Nonparametric regression estimation with general parametric error covariance: a more efficient two-step estimator

Recently Martins-Filho and Yao (J Multivar Anal 100:309–333, 2009) have proposed a two-step estimator of nonparametric regression function with parametric error covariance and demonstrate that it is more efficient than the usual LLE. In the present paper we demonstrate that MY’s estimator can be further improved. First, we extend MY’s estimator to the multivariate case, and also establish the asymptotic theorem for the slope estimators; second, we propose a more efficient two-step estimator for nonparametric regression function with general parametric error covariance, and develop the corresponding asymptotic theorems. Monte Carlo study shows the relative efficiency loss of MY’s estimator in comparison with our estimator in nonparametric regression with either AR(2) errors or heteroskedastic errors. Finally, in an empirical study we apply the proposed estimator to estimate the public capital productivity to illustrate its performance in a real data setting.     

Estimation of Binary Response Models With Endogenous Regressors

This paper reviews six approaches to binary response (y₁) structural forms with an endogenous regressor y₂: (i) the two‐stage least squares estimator‐like substitution approach, (ii) the control function approach, (iii) the system reduced‐form approach, (iv) the artificial instrumental regressor approach, (v) the transformed‐response instrumental variable estimator approach and (vi) the classical maximum likelihood estimator approach. The applicability of the six methods differs greatly, depending on whether y₂ is a continuously distributed random variable or a discrete transformation of a latent . We conduct a real‐data‐based simulation study, and provide an empirical illustration. Our overall recommendation is using (i) and (ii), as the others have undesirable features such as analytic complexity in (iii), computational difficulty in (iv) and (vi), and poor finite‐sample performance in (v).     

On the Inconsistency of Instrumental Variables Estimators for the Coefficients of Certain Dummy Variables

In this paper we consider the asymptotic properties of the Instrumental Variables (IV) estimator of the parameters in a linear regression model with some random regressors, and other regressors that are dummy variables. The latter have the special property that the number of non-zero values is fixed, and does not increase with the sample size. We prove that the IV estimator of the coefficient vector for the dummy variables is inconsistent, while that for the other regressors is weakly consistent under standard assumptions. However, the usual estimator for the asymptotic covariance matrix of the I.V. estimator for all of the coefficients retains its usual consistency. The t-test statistics for the dummy variable coefficients are still asymptotically standard normal, despite the inconsistency of the associated IV coefficient estimator. These results extend the earlier results of Hendry and Santos (Oxf Bull Econ Stat 67:571–595, 2005), which relate to a fixed-regressor model, in which the dummy variables are non-zero for just a single observation, and OLS estimation is used.

     

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.     

An Improved Two‐step Regularization Scheme for Spot Volatility Estimation

We are concerned with the problem of spot volatility estimation in the presence of microstructure noise. We introduce an estimator based on the technique of multi‐step regularization. A preliminary form for such an estimator was proposed in Ogawa (2008) and was shown to work in a real‐time manner. However, the main drawback of this scheme is that it needs a lot of observation data. The aim of the present paper is to introduce an improvement to this scheme, such that the modified estimator can work more efficiently and with a data set of smaller size. The technical aspects of implementation of the proposed scheme and its performance on simulated data are analysed. The scheme is tested against other spot volatility estimators, namely a realized volatility type estimator, the Fourier estimator and three kernel estimators.     

Optimal Time Interval Selection in Long-Run Correlation Estimation

This paper presents an asymptotically optimal time interval selection criterion for the long-run correlation block estimator (Bartlett kernel estimator) based on the Newey–West and Andrews–Monahan approaches. An alignment criterion that enhances finite-sample performance is also proposed. The procedure offers an optimal alternative to the customary practice in finance and economics of heuristically or arbitrarily choosing time intervals or lags in correlation studies. A Monte Carlo experiment using parameters derived from Dow Jones returns data confirms that the procedure can be MSE-superior to alternatives such as aggregation over arbitrary time intervals, parametric VAR, and Newey–West covariance matrix estimation with automatic lag selection.

