Testing functional inequalities

This paper develops tests for inequality constraints of nonparametric regression functions. The test statistics involve a one-sided version of _Lp$L_p$-type functionals of kernel estimators (1≤p<∞)$(1\le p<\infty )$. Drawing on the approach of Poissonization, this paper establishes that the tests are asymptotically distribution free, admitting asymptotic normal approximation. In particular, the tests using the standard normal critical values have asymptotically correct size and are consistent against general fixed alternatives. Furthermore, we establish conditions under which the tests have nontrivial local power against Pitman local alternatives. Some results from Monte Carlo simulations are presented.     

Risk-parameter estimation in volatility models

This paper introduces the concept of risk parameter in conditional volatility models of the form _?t=_σt(_θ0)_ηt${?}_t={\sigma }_t\left({\theta }_0\right){\eta }_t$ and develops statistical procedures to estimate this parameter. For a given risk measure r$r$, the risk parameter is expressed as a function of the volatility coefficients _θ0${\theta }_0$ and the risk, r(_ηt)$r\left({\eta }_t\right)$, of the innovation process. A two-step method is proposed to successively estimate these quantities. An alternative one-step approach, relying on a reparameterization of the model and the use of a non Gaussian QML, is proposed. Asymptotic results are established for smooth risk measures, as well as for the Value-at-Risk (VaR). Asymptotic comparisons of the two approaches for VaR estimation suggest a superiority of the one-step method when the innovations are heavy-tailed. For standard GARCH models, the comparison only depends on characteristics of the innovations distribution, not on the volatility parameters. Monte-Carlo experiments and an empirical study illustrate the superiority of the one-step approach for financial series.     

Identification and estimation of nonlinear dynamic panel data models with unobserved covariates

This paper considers nonparametric identification of nonlinear dynamic models for panel data with unobserved covariates. Including such unobserved covariates may control for both the individual-specific unobserved heterogeneity and the endogeneity of the explanatory variables. Without specifying the distribution of the initial condition with the unobserved variables, we show that the models are nonparametrically identified from two periods of the dependent variable _Yit$Y_{it}$ and three periods of the covariate _Xit$X_{it}$. The main identifying assumptions include high-level injectivity restrictions and require that the evolution of the observed covariates depends on the unobserved covariates but not on the lagged dependent variable. We also propose a sieve maximum likelihood estimator (MLE) and focus on two classes of nonlinear dynamic panel data models, i.e., dynamic discrete choice models and dynamic censored models. We present the asymptotic properties of the sieve MLE and investigate the finite sample properties of these sieve-based estimators through a Monte Carlo study. An intertemporal female labor force participation model is estimated as an empirical illustration using a sample from the Panel Study of Income Dynamics (PSID).     

Nonlinearity,nonstationarity, and spurious forecasts

Implications of nonlinearity, nonstationarity, and misspecification are considered from a forecasting perspective. Our model allows for small departures from the martingale difference sequence hypothesis by including a nonlinear component, formulated as a general, integrable transformation of the I(1)$I(1)$ predictor. We assume that the true generating mechanism is unknown to the econometrician and he is therefore forced to use some approximating functions. It is shown that in this framework the linear regression techniques lead to spurious forecasts. Improvements of the forecast accuracy are possible with properly chosen nonlinear transformations of the predictor. The paper derives the limiting distribution of the forecasts’ mean squared error (MSE). In the case of square integrable approximants, it depends on the _L2$L_2$-distance between the nonlinear component and approximating function. Optimal forecasts are available for a given class of approximants.     

Higher-order improvements of the parametric bootstrap for long-memory Gaussian processes

This paper determines coverage probability errors of both delta method and parametric bootstrap confidence intervals (CIs) for the covariance parameters of stationary long-memory Gaussian time series. CIs for the long-memory parameter _d0$d_0$ are included. The results establish that the bootstrap provides higher-order improvements over the delta method. Analogous results are given for tests. The CIs and tests are based on one or other of two approximate maximum likelihood estimators. The first estimator solves the first-order conditions with respect to the covariance parameters of a “plug-in” log-likelihood function that has the unknown mean replaced by the sample mean. The second estimator does likewise for a plug-in Whittle log-likelihood.     

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.     

Coherent risk measures in general economic models and price bubbles

In this article we study coherent risk measures in general economic models where the set of financial positions is an ordered Banach space E$E$ and the safe asset an order unit _x0$x_0$ of E$E$. First we study some properties of risk measures. We show that the set of normalized (with respect to _x0$x_0$) price systems is weak star compact and by using this result we prove a maximum attainment representation theorem which improves the one of Jaschke and Küchler (2001). Also we study how a risk measure changes under different safe assets and we show a kind of equivalence between these risk measures. In the sequel we study subspaces of E$E$ consisting of financial positions of risk greater or equal to zero and we call these subspaces unsure. We find some criteria and we give examples of these subspaces. In the last section, we combine the unsure subspaces with the theory of price-bubbles of Gilles and LeRoy (1992).     

