and application of new methods which use the tempered stable distributions in:

Static analysis (Objective 1): The assumption here is that the uncertainties are not changing over time. Works will initially concentrate on the methodological problems of the quantification of uncertainty, progressing to the analysis of characteristics of such uncertainties and developing new methods of estimation and hypothesis testing. The empirical part will consist of the analysis of such uncertainties in world inflation and exchange rates between major currencies.

Dynamic analysis (Objective 2): Within this objective we analyse the ex-post time dependencies in dynamic models constructed on the basis of the methods researched within Objective 1. More specifically the approach of analysis of univariate time series under the assumption of time dependence and incorporating distributions of uncertainties will be proposed here and applied to modelling inflationary, interest and exchange rates processes.

Multivariate analysis (Objective 3): Here we generalise the methods developed within Objective 1 in such a way that they could be used for the analysis of uncertainties jointly for a number of periods. These methods will be applied for constructing methodologically innovative probabilistic forecasts of inflation in OECD countries, deviations from target (or equilibrium) interest rates and forecasting of turning points and continuations of tendencies in macroeconomic development. The project will lead to publications of papers in academic journals, development of software which could be used for teaching and further research and delivering fully elaborated methods and computational algorithms to end users (banks and government bodies).