     

A THREE‐STEP PROCEDURE FOR ESTIMATING AND TESTING COINTEGRATED ARMAX MODELS*

To deal with a variety of inferential problems on non‐stationary cointegrated time series, this paper proposes a computationally feasible method based on the Whittle likelihood and examines its performance. For the empirical application of our method, the paper investigates three sets of Japanese and US monetary and financial time‐series data. To evaluate the p‐value of the likelihood ratio statistic, we propose an approximation procedure based on the gamma distribution and the accompanying Laguerre expansion for reducing the computational burden. We also provide a numerical procedure for the asymptotic covariance matrix of the Whittle estimator.     

Moment Estimation of the Probit Model with an Endogenous Continuous Regressor

We propose a generalized method of moments (GMM) estimator with optimal instruments for a probit model that includes a continuous endogenous regressor. This GMM estimator incorporates the probit error and the heteroscedasticity of the error term in the first‐stage equation in order to construct the optimal instruments. The estimator estimates the structural equation and the first‐stage equation jointly and, based on this joint moment condition, is efficient within the class of GMM estimators. To estimate the heteroscedasticity of the error term of the first‐stage equation, we use the k‐nearest neighbour (k‐nn) non‐parametric estimation procedure. Our Monte Carlo simulation shows that in the presence of heteroscedasticity and endogeneity, our GMM estimator outperforms the two‐stage conditional maximum likelihood estimator. Our results suggest that in the presence of heteroscedasticity in the first‐stage equation, the proposed GMM estimator with optimal instruments is a useful option for researchers.     

Capital adjustment cost and inconsistency in income‐based dynamic panel models with fixed effects

After the seminal work of Nickell (1981), a vast literature demonstrates the inconsistency of ‘conditional convergence’ estimator in income‐based dynamic panel models with fixed effects when the time horizon (T) is short but the sample of countries (N) is large. Less attention is given to the economic root of inconsistency of the fixed effects estimator when T is also large. Using a variant of the Ramsey growth model with long‐run adjustment cost of capital, we demonstrate that the fixed effects estimator of such models could be inconsistent when T is large. This inconsistency arises because of the long‐run adjustment cost of capital which gives rise to a negative moving average coefficient in the error term. Income convergence will be thus overestimated. We theoretically characterize the order of this inconsistency. Our Monte Carlo simulation demonstrates that the size of the bias is substantial and it is greater in economies with higher capital adjustment costs. We show that the use of instrumental variables that take into account the presence of the negative moving average term in the error will overcome this bias.     

A necessary and sufficient condition for the dominance of an improved family of estimators in linear regression models

This article presents a necessary and sufficient condition for the dominance, with respect to the risk under a general quadratic loss function, of the double k-class estimators (characterized by non-stochastic scalars) over the least squares estimator of coefficients in linear regression models.     

ASYMPTOTIC NORMALITY OF A NONPARAMETRIC INSTRUMENTAL VARIABLES ESTIMATOR*

This article gives conditions under which the nonparametric instrumental variables estimator of Hall and Horowitz (Annals of Statistics 33 (December 2005), 2904–2929) is asymptotically normally distributed. With sufficiently large samples, the asymptotic normality result can be used to form confidence intervals for the unknown function that is estimated by the Hall–Horowitz procedure. The article reports the results of a Monte Carlo investigation of the finite‐sample coverage probabilities of the confidence intervals.     

Fixed effects models with time invariant variables: a theoretical note

This paper demonstrates the equivalence between a consistent two-stage GLS estimator and the pooled OLS estimator of the coefficients on time invariant covariates in an unbalanced FE panel. In general the estimated standard errors differ between these two procedures.     

