The maximum entropy moment matrix with missing values

The occurrence of missing values for one or several variables has the effect of adding a ridge along the diagonal of their maximum entropy (ME) covariance matrix. This is a second ridge in addition to the usual ridge of the ME covariance matrix.     

Economic benefits of using realized covariance forecasts in risk-based portfolios

This article examines the economic benefit of using the realized covariance matrix forecasts, for constructing the risk-based portfolios. We use the two-scale realized covariance estimator (TSC), the jump robust two-scale realized covariance estimator (RTSC) and the realized bipower covariance estimator (BPC), to forecast the daily realized covariance matrix. Using these covariance matrix forecasts, we implement three risk-based portfolios: the global minimum variance portfolio, the equal risk contribution portfolio and the most diversified portfolio. There is evidence that the portfolio performance improves by using TSC or RTSC estimators as compared to the daily-returns-based estimator. The performance gains are robust to the choice of risk-based portfolio strategy, the degree of investor’s relative risk-aversion, the market conditions and the choice of time intervals.     

Forecast of realized covariance matrix based on asymptotic distribution of the LU decomposition with an application for balancing minimum variance portfolio

We derive the asymptotic distribution for the LU decomposition, that is, the Cholesky decomposition, of realized covariance matrix. Distributional properties are combined with an existing generalized heterogeneous autoregressive (GHAR) method for forecasting realized covariance matrix, which will be referred to as a generalized HARQ (GHARQ) method. An out-of-sample forecast comparison of a real data set shows that the proposed GHARQ method outperforms other existing methods in terms of optimizing the variances of portfolios.     

Principal Component Models for Generating Large GARCH Covariance Matrices

The implementation of multivariate GARCH models in more than a few dimensions is extremely difficult: because the model has many parameters, the likelihood function becomes very flat, and consequently the optimization of the likelihood becomes practicably impossible. There is simply no way that full multivariate GARCH models can be used to estimate directly the very large covariance matrices that are required to net all the risks in a large trading book. This paper begins by describing the principal component GARCH or 'orthogonal GARCH' (O-GARCH) model for generating large GARCH covariance matrices that was first introduced in Alexander and Chibumba (1996) and subsequently developed in Alexander (2000, 2001b). The O-GARCH model is an accurate and efficient method for generating large covariance matrices that only requires the estimation of univariate GARCH models. Hence, it has many practical advantages, for example in value–at–risk models. It works best in highly correlated systems, such as term structures. The purpose of this paper is to show that, if sufficient care is taken with the initial calibration of the model, equities and foreign exchange rates can also be included in one large covariance matrix. Simple conditions for the final covariance matrix to be positive semi-definite are derived.
(J.E.L.: C32, C53, G19, G21, G28).     

On the Fisher information matrix of a vector ARMA process

We derive a neat and compact representation of the asymptotic Fisher information matrix of a vector ARMA process. Its inverse can be used immediately as the asymptotic covariance matrix of the Gaussian maximum likelihood estimator. We also provide the robust sandwich covariance estimator when the process is non-Gaussian.     

A generalized specification test

In a non-linear parametric setting, a class of specification tests developed by Hausman (1978) is extended to accomodate a singular covariance matrix. An application to limited information tests for the exogeneity of instrumental variables is presented.     

S-values and Bayesian weighted all-subsets regressions

This paper compares and contrasts Bayesian variable-exclusion methods proposed by Eduardo Ley and coauthors with methods proposed by Raftery and Sala-i-Martin et al. and with the s-values proposed by myself. A distinction is drawn between estimation uncertainty which is the focus of Ley׳s research and model ambiguity which arises in Ley׳s work and is the focus of my own recent proposal. The discussion is organized around the prior covariance matrix, which needs to be diagonal to support all-subsets regressions. The basic question addressed here is: what aspects of the prior covariance matrix can be taken as known, what aspects can be estimated and what aspects require a sensitivity analysis because they are neither known nor estimable. When diagonality is in doubt, we are more-or-less forced into a model ambiguity sensitivity mode because the data are never rich enough credibly to estimate the full prior covariance matrix. When diagonality is assumed, the data evidence, though very limited, can help to estimate the diagonal elements, but this literature has not yet produced a compelling conventional treatment which will necessarily include both estimation uncertainty and model ambiguity as they relate both to the diagonal values and to the rest of the prior covariance matrix. But there has been a lot of progress.     

A latent dynamic factor approach to forecasting multivariate stock market volatility

This paper proposes a latent dynamic factor model for high-dimensional realized covariance matrices of stock returns. The approach is based on the matrix logarithm and combines common latent factors driven by HAR processes and idiosyncratic autoregressive dynamics. The model accounts for positive definiteness of covariance matrices without imposing parametric restrictions. Simulated Bayesian parameter estimates are obtained using basic Markov chain Monte Carlo methods. An empirical application to 5-dimensional and 30-dimensional realized covariance matrices shows remarkably good forecasting results, in-sample and out-of-sample.     

Finite-sample size distortion of the AESTAR unit root test: GARCH,corrected variance–covariance matrix estimators and adjusted critical values

The behaviour of the asymmetric exponential smooth transition autoregressive (AESTAR) unit root test, which allows for asymmetric and nonlinear reversion to equilibrium, is examined in the presence of generalized autoregressive conditional heteroscedasticity (GARCH). It is found that while the test is relatively robust in the presence of ‘low volatility’ GARCH processes, it exhibits substantial size distortion when large values of the volatility parameter are considered. Attempted resolution via the routine application of heteroscedasticity consistent (or ‘corrected’) covariance matrix estimators (HCCMEs) is shown to result in overwhelming size distortion due to their impact upon the finite-sample distribution of the underlying test statistic. However, application of a corrected HCCME, in combination with critical values derived specifically under its use, results in the control of test size. Analogous results for the Dickey–Fuller (1979) test are presented to permit comparison with a test considering linear, symmetric adjustment. It is found that the AESTAR test is subject to far greater distortion than its linear, symmetric alternative. In summary, the results indicate that caution must be exercised when applying the AESTAR test to macroeconomic and financial time series, particularly if routine application of corrected covariance matrix estimators occurs.     

