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Book part
Publication date: 29 March 2006

Peter A. Zadrozny

A univariate GARCH(p,q) process is quickly transformed to a univariate autoregressive moving-average process in squares of an underlying variable. For positive integer m…

Abstract

A univariate GARCH(p,q) process is quickly transformed to a univariate autoregressive moving-average process in squares of an underlying variable. For positive integer m, eigenvalue restrictions have been proposed as necessary and sufficient restrictions for existence of a unique mth moment of the output of a univariate GARCH process or, equivalently, the 2mth moment of the underlying variable. However, proofs in the literature that an eigenvalue restriction is necessary and sufficient for existence of unique 4th or higher even moments of the underlying variable, are either incorrect, incomplete, or unnecessarily long. Thus, the paper contains a short and general proof that an eigenvalue restriction is necessary and sufficient for existence of a unique 4th moment of the underlying variable of a univariate GARCH process. The paper also derives an expression for computing the 4th moment in terms of the GARCH parameters, which immediately implies a necessary and sufficient inequality restriction for existence of the 4th moment. Because the inequality restriction is easily computed in a finite number of basic arithmetic operations on the GARCH parameters and does not require computing eigenvalues, it provides an easy means for computing “by hand” the 4th moment and for checking its existence for low-dimensional GARCH processes. Finally, the paper illustrates the computations with some GARCH(1,1) processes reported in the literature.

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Econometric Analysis of Financial and Economic Time Series
Type: Book
ISBN: 978-0-76231-274-0

Book part
Publication date: 19 October 2020

Hon Ho Kwok

This chapter develops a set of two-step identification methods for social interactions models with unknown networks, and discusses how the proposed methods are connected to the…

Abstract

This chapter develops a set of two-step identification methods for social interactions models with unknown networks, and discusses how the proposed methods are connected to the identification methods for models with known networks. The first step uses linear regression to identify the reduced forms. The second step decomposes the reduced forms to identify the primitive parameters. The proposed methods use panel data to identify networks. Two cases are considered: the sample exogenous vectors span Rn (long panels), and the sample exogenous vectors span a proper subspace of Rn (short panels). For the short panel case, in order to solve the sample covariance matrices’ non-invertibility problem, this chapter proposes to represent the sample vectors with respect to a basis of a lower-dimensional space so that we have fewer regression coefficients in the first step. This allows us to identify some reduced form submatrices, which provide equations for identifying the primitive parameters.

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Review of Marketing Research
Type: Book
ISBN: 978-0-85724-723-0

Book part
Publication date: 1 June 2022

Monica Billio, Roberto Casarin and Fausto Corradin

This chapter studies the effects of the COVID-19 pandemic on the economic structure of the US and EU economies by measuring its impact on some reference macro-economic variables…

Abstract

This chapter studies the effects of the COVID-19 pandemic on the economic structure of the US and EU economies by measuring its impact on some reference macro-economic variables. We use a factor model approach on a set of variables available at different frequencies (daily, weekly, monthly, and quarterly) and provide evidence of instability in the primary factors driving the economy. A sequential analysis of the factors allows us to evaluate the model's forecasting performance and extract some instability measures based on the factor model's eigenvalues. Finally, we show how to use COVID-related variables, such as policy, economic, and health indicators, to compute conditional forecasts with factor models, and perform a scenario analysis on the variables of interest to understand economic instability.

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The Economics of COVID-19
Type: Book
ISBN: 978-1-80071-694-0

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Book part
Publication date: 18 January 2022

Cheng Hsiao, Yan Shen and Qiankun Zhou

Panel data provide the possibilities of estimating individual treatment effects for multiple individuals. Two issues are considered: (1) differences in the estimated individual…

Abstract

Panel data provide the possibilities of estimating individual treatment effects for multiple individuals. Two issues are considered: (1) differences in the estimated individual treatment effects are due to heterogeneity or a chance mechanism? (2) what is the best way to estimate the average treatment effects? Testing and aggregation methods are suggested. Monte Carlo simulations are also conducted to shed light on these two issues. An empirical analysis on the involvement of underground organization in China’s Peer-to-Peer (P2P) activities through the “anti-gang” campaign is also provided.

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Essays in Honor of M. Hashem Pesaran: Panel Modeling, Micro Applications, and Econometric Methodology
Type: Book
ISBN: 978-1-80262-065-8

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Book part
Publication date: 4 September 2023

Stephen E. Spear and Warren Young

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Overlapping Generations: Methods, Models and Morphology
Type: Book
ISBN: 978-1-83753-052-6

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Economic Complexity
Type: Book
ISBN: 978-0-44451-433-2

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Functional Structure Inference
Type: Book
ISBN: 978-0-44453-061-5

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Explaining Unemployment: Econometric Models for the Netherlands
Type: Book
ISBN: 978-1-84950-847-6

Book part
Publication date: 21 November 2014

Jin Seo Cho and Halbert White

We provide a new characterization of the equality of two positive-definite matrices A and B, and we use this to propose several new computationally convenient statistical tests…

Abstract

We provide a new characterization of the equality of two positive-definite matrices A and B, and we use this to propose several new computationally convenient statistical tests for the equality of two unknown positive-definite matrices. Our primary focus is on testing the information matrix equality (e.g. White, 1982, 1994). We characterize the asymptotic behavior of our new trace-determinant information matrix test statistics under the null and the alternative and investigate their finite-sample performance for a variety of models: linear regression, exponential duration, probit, and Tobit. The parametric bootstrap suggested by Horowitz (1994) delivers critical values that provide admirable level behavior, even in samples as small as n = 50. Our new tests often have better power than the parametric-bootstrap version of the traditional IMT; when they do not, they nevertheless perform respectably.

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Essays in Honor of Peter C. B. Phillips
Type: Book
ISBN: 978-1-78441-183-1

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