Asymptotic inference for a one-dimensional simultaneous autoregressive model

A nonstationary simultaneous autoregressive model \({X^{(n)}_k=\alpha \Big(X^{(n)}_{k-1}+X^{(n)}_{k+1}\Big)+\varepsilon_k, k=1, 2, \ldots , n-1}\), is investigated, where \({X^{(n)}_0}\) and \({X^{(n)}_n}\) are given random variables. It is shown that in the unstable case α = 1/2 the least squares estimator of the autoregressive parameter converges to a functional of a standard Wiener process with a rate of convergence n ², while in the stable situation |α| < 1/2 the estimator is biased but asymptotically normal with a rate n ^1/2.     

Half‐panel jackknife fixed‐effects estimation of linear panels with weakly exogenous regressors

This paper considers estimation and inference in linear panel regression models with lagged dependent variables and/or other weakly exogenous regressors when N (the cross‐section dimension) is large relative to T (the time series dimension). It allows for fixed and time effects (FE‐TE) and derives a general formula for the bias of the FE‐TE estimator which generalizes the well‐known Nickell bias formula derived for the pure autoregressive dynamic panel data models. It shows that in the presence of weakly exogenous regressors inference based on the FE‐TE estimator will result in size distortions unless N/T is sufficiently small. To deal with the bias and size distortion of the FE‐TE estimator the use of a half‐panel jackknife FE‐TE estimator is considered and its asymptotic distribution is derived. It is shown that the bias of the half‐panel jackknife FE‐TE estimator is of order T^?2, and for valid inference it is only required that N/T³→0, as N,T→∞ jointly. Extension to unbalanced panel data models is also provided. The theoretical results are illustrated with Monte Carlo evidence. It is shown that the FE‐TE estimator can suffer from large size distortions when N>T, with the half‐panel jackknife FE‐TE estimator showing little size distortions. The use of half‐panel jackknife FE‐TE estimator is illustrated with two empirical applications from the literature.     

Exact local Whittle estimation of fractionally cointegrated systems

Semiparametric estimation of a bivariate fractionally cointegrated system is considered. We propose a two-step procedure that accommodates both (asymptotically) stationary (δ<1/2)$(\delta <1/2)$ and nonstationary (δ≥1/2)$(\delta \ge 1/2)$ stochastic trend and/or equilibrium error. A tapered version of the local Whittle estimator of Robinson (2008) is used as the first-stage estimator, and the second-stage estimator employs the exact local Whittle approach of Shimotsu and Phillips (2005). The consistency and asymptotic distribution of the two-step estimator are derived. The estimator of the memory parameters has the same Gaussian asymptotic distribution in both the stationary and the nonstationary case. The convergence rate and the asymptotic distribution of the estimator of the cointegrating vector are affected by the difference between the memory parameters. Further, the estimator has a Gaussian asymptotic distribution when the difference between the memory parameters is less than 1/2.     

One-stage estimation of the effects of operational conditions and practices on productive performance: asymptotically normal and efficient,root-<Emphasis Type="Italic">n</Emphasis> consistent StoNEZD method

Understanding the effects of operational conditions and practices on productive efficiency can provide valuable economic and managerial insights. The conventional approach is to use a two-stage method where the efficiency estimates are regressed on contextual variables representing the operational conditions. The main problem of the two-stage approach is that it ignores the correlations between inputs and contextual variables. To address this shortcoming, we build on the recently developed regression interpretation of data envelopment analysis (DEA) to develop a new one-stage semi-nonparametric estimator that combines the nonparametric DEA-style frontier with a regression model of the contextual variables. The new method is referred to as stochastic semi-nonparametric envelopment of z variables data (StoNEZD). The StoNEZD estimator for the contextual variables is shown to be statistically consistent under less restrictive assumptions than those required by the two-stage DEA estimator. Further, the StoNEZD estimator is shown to be unbiased, asymptotically efficient, asymptotically normally distributed, and converge at the standard parametric rate of order n ^−1/2. Therefore, the conventional methods of statistical testing and confidence intervals apply for asymptotic inference. Finite sample performance of the proposed estimators is examined through Monte Carlo simulations.     

