Asymptotics of the principal components estimator of large factor models with weakly influential factors

This paper introduces a drifting-parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components (PC) estimator in situations when the factors’ explanatory power does not strongly dominate the explanatory power of the cross-sectionally and temporally correlated idiosyncratic terms. Under our asymptotics, the PC estimator is inconsistent. We find explicit formulae for the amount of the inconsistency, and propose an estimator of the number of factors for which the PC estimator works reasonably well. For the special case when the idiosyncratic terms are cross-sectionally but not temporally correlated (or vice versa), we show that the coefficients in the OLS regressions of the PC estimates of factors (loadings) on the true factors (true loadings) are asymptotically normal, and find explicit formulae for the corresponding asymptotic covariance matrix. We explain how to estimate the parameters of the derived asymptotic distributions. Our Monte Carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n$n$ and T$T$. We apply our theoretical results to test a hypothesis about the factor content of the US stock return data.     

Minimum Distance Estimation and Testing of DSGE Models from Structural VARs*

The aim of this paper is to complement the minimum distance estimation–structural vector autoregression approach when the weighting matrix is not optimal. In empirical studies, this choice is motivated by stochastic singularity or collinearity problems associated with the covariance matrix of impulse response functions. Consequently, the asymptotic distribution cannot be used to test the economic model's fit. To circumvent this difficulty, we propose a simple simulation method to construct critical values for the test statistics. An empirical application with US data illustrates the proposed method.     

Properties and estimation of asymmetric exponential power distribution

The new distribution class, Asymmetric Exponential Power Distribution (AEPD), proposed in this paper generalizes the class of Skewed Exponential Power Distributions (SEPD) in a way that in addition to skewness introduces different decay rates of density in the left and right tails. Our parametrization provides an interpretable role for each parameter. We derive moments and moment-based measures: skewness, kurtosis, expected shortfall. It is demonstrated that a maximum entropy property holds for the AEPD distributions. We establish consistency, asymptotic normality and efficiency of the maximum likelihood estimators over a large part of the parameter space by dealing with the problems created by non-smooth likelihood function and derive explicit analytical expressions of the asymptotic covariance matrix; where the results apply to the SEPD class they enlarge on the current literature. Also we give a convenient stochastic representation of the distribution; our Monte Carlo study illustrates the theoretical results. We also provide some empirical evidence for the usefulness of employing AEPD errors in GARCH type models for predicting downside market risk of financial assets.     

Testing the parametric form of the volatility in continuous time diffusion models—a stochastic process approach

We present new tests for the form of the volatility function which are based on stochastic processes of the integrated volatility. We prove weak convergence of these processes to centered processes whose conditional distributions are Gaussian. In the case of testing for a constant volatility the limiting process are standard Brownian bridges. As a consequence an asymptotic distribution free test and bootstrap tests (for testing of a general parametric form) can easily be implemented. It is demonstrated that the new tests are more than the currently available procedures. The new approach is also demonstrated by means of a simulation study.     

Unit root testing via the stationary bootstrap

A nonparametric, residual-based stationary bootstrap procedure is proposed for unit root testing in a time series. The procedure generates a pseudoseries which mimics the original, but ensures the presence of a unit root. Unlike many others in the literature, the proposed test is valid for a wide class of weakly dependent processes and is not based on parametric assumptions on the data-generating process. Large sample theory is developed and asymptotic validity is shown via a bootstrap functional central limit theorem. The case of a least squares statistic is discussed in detail, including simulations to investigate the procedure's finite sample performance.     

Pre-averaging estimators of the ex-post covariance matrix in noisy diffusion models with non-synchronous data

We show how pre-averaging can be applied to the problem of measuring the ex-post covariance of financial asset returns under microstructure noise and non-synchronous trading. A pre-averaged realised covariance is proposed, and we present an asymptotic theory for this new estimator, which can be configured to possess an optimal convergence rate or to ensure positive semi-definite covariance matrix estimates. We also derive a noise-robust Hayashi–Yoshida estimator that can be implemented on the original data without prior alignment of prices. We uncover the finite sample properties of our estimators with simulations and illustrate their practical use on high-frequency equity data.     

Goodness-of-fit tests based on series estimators in nonparametric instrumental regression

This paper proposes several tests of restricted specification in nonparametric instrumental regression. Based on series estimators, test statistics are established that allow for tests of the general model against a parametric or nonparametric specification as well as a test of exogeneity of the vector of regressors. The tests’ asymptotic distributions under correct specification are derived and their consistency against any alternative model is shown. Under a sequence of local alternative hypotheses, the asymptotic distributions of the tests are derived. Moreover, uniform consistency is established over a class of alternatives whose distance to the null hypothesis shrinks appropriately as the sample size increases. A Monte Carlo study examines finite sample performance of the test statistics.     

