Paolo Paruolo

ON THE DETERMINATION OF INTEGRATION INDICES IN I(2) SYSTEMS

Abstract This paper proposes estimators of the numbers of common components integrated of a given order in a VAR system, possibly in the presence of a linear trend. The estimators are based on sequences of tests in a two-stage analysis proposed in Johansen (1995), which involves only regression and reduced rank regression. The asymptotic distributions of the test statistics belong to the Limiting Gaussian Functional family and are tabulated by simulation. The proposed estimators select the correct integration indices with limiting probability equal to the complement to 1 of the size of each test in the sequence. The statistical analysis allows also to test for the presence of a linear trend; the combined procedure presents similar asymptotic properties. The finite-sample behavior of the above statistics is studied through a small Monte Carlo experiment. An application to a UK money demand system illustrates the proposed procedures.

Ratings and rankings: Voodoo or Science?

Summary.  Composite indicators aggregate a set of variables by using weights which are understood to reflect the variables’ importance in the index. We propose to measure the importance of a given variable within existing composite indicators via Karl Pearsons ‘correlation ratio’; we call this measure the ‘main effect’. Because socio-economic variables are heteroscedastic and correlated, relative nominal weights are hardly ever found to match relative main effects; we propose to summarize their discrepancy with a divergence measure. We discuss to what extent the mapping from nominal weights to main effects can be inverted. This analysis is applied to six composite indicators, including the human development index and two popular league tables of university performance. It is found that in many cases the declared importance of single indicators and their main effect are very different, and that the data correlation structure often prevents developers from obtaining the stated importance, even when modifying the nominal weights in the set of non-negative numbers with unit sum.

Asymptotic Inference on the Moving Average Impact Matrix in Cointegrated 1(1) VAR Systems

This paper addresses the problem of inference on the moving average impact matrix and on its row and column spaces in cointegrated 1(1) VAR processes. The choice of bases (i.e., the identification) of these spaces, which is of interest in the definition of the common trend structure of the system, is discussed. Maximum likelihood estimators and their asymptotic distributions are derived, making use of a relation between properly normalized bases of orthogonal spaces, a result that may be of separate interest. Finally, Wald-type tests are given, and their use in connection with existing likelihood ratio tests is discussed.

ASYMPTOTIC EFFICIENCY OF THE TWO STAGE ESTIMATOR IN I (2) SYSTEMS

This paper derives the distribution of the two stage estimator of cointegrating parameters in I(2) systems, abbreviated 2SI2, under several assumptions regarding the drift of the process. The asymptotic distribution is compared with that of the maximum likelihood (ML) estimator derived in Johansen (1997, Scandinavian Journal of Statistics 24, 433–462). It is found that the two asymptotic distributions are the same, thus showing that the 2SI2 estimator is asymptotically as efficient as ML.

Weights and Importance in Composite Indicators: Closing the Gap

Highlights • Composite indicators are widely used in sustainable development and elsewhere.• The effect of weights used in aggregating indicators is complex.• Three tools are presented which help developers and users to investigate effects of weights.• Case studies related to sustainable development demonstrate the benefits.

The Likelihood Ratio Test for the Rank of a Cointegration Submatrix

This paper proposes a likelihood ratio test for rank deficiency of a submatrix of the cointegrating matrix. Special cases of the test include the one of invalid normalization in systems of cointegrating equations, the feasibility of permanent-transitory decompositions and of subhypotheses related to neutrality and long-run Granger noncausality. The proposed test has a chi-squared limit distribution and indicates the validity of the normalization with probability one in the limit, for valid normalizations. The asymptotic properties of several derived estimators of the rank are also discussed. It is found that a testing procedure that starts from the hypothesis of minimal rank is preferable. Copyright 2006 Blackwell Publishing Ltd.

Structured Multivariate Volatility Models

This paper proposes structured parametrizations for multivariate volatility models, which use spatial weight matrices induced by economic proximity. These structured specifications aim at solving the curse of dimensionality problem, which limits feasibility of model-estimation to small cross-sections for unstructured models. Structured parametrizations possess the following four desirable properties: i) they are flexible, allowing for covariance spill-over; ii) they are parsimonious, being characterized by a number of parameters that grows only linearly with the cross-section dimension; iii) model parameters have a direct economic interpretation that reflects the chosen notion of economic classification; iv) model-estimation computations are faster than for unstructured specifications. We give examples of structured specifications for multivariate GARCH models as well as for Stochastic- and Realized-Volatility models. The paper also discusses how to construct spatial weight matrices that are time-varying and possibly derived from a set of covariates.

On Monte Carlo estimation of relative power

This paper derives standard errors for Monte Carlo (MC) estimators of (relative) power of tests when the critical values under the null have also been estimated. This situation is common, for example, in unit root and cointegration (CI) tests. The associated issue of MC design is discussed. The results are illustrated on likelihood-based tests for CI rank determination.

A Reduced Rank Regression Approach to Tests of Asset Pricing

Both the Arbitrage Pricing Theory (APT) and the Capital Asset Pricing Model (CAPM) place restrictions of the cross sectional variation of conditional expectations of asset returns and of macro-indicators. The authors show that these restrictions imposed on the reference statistical models lead to special cases of the reduced rank regression model. The maximum likelihood problem is solved by canonical correlation analysis. Likelihood ratio tests about the number of factors underlying stock returns are straightforward to calculate, thus allowing to discriminate between competing financial theories. Moreover LR tests on the relevance of each macroeconomic indicator within a chosen model can be implemented. Some of the tests are illustrated by an application to Italian stock market data. Copyright 1997 by Blackwell Publishing Ltd

Proximity-Structured Multivariate Volatility Models

In many multivariate volatility models, the number of parameters increases faster than the cross-section dimension, hence creating a curse of dimensionality problem. This paper discusses specification and identification of structured parameterizations based on weight matrices induced by economic proximity. It is shown that structured specifications can mitigate or even solve the curse of dimensionality problem. Identification and estimation of structured specifications are analyzed, rank and order conditions for identification are given and the specification of weight matrices is discussed. Several structured specifications compare well with alternatives in modelling conditional covariances of six returns from the New York Stock Exchange.