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Likelihood-Based Tests for Common and Idiosyncratic Unit Roots in the Exact Factor Model
Uppsala University, Disciplinary Domain of Humanities and Social Sciences, Faculty of Social Sciences, Department of Statistics.
2013 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Dynamic panel data models are widely used by econometricians to study over time the economics of, for example, people, firms, regions, or countries, by pooling information over the cross-section. Though much of the panel research concerns inference in stationary models, macroeconomic data such as GDP, prices, and interest rates are typically trending over time and require in one way or another a nonstationary analysis. In time series analysis it is well-established how autoregressive unit roots give rise to stochastic trends, implying that random shocks to a dynamic process are persistent rather than transitory. Because the implications of, say, government policy actions are fundamentally different if shocks to the economy are lasting than if they are temporary, there are now a vast number of univariate time series unit root tests available. Similarly, panel unit root tests have been designed to test for the presence of stochastic trends within a panel data set and to what degree they are shared by the panel individuals. Today, growing data certainly offer new possibilities for panel data analysis, but also pose new problems concerning double-indexed limit theory, unobserved heterogeneity, and cross-sectional dependencies. For example, economic shocks, such as technological innovations, are many times global and make national aggregates cross-country dependent and related in international business cycles.

Imposing a strong cross-sectional dependence, panel unit root tests often assume that the unobserved panel errors follow a dynamic factor model. The errors will then contain one part which is shared by the panel individuals, a common component, and one part which is individual-specific, an idiosyncratic component. This is appealing from the perspective of economic theory, because unobserved heterogeneity may be driven by global common shocks, which are well captured by dynamic factor models. Yet, only a handful of tests have been derived to test for unit roots in the common and in the idiosyncratic components separately. More importantly, likelihood-based methods, which are commonly used in classical factor analysis, have been ruled out for large dynamic factor models due to the considerable number of parameters.

This thesis consists of four papers where we consider the exact factor model, in which the idiosyncratic components are mutually independent, and so any cross-sectional dependence is through the common factors only. Within this framework we derive some likelihood-based tests for common and idiosyncratic unit roots. In doing so we address an important issue for dynamic factor models, because likelihood-based tests, such as the Wald test, the likelihood ratio test, and the Lagrange multiplier test, are well-known to be asymptotically most powerful against local alternatives.

Our approach is specific-to-general, meaning that we start with restrictions on the parameter space that allow us to use explicit maximum likelihood estimators. We then proceed with relaxing some of the assumptions, and consider a more general framework requiring numerical maximum likelihood estimation. By simulation we compare size and power of our tests with some established panel unit root tests. The simulations suggest that the likelihood-based tests are locally powerful and in some cases more robust in terms of size.

Place, publisher, year, edition, pages
Uppsala: Acta Universitatis Upsaliensis, 2013. , 51 p.
Series
Digital Comprehensive Summaries of Uppsala Dissertations from the Faculty of Social Sciences, ISSN 1652-9030 ; 90
Keyword [en]
panel unit root, dynamic factors, maximum likelihood, Lagrange multiplier, likelihood ratio, factor analysis
National Category
Social Sciences
Research subject
Statistics
Identifiers
URN: urn:nbn:se:uu:diva-207296ISBN: 978-91-554-8754-6 (print)OAI: oai:DiVA.org:uu-207296DiVA: diva2:647586
Public defence
2013-11-08, Hörsal 2, Ekonomikum, Kyrkogårdsgatan 10, Uppsala, 10:00 (English)
Opponent
Supervisors
Projects
Solving Macroeconomic Problems Using Non-Stationary Panel Data
Available from: 2013-10-17 Created: 2013-09-11 Last updated: 2014-01-23
List of papers
1. An LM-type Test for Idiosyncratic Unit Roots in the Exact Factor Model with Nonstationary Common Shocks
Open this publication in new window or tab >>An LM-type Test for Idiosyncratic Unit Roots in the Exact Factor Model with Nonstationary Common Shocks
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider an exact factor model with integrated factors and propose an LM-type test for unit roots in the idiosyncratic component. We show that, for a fixed number of panel individuals (N) and when the number of time points (T) tends to infinity, the limiting distribution of the LM-type statistic is a weighted sum of independent chi-square-one variables, and when T tends to infinity followed by N tending to infinity, the limiting distribution is standard normal. The test is derived under assumptions that are restrictive just enough to be able to rely on explicit maximum likelihood estimators, and should contribute to the challenging task of deriving likelihood or quasi-likelihood based unit root tests in dynamic factor models.

