THIS SHOCK IS DIFFERENT: ESTIMATION AND INFERENCE IN MISSPECIFIED TWO-WAY FIXED EFFECTS PANEL REGRESSIONS

Summary This paper studies the asymptotic properties of standard panel data estimators in a simple panel regression model with random error component disturbances. Both the regressor and the remainder disturbance term are assumed to be autoregressive and possibly non-stationary. Asymptotic distributions are derived for the standard panel data estimators including ordinary least squares (OLS), fixed effects (FE), first-difference (FD) and generalized least squares (GLS) estimators when both T and n are large. We show that all the estimators have asymptotic normal distributions and have different convergence rates dependent on the non-stationarity of the regressors and the remainder disturbances. We show using Monte Carlo experiments that the loss in efficiency of the OLS, FE and FD estimators relative to true GLS can be substantial.

We investigate a bracketing property that purports to yield upper- and lower bounds on the treatment effects obtained from a fixed effects (FE) and lagged dependent variable (LDV) model. Referencing both analytical results and a Monte Carlo simulation, we explore the conditions under which the bracketing property holds, confirming this to be the case when the data generating process (DGP) is characterized by either unobserved heterogeneity or feedback effects from a lagged dependent variable. However, when the DGP is characterized by both features simultaneously, we find that bracketing of the treatment effect only holds under certain conditions—but not in general. Practitioners can nevertheless obtain the lower bound estimate by referencing a model that includes both FE and an LDV. While the Nickell bias in the coefficient of the LDV is known to be of order $1/T$ , we show that the Nickell-type bias in the estimator of the treatment effect is of order $1/T^2$ .

In the application of extreme value analysis it is usually assumed that the size of the samples from which the extreme values are obtained is sufficiently large for the asymptotic extreme value distribution to be used. The necessary sample size depends upon the population distribution and this is generally not known; but assuming a Weibull distribution, which is often fitted to wind speed and wave height data, it is shown that the rate of convergence is rapid and that the asymptotic distribution may be used for a sample size as small as ten. An ‘exponential approximation’ for the distribution of maxima is sometimes confused with the extreme value distribution. This approximate form is derived for the Weibull distribution and the essential difference between it and the asymptotic extreme value distribution is explained.

In the application of extreme value analysis it is usually assumed that the size of the samples from which the extreme values are obtained is sufficiently large for the asymptotic extreme value distribution to be used. The necessary sample size depends upon the population distribution and this is generally not known; but assuming a Weibull distribution, which is often fitted to wind speed and wave height data, it is shown that the rate of convergence is rapid and that the asymptotic distribution may be used for a sample size as small as ten. An ‘exponential approximation’ for the distribution of maxima is sometimes confused with the extreme value distribution. This approximate form is derived for the Weibull distribution and the essential difference between it and the asymptotic extreme value distribution is explained.

The contribution of this paper is two-fold. First, we derive the asymptotic null distribution of the familiar augmented Dickey-Fuller [ADF] statistics in the case where the shocks follow a linear process driven by in…nite variance innovations. We show that these distributions are free of serial correlation nuisance parameters but depend on the tail index of the in…nite variance process. These distributions are shown to coincide with the corresponding results for the case where the shocks follow a …nite autoregression, provided the lag length in the ADF regression satis…es the same o(T1=3) rate condition as is required in the …nite variance case. In addition, we establish the rates of consistency and (where they exist) the asymptotic distributions of the ordinary least squares sieve estimates from the ADF regression. Given the dependence of their null distributions on the unknown tail index, our second contribution is to explore sieve wild bootstrap implementations of the ADF tests. Under the assumption of symmetry, we demonstrate the asymptotic validity (bootstrap consistency) of the wild bootstrap ADF tests. This is done by establishing that (conditional on the data) the wild bootstrap ADF statistics attain the same limiting distribution as that of the original ADF statistics taken conditional on the magnitude of the innovations.

