Matrix Differential Calculus with Applications in Statistics and Econometrics

The commutation matrix $K$ is defined as a square matrix containing only zeroes and ones. Its main properties are that it transforms vecA into vecA', and that it reverses the order of a Kronecker product. An analytic expression for $K$ is given and many further properties are derived. Subsequently, these properties are applied to some problems connected with the normal distribution. The expectation is derived of $\varepsilon' A\varepsilon\cdot\varepsilon' B\varepsilon\cdot\varepsilon'C\varepsilon$, where $\varepsilon \sim N(0, V)$, and $A, B, C$ are symmetric. Further, the expectation and covariance matrix of $x \otimes y$ are found, where $x$ and $y$ are normally distributed dependent variables. Finally, the variance matrix of the (noncentral) Wishart distribution is derived.

The commutation matrix: Some properties and applications

Publisher Summary This chapter provides a unified continuous-time, frictionless, no-arbitrage framework for systematically categorizing the various volatility concepts, measurement procedures, and modeling procedures. Volatility has been one of the most active areas of research in empirical finance and time series econometrics during the past decade. It defines different volatility concepts; the notional volatility corresponding to the sample-path return variability over a fixed time interval; the expected volatility over a fixed time interval; and the instantaneous volatility corresponding to the strength of the volatility process at a point in time. The parametric procedures rely on explicit functional form assumptions regarding the expected and/or instantaneous volatility. In the discrete-time ARCH class of models, the expectations are formulated in terms of directly observable variables, while the discrete and continuous-time stochastic volatility models involve latent state variable(s). The nonparametric procedures are generally free from such functional form assumptions and hence afford estimates of notional volatility that are flexible yet consistent (as the sampling frequency of the underlying returns increases). The nonparametric procedures include ARCH filters and smoothers designed to measure the volatility over infinitesimally short horizons, as well as the recently popularized realized volatility measures for fixed-length time intervals.

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Parametric and Nonparametric Volatility Measurement

A large literature over several decades reveals both extensive concern with the question of time-varying betas and an emerging consensus that betas are in fact time-varying, leading to the prominence of the conditional CAPM. Set against that background, we assess the dynamics in realized betas, vis-a-vis the dynamics in the underlying realized market variance and individual equity covariances with the market. Working in the recently popularized framework of realized volatility, we are led to a framework of nonlinear fractional cointegration: although realized variances and covariances are very highly persistent and well approximated as fractionally integrated, realized betas, which are simple nonlinear functions of those realized variances and covariances, are less persistent and arguably best modeled as stationary I(0) processes. We conclude by drawing implications for asset pricing and portfolio management.

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Realized Beta: Persistence and Predictability

This chapter gives a tour through the empirical analysis of univariate GARCH models for financial time series with stops along the way to discuss various practical issues associated with model specification, estimation, diagnostic evaluation and forecasting.

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Practical Issues in the Analysis of Univariate GARCH Models

A reinterpretation of the tests of overidentifying restrictions

Refined asymptotic methods are used to produce degrees-of-freedom- adjusted Edgeworth and Cornish-Fisher size corrections of the t and F testing procedures for the parameters of a S.U.R. model with serially correlated errors. The corrected tests follow the Student-t and F distributions, respectively, with an approximation error of order Oτ3$$O\left({{\tau ^3}} \right)$$, where τ=1/T$$\tau = 1/\sqrt T $$ and T is the number of time observations. Monte Carlo simulations provide evidence that the size corrections suggested hereby have better finite sample properties, compared to the asymptotic testing procedures (either standard or Edgeworth corrected), which do not adjust for the degrees of freedom.

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Size corrected Significance Tests in Seemingly Unrelated Regressions with Autocorrelated Errors

Alternative size corrections for some GLS test statistics the case of the AR(1) model

A generalization of Zellner's SUR model is derived for sets of seemingly unrelated systems of econometric equations. The resulting structural form – worked out for a set of Cowles Commission-type simultaneous equations systems – is general enough to include any SUR-type or panel-type specification of systems of econometric equations with contemporaneously correlated errors. Maximum estimation efficiency is obtained by treating all the individual subsystems at once rather than in a subsystem-by-subsystem fashion.

Seemingly unrelated systems of econometric equations

The Refinement of Econometric Estimation and Test Procedures: Cornish-Fisher Size Corrected t and F Statistics for the Linear Regression Model with Heteroscedastic Errors

Spyridon D. Symeonides

Papers

A reinterpretation of the tests of overidentifying restrictions

Size corrected Significance Tests in Seemingly Unrelated Regressions with Autocorrelated Errors

Alternative size corrections for some GLS test statistics the case of the AR(1) model

Seemingly unrelated systems of econometric equations

The Refinement of Econometric Estimation and Test Procedures: Cornish-Fisher Size Corrected t and F Statistics for the Linear Regression Model with Heteroscedastic Errors