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.     

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.     

Quasi-maximum likelihood estimation of volatility with high frequency data

This paper investigates the properties of the well-known maximum likelihood estimator in the presence of stochastic volatility and market microstructure noise, by extending the classic asymptotic results of quasi-maximum likelihood estimation. When trying to estimate the integrated volatility and the variance of noise, this parametric approach remains consistent, efficient and robust as a quasi-estimator under misspecified assumptions. Moreover, it shares the model-free feature with nonparametric alternatives, for instance realized kernels, while being advantageous over them in terms of finite sample performance. In light of quadratic representation, this estimator behaves like an iterative exponential realized kernel asymptotically. Comparisons with a variety of implementations of the Tukey–Hanning₂ kernel are provided using Monte Carlo simulations, and an empirical study with the Euro/US Dollar future illustrates its application in practice.     

Semiparametric estimation of Markov decision processes with continuous state space

We propose a general two-step estimator for a popular Markov discrete choice model that includes a class of Markovian games with continuous observable state space. Our estimation procedure generalizes the computationally attractive methodology of Pesendorfer and Schmidt-Dengler (2008) that assumed finite observable states. This extension is non-trivial as the policy value functions are solutions to some type II integral equations. We show that the inverse problem is well-posed. We provide a set of primitive conditions to ensure root-T consistent estimation for the finite dimensional structural parameters and the distribution theory for the value functions in a time series framework.     

Semiparametric robust estimation of truncated and censored regression models

Many estimation methods of truncated and censored regression models such as the maximum likelihood and symmetrically censored least squares (SCLS) are sensitive to outliers and data contamination as we document. Therefore, we propose a semiparametric general trimmed estimator (GTE) of truncated and censored regression, which is highly robust but relatively imprecise. To improve its performance, we also propose data-adaptive and one-step trimmed estimators. We derive the robust and asymptotic properties of all proposed estimators and show that the one-step estimators (e.g., one-step SCLS) are as robust as GTE and are asymptotically equivalent to the original estimator (e.g., SCLS). The finite-sample properties of existing and proposed estimators are studied by means of Monte Carlo simulations.     

Efficient estimation of a multivariate multiplicative volatility model

We propose a multivariate generalization of the multiplicative volatility model of Engle and Rangel (2008), which has a nonparametric long run component and a unit multivariate GARCH short run dynamic component. We suggest various kernel-based estimation procedures for the parametric and nonparametric components, and derive the asymptotic properties thereof. For the parametric part of the model, we obtain the semiparametric efficiency bound. Our method is applied to a bivariate stock index series. We find that the univariate model of Engle and Rangel (2008) appears to be violated in the data whereas our multivariate model is more consistent with the data.     

Institutions and Wage Determination: a Multi‐country Approach*

We present an empirical analysis of the determinants of labour cost in OECD countries, with particular reference to the impact of labour market institutions from 1960 to 1994. We show that labour market regulations can explain a large part of the labour cost rise in OECD countries in the last few decades once we control for productivity. These results are consistent with the findings of a companion paper [ Nickell, Nunziata and Ochel (2005) Economic Journal, Vol. 115, pp. 1–27] where the effects of institutions on unemployment are examined. We present also a discussion of the econometric issues related to the estimation of a macro pooled model like ours. We focus, among other things, on the hypothesis of poolability and on the cointegration properties of the model. The explanatory power of the model is finally tested by means of a series of country by country dynamic simulations.     

Distribution free estimation of heteroskedastic binary response models using Probit/Logit criterion functions

In this paper estimators for distribution free heteroskedastic binary response models are proposed. The estimation procedures are based on relationships between distribution free models with a conditional median restriction and parametric models (such as Probit/Logit) exhibiting (multiplicative) heteroskedasticity. The first proposed estimator is based on the observational equivalence between the two models, and is a semiparametric sieve estimator (see, e.g. Gallant and Nychka (1987), Ai and Chen (2003) and Chen et al. (2005)) for the regression coefficients, based on maximizing standard Logit/Probit criterion functions, such as NLLS and MLE. This procedure has the advantage that choice probabilities and regression coefficients are estimated simultaneously. The second proposed procedure is based on the equivalence between existing semiparametric estimators for the conditional median model (,  and ) and the standard parametric (Probit/Logit) NLLS estimator. This estimator has the advantage of being implementable with standard software packages such as Stata. Distribution theory is developed for both estimators and a Monte Carlo study indicates they both perform well in finite samples.     