Despite the long-standing discussion on fixed effects (FE) and random effects (RE) models, how and under what conditions both methods can eliminate unmeasured confounding bias has not yet been widely understood in practice. Using a simple pretest-posttest design in a linear setting, this paper translates the conventional algebraic formalization of FE and RE models into causal graphs and provides intuitively accessible graphical explanations about their data-generating and bias-removing processes. The proposed causal graphs highlight that FE and RE models consider different data-generating models. RE models presume a data-generating model that is identical to a randomized controlled trial, while FE models allow for unobserved time-invariant treatment-outcome confounding. Augmenting regular causal graphs that describe data-generating processes by adding the computational structures of FE and RE estimators, the paper visualizes how FE estimators (gain score and deviation score estimators) and RE estimators (quasi-deviation score estimators) offset unmeasured confounding bias. In contrast to standard regression or matching estimators that reduce confounding bias by blocking non-causal paths via conditioning, FE and RE estimators offset confounding bias by deliberately creating new non-causal paths and associations of opposite sign. Though FE and RE estimators are similar in their bias-offsetting mechanisms, the augmented graphs reveal their subtle differences that can result in different biases in observational studies.

The impact of the choice of the lag length on tests for the number of cointegration relations in a vector autoregressive (VAR) process is investigated. It is shown that the asymptotic distribution of likelihood ratio (LR) tests for the cointegrating rank remains unchanged if the true data generation process (DGP) is of finite order and a consistent model selection criterion is used for choosing the lag length. A similar result also holds if the true DGP is an in finite order VAR. In a simulation study we find that small sample power and size of LR cointegration tests strongly depend on the choice of the lag order.

The additive hazard model, which focuses on risk differences rather than risk ratios, has been widely applied in practice. In this paper, we consider an additive hazard model with varying coefficients to analyze recurrent events data. The model allows for both varying and constant coefficients. We first propose an estimating equation-based approach with spline basis smoothing for all functional coefficients. Then, we provide theoretical justifications for the resulting estimates, including consistency, rate of convergence, and asymptotic distribution. Furthermore, we construct a Cramér-von Mises test procedure to investigate whether the functional coefficients should be treated as constant, and its asymptotic null distribution is also derived. Extensive simulation experiments are conducted to evaluate the finite-sample performance of the proposed approaches. A Chronic Granulotamous Disease data set was analyzed to illustrate our methodology.

The maximum likelihood estimator of a nondecreasing regression function with normally distributed errors has been considered in the literature. Its asymptotic distribution at a point is related to a solution of the heat equation, and its rate of convergence to the underlying regression function is of order $n^{-1/3}$. This estimator can be modified by grouping adjacent observations and then "isotonizing" the corresponding means. It is shown that the resulting estimator has an asymptotic normal distribution for certain group sizes and its rate of convergence is of order $n^{-2/5}$. The results of a simulation study for small sample sizes are presented and grouping procedures are discussed.

Motivated by the analysis of longitudinal neuroimaging studies, we study the longitudinal functional linear regression model under asynchronous data setting for modeling the association between clinical outcomes and functional (or imaging) covariates. In the asynchronous data setting, both covariates and responses may be measured at irregular and mismatched time points, posing methodological challenges to existing statistical methods. We develop a kernel weighted loss function with roughness penalty to obtain the functional estimator and derive its representer theorem. The rate of convergence, a Bahadur representation, and the asymptotic pointwise distribution of the functional estimator are obtained under the reproducing kernel Hilbert space framework. We propose a penalized likelihood ratio test to test the nullity of the functional coefficient, derive its asymptotic distribution under the null hypothesis, and investigate the separation rate under the alternative hypotheses. Simulation studies are conducted to examine the finite-sample performance of the proposed procedure. We apply the proposed methods to the analysis of multitype data obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) study, which reveals significant association between 21 regional brain volume density curves and the cognitive function. Data used in preparation of this paper were obtained from the ADNI database (adni.loni.usc.edu).