A class of multiattribute utility functions

A function u(z) is a utility function if u′(z) > 0. It is called risk averse if we also have u′′(z) < 0. Some authors, however, require that u ⁽ⁱ⁾(z) > 0 if i is odd and u ⁽ⁱ⁾(z) < 0 if i is even. The notion of a multiattribute utility function can be defined by requiring that it is increasing in each variable and concave as an s-variate function. A stronger condition, similar to the one in case of a univariate utility function, requires that, in addition, all partial derivatives of total order m should be positive if m is odd and negative if m is even. In this paper, we present a class of functions in analytic form such that each of them satisfies this stronger condition. We also give sharp lower and upper bounds for E[u(X ₁,... , X _s )] under moment information with respect to the joint probability distribution of the random variables X ₁,... , X _s assumed to be discrete and representing wealths. Partially supported by OTKA grants F-046309 and T-047340 in Hungary.     

Chaos in East European black market exchange rates

In this paper we test for deterministic chaos in seven East European black market exchange rates, using Koedijk and Kool's (1992, Journal of Business and Economic Statistics, 10, 83-96) monthly data from January 1955 through May 1990. In doing so we use three (non-parametric) inference methods, the BDS (Brocket al., 1996, Econometric Reviews, 15, 197-235) test for whiteness, the Lyapunov exponent estimator of Nychkaet al.(1992, Journal of the Royal Statistical Society, 12, 135-136) as well as the Lyapunov exponent estimator of Gencay and Dechert (1992, Physica D, 59, 142-157). We find some consistency in inference across methods, and we conclude, based on the Nychkaet al.(1992) estimator, that there is evidence consistent with a chaotic non-linear generation process in only two out of seven series.     

On Calculation of the Extended Gini Coefficient

The conventional formula for estimating the extended Gini coefficient is a covariance formula provided by Lerman and Yitzhaki (1989). We suggest an alternative estimator, obtained by approximating the Lorenz curve by a series of linear segments. In a Monte Carlo experiment designed to assess the relative bias and efficiency of the two estimators, we find that, when using grouped data with 20 or fewer groups, our new estimator has less bias and lower mean squared error than the covariance estimator. When individual observations are used, or the number of groups is 30 or more, there is little or no difference in the performance of the two estimators.     

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.     

Numerical Computation of Equilibrium Bid Functions in a First-Price Auction with Heterogeneous Risk Attitudes

We use numerical methods to compute Nash equilibrium (NE) bid functions for four agents bidding in a first-price auction. Each bidderi is randomly assigned:r _i [0,r _max], where 1 –r _i is the Arrow-Pratt measure of constant relative risk aversion. Eachr _i is independently drawn from the cumulative distribution function (·), a beta distribution on [0,r _max]. For various values of the maximum propensity to seek risk,r _max, the expected value of any bidder's risk characteristic,E (r _i ), and the probability that any bidder is risk seeking,P (r _i > 1), we determine the nonlinear characteristics of the (NE) bid functions.     

Dividend policy and earnings: a study of short- and long-term causality

This research examines whether earnings per share (EPS) and dividends per share (DPS) exhibit a short and long causality. The data employed in this study consist of quarterly EPS and DPS for 28 of the DJIA companies obtained from Bloomberg over a recent 10-year period. The companies under investigation all have EPS and DPS data available over the period studied. Dividends are generally paid out of earnings. The amount and timing of the dividend paid is a function of the respective company’s dividend policy. Therefore, the EPS_t can be expressed in terms of the DPS_t as follows: EPS_t = αDPS_t where α is a nonnegative constant. The equation suggests that there is a linear relationship between the EPS_t and the DPS_t. The results of this study indicate that bi-directional causality exists for some of the companies.     

The jackknife and heteroskedasticity: Consistent variance estimation for regression models

The weighted jackknife leads to a consistent estimator for the covariance matrix of the least squares estimators of the parameters in a regression model. In this note we show that this estimator has a simple relationship to the White estimator which is widely used in econometrics.