Substitution and complementarity between managers and subordinates: Evidence from British football

We use data on British football managers and teams over the 1994-2007 period to study substitution and complementarity between leaders and subordinates. We find for the Premier League (the highest level of competition) that, other things being equal, managers who themselves played at a higher level raise the productivity of less-skilled teams by more than that of highly skilled teams. This is consistent with the hypothesis that one function of a top manager is to communicate to subordinates the skills needed to succeed, since less skilled players have more to learn. However, we also find that managers with more accumulated professional managing experience raise the productivity of talented players by more than that of less-talented players. This is consistent with the hypothesis that a further function of successful managers in high-performance workplaces is to manage the egos of elite workers. Such a function is potentially more important the more accomplished the workers are-as indicated, in our data, by teams with greater payrolls.     

Partial maximum likelihood estimation of spatial probit models

This paper analyzes spatial Probit models for cross sectional dependent data in a binary choice context. Observations are divided by pairwise groups and bivariate normal distributions are specified within each group. Partial maximum likelihood estimators are introduced and they are shown to be consistent and asymptotically normal under some regularity conditions. Consistent covariance matrix estimators are also provided. Estimates of average partial effects can also be obtained once we characterize the conditional distribution of the latent error. Finally, a simulation study shows the advantages of our new estimation procedure in this setting. Our proposed partial maximum likelihood estimators are shown to be more efficient than the generalized method of moments counterparts.     

Dynamic estimation of volatility risk premia and investor risk aversion from option-implied and realized volatilities

This paper proposes a method for constructing a volatility risk premium, or investor risk aversion, index. The method is intuitive and simple to implement, relying on the sample moments of the recently popularized model-free realized and option-implied volatility measures. A small-scale Monte Carlo experiment confirms that the procedure works well in practice. Implementing the procedure with actual S&P500 option-implied volatilities and high-frequency five-minute-based realized volatilities indicates significant temporal dependencies in the estimated stochastic volatility risk premium, which we in turn relate to a set of macro-finance state variables. We also find that the extracted volatility risk premium helps predict future stock market returns.     

Adaptive estimation of the dynamics of a discrete time stochastic volatility model

This paper is concerned with the discrete time stochastic volatility model _Yi=exp(_Xi/2)_ηi$Y_i=exp(X_i/2){\eta }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where only (_Yi)$\left(Y_i\right)$ is observed. The model is rewritten as a particular hidden model: _Zi=_Xi+_εi$Z_i=X_i+{\epsilon }_i$, _Xi+1=b(_Xi)+σ(_Xi)_ξi+1$X_{i+1}=b\left(X_i\right)+\sigma \left(X_i\right){\xi }_{i+1}$, where (_ξi)$\left({\xi }_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent sequences of i.i.d. noise. Moreover, the sequences (_Xi)$\left(X_i\right)$ and (_εi)$\left({\epsilon }_i\right)$ are independent and the distribution of ε$\epsilon$ is known. Then, our aim is to estimate the functions b$b$ and σ²${\sigma }^2$ when only observations _Z1,…,_Zn$Z_1,\dots ,Z_n$ are available. We propose to estimate bf$bf$ and (b²+σ²)f$(b^2+{\sigma }^2)f$ and study the integrated mean square error of projection estimators of these functions on automatically selected projection spaces. By ratio strategy, estimators of b$b$ and σ²${\sigma }^2$ are then deduced. The mean square risk of the resulting estimators are studied and their rates are discussed. Lastly, simulation experiments are provided: constants in the penalty functions defining the estimators are calibrated and the quality of the estimators is checked on several examples.     

Estimating a tournament model of intra-firm wage differentials

We consider the estimation of a tournament model with moral hazard (based on Rosen (1986), AER)) when only aggregate data on intra-firm employment levels and salaries are available. Equilibrium restrictions of the model allow us to recover parameters of interest, including equilibrium effort levels in each hierarchical stage of the firm. We illustrate our estimation procedures using data from major retail chains in the US. We find that only a fraction of the wage differential directly compensates workers for higher effort levels, implying that a large portion of the differentials arises to maintain incentives at lower rungs of the retailers.     

Endogenous selection or treatment model estimation

In a sample selection or treatment effects model, common unobservables may affect both the outcome and the probability of selection in unknown ways. This paper shows that the distribution function of potential outcomes, conditional on covariates, can be identified given an observed variable V$V$ that affects the treatment or selection probability in certain ways and is conditionally independent of the error terms in a model of potential outcomes. Selection model estimators based on this identification are provided, which take the form of simple weighted averages, GMM, or two stage least squares. These estimators permit endogenous and mismeasured regressors. Empirical applications are provided to estimation of a firm investment model and a schooling effects on wages model.     

