Local inference for locally stationary time series based on the empirical spectral measure

The time varying empirical spectral measure plays a major role in the treatment of inference problems for locally stationary processes. The properties of the empirical spectral measure and related statistics are studied — both when its index function is fixed or when dependent on the sample size. In particular we prove a general central limit theorem. Several applications and examples are given including semiparametric Whittle estimation, local least squares estimation and spectral density estimation.     

Nonparametric spatial regression under near-epoch dependence

This paper establishes asymptotic normality and uniform consistency with convergence rates of the local linear estimator for spatial near-epoch dependent (NED) processes. The class of the NED spatial processes covers important spatial processes, including nonlinear autoregressive and infinite moving average random fields, which generally do not satisfy mixing conditions. Apart from accommodating a larger class of dependent processes, the proposed asymptotic theory allows for triangular arrays of heterogeneous random fields located on unevenly spaced lattices and sampled over regions of arbitrary configuration. All these features make the results applicable in a wide range of empirical settings.     

Estimation of copula-based semiparametric time series models

This paper studies the estimation of a class of copula-based semiparametric stationary Markov models. These models are characterized by nonparametric marginal distributions and parametric copula functions, while the copulas capture all the scale-free temporal dependence of the processes. Simple estimators of the marginal distribution and the copula parameter are provided, and their asymptotic properties are established under easily verifiable conditions. These results are used to obtain root-n consistent and asymptotically normal estimators of important features of the transition distribution such as the (nonlinear) conditional moments and conditional quantiles. The semiparametric conditional quantile estimators are automatically monotonic across quantiles, which is attractive for portfolio conditional value-at-risk calculations.     

Statistical inference on regression with spatial dependence

Central limit theorems are developed for instrumental variables estimates of linear and semiparametric partly linear regression models for spatial data. General forms of spatial dependence and heterogeneity in explanatory variables and unobservable disturbances are permitted. We discuss estimation of the variance matrix, including estimates that are robust to disturbance heteroscedasticity and/or dependence. A Monte Carlo study of finite-sample performance is included. In an empirical example, the estimates and robust and non-robust standard errors are computed from Indian regional data, following tests for spatial correlation in disturbances, and nonparametric regression fitting. Some final comments discuss modifications and extensions.     

Asymptotic theory for nonparametric regression with spatial data

Nonparametric regression with spatial, or spatio-temporal, data is considered. The conditional mean of a dependent variable, given explanatory ones, is a nonparametric function, while the conditional covariance reflects spatial correlation. Conditional heteroscedasticity is also allowed, as well as non-identically distributed observations. Instead of mixing conditions, a (possibly non-stationary) linear process is assumed for disturbances, allowing for long range, as well as short-range, dependence, while decay in dependence in explanatory variables is described using a measure based on the departure of the joint density from the product of marginal densities. A basic triangular array setting is employed, with the aim of covering various patterns of spatial observation. Sufficient conditions are established for consistency and asymptotic normality of kernel regression estimates. When the cross-sectional dependence is sufficiently mild, the asymptotic variance in the central limit theorem is the same as when observations are independent; otherwise, the rate of convergence is slower. We discuss the application of our conditions to spatial autoregressive models, and models defined on a regular lattice.     

Asymptotics for Autocovariances and Integrated Periodograms for Linear Processes Observed at Lower Frequencies

One of the most frequently used class of processes in time series analysis is the one of linear processes. For many statistical quantities, among them sample autocovariances and sample autocorrelations, central limit theorems are available in the literature. We investigate classical linear processes under a nonstandard observation pattern; namely, we assume that we are only able to observe the linear process at a lower frequency. It is shown that such observation pattern destroys the linear structure of the observations and leads to substantially different asymptotic results for standard statistical quantities. Central limit theorems are given for sample autocovariances and sample autocorrelations as well as more general integrated periodograms and ratio statistics. Moreover, for specific autoregressive processes, the possibilities to estimate the parameters of the underlying autoregression from lower frequency observations are addressed. Finally, we suggest for autoregressions of order 2 a valid bootstrap procedure. A small simulation study demonstrates the performance of the bootstrap proposal for finite sample size.     

Estimation of fractional integration under temporal aggregation

A result characterizing the effect of temporal aggregation in the frequency domain is known for arbitrary stationary processes and generalized for difference-stationary processes here. Temporal aggregation includes cumulation of flow variables as well as systematic (or skip) sampling of stock variables. Next, the aggregation result is applied to fractionally integrated processes. In particular, it is investigated whether typical frequency domain assumptions made for semiparametric estimation and inference are closed with respect to aggregation. With these findings it is spelled out, which estimators remain valid upon aggregation under which conditions on bandwidth selection.     

