Money and inflation in Germany: A cointegration analysis

This paper analyses the relation between money and inflation in Germany in a cost-push/demand-pull model of an open small economy by means of cointegration methods. The full-information-maximum-likelihood method of Johansen as well as structural methods are applied to datasubsets and the full data set. The focus of the paper is on tests for overidentifying restrictions and for weak and strong exogeneity within these data sets. The result of the paper is that the money stock, the price level and gross national product are endogenous whereas the interest rate and the real import price are both weakly and strongly exogenous. By means of the price cointegration relation we illustrate how monetary targeting should react to imported inflation.     

The stability of German money demand: Tests of the cointegration relation

The Stability of German Money Demand: Tests of the Cointegration Relation. — In this study, two money demand functions are specified in single equation error correction as well as triangular error correction form involving real M1 (M3), real GNP, a short-and a long-term interest rate. Using various tests, it is shown that there may be a cointegration relation even after the German Monetary Union (GMU) was established in 1990. But the long-run coefficients of GNP and the interest rate probably have a structural break in 1973, when the Bundesbank changed its monetary regime, and in 1990, when the GMU was formed. The tests support weak exogeneity of real GNP and the interest rate.     

On the properties of the coefficient of determination in regression models with infinite variance variables

We examine the asymptotic properties of the coefficient of determination, R²$R^2$, in models with α-stable$\alpha \text{-stable}$   random variables. If the regressor and error term share the same index of stability α<2$\alpha <2$, we show that the R²$R^2$  statistic does not converge to a constant but has a nondegenerate distribution on the entire [0,1]$[0,1]$ interval. We provide closed-form expressions for the cumulative distribution function and probability density function of this limit random variable, and we show that the density function is unbounded at 0 and 1. If the indices of stability of the regressor and error term are unequal, we show that the coefficient of determination converges in probability to either 0 or 1, depending on which variable has the smaller index of stability, irrespective of the value of the slope coefficient. In an empirical application, we revisit the Fama and MacBeth (1973) two-stage regression and demonstrate that in the infinite-variance case the R²$R^2$  statistic of the second-stage regression converges to 0 in probability even if the slope coefficient is nonzero. We deduce that a small value of the R²$R^2$  statistic should not, in itself, be used to reject the usefulness of a regression model.     

Exact inference and optimal invariant estimation for the stability parameter of symmetric α-stable distributions

Hill estimation (Hill, 1975), the most widespread method for estimating tail thickness of heavy-tailed financial data, suffers from two drawbacks. One is that the optimal number of tail observations to use in the estimation is a function of the unknown tail index being estimated, which diminishes the empirical relevance of the Hill estimation. The other is that the hypothesis test of the underlying data lying in the domain of attraction of an α-stable law (α < 2) or of a normal law (α ≥ 2) for finite samples, is performed on the basis of the asymptotic distribution, which can be different from those for finite samples. In this paper, using the Monte Carlo technique, we propose an exact test method for the stability parameter of α-stable distributions which is based on the Hill estimator, yet is able to provide exact confidence intervals for finite samples. Our exact test method automatically includes an estimation procedure which does not need the assumption of a known number of observations on the distributional tail. Empirical applications demonstrate the advantages of our new method in comparison with the Hill estimation.