Estimating dynamic panel data discrete choice models with fixed effects

This paper considers the estimation of dynamic binary choice panel data models with fixed effects. It is shown that the modified maximum likelihood estimator (MMLE) used in this paper reduces the order of the bias in the maximum likelihood estimator from O(T^-1) to O(T^-2), without increasing the asymptotic variance. No orthogonal reparametrization is needed. Monte Carlo simulations are used to evaluate its performance in finite samples where T is not large. In probit and logit models containing lags of the endogenous variable and exogenous variables, the estimator is found to have a small bias in a panel with eight periods. A distinctive advantage of the MMLE is its general applicability. Estimation and relevance of different policy parameters of interest in this kind of models are also addressed.     

Estimation in the first-order moving average model through the finite autoregressive approximation: Some asymptotic results

To estimate α in the model y_t = u_t+αu_t?1, we consider a proposal by Durbin (Biometrika, 1969). It consists in fitting an autoregression of order k to the data, and deriving from there an estimate α^. The probability limit and the variance of the limiting normal distribution of α^ are presented and discussed in detail, when the sample size T → ∞, but k remains fixed. The differences between the resulting values and those corresponding to the maximum likelihood estimator are exponentially decreasing functions of k. Several modifications of the estimator are discussed and found consistent, but asymptotically inefficient.     

An approximation to the finite sample distribution of Zellner's seemingly unrelated regression estimator

A multiple equation model of the seemingly unrelated regressions type is considered. We derive an Edgeworth expansion up to 0(T^-1), where T is the sample size, of the finite sampls distribution function of the seemingly unrelated regression estimator of the parameters in thie model. We examine the two-equation case where our results can be related to exact theory in the special case of orthogonal exogenous variables and we take as a particular numerical example Zellner's original application to micro-investment functions.     

Several efficient two-step estimators for the dynamic simultaneous equations model with autoregressive disturbances

Several asymptotically efficient methods are suggested on both the full and the limited information approach to estimate the simultaneous equations model in which the lagged endogenous variables and the autoregressive disturbances coexist. They are two-step procedures and do not involve iterations. A method is suggested also for the case where any portion of the autoregressive parameter matrix is specified to be zero. Since the consistency and efficiency depend upon the asymptotic, local identifiability, the necessary and sufficient condition is derived for it. It does not depend on the exclusion of the lagged endogenous variables.     

Discrete choice modeling with nonstationary panels applied to exchange rate regime choice

This paper develops a regression limit theory for discrete choice nonstationary panels with large cross section (N) and time series (T) dimensions. Some results emerging from this theory are directly applicable in the wider context of M-estimation. This includes an extension of work by Wooldridge [Wooldridge, J.M., 1994. Estimation and Inference for Dependent Processes. In: Engle, R.F., McFadden, D.L. (Eds.). Handbook of Econometrics, vol. 4, North-Holland, Amsterdam] on the limit theory of local extremum estimators to multi-indexed processes in nonlinear nonstationary panel data models.It is shown that the maximum likelihood (ML) estimator is consistent without an incidental parameters problem and has a limit theory with a fast rate of convergence N^1/2T^3/4 (in the stationary case, the rate is N^1/2T^1/2) for the regression coefficients and thresholds, and a normal limit distribution. In contrast, the limit distribution is known to be mixed normal in time series modeling, as shown in [Park, J.Y., Phillips, P.C.B., 2000, Nonstationary binary choice. Econometrica, 68, 1249–1280] (hereafter PP), and [Phillips, P.C.B., Jin, S., Hu, L., 2007. Nonstationary discrete choice: A corrigendum and addendum. Journal of Econometrics 141(2), 1115–1130] (hereafter, PJH).The approach is applied to exchange rate regime choice by monetary authorities, and we provide an analysis of the empirical phenomenon known as “fear of floating”.     

Nonstationary nonlinear heteroskedasticity in regression

This paper considers the regression with errors having nonstationary nonlinear heteroskedasticity. For both the usual stationary regression and the nonstationary cointegrating regression, we develop the asymptotic theories for the least squares methods in the presence of conditional heterogeneity given as a nonlinear function of an integrated process. In particular, we show that the nonstationarity of volatility in the regression errors may induce spuriousness of the underlying regression, if excessive nonstationary volatility is present in the errors. Mild nonstationary volatilities do not render the underlying regression spurious, but their presence makes the least squares estimator asymptotically biased and inefficient and the usual chi-square test invalid.     