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.     

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$\rho$.     

Generalized spectral tests for the martingale difference hypothesis

This article proposes a test for the martingale difference hypothesis (MDH) using dependence measures related to the characteristic function. The MDH typically has been tested using the sample autocorrelations or in the spectral domain using the periodogram. Tests based on these statistics are inconsistent against uncorrelated non-martingales processes. Here, we generalize the spectral test of Durlauf (1991) for testing the MDH taking into account linear and nonlinear dependence. Our test considers dependence at all lags and is consistent against general pairwise nonparametric Pitman's local alternatives converging at the parametric rate n^-1/2,$n^{-1/2},$ with n$n$ the sample size. Furthermore, with our methodology there is no need to choose a lag order, to smooth the data or to formulate a parametric alternative. Our approach could be extended to specification testing of the conditional mean of possibly nonlinear models. The asymptotic null distribution of our test depends on the data generating process, so a bootstrap procedure is proposed and theoretically justified. Our bootstrap test is robust to higher order dependence, in particular to conditional heteroskedasticity. A Monte Carlo study examines the finite sample performance of our test and shows that it is more powerful than some competing tests. Finally, an application to the S&P 500 stock index and exchange rates highlights the merits of our approach.     

Optimal statistical decisions about some alternative financial models

We study Neyman–Pearson testing and Bayesian decision making based on observations of the price dynamics (_Xt:t∈[0,T])$(X_t:t\in [0,T])$ of a financial asset, when the hypothesis is the classical geometric Brownian motion with a given constant growth rate and the alternative is a different random diffusion process with a given, possibly price-dependent, growth rate. Examples of asset price observations are introduced and used throughout the paper to demonstrate the applicability of the theory. By a rigorous mathematical approach, we obtain exact formulae and bounds for the most common statistical characteristics of testing and decision making, such as the power of test (type II error probability), the Bayes factor and its moments (power divergences), and the Bayes risk or Bayes error. These bounds can be much more easily evaluated than the exact formulae themselves and, consequently, they are useful for practical applications. An important theoretical conclusion of this paper is that for the class of alternatives considered   neither the risk nor the errors converge to zero faster than exponentially in the observation time T$T$. We illustrate in concrete decision situations that the actual rate of convergence is well approximated by the bounds given in the paper.     

The cointegrated vector autoregressive model with general deterministic terms

In the cointegrated vector autoregression (CVAR) literature, deterministic terms have until now been analyzed on a case-by-case, or as-needed basis. We give a comprehensive unified treatment of deterministic terms in the additive model $X_t=\gamma Z_t+Y_t$, where $Z_t$ belongs to a large class of deterministic regressors and $Y_t$ is a zero-mean CVAR. We suggest an extended model that can be estimated by reduced rank regression, and give a condition for when the additive and extended models are asymptotically equivalent, as well as an algorithm for deriving the additive model parameters from the extended model parameters. We derive asymptotic properties of the maximum likelihood estimators and discuss tests for rank and tests on the deterministic terms. In particular, we give conditions under which the estimators are asymptotically (mixed) Gaussian, such that associated tests are ${\chi }^2$-distributed.     

Randomized experiments from non-random selection in U.S. House elections

This paper establishes the relatively weak conditions under which causal inferences from a regression–discontinuity (RD) analysis can be as credible as those from a randomized experiment, and hence under which the validity of the RD design can be tested by examining whether or not there is a discontinuity in any pre-determined (or “baseline”) variables at the RD threshold. Specifically, consider a standard treatment evaluation problem in which treatment is assigned to an individual if and only if V>_v0$V>v_0$, but where _v0$v_0$ is a known threshold, and V is observable. V can depend on the individual's characteristics and choices, but there is also a random chance element: for each individual, there exists a well-defined probability distribution for V  . The density function—allowed to differ arbitrarily across the population—is assumed to be continuous. It is formally established that treatment status here is as good as randomized in a local neighborhood of V=_v0$V=v_0$. These ideas are illustrated in an analysis of U.S. House elections, where the inherent uncertainty in the final vote count is plausible, which would imply that the party that wins is essentially randomized among elections decided by a narrow margin. The evidence is consistent with this prediction, which is then used to generate “near-experimental” causal estimates of the electoral advantage to incumbency.