Dominating estimators for minimum-variance portfolios

In this paper, we derive two shrinkage estimators for minimum-variance portfolios that dominate the traditional estimator with respect to the out-of-sample variance of the portfolio return. The presented results hold for any number of assets d≥4$d\ge 4$ and number of observations n≥d+2$n\ge d+2$. The small-sample properties of the shrinkage estimators as well as their large-sample properties for fixed d$d$ but n→∞$n\to \infty$ and n,d→∞$n,d\to \infty$ but n/d→q≤∞$n/d\to q\le \infty$ are investigated. Furthermore, we present a small-sample test for the question of whether it is better to completely ignore time series information in favor of naive diversification.     

Estimating fixed-effect panel stochastic frontier models by model transformation

Traditional panel stochastic frontier models do not distinguish between unobserved individual heterogeneity and inefficiency. They thus force all time-invariant individual heterogeneity into the estimated inefficiency. Greene (2005) proposes a true fixed-effect stochastic frontier model which, in theory, may be biased by the incidental parameters problem. The problem usually cannot be dealt with by model transformations owing to the nonlinearity of the stochastic frontier model. In this paper, we propose a class of panel stochastic frontier models which create an exception. We show that first-difference and within-transformation can be analytically performed on this model to remove the fixed individual effects, and thus the estimator is immune to the incidental parameters problem. Consistency of the estimator is obtained by either N→∞$N\to \infty$ or T→∞$T\to \infty$, which is an attractive property for empirical researchers.     

Cointegrating rank selection in models with time-varying variance

Reduced rank regression (RRR) models with time varying heterogeneity are considered. Standard information criteria for selecting cointegrating rank are shown to be weakly consistent in semiparametric RRR models in which the errors have general nonparametric short memory components and shifting volatility provided the penalty coefficient _Cn→∞$C_n\to \infty$ and _Cn/n→0$C_n/n\to 0$ as n→∞$n\to \infty$. The AIC criterion is inconsistent and its limit distribution is given. The results extend those in Cheng and Phillips (2009a) and are useful in empirical work where structural breaks or time evolution in the error variances is present. An empirical application to exchange rate data is provided.     

On bootstrapping panel factor series

This paper studies the asymptotic validity of sieve bootstrap for nonstationary panel factor series. Two main results are shown. Firstly, a bootstrap Invariance Principle is derived pointwise in i$i$, obtaining an upper bound for the order of truncation of the AR polynomial that depends on n$n$ and T$T$. Consistent estimation of the long run variances is also studied for (n,T)→∞$(n,T)\to \infty$. Secondly, joint bootstrap asymptotics is also studied, investigating the conditions under which the bootstrap is valid. In particular, the extent of cross sectional dependence which can be allowed for is investigated. Whilst we show that, for general forms of cross dependence, consistent estimation of the long run variance (and therefore validity of the bootstrap) is fraught with difficulties, however we show that “one-cross-sectional-unit-at-a-time” resampling schemes yield valid bootstrap based inference under weak forms of cross-sectional dependence.     

On the distribution of estimated technical efficiency in stochastic frontier models

We consider a stochastic frontier model with error ε=v−u$\epsilon =v-u$, where v$v$ is normal and u$u$ is half normal. We derive the distribution of the usual estimate of u,E(u|ε)$u,E\left(u\right|\epsilon )$. We show that as the variance of v$v$ approaches zero, E(u|ε)−u$E\left(u\right|\epsilon )-u$ converges to zero, while as the variance of v$v$ approaches infinity, E(u|ε)$E\left(u\right|\epsilon )$ converges to E(u)$E\left(u\right)$. We graph the density of E(u|ε)$E\left(u\right|\epsilon )$ for intermediate cases. To show that E(u|ε)$E\left(u\right|\epsilon )$ is a shrinkage of u towards its mean, we derive and graph the distribution of E(u|ε)$E\left(u\right|\epsilon )$ conditional on u$u$. We also consider the distribution of estimated inefficiency in the fixed-effects panel data setting.     