: This document assesses the feasibility of a goodness-of-fit test based on an integral of the weighted squared modulus of the discrepancy between the sample and population characteristic functions. The resulting statistic is therefore analogous to the Cramer-von Mises statistic and is shown to reduce to it as a special case. A number of properties of the test have been derived, including the asymptotic null distribution of its statistic. It is shown that under mild regularity conditions the test is consistent. A number of approximations to the null distribution of the test statistic are considered, and are found to be successful in simplifying its application without due loss of accuracy. Keywords: Sample characteristic function; Goodness-of-fit; Weighted sum of chi-squared variates; Consistency; Asymptotic distribution; Rate of convergence; Cumulants. (Author)

In this paper, we propose a varying coefficient panel data model with unobservable multiple interactive fixed effects that are correlated with the regressors. We approximate each coefficient function by B-spline, and propose a robust nonlinear iteration scheme based on the least squares method to estimate the coefficient functions of interest. We also establish the asymptotic theory of the resulting estimators under certain regularity assumptions, including the consistency, the convergence rate and the asymptotic distribution. Furthermore, we develop a least squares dummy variable method to study an important special case of the proposed model: the varying coefficient panel data model with additive fixed effects. To construct the pointwise confidence intervals for the coefficient functions, a residual-based block bootstrap method is proposed to reduce the computational burden as well as to avoid the accumulative errors. Simulation studies and a real data analysis are also carried out to assess the performance of our proposed methods.

The negative-binomial sum is an extension of a geometric sum. It has been arisen from the necessity to resolve practical problems in telecommunications, network analysis, stochastic finance and insurance mathematics, etc. Up to the present, the topics related to negative-binomial sums like asymptotic distributions and rates of convergence have been investigated by many mathematicians. However, in a lot of various situations, the results concerned the rates of convergence for negative-binomial sums are still restrictive. The main purpose of this paper is to establish some weak limit theorems for negative-binomial sums of independent, identically distributed (i.i.d.) random variables via Gnedenko's Transfer Theorem originated by Gnedenko and Fahim (1969). Using Zolotarev's probability metric, the rate of convergence in weak limit theorems for negativebinomial sum are established. The received results are the rates of convergence in weak limit theorem for partial sum of i.i.d random variables related to symmetric stable distribution (Theorem 1), and asymptotic distribution together with the convergence rates for negative-binomial sums of i.i.d. random variables concerning to symmetric Linnik laws and Generalized Linnik distribution (Theorem 2 and Theorem 3). Based on the results of this paper, the analogous results for geometric sums of i.i.d. random variables will be concluded as direct consequences. However, the article has just been solved for the case of 1 <a < 2; it is quite hard to estimate in the case of a 2 (0;1) via the Zolotarev's probability metric.
 Mathematics Subject Classification 2010: 60G50; 60F05; 60E07.

This paper studies the asymptotic properties of standard panel data estimators in a simple panel regression model with error component disturbances. Both the regressor and the remainder disturbance term are assumed to be autoregressive and possibly non-stationary. Asymptotic distributions are derived for the standard panel data estimators including ordinary least squares, fixed effects, first-difference, and generalized least squares (GLS) estimators when both T and n are large. We show that all the estimators have asymptotic normal distributions and have different convergence rates dependent on the non-stationarity of the regressors and the remainder disturbances. We show using Monte Carlo experiments that the loss in efficiency of the OLS, FE and FD estimators relative to true GLS can be substantial.

For a stochastic differential equation(SDE) driven by a fractional Brownian motion(fBm) with Hurst parameter H>12, it is known that the existing (naive) Euler scheme has the rate of convergence n1−2H. Since the limit H→12 of the SDE corresponds to a Stratonovich SDE driven by standard Brownian motion, and the naive Euler scheme is the extension of the classical Euler scheme for Itô SDEs for H=12, the convergence rate of the naive Euler scheme deteriorates for H→12. In this paper we introduce a new (modified Euler) approximation scheme which is closer to the classical Euler scheme for Stratonovich SDEs for H=12, and it has the rate of convergence γ−1n, where γn=n2H−1/2 when H<34, γn=n/logn−−−−√ when H=34 and γn=n if H>34. Furthermore, we study the asymptotic behavior of the fluctuations of the error. More precisely, if {Xt,0≤t≤T} is the solution of a SDE driven by a fBm and if {Xnt,0≤t≤T} is its approximation obtained by the new modified Euler scheme, then we prove that γn(Xn−X) converges stably to the solution of a linear SDE driven by a matrix-valued Brownian motion, when H∈(12,34]. In the case H>34, we show the Lp convergence of n(Xnt−Xt), and the limiting process is identified as the solution of a linear SDE driven by a matrix-valued Rosenblatt process. The rate of weak convergence is also deduced for this scheme. We also apply our approach to the naive Euler scheme.