A fast resample method for parametric and semiparametric models

We propose a fast resample method for two step nonlinear parametric and semiparametric models, which does not require recomputation of the second stage estimator during each resample iteration. The fast resample method directly exploits the score function representations computed on each bootstrap sample, thereby reducing computational time considerably. This method is used to approximate the limit distribution of parametric and semiparametric estimators, possibly simulation based, that admit an asymptotic linear representation. Monte Carlo experiments demonstrate the desirable performance and vast improvement in the numerical speed of the fast bootstrap method.     

A central limit theorem for endogenous locations and complex spatial interactions

We provide a new central limit theorem (CLT) for spatial processes under weak conditions that are plausible for many economic applications in which location is endogenous. In particular, our CLT is designed for problems that have some, but not necessarily all, of the following features: (i) Agents choose the locations of observations to maximize profits, welfare, or some other objective. (ii) The objects that are chosen (e.g., stores or brands) interact with one another. For example, they can be substitutes or complements. (iii) Interaction can be complex. In particular, interaction between i and j need not depend only on the distance between the locations of i and j, but can also depend on distance to or location of other observations k, or possibly on the number of other such observations.     

Semiparametric inference in a GARCH-in-mean model

A new semiparametric estimator for an empirical asset pricing model with general nonparametric risk-return tradeoff and GARCH-type underlying volatility is introduced. Based on the profile likelihood approach, it does not rely on any initial parametric estimator of the conditional mean function, and it is under stated conditions consistent, asymptotically normal, and efficient, i.e., it achieves the semiparametric lower bound. A sampling experiment provides finite sample comparisons with the parametric approach and the iterative semiparametric approach with parametric initial estimate of Conrad and Mammen (2008). An application to daily stock market returns suggests that the risk-return relation is indeed nonlinear.     

Limit theory for panel data models with cross sectional dependence and sequential exogeneity

The paper derives a general Central Limit Theorem (CLT) and asymptotic distributions for sample moments related to panel data models with large n$n$. The results allow for the data to be cross sectionally dependent, while at the same time allowing the regressors to be only sequentially rather than strictly exogenous. The setup is sufficiently general to accommodate situations where cross sectional dependence stems from spatial interactions and/or from the presence of common factors. The latter leads to the need for random norming. The limit theorem for sample moments is derived by showing that the moment conditions can be recast such that a martingale difference array central limit theorem can be applied. We prove such a central limit theorem by first extending results for stable convergence in Hall and Heyde (1980) to non-nested martingale arrays relevant for our applications. We illustrate our result by establishing a generalized estimation theory for GMM estimators of a fixed effect panel model without imposing i.i.d. or strict exogeneity conditions. We also discuss a class of Maximum Likelihood (ML) estimators that can be analyzed using our CLT.     

Extreme returns and intensity of trading

Asymmetric information models of market microstructure claim that variables such as trading intensity are proxies for latent information on the value of financial assets. We consider the interval‐valued time series (ITS) of low/high returns and explore the relationship between these extreme returns and the intensity of trading. We assume that the returns (or prices) are generated by a latent process with some unknown conditional density. At each period of time, from this density, we have some random draws (trades) and the lowest and highest returns are the realized extreme observations of the latent process over the sample of draws. In this context, we propose a semiparametric model of extreme returns that exploits the results provided by extreme value theory. If properly centered and standardized extremes have well‐defined limiting distributions, the conditional mean of extreme returns is a nonlinear function of the conditional moments of the latent process and of the conditional intensity of the process that governs the number of draws. We implement a two‐step estimation procedure. First, we estimate parametrically the regressors that will enter into the nonlinear function, and in a second step we estimate nonparametrically the conditional mean of extreme returns as a function of the generated regressors. Unlike current models for ITS, the proposed semiparametric model is robust to misspecification of the conditional density of the latent process. We fit several nonlinear and linear models to the 5‐minute and 1‐minute low/high returns to seven major banks and technology stocks, and find that the nonlinear specification is superior to the current linear models and that the conditional volatility of the latent process and the conditional intensity of the trading process are major drivers of the dynamics of extreme returns.     

Model specification test with correlated but not cointegrated variables

Many macroeconomic and financial variables show highly persistent and correlated patterns but are not necessarily cointegrated. Recently,  Sun et al. (2011) propose using a semiparametric varying coefficient approach to capture correlations between integrated but non cointegrated variables. Due to the complication arising from the integrated disturbance term and the semiparametric functional form, consistent estimation of such a semiparametric model requires stronger conditions than usually needed for consistent estimation for a linear (spurious) regression model, or a semiparametric varying coefficient model with a stationary disturbance. Therefore, it is important to develop a testing procedure to examine for a given data set, whether linear relationship holds or not, while allowing for the disturbance being an integrated process. In this paper we propose two test statistics for detecting linearity against semiparametric varying coefficient alternative specification. Monte Carlo simulations are used to examine the finite sample performances of the proposed tests.     