High dimensional covariance matrix estimation using a factor model

High dimensionality comparable to sample size is common in many statistical problems. We examine covariance matrix estimation in the asymptotic framework that the dimensionality p$p$ tends to ∞$\infty$ as the sample size n$n$ increases. Motivated by the Arbitrage Pricing Theory in finance, a multi-factor model is employed to reduce dimensionality and to estimate the covariance matrix. The factors are observable and the number of factors K$K$ is allowed to grow with p$p$. We investigate the impact of p$p$ and K$K$ on the performance of the model-based covariance matrix estimator. Under mild assumptions, we have established convergence rates and asymptotic normality of the model-based estimator. Its performance is compared with that of the sample covariance matrix. We identify situations under which the factor approach increases performance substantially or marginally. The impacts of covariance matrix estimation on optimal portfolio allocation and portfolio risk assessment are studied. The asymptotic results are supported by a thorough simulation study.     

Spatial correlation robust inference with errors in location or distance

This paper presents results from a Monte Carlo study concerning inference with spatially dependent data. We investigate the impact of location/distance measurement errors upon the accuracy of parametric and nonparametric estimators of asymptotic variances. Nonparametric estimators are quite robust to such errors, method of moments estimators perform surprisingly well, and MLE estimators are very poor. We also present and evaluate a specification test based on a parametric bootstrap that has good power properties for the types of measurement error we consider.     

Efficiency. of infinite dimensional M- estimators

It is well-known that maximum likelihood estimators are asymptotically normal with covariance equal to the inverse Fisher information in smooth, finite dimensional parametric models. Thus they are asymptotically efficient. A similar phenomenon has been observed for certain infinite dimensional parameter spaces. We give a simple proof of efficiency, starting from a theorem on asymptotic normality of infinite dimensional M -estimators. The proof avoids the explicit calculation of the Fisher information. We also address Hadamard differentiability of the corresponding M -functionals.     

Matrix exponential GARCH

I propose a new multivariate GARCH specification that maintains positive definiteness of the conditional covariance matrix. The idea is to specify the dynamics in the matrix logarithm of the conditional covariance. Because the matrix exponential transformation ensures positive definiteness, the dynamics can be specified without the positive definiteness constraint. This affords a variety of specifications and, in particular, we can specify element-by-element the dynamics of the matrix logarithm. I discuss specifications with leverage effects, estimation with multivariate Gaussian and t-distributions, and diagnostics that evaluate the appropriateness of the matrix exponential specification.     

Nonparametric cointegration analysis of fractional systems with unknown integration orders

In this paper a nonparametric variance ratio testing approach is proposed for determining the cointegration rank in fractionally integrated systems. The test statistic is easily calculated without prior knowledge of the integration order of the data, the strength of the cointegrating relations, or the cointegration vector(s). The latter property makes it easier to implement than regression-based approaches, especially when examining relationships between several variables with possibly multiple cointegrating vectors. Since the test is nonparametric, it does not require the specification of a particular model and is invariant to short-run dynamics. Nor does it require the choice of any smoothing parameters that change the test statistic without being reflected in the asymptotic distribution. Furthermore, a consistent estimator of the cointegration space can be obtained from the procedure. The asymptotic distribution theory for the proposed test is non-standard but easily tabulated or simulated. Monte Carlo simulations demonstrate excellent finite sample properties, even rivaling those of well-specified parametric tests. The proposed methodology is applied to the term structure of interest rates, where, contrary to both fractional- and integer-based parametric approaches, evidence in favor of the expectations hypothesis is found using the nonparametric approach.     

Local Asymptotic Power of Breitung's Test*

In this article, we derive the local asymptotic power function of the unit root test proposed by Breitung [Journal of Econometrics (2002) Vol. 108, pp. 343–363]. Breitung's test is a non‐parametric test and is free of nuisance parameters. We compare the local power curve of the Breitungs’ test with that of the Dickey–Fuller test. This comparison is in fact a quantification of the loss of power that one has to accept when applying a non‐parametric test.     

Asymptotic properties of estimators for the volume fractions of jointly stationary random sets

In the present paper, we show how a consistent estimator can be derived for the asymptotic covariance matrix of stationary 0–1-valued vector fields in R ^d , whose supports are jointly stationary random closed sets. As an example, which is of particular interest for statistical applications, we consider jointly stationary random closed sets associated with the Boolean model in R ^d such that the components indicate the frequency of coverage by the single grains of the Boolean model. For this model, a representation formula for the entries of the covariance matrix is obtained.     

Optimum allocation in multivariate stratified random sampling: stochastic matrix mathematical programming

The allocation problem for multivariate stratified random sampling as a problem of stochastic matrix integer mathematical programming is considered, minimizing the estimated covariance matrix of estimated means subject to fixed cost or fixed total sample size. With these aims the asymptotic normality of sample covariance matrices for each strata is established. Some alternative approaches are suggested for its solution. An example is solved by applying the proposed techniques.     