Third-order optimum properties of estimator-sequences

SequencesT ⁽ⁿ⁾ ,n∈N, are considered, whereT ⁽ⁿ⁾ estimates a vector parameter ?∈R ^p from an i.i.d. sample of sizen, and such sequences are compared on the basis of their risks ∫L(n ^1/2(T ⁿ (x)?θ))P _θ ⁿ (dx) relative to loss functionsL:R ^p →R. A characterization is given for sequencesT ^*(n) ,n∈N, which generate an essentially complete class in the following sense: For any sequenceT ⁽ⁿ⁾ ,n∈N, there exist functions Φ _n ,n∈N, such that forn→∞ we have $$\begin{gathered} \smallint L (n^{1/2} (T^{*(n)} + n^{ - 1} \Phi _n (T^{*(n)} ) - \theta )) dP_\theta ^n \leqslant \hfill \\ \leqslant \smallint L (n^{1/2} (T^{(n)} - \theta )) dP_\theta ^n + o (n^{ - 1} ), \hfill \\ \end{gathered} $$ for every ? and everyL satisfying certain conditions. If the estimator-sequences are compared by their risks ∫W(T ⁽ⁿ⁾ d P _θ ⁿ ,θ) with respect to loss functionsW:R ^p ×Θ→R then a similar result on asymptotically complete classes is valid. The results are obtained under the assumption thatT ^*(n) ,n∈N, andT ⁽ⁿ⁾ ,n∈N, admit stochastic expansions which are sufficiently regular, that the loss functionsL andW are sufficiently smooth and bounded by polynomials, and that the estimator-sequences have asymptotically bounded moments; the latter condition is not needed for bounded functionsL andW.     

To Pool or Not to Pool: Revisited

This paper provides a new comparative analysis of pooled least squares and fixed effects (FE) estimators of the slope coefficients in the case of panel data models when the time dimension (T) is fixed while the cross section dimension (N) is allowed to increase without bounds. The individual effects are allowed to be correlated with the regressors, and the comparison is carried out in terms of an exponent coefficient, δ, which measures the degree of pervasiveness of the FE in the panel. The use of δ allows us to distinguish between poolability of small N dimensional panels with large T from large N dimensional panels with small T. It is shown that the pooled estimator remains consistent so long as δ<1, and is asymptotically normally distributed if δ<1/2, for a fixed T and as N→∞. It is further shown that when δ<1/2, the pooled estimator is more efficient than the FE estimator. We also propose a Hausman type diagnostic test of δ<1/2 as a simple test of poolability, and propose a pretest estimator that could be used in practice. Monte Carlo evidence supports the main theoretical findings and gives some indications of gains to be made from pooling when δ<1/2.     

Simple and powerful GMM over-identification tests with accurate size

Based on the series long run variance estimator, we propose a new class of over-identification tests that are robust to heteroscedasticity and autocorrelation of unknown forms. We show that when the number of terms used in the series long run variance estimator is fixed, the conventional J statistic, after a simple correction, is asymptotically F-distributed. We apply the idea of the F-approximation to the conventional kernel-based J tests. Simulations show that the J^∗ tests based on the finite sample corrected J statistic and the F-approximation have virtually no size distortion, and yet are as powerful as the standard J tests.     

On a two-step estimation of a multivariate logit model

In this article the author studies the properties of the two-step estimation method proposed by Domencich and McFadden (Urban Travel Demand, North-Holland, 1975) for a multivariate logit model and shows that it is consistent but asymptotically less efficient than the maximum likelihood estimator. Its computation, however, can be considerably simpler than that of the maximum likelihood estimator, especially in models involving several dependent variables.     

Optimal inference for instrumental variables regression with non-Gaussian errors

This paper is concerned with inference on the coefficient on the endogenous regressor in a linear instrumental variables model with a single endogenous regressor, nonrandom exogenous regressors and instruments, and i.i.d. errors whose distribution is unknown. It is shown that under mild smoothness conditions on the error distribution it is possible to develop tests which are “nearly” efficient in the sense of Andrews et al. (2006) when identification is weak and consistent and asymptotically optimal when identification is strong. In addition, an estimator is presented which can be used in the usual way to construct valid (indeed, optimal) confidence intervals when identification is strong. The estimator is of the two stage least squares variety and is asymptotically efficient under strong identification whether or not the errors are normal.     