Estimation and inference in unstable nonlinear least squares models

There is compelling evidence that many macroeconomic and financial variables are not generated by linear models. This evidence is based on testing linearity against either smooth nonlinearity or piece-wise linearity, but there is no framework that encompasses both. This paper provides an econometric framework that allows for both breaks and smooth nonlinearity in between breaks. We estimate the unknown break-dates simultaneously with other parameters via nonlinear least-squares. Using new central limit results for nonlinear processes, we provide inference methods on break-dates and parameter estimates and several instability tests. We illustrate our methods via simulated and empirical smooth transition models with breaks.     

Asymptotics for duration-driven long range dependent processes

We consider processes with second order long range dependence resulting from heavy tailed durations. We refer to this phenomenon as duration-driven long range dependence (DDLRD), as opposed to the more widely studied linear long range dependence based on fractional differencing of an i.i.d. process. We consider in detail two specific processes having DDLRD, originally presented in Taqqu and Levy [1986. Using renewal processes to generate long-range dependence and high variability. Dependence in Probability and Statistics. Birkhauser, Boston, pp. 73–89], and Parke [1999. What is fractional integration? Review of Economics and Statistics 81, 632–638]. For these processes, we obtain the limiting distribution of suitably standardized discrete Fourier transforms (DFTs) and sample autocovariances. At low frequencies, the standardized DFTs converge to a stable law, as do the standardized sample autocovariances at fixed lags. Finite collections of standardized sample autocovariances at a fixed set of lags converge to a degenerate distribution. The standardized DFTs at high frequencies converge to a Gaussian law. Our asymptotic results are strikingly similar for the two DDLRD processes studied. We calibrate our asymptotic results with a simulation study which also investigates the properties of the semiparametric log periodogram regression estimator of the memory parameter.     

Semiparametric hedonic price models: assessing the effects of agricultural nonpoint source pollution

In the area of environmental analysis using hedonic price models, we investigate the performance of various nonparametric and semiparametric specifications. The proposed model specifications are made up of two parts: a linear component for house characteristics and a non‐(semi)parametric component representing the nonlinear influence of environmental indicators on house prices. We adopt a general‐to‐specific search procedure, based on recent specification tests comparing the proposed specifications with a fully nonparametric benchmark model, to select the best model specification. An application of these semiparametric models to rural districts indicates that pollution resulting from intensive livestock farming has a significant nonlinear impact on house prices. Copyright © 2008 John Wiley & Sons, Ltd.     

Central limit theorems and uniform laws of large numbers for arrays of random fields

Over the last decades, spatial-interaction models have been increasingly used in economics. However, the development of a sufficiently general asymptotic theory for nonlinear spatial models has been hampered by a lack of relevant central limit theorems (CLTs), uniform laws of large numbers (ULLNs) and pointwise laws of large numbers (LLNs). These limit theorems form the essential building blocks towards developing the asymptotic theory of M-estimators, including maximum likelihood and generalized method of moments estimators. The paper establishes a CLT, ULLN, and LLN for spatial processes or random fields that should be applicable to a broad range of data processes.     

Increasing risk: Dynamic mean-preserving spreads

We extend the celebrated Rothschild and Stiglitz (1970) definition of Mean-Preserving Spreads to a dynamic framework. We adapt the original integral conditions to transition probability densities, and give sufficient conditions for their satisfaction. We then focus on a class of nonlinear scalar diffusion processes, the super-diffusive ballistic process, and prove that it satisfies the integral conditions. We further prove that this class is unique among Brownian bridges. This class of processes can be generated by a random superposition of linear Markov processes with constant drifts. This exceptionally simple representation enables us to systematically revisit, by means of the properties of dynamic mean-preserving spreads, workhorse economic models originally based on White Gaussian Noise. A selection of four examples is presented and explicitly solved.     

The Missing Link Between the Theory and Empirics of Path Dependence: Conceptual Clarification,Testability Issue,and Methodological Implications

Path dependence is a central construct in organizational research, used to describe a mechanism that connects the past and the future in an abstract way. However, across institutional, technology, and strategy literatures, it remains unclear why path dependence sometimes occurs and sometimes not, why it sometimes lead to inefficient outcomes and sometimes not, how it differs from mere increasing returns, and how scholars can empirically support their claims on path dependence. Hence, path dependence is not yet a theory since it does not causally relate identified variables in a systematized manner. Instead, the existing literature tends to conflate path dependence as a process (i.e. history unfolding in a self‐reinforcing manner) and as an outcome (i.e. a persisting state of the world with specific properties, called ‘lock‐in’). This paper contributes theoretically and methodologically to tackling these issues by: (1) providing a formal definition of path dependence that disentangles process and outcome, and identifies the necessary conditions for path dependence; (2) distinguishing clearly between path dependence and other ‘history matters’ kinds of mechanisms; and (3) specifying the missing link between theoretical and empirical path dependence. In particular, we suggest moving away from historical case studies of supposedly path‐dependent processes to focus on more controlled research designs such as simulations, experiments, and counterfactual investigation.