A family of empirical likelihood functions and estimators for the binary response model

The minimum discrimination information principle is used to identify an appropriate parametric family of probability distributions and the corresponding maximum likelihood estimators for binary response models. Estimators in the family subsume the conventional logit model and form the basis for a set of parametric estimation alternatives with the usual asymptotic properties. Sampling experiments are used to assess finite sample performance.     

Efficient estimation of the semiparametric spatial autoregressive model

Efficient semiparametric and parametric estimates are developed for a spatial autoregressive model, containing non-stochastic explanatory variables and innovations suspected to be non-normal. The main stress is on the case of distribution of unknown, nonparametric, form, where series nonparametric estimates of the score function are employed in adaptive estimates of parameters of interest. These estimates are as efficient as the ones based on a correct form, in particular they are more efficient than pseudo-Gaussian maximum likelihood estimates at non-Gaussian distributions. Two different adaptive estimates are considered, relying on somewhat different regularity conditions. A Monte Carlo study of finite sample performance is included.     

Unionisation structures, productivity and firm performance: New insights from a heterogeneous firm model

This paper introduces collective bargaining at the firm and at the sector level into the heterogeneous firm model of Melitz and Ottaviano (Melitz, M. J., Ottaviano, G. I. P., 2008. Market size, trade, and productivity. Review of Economic Studies 75 (1), 295-316). It then analyses how the two bargaining regimes change aggregate industry productivity and firm performance relative to a competitive labour market. While sector-level bargaining forces the least productive firms to exit and thus increases average productivity relative to the competitive benchmark, firm-level bargaining allows less productive firms to stay in the market and thus reduces average productivity. Sector-level bargaining also results in higher average output and profit levels than either firm-level bargaining or a competitive labour market. The paper also shows that the choice between sector- and firm-level bargaining can involve a trade-off between product variety and product prices: Not only the average price level but also product variety tends to be lower under sector-level bargaining than under firm-level bargaining.     

Inference on an extended Roy model,with an application to schooling decisions in France

This paper considers the identification and estimation of an extension of Roy’s model (1951) of sectoral choice, which includes a non-pecuniary component in the selection equation and allows for uncertainty on potential earnings. We focus on the identification of the non-pecuniary component, which is key to disentangling the relative importance of monetary incentives versus preferences in the context of sorting across sectors. By making the most of the structure of the selection equation, we show that this component is point identified from the knowledge of the covariate effects on earnings, as soon as one covariate is continuous. Notably, and in contrast to most results on the identification of Roy models, this implies that identification can be achieved without any exclusion restriction nor large support condition on the covariates. As a by-product, bounds are obtained on the distribution of the ex ante   monetary returns. We propose a three-stage semiparametric estimation procedure for this model, which yields root-n$n$ consistent and asymptotically normal estimators. Finally, we apply our results to the educational context, by providing new evidence from French data that non-pecuniary factors are a key determinant of higher education attendance decisions.     

Simplified Implementation of the Heckman Estimator of the Dynamic Probit Model and a Comparison with Alternative Estimators*

This paper presents a convenient shortcut method for implementing the Heckman estimator of the dynamic random effects probit model and other dynamic nonlinear panel data models using standard software. It then compares the estimators proposed by Heckman, Orme and Wooldridge, based on three alternative approximations, first in an empirical model for the probability of unemployment and then in a set of simulation experiments. The results indicate that none of the three estimators dominates the other two in all cases. In most cases, all three estimators display satisfactory performance, except when the number of time periods is very small.     

A collective labor supply model with complementarities in leisure: Identification and estimation by means of panel data

We present a novel identification strategy for a collective labor supply model that allows for complementarities in leisure (i.e., individuals may enjoy leisure more in company of their spouse). Individual preferences and the Pareto weights (which capture the intra-household bargaining process) are identified by making use of panel data with couples and individuals who became a widow(er) in the observation period, along with the assumption that an individual's preferences can only change in a particular manner after the spouse's death. The change in preferences comes from changes in observable variables that can be controlled for (like mental health) and from the loss of the possibility to jointly enjoy leisure after the couple's dissolution. We apply the model to American households coming from the first nine waves of the Health and Retirement Study (1992-2008) and show that complementarities in leisure are indeed important when modeling spouses' labor supply choices.