Functional-coefficient models for nonstationary time series data

This paper studies functional coefficient regression models with nonstationary time series data, allowing also for stationary covariates. A local linear fitting scheme is developed to estimate the coefficient functions. The asymptotic distributions of the estimators are obtained, showing different convergence rates for the stationary and nonstationary covariates. A two-stage approach is proposed to achieve estimation optimality in the sense of minimizing the asymptotic mean squared error. When the coefficient function is a function of a nonstationary variable, the new findings are that the asymptotic bias of its nonparametric estimator is the same as the stationary covariate case but convergence rate differs, and further, the asymptotic distribution is a mixed normal, associated with the local time of a standard Brownian motion. The asymptotic behavior at boundaries is also investigated.     

The coefficient of determination and simultaneous equation systems

In this paper we show that the Carter-Nagar (1977) R²'s for single structural equations and systems are in fact R² for the reduced form where the partially restricted reduced form estimation method is employed. We also show that the results of McElroy (1977) may be used to derive the Carter-Nagar system measure. If the reduced form equations are estimated by Kakwani's (1975) k-class reduced form estimator a new R² may be defined which is shown to be asymptotically equivalent to the Carter-Nagar measure.     

Asymptotics of the principal components estimator of large factor models with weakly influential factors

This paper introduces a drifting-parameter asymptotic framework to derive accurate approximations to the finite sample distribution of the principal components (PC) estimator in situations when the factors’ explanatory power does not strongly dominate the explanatory power of the cross-sectionally and temporally correlated idiosyncratic terms. Under our asymptotics, the PC estimator is inconsistent. We find explicit formulae for the amount of the inconsistency, and propose an estimator of the number of factors for which the PC estimator works reasonably well. For the special case when the idiosyncratic terms are cross-sectionally but not temporally correlated (or vice versa), we show that the coefficients in the OLS regressions of the PC estimates of factors (loadings) on the true factors (true loadings) are asymptotically normal, and find explicit formulae for the corresponding asymptotic covariance matrix. We explain how to estimate the parameters of the derived asymptotic distributions. Our Monte Carlo analysis suggests that our asymptotic formulae and estimators work well even for relatively small n$n$ and T$T$. We apply our theoretical results to test a hypothesis about the factor content of the US stock return data.     

A consistent nonparametric test for nonlinear causality—Specification in time series regression

Since the pioneering work by Granger (1969), many authors have proposed tests of causality between economic time series. Most of them are concerned only with “linear causality in mean”, or if a series linearly affects the (conditional) mean of the other series. It is no doubt of primary interest, but dependence between series may be nonlinear, and/or not only through the conditional mean. Indeed conditional heteroskedastic models are widely studied recently. The purpose of this paper is to propose a nonparametric test for possibly nonlinear causality. Taking into account that dependence in higher order moments are becoming an important issue especially in financial time series, we also consider a test for causality up to the Kth conditional moment. Statistically, we can also view this test as a nonparametric omitted variable test in time series regression. A desirable property of the test is that it has nontrivial power against T^1/2-local alternatives, where T is the sample size. Also, we can form a test statistic accordingly if we have some knowledge on the alternative hypothesis. Furthermore, we show that the test statistic includes most of the omitted variable test statistics as special cases asymptotically. The null asymptotic distribution is not normal, but we can easily calculate the critical regions by simulation. Monte Carlo experiments show that the proposed test has good size and power properties.     

Estimation of a dynamic system of seemingly unrelated regressions with autoregressive disturbances

This paper examines the small sample properties of the asymptotically efficient estimator due to Hatanaka (1976) and Dhrymes and Taylor (1976) applied to a system of seemingly unrelated regressions characterized by both autoregressive disturbances and lagged endogenous variables. The results of several Monte Carlo experiments suggest that, in general, this estimation procedure performs well in samples of modest size. Two important situations in which the results are mixed are also reported.