In this aticle, the extension of Nelson's (1991) univariate EGARCH model to the multivariate version has been reexamined and compared with the existing one given by Koutmos and Booth (1995). The magnitude and sign of standardized innovations have been constrained in Koutmos and Booth's multivariate EGARCH model, but not in the actual multivariate EGARCH model. The constraints imposed on Koutmos and Booth's EGARCH model may lead to inaccurate parameter estimates. Since the actual multivariate EGARCH model obtained is more general, and can produce more accurate inferential results, we suggest that the actual multivariate EGARCH model be used in future financial empirical studies.
In the present article, we demonstrates the use of SAS PROC CALIS to fit various types of Level-1 error covariance structures of latent growth models (LGM). Advantages of the SEM approach, on which PROC CALIS is based, include the capabilities of modeling the change over time for latent constructs, measured by multiple indicators; embedding LGM into a larger latent variable model; incorporating measurement models for latent predictors; and better assessing model fit and the flexibility in specifying error covariance structures. The strength of PROC CALIS is always accompanied with technical coding work, which needs to be specifically addressed. We provide a tutorial on the SAS syntax for modeling the growth of a manifest variable and the growth of a latent construct, focusing the documentation on the specification of Level-1 error covariance structures. Illustrations are conducted with the data generated from two given latent growth models. The coding provided is helpful when the growth model has been well determined and the Level-1 error covariance structure is to be identified.Keywords Error covariance structure . Latent growth model . Structural equation modeling The latent growth model (LGM) plays an important role in repeated measure analysis over a limited number of occasions in large samples (e.g., Meredith & Tisak, 1990;Muthén & Khoo, 1998; Preacher, Wichman, MacCallum, & Briggs, 2008, p. 12). The model can not only characterize intraindividual (within subjects) change over time, but also examine interindividual (between subjects) difference by means of random growth coefficients, and is a typical application of hierarchical linear modeling (HLM). The within-subjects errors over time and the between-subjects errors are conventionally referred to as "Level-1" and "Level-2" errors, respectively. LGM can also be handled by using structural equation modeling (SEM) (e.g., Bauer, 2003;Bollen & Curran, 2006;Chan, 1998;Curran, 2003;Duncan, Duncan, & Hops, 1996; Mehta & Neal 2005;Meredith & Tisak, 1990; Willet & Sayer, 1994). SEM and HLM stem from different statistical theory, and each has developed its own terminology and standard ways of framing research questions. However, there exists much overlap between the two methodologies under some circumstances. Typically, when a two-level data structure arises from the repeated observations of a variable over time for a set of individuals (so that time is hierarchically nested within each individual), SEM is analytically equivalent
It has been pointed out in the literature that misspecification of the level-1 error covariance structure in latent growth modeling (LGM) has detrimental impacts on the inferences about growth parameters. Since correct covariance structure is difficult to specify by theory, the identification needs to rely on a specification search, which, however, is not systematically addressed in the literature. In this study, we first discuss characteristics of various covariance structures and their nested relations, based on which we then propose a systematic approach to facilitate identifying a plausible covariance structure. A test for stationarity of an error process and the sequential chi-square difference test are conducted in the approach. Preliminary simulation results indicate that the approach performs well when sample size is large enough. The approach is illustrated with empirical data. We recommend that the approach be used in LGM empirical studies to improve the quality of the specification of the error covariance structure.
In the present study, we discuss reliability, consistency, and method specificity based on the CT-C (M−1) model, which provides clear definitions of trait and method factors and can facilitate parameter estimation. Properties of the reliability coefficient, the consistency coefficient, and the method-specificity coefficient of the summated score for a trait factor are addressed. The consistency coefficient and the method-specificity coefficient are both functions of the number of items, the average item consistency, and the average item method specificity. The usefulness of the findings is demonstrated in an alternative approach proposed for scale reduction. The approach, taking into account both traits and methods, helps identify the items leading to the maximum of convergent validity or method effects. The approach, illustrated with a simulated data set, is recommended for scale development based on multitraitmultimethod designs.
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