Lorenzo Trapani

Asymptotics for Panel Models with Common Shocks

This article develops a novel asymptotic theory for panel models with common shocks. We assume that contemporaneous correlation can be generated by both the presence of common regressors among units and weak spatial dependence among the error terms. Several characteristics of the panel are considered: cross-sectional and time-series dimensions can either be fixed or large; factors can either be observable or unobservable; the factor model can describe either a cointegration relationship or a spurious regression, and we also consider the stationary case. We derive the rate of convergence and the limit distributions for the ordinary least square (OLS) estimates of the model parameters under all the aforementioned cases.

Detecting Common Longevity Trends by a Multiple Population Approach

Recently the interest in the development of country and longevity risk models has been growing. The investigation of long-run equilibrium relationships could provide valuable information about the factors driving changes in mortality, in particular across ages and across countries. In order to investigate cross-country common longevity trends, tools to quantify, compare, and model the strength of dependence become essential. On one hand, it is necessary to take into account either the dependence for adjacent age groups or the dependence structure across time in a single population setting—a sort of intradependence structure. On the other hand, the dependence across multiple populations, which we describe as interdependence, can be explored for capturing common long-run relationships between countries. The objective of our work is to produce longevity projections by taking into account the presence of various forms of cross-sectional and temporal dependencies in the error processes of multiple populations, considering mortality data from different countries. The algorithm that we propose combines model-based predictions in the Lee-Carter (LC) framework with a bootstrap procedure for dependent data, and so both the historical parametric structure and the intragroup error correlation structure are preserved. We introduce a model which applies a sieve bootstrap to the residuals of the LC model and is able to reproduce, in the sampling, the dependence structure of the data under consideration. In the current article, the algorithm that we build is applied to a pool of populations by using ideas from panel data; we refer to this new algorithm as the Multiple Lee-Carter Panel Sieve (MLCPS). We are interested in estimating the relationship between populations of similar socioeconomic conditions. The empirical results show that the MLCPS approach works well in the presence of dependence.

Testing for instability in covariance structures

We propose a test for the stability over time of the covariance matrix of multivariate time series. The analysis is extended to the eigensystem to ascertain changes due to instability in the eigenvalues and/or eigenvectors. Using strong Invariance Principles and Law of Large Numbers, we normalise the CUSUM-type statistics to calculate their supremum over the whole sample. The power properties of the test versus alternative hypotheses, including also the case of breaks close to the beginning/end of sample are investigated theoretically and via simulation. We extend our theory to test for the stability of the covariance matrix of a multivariate regression model. The testing procedures are illustrated by studying the stability of the principal components of the term structure of 18 US interest rates.

A randomised sequential procedure to determine the number of factors

ABSTRACT This article proposes a procedure to estimate the number of common factors k in a static approximate factor model. The building block of the analysis is the fact that the first k eigenvalues of the covariance matrix of the data diverge, while the others stay bounded. On the grounds of this, we propose a test for the null that the ith eigenvalue diverges, using a randomized test statistic based directly on the estimated eigenvalue. The test only requires minimal assumptions on the data, and no assumptions are required on factors, loadings or idiosyncratic errors. The randomized tests are then employed in a sequential procedure to determine k. Supplementary materials for this article are available online.

On the asymptotic t-test for large nonstationary panel models

The asymptotic t-test for the long-run average in a heterogeneous nonstationary panel model is derived. The asymptotics of the Least Squares Dummy Variable (LSDV) and of the Pooled-OLS (POLS) estimators for the slope parameter are studied under various circumstances (serial correlation, strong cross-sectional dependence in the errors and in the regressors and mixed stationary/nonstationary errors) and a modified estimator of the asymptotic variance is derived. The asymptotic variance is computed up to a simple transformation of the residual and no nuisance parameters need to be estimated. The resulting t-statistics are shown to have a standard normal limiting distribution. Asymptotic tests based on the standardized version of the t-statistic are shown to have good power properties, and the correct size, even for n as small as 25.

Testing for Exogeneity in Cointegrated Panels

This paper proposes a test for the null that, in a cointegrated panel, the long-run correlation between the regressors and the error term is different from zero. As is well known, in such case the OLS estimator is T-consistent, whereas it is math formula-consistent when there is no endogeneity. Other estimators can be employed, such as the FM-OLS, that are math formula-consistent irrespective of whether exogeneity is present or not. Using the difference between the former and the latter estimator, we construct a test statistic which diverges at a rate math formula under the null of endogeneity, whilst it is bounded under the alternative of exogeneity, and employ a randomization approach to carry out the test. Monte Carlo evidence shows that the test has the correct size and good power.

On Bootstrapping Panel Factor Series - Extended Version

This paper studies the asymptotic validity of sieve bootstrap for nonstationary panel factor series. Two main results are shown. Firstly, a bootstrap Invariance Principle is derived pointwise in i, obtaining an upper bound for the order of truncation of the AR polynomial that depends on n and T. Consistent estimation of the long run variances is also studied for (n,T)→∞. Secondly, joint bootstrap asymptotics is also studied, investigating the conditions under which the bootstrap is valid. The extent of cross sectional dependence which can be allowed for is investigated, showing that, in the presence of strong cross dependence, consistent estimation of the long run variance (and therefore validity of the bootstrap) is no longer possible. The paper also considers extensions to the case of a mixture of stationary and nonstationary common factors.

Common stochastic trends and aggregation in heterogeneous panels

In nonstationary heterogeneous panels where the number of units is finite and where each unit cointegrates, a large number of conditions needs to be satisfied for cointegration to be preserved in the aggregate relationship. In reality, the conditions most likely will not hold. This paper takes a closer look at what happens when the conditions are violated. In this case, the question of whether an aggregate relationship is observationally equivalent to a cointegrating equation is of particular interest. We derive a measure of the degree of noncointegration of the aggregate estimates, and we explore its asymptotic properties.

Testing for Breaks in Cointegrated Panels with Common and Idiosyncratic Stochastic Trends

In this paper, we develop tests for structural change in cointegrated panel regressions with common and idiosyncratic trends. We consider both the cases of observable and nonobservable common trends, deriving a Functional Central Limit Theorem for the partial sample estimators under the null of no break. We show that tests based on sup-Wald statistics are powerful versus breaks of size , also proving that power is present when the time of change differs across units and when only some units have a break. Our framework is extended to the case of cross correlated regressors and endogeneity. Monte Carlo evidence shows that the tests have the correct size and good power properties.

Asymptotics for Panel Models with Common Shocks - Extended Version

This paper develops a novel asymptotic theory for panel models with common shocks. We assume that contemporaneous correlation can be generated by both the presence of common regressors among units and weak spatial dependence among the error terms. Several characteristics of the panel are considered: cross-sectional and time-series dimensions can either be fixed or large; factors can either be observable or unobservable; the factor model can describe either a cointegration relationship or a spurious regression, and we also consider the stationary case. We derive the rate of convergence and the limit distributions for the ordinary least squares (OLS) estimates of the model parameters under all the aforementioned cases.