Efficient tests for the presence of a pair of complex conjugate unit roots in real time series

We introduce a framework which allows us to draw a clear parallel between the test for the presence of seasonal unit roots and that for unit root at frequency 0 (or π$\pi$). It relies on the properties of the complex conjugate integrated of order one processes which are implicitly at work in the real time series. In the same framework as that of Phillips and Perron (Biometrica 75 (1988) 335), we derive tests for the presence of a pair of conjugate complex unit roots. The asymptotic distribution we obtain are formally close to those derived by these authors but expressed with complex Wiener processes. We then introduce sequences of near-integrated processes which allow us to study the local-to-unity asymptotic of the above test statistics. We state a result on the weak convergence of the partial sum of complex near-random walks which leads to complex Orstein–Uhlenbeck processes. Drawing on Elliott et al. (Econometrica 64 (1996) 813) we then study the design of point-optimal invariant test procedures and compute their envelope employing local-to-unity asymptotic approximations. This leads us to introduce new feasible and near efficient seasonal unit root tests. Their finite sample properties are investigated and compared with the different test procedures already available (J. Econometrics 44 (1991) 215; 62 (1994) 415; 85 (1998) 269) and those introduced in the first part of the paper.     

A martingale approach for testing diffusion models based on infinitesimal operator

I develop an omnibus specification test for diffusion models based on the infinitesimal operator. The infinitesimal operator based identification of the diffusion process is equivalent to a “martingale hypothesis” for the processes obtained by a transformation of the original diffusion model. My test procedure is then constructed by checking the “martingale hypothesis” via a multivariate generalized spectral derivative based approach that delivers a N(0,1) asymptotical null distribution for the test statistic. The infinitesimal operator of the diffusion process is a closed-form function of drift and diffusion terms. Consequently, my test procedure covers both univariate and multivariate diffusion models in a unified framework and is particularly convenient for the multivariate case. Moreover, different transformed martingale processes contain separate information about the drift and diffusion specifications. This motivates me to propose a separate inferential test procedure to explore the sources of rejection when a parametric form is rejected. Simulation studies show that the proposed tests have reasonable size and excellent power performance. An empirical application of my test procedure using Eurodollar interest rates finds that most popular short-rate models are rejected and the drift misspecification plays an important role in such rejections.     

The conditional autoregressive Wishart model for multivariate stock market volatility

We propose a Conditional Autoregressive Wishart (CAW) model for the analysis of realized covariance matrices of asset returns. Our model assumes an autoregressive moving average structure for the scale matrix of the Wishart distribution. It accounts for positive definiteness of covariance matrices without imposing parametric restrictions, and can be estimated by Maximum Likelihood. We also propose extensions of the CAW model obtained by including a Mixed Data Sampling (MIDAS) component and Heterogeneous Autoregressive (HAR) dynamics for long-run fluctuations. The CAW models are applied to realized variances and covariances for five New York Stock Exchange stocks.     

Generalized reduced rank tests using the singular value decomposition

We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson [Annals of Mathematical Statistics (1951), 22, 327–351] sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald [Journal of the American Statistical Association (1996), 91, 1301–1309] and Gill and Lewbel [Journal of the American Statistical Association (1992), 87, 766–776] a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith [Econometric Theory (2000), 16, 151–175] usage of numerical optimization for the objective function statistic of Cragg and Donald [Journal of Econometrics (1997), 76, 223–250] and ignoring the non-negativity restriction on the singular values in Ratsimalahelo [2002, Rank test based on matrix perturbation theory. Unpublished working paper, U.F.R. Science Economique, University de Franche-Comté]. In the non-stationary cointegration case, the limiting distribution of the new rank statistic is identical to that of the Johansen trace statistic.     

Inference on inequality from household survey data

I develop a theory of asymptotic inference for the Lorenz curve and the Gini coefficient for testing economic inequality when the data come from stratified and clustered household surveys with large number of clusters per stratum. Using the asymptotic framework of Bhattacharya [Asymptotic Inference from multi-stage surveys. Journal of Econometrics 126(1), 145–171], I derive a weak convergence result for the continuously-indexed Lorenz process even when the underlying density is not uniformly bounded away from zero. I provide analytical formulae for the asymptotic covariance functions that are corrected for both stratification and clustering and develop consistent tests for Lorenz dominance. Inference on the Gini coefficient follows as a corollary. The methods are applied to per capita household expenditure data from the complexly designed Indian national sample survey to test for changes in inequality before and after the reforms of the early 1990s. Ignoring the survey design is seen to produce qualitatively different results, especially in the urban sector where the population sorts more completely into rich and poor neighborhoods.