Bounding endogenous regressor coefficients using moment inequalities and generalized instruments

The main approach to deal with regressor endogeneity is instrumental variable estimator (IVE), where an instrumental variable (IV) m is required to be uncorrelated to the regression model error term u (COR(m,u)=0) and correlated to the endogenous regressor. If COR(m,u)≠0 is likely, then m gets discarded. But even when COR(m,u)≠0, often one has a good idea on the sign of COR(m,u). This article shows how to make use of the sign information on COR(m,u) to obtain an one‐sided bound on the endogenous regressor coefficient, calling m a ‘generalized instrument’ or ‘generalized instrumental variable (GIV)’. If there are two GIV's m₁ and m₂, then a two‐sided bound or an improved one‐sided bound can be obtained. Our approach is simple, needing only IVE; no non‐parametrics, nor any ‘tuning constants’. Specifically, the usual IVE is carried out, and the only necessary modification is that the estimate for the endogenous regressor coefficient is interpreted as a lower/upper bound depending on the prior notion on the sign of COR(m,u) and some estimable moment. A real data application is done to Korean household data with two or more children to illustrate our approach for the issue of child quantity–quality trade‐off.     

Spurious Fixed Effects Regression*

This article shows that spurious regression results can occur for a fixed effects model with weak time series variation in the regressor and/or strong time series variation in the regression errors when the first‐differenced and Within‐OLS estimators are used. Asymptotic properties of these estimators and the related t‐tests and model selection criteria are studied by sending the number of cross‐sectional observations to infinity. This article shows that the first‐differenced and Within‐OLS estimators diverge in probability, that the related t‐tests are inconsistent, that R²s converge to zero in probability and that AIC and BIC diverge to ?∞ in probability. The results of the article warn that one should not jump to the use of fixed effects regressions without considering the degree of time series variations in the data.     

Identifiability of mean‐reverting measurement error with instrumental variable

In the context where one main regressor is measured with error and at least one instrumental variable is available for the correction of measurement error, this paper provides, to the best of our knowledge, a first point‐identification result on the variance of measurement error, the variance of latent variable, and their covariance. We show that the parameters are identified if the regression model is not de facto linear. We illustrate the method in an application to identify mean‐reverting measurement error, a typical issue in reported income where the measurement error of income is negatively correlated with the true income.     

Further Evidence on the Statistical Properties of Real GNP*

The well‐known lack of power of unit‐root tests has often been attributed to the short length of macroeconomic variables and also to data‐generating processes (DGPs) departing from the I(1)–I(0) models. This paper shows that by using long spans of annual real gross national product (GNP) and GNP per capita (133 years), high power can be achieved, leading to the rejection of both the unit‐root and the trend‐stationary hypothesis. More flexible representations are then considered, namely, processes containing structural breaks (SB) and fractional orders of integration (FI). Economic justification for the presence of these features in GNP is provided. It is shown that both FI and SB formulations are in general preferred to the autoregressive integrated moving average (ARIMA) [I(1) or I(0)] formulations. As a novelty in this literature, new techniques are applied to discriminate between FI and SB. It turns out that the FI specification is preferred, implying that GNP and GNP per capita are non‐stationary, highly persistent but mean‐reverting series. Finally, it is shown that the results are robust when breaks in the deterministic component are allowed for in the FI model. Some macroeconomic implications of these findings are also discussed.     

k‐Nearest neighbors local linear regression for functional and missing data at random

We combine the k‐Nearest Neighbors (kNN) method to the local linear estimation (LLE) approach to construct a new estimator (LLE‐kNN) of the regression operator when the regressor is of functional type and the response variable is a scalar but observed with some missing at random (MAR) observations. The resulting estimator inherits many of the advantages of both approaches (kNN and LLE methods). This is confirmed by the established asymptotic results, in terms of the pointwise and uniform almost complete consistencies, and the precise convergence rates. In addition, a numerical study (i) on simulated data, then (ii) on a real dataset concerning the sugar quality using fluorescence data, were conducted. This practical study clearly shows the feasibility and the superiority of the LLE‐kNN estimator compared to competitive estimators.     

Properties of nonlinear transformations of fractionally integrated processes

This paper shows that the properties of nonlinear transformations of a fractionally integrated process strongly depend on whether the initial series is stationary or not. Transforming a stationary Gaussian I(d) process with d>0 leads to a long-memory process with the same or a smaller long-memory parameter depending on the Hermite rank of the transformation. Any nonlinear transformation of an antipersistent Gaussian I(d) process is I(0)). For non-stationary I(d) processes, every polynomial transformation is non-stationary and exhibits a stochastic trend in mean and in variance. In particular, the square of a non-stationary Gaussian I(d) process still has long memory with parameter d, whereas the square of a stationary Gaussian I(d) process shows less dependence than the initial process. Simulation results for other transformations are also discussed.     

Threshold Effects in Cointegrating Relationships*

In this paper, we introduce threshold‐type nonlinearities within a single‐equation cointegrating regression model and propose a testing procedure for testing the null hypothesis of linear cointegration vs. cointegration with threshold effects. Our framework allows the modelling of long‐run equilibrium relationships that may change according to the magnitude of a threshold variable assumed to be stationary and ergodic, and thus constitutes an attempt to deal econometrically with the potential presence of multiple equilibria. The framework is flexible enough to accommodate regressor endogeneity and serial correlation.     

Testing for Multicointegration in Panel Data with Common Factors*

This paper addresses the concept of multicointegration in a panel data framework and builds upon the panel data cointegration procedures developed in Pedroni [Econometric Theory (2004), Vol. 20, pp. 597–625]. When individuals are either cross‐section independent, or cross‐section dependence can be removed by cross‐section demeaning, our approach can be applied to the wider framework of mixed I(2) and I(1) stochastic processes. The paper also deals with the issue of cross‐section dependence using approximate common‐factor models. Finite sample performance is investigated through Monte Carlo simulations. Finally, we illustrate the use of the procedure investigating an inventories, sales and production relationship for a panel of US industries.     

Spurious Regression and Trending Variables*

This paper analyses the asymptotic and finite‐sample implications of different types of non‐stationary behaviour among the dependent and explanatory variables in a linear spurious regression model. We study cases when the non‐stationarity in the dependent and explanatory variables is deterministic as well as stochastic. In particular, we derive the order in probability of the t‐statistic in a spurious regression equation under a variety of empirically relevant data generation processes, and show that the spurious regression phenomenon is present in all cases when both dependent and explanatory variables behave in a non‐stationary way. Simulation experiments confirm our asymptotic