Treatment effects in sample selection models and their nonparametric estimation

In a sample-selection model with the ‘selection’ variable Q$Q$ and the ‘outcome’ variable Y^∗$Y^{\ast }$, Y^∗$Y^{\ast }$ is observed only when Q=1$Q=1$. For a treatment D$D$ affecting both Q$Q$ and Y^∗$Y^{\ast }$, three effects are of interest: ‘participation  ’ (i.e., the selection) effect of D$D$ on Q$Q$, ‘visible performance  ’ (i.e., the observed outcome) effect of D$D$ on Y≡QY^∗$Y\equiv Q{Y}^{\ast }$, and ‘invisible performance  ’ (i.e., the latent outcome) effect of D$D$ on Y^∗$Y^{\ast }$. This paper shows the conditions under which the three effects are identified, respectively, by the three corresponding mean differences of Q$Q$, Y$Y$, and Y|Q=1$Y|Q=1$ (i.e., Y^∗|Q=1$Y^{\ast }|Q=1$) across the control (D=0$D=0$) and treatment (D=1$D=1$) groups. Our nonparametric estimators for those effects adopt a two-sample framework and have several advantages over the usual matching methods. First, there is no need to select the number of matched observations. Second, the asymptotic distribution is easily obtained. Third, over-sampling the control/treatment group is allowed. Fourth, there is a built-in mechanism that takes into account the ‘non-overlapping support problem’, which the usual matching deals with by choosing a ‘caliper’. Fifth, a sensitivity analysis to gauge the presence of unobserved confounders is available. A simulation study is conducted to compare the proposed methods with matching methods, and a real data illustration is provided.     

Panel unit root tests in the presence of a multifactor error structure

This paper extends the cross-sectionally augmented panel unit root test (CIPS$CIPS$) proposed by Pesaran (2007) to the case of a multifactor error structure, and proposes a new panel unit root test based on a simple average of cross-sectionally augmented Sargan–Bhargava statistics (CSB$CSB$). The basic idea is to exploit information regarding the m$m$ unobserved factors that are shared by k$k$ observed time series in addition to the series under consideration. Initially, we develop the tests assuming that m⁰$m^0$, the true number of factors, is known and show that the limit distribution of the tests does not depend on any nuisance parameters, so long as k≥m⁰−1$k\ge m^0-1$. Small sample properties of the tests are investigated by Monte Carlo experiments and are shown to be satisfactory. Particularly, the proposed CIPS$CIPS$ and CSB$CSB$ tests have the correct size for all   combinations of the cross section (N$N$) and time series (T$T$) dimensions considered. The power of both tests rises with N$N$ and T$T$, although the CSB$CSB$ test performs better than the CIPS$CIPS$ test for smaller sample sizes. The various testing procedures are illustrated with empirical applications to real interest rates and real equity prices across countries.     

The first-order approach when the cost of effort is money

We provide sufficient conditions for the first-order approach in the principal-agent problem when the agent’s utility has the nonseparable form u(y−c(a))$u(y-c\left(a\right))$ where y$y$ is the contractual payoff and c(a)$c\left(a\right)$ is the money cost of effort. We first consider a decision-maker facing prospects which cost c(a)$c\left(a\right)$ and with distributions of returns y$y$ that depend on a$a$. The decision problem is shown to be concave if the primitive of the cdf of returns is jointly convex in a$a$ and y$y$, a condition we call Concavity of the Cumulative Quantile (CCQ) and which is satisfied by many common distributions. Next we apply CCQ to the distribution of outcomes (or their likelihood-ratio transforms) in the principal-agent problem and derive restrictions on the utility function that validate the first-order approach. We also discuss another condition, log-convexity of the distribution, and show that it allows binding limited liability constraints, which CCQ does not.     

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.     

Quasi ML estimation of the panel AR(1) model with arbitrary initial conditions

In this paper we show that the Quasi ML estimation method yields consistent Random and Fixed Effects estimators for the autoregression parameter ρ$\rho$ in the panel AR(1) model with arbitrary initial conditions and possibly time-series heteroskedasticity even when the error components are drawn from heterogeneous distributions. We investigate both analytically and by means of Monte Carlo simulations the properties of the QML estimators for ρ$\rho$. The RE(Q)MLE for ρ$\rho$ is asymptotically at least as robust to individual heterogeneity and, when the data are i.i.d. and normal, at least as efficient as the FE(Q)MLE for ρ$\rho$. Furthermore, the QML estimators for ρ$\rho$ only suffer from a ‘weak moment conditions’ problem when ρ$\rho$ is close to one if the cross-sectional average of the variances of the errors is (almost) constant over time, e.g. under time-series homoskedasticity. However, in this case the QML estimators for ρ$\rho$ are still consistent when ρ$\rho$ is local to or equal to one although they converge to a non-normal possibly asymmetric distribution at a rate that is lower than N^1/2$N^{1/2}$ but at least N^1/4$N^{1/4}$. Finally, we study the finite sample properties of two types of estimators for the standard errors of the QML estimators for ρ$\rho$, and the bounds of QML based confidence intervals for ρ$\rho$.     