Bartlett adjustments for systems of linear equations with linear restrictions

Simple matrix formulae are derived for calculating a Bartlett adjustment to the likelihood ratio test statistic for testing linear parameter restrictions in a system of linear equations. For the special case of column and/or row restrictions on the matrix of coefficients the adjustment is a simple function of matrix dimensions being invariant to sample observations and the error covariance matrix. An example of testing for homogeneity and symmetry in a demand system is given.     

Time Varying Structural Vector Autoregressions and Monetary Policy

Monetary policy and the private sector behaviour of the U.S. economy are modelled as a time varying structural vector autoregression, where the sources of time variation are both the coefficients and the variance covariance matrix of the innovations. The paper develops a new, simple modelling strategy for the law of motion of the variance covariance matrix and proposes an efficient Markov chain Monte Carlo algorithm for the model likelihood/posterior numerical evaluation. The main empirical conclusions are: (1) both systematic and non-systematic monetary policy have changed during the last 40 years|in particular, systematic responses of the interest rate to inflation and unemployment exhibit a trend toward a more aggressive behaviour, despite remarkable oscillations; (2) this has had a negligible effect on the rest of the economy. The role played by exogenous non-policy shocks seems more important than interest rate policy in explaining the high inflation and unemployment episodes in recent U.S. economic history.     

A simple and efficient estimation method for a non-linear demand system

A least-squares-type estimation method appears to be about as efficient as maximum likelihood with a known contemporaneous covariance matrix.     

Simultaneously determined,time-varying hedge ratios in the soybean complex

Time-varying hedge ratios are derived which account for the dynamic characteristics of prices in the soybean complex. A multivariate generalized autogressive heteroskedastic (MGARCH) model, along with other conditional models, is used to specify the relevant covariance matrix. While the time-varying representations of the variance matrix are statistically appropriateex anteand ex posthedging effectiveness indicate that they provide minimal gain to hedging in terms of mean return and reduction in variance over a constant conditional procedure. Whether similar findings arise from other applications of GARCH models to optimal hedging is a question for further research.     

CONSISTENT ESTIMATION OF A LIMITING COVARIANCE MATRIX

This paper provides a consistent and positive semi-definite estimator of the limiting covariance matrix of a nonlinear instrumental variable estimator for a nonlinear simultaneous equation model with selectivity studied in Sapra (1989).     

A new test for linear inequality constraints when the variance–covariance matrix depends on the unknown parameters

We extend Hansen’s (2005) test to testing hypotheses involving general inequality constraints where the variance–covariance matrix of the functions in the constraints depends on the unknown parameters. The test can be applied to a wider class of problems than Wolak’s (1991).     

SEMINONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION OF CONDITIONAL MOMENT RESTRICTION MODELS*

This article studies estimation of a conditional moment restriction model with the seminonparametric maximum likelihood approach proposed by Gallant and Nychka (Econometrica 55 (March 1987), 363–90). Under some sufficient conditions, we show that the estimator of the finite dimensional parameter θ is asymptotically normally distributed and attains the semiparametric efficiency bound and that the estimator of the density function is consistent under L₂ norm. Some results on the convergence rate of the estimated density function are derived. An easy to compute covariance matrix for the asymptotic covariance of the θ estimator is presented.     

Sources of employment fluctuations in Taiwan's industries and regions

This empirical study explores the sources of employment fluctuations in Taiwan's industries and regions over the period 1978 to 2004. The quarterly growth rates of employment in nine industries and four regions are modelled with a structural vector autoregression (VAR), and the employment shocks are measured by VAR residuals. The covariance matrix of the VAR residuals is decomposed using system estimation method that selects the parameters to make the error model close to the covariance matrix and, in turn, to estimate the relative importance of national as well as industry-specific and region-specific shocks. The empirical results show that industry-specific shocks account for the major fluctuations in industries and regions. On average, about 83.95% of an industry's cyclical variations and 56.28% of the volatility in a region may be attributed to industry-specific shocks. National shocks account for little employment volatility in industries. Only the finance and personal service industries are highly sensitive to national shocks.     

Maximum likelihood estimation of sum-constrained linear models with insufficient observations

A maximum likelihood procedure for estimating sum-constrained linear models is presented, which seems to provide a good balance between excessive observational requirements on the one hand and an unduly restrictive specification of the contemporaneous covariance matrix on the other.     

A Note on Covariance Matrix Estimation in Quantile Regressions

This note discusses some issues related to bandwidth selection based on moment expansions of the mean squared error （MSE） of the regression quantile estimator. We use higher order expansions to provide a way to distinguish among asymptotically equivalent nonparametric estimators. We derive approximations to the （standardized） MSE of the covariance matrix estimation. This facilitates a comparison of different estimators at the second order level, where differences do occur and depend on the bandwidth choice. A method of bandwidth selection is defined by minimizing the second order effect in the mean squared error.     

Hypothesis testing in regression models with AR(1) errors and a lagged dependent variable: Bootstrapping and artificial regression compared

Serially correlated errors in dynamic models render the standard conditional estimator of the covariance matrix inconsistent. A Monte Carlo experiment confirms that the downward bias in the conventional variance estimator also exists in small samples. The results favour a consistent estimator based on an artificial regression (suggested by Davidson and Mackinnon) over bootstrapping the distribution of parameter estimates.