Keyword
Panel unit root, Dynamic factors, Maximum likelihood, Lagrange multiplier
National Category
Social Sciences
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-207291 (URN)
Projects
Solving Macroeconomic Problems Using Non-Stationary Panel Data
Available from: 2013-09-11 Created: 2013-09-11 Last updated: 2015-01-23
2. On a Likelihood Ratio Test for Idiosyncratic Unit Roots in the Exact Factor Model with Nonstationary Common Shocks
Open this publication in new window or tab >>On a Likelihood Ratio Test for Idiosyncratic Unit Roots in the Exact Factor Model with Nonstationary Common Shocks
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider an exact factor model with unobservable common stochastic trends imposed by nonstationary factors, and study, by simulation, the power of the likelihood ratio test for unit roots in the idiosyncratic components. The power of the test is compared with the analogous Lagrange multiplier test and the Fisher-type test proposed by Bai and Ng (A PANIC attack on unit roots and cointegration, Econometrica,72, 1127–1177, 2004). The results suggest that the benefit of the likelihood ratio test is in panels with a small cross-section.

Keyword
Panel unit root, Dynamic factors, Maximum likelihood, Likelihood ratio
National Category
Social Sciences
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-207293 (URN)
Projects
Solving Macroeconomic Problems Using Non-Stationary Panel Data
Available from: 2013-09-11 Created: 2013-09-11 Last updated: 2014-01-23
3. LM-type Tests For Idiosyncratic and Common Unit Roots in the Exact Factor Model with AR(1) Dynamics
Open this publication in new window or tab >>LM-type Tests For Idiosyncratic and Common Unit Roots in the Exact Factor Model with AR(1) Dynamics
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Recent developments within the panel unit-root literature have illustrated how the exact factor model serves as a parsimonious framework and allows for consistent maximum likelihood inference even when it is misspecified contra the more general approximate factor model. In this paper we consider an exact factor model with AR(1) dynamics and propose LM-type tests for idiosyncratic and common unit roots. We derive the asymptotic distributions and carry out simulations to investigate size and power of the tests in finite samples, as well as compare the performance with some existing tests.

Keyword
Panel unit root, Dynamic factors, Maximum likelihood, Lagrange multiplier
National Category
Social Sciences
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-207294 (URN)
Projects
Solving Macroeconomic Problems Using Non-Stationary Panel Data
Available from: 2013-09-11 Created: 2013-09-11 Last updated: 2016-03-30
4. An LM-type Test for Idiosyncratic Unit Roots in the Exact Factor Model under Misspecification
Open this publication in new window or tab >>An LM-type Test for Idiosyncratic Unit Roots in the Exact Factor Model under Misspecification
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We consider an exact factor model and derive a Lagrange multiplier-type test for unit roots in the idiosyncratic components. The asymptotic distribution of the statistic is derived under the misspecification that the differenced factors are white noise. We prove that the asymptotic distribution is independent of the distribution of the factors, and that the factors are allowed to be integrated, cointegrate, or be stationary. In a simulation study, size and power is compared with some popular second generation panel unit root tests. The simulations suggest that our statistic is well-behaved in terms of size and that it is powerful and robust in comparison with existing tests.

Keyword
Panel unit root, Dynamic factors, Maximum likelihood, Lagrange multiplier
National Category
Social Sciences
Research subject
Statistics
Identifiers
urn:nbn:se:uu:diva-207295 (URN)
Projects
Solving Macroeconomic Problems Using Non-Stationary Panel Data
Available from: 2013-09-11 Created: 2013-09-11 Last updated: 2014-01-23

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Output format
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