Estimating covariation: Epps effect,microstructure noise

This paper is about how to estimate the integrated covariance _〈X,Y〉T${〈X,Y〉}_T$ of two assets over a fixed time horizon [0,T]$[0,T]$, when the observations of X$X$ and Y$Y$ are “contaminated” and when such noisy observations are at discrete, but not synchronized, times. We show that the usual previous-tick covariance estimator is biased, and the size of the bias is more pronounced for less liquid assets. This is an analytic characterization of the Epps effect. We also provide the optimal sampling frequency which balances the tradeoff between the bias and various sources of stochastic error terms, including nonsynchronous trading, microstructure noise, and time discretization. Finally, a two scales covariance estimator is provided which simultaneously cancels (to first order) the Epps effect and the effect of microstructure noise. The gain is demonstrated in data.     

Identification and nonparametric estimation of a transformed additively separable model

Let r(x,z)$r(x,z)$ be a function that, along with its derivatives, can be consistently estimated nonparametrically. This paper discusses the identification and consistent estimation of the unknown functions H$H$, M$M$, G$G$ and F$F$, where r(x,z)=H[M(x,z)]$r(x,z)=H\left[M(x,z)\right]$, M(x,z)=G(x)+F(z)$M(x,z)=G\left(x\right)+F\left(z\right)$, and H$H$ is strictly monotonic. An estimation algorithm is proposed for each of the model’s unknown components when r(x,z)$r(x,z)$ represents a conditional mean function. The resulting estimators use marginal integration to separate the components G$G$ and F$F$. Our estimators are shown to have a limiting Normal distribution with a faster rate of convergence than unrestricted nonparametric alternatives. Their small sample performance is studied in a Monte Carlo experiment. We apply our results to estimate generalized homothetic production functions for four industries in the Chinese economy.     

One-step R-estimation in linear models with stable errors

Classical estimation techniques for linear models either are inconsistent, or perform rather poorly, under α$\alpha$-stable error densities; most of them are not even rate-optimal. In this paper, we propose an original one-step R-estimation method and investigate its asymptotic performances under stable densities. Contrary to traditional least squares, the proposed R-estimators remain root-n$n$ consistent (the optimal rate) under the whole family of stable distributions, irrespective of their asymmetry and tail index. While parametric stable-likelihood estimation, due to the absence of a closed form for stable densities, is quite cumbersome, our method allows us to construct estimators reaching the parametric efficiency bounds associated with any prescribed values (_α0,_b0)$({\alpha }_0,b_0)$ of the tail index α$\alpha$ and skewness parameter b$b$, while preserving root-n$n$ consistency under any (α,b)$(\alpha ,b)$ as well as under usual light-tailed densities. The method furthermore avoids all forms of multidimensional argmin computation. Simulations confirm its excellent finite-sample performances.     

Semiparametric binary regression models under shape constraints with an application to Indian schooling data

We consider estimation of the regression function in a semiparametric binary regression model defined through an appropriate link function (with emphasis on the logistic link) using likelihood-ratio based inversion. The dichotomous response variable Δ$\Delta$ is influenced by a set of covariates that can be partitioned as (X,Z)$(X,Z)$ where Z$Z$ (real valued) is the covariate of primary interest and X$X$ (vector valued) denotes a set of control variables. For any fixed X$X$, the conditional probability of the event of interest (Δ=1$\Delta =1$) is assumed to be a non-decreasing function of Z$Z$. The effect of the control variables is captured by a regression parameter β$\beta$. We show that the baseline conditional probability function (corresponding to X=0$X=0$) can be estimated by isotonic regression procedures and develop a likelihood ratio based method for constructing asymptotic confidence intervals for the conditional probability function (the regression function) that avoids the need to estimate nuisance parameters. Interestingly enough, the calibration of the likelihood ratio based confidence sets for the regression function no longer involves the usual χ²${\chi }^2$ quantiles, but those of the distribution of a new random variable that can be characterized as a functional of convex minorants of Brownian motion with quadratic drift. Confidence sets for the regression parameter β$\beta$ can however be constructed using asymptotically χ²${\chi }^2$ likelihood ratio statistics. The finite sample performance of the methods are assessed via a simulation study. The techniques of the paper are applied to data sets on primary school attendance among children belonging to different socio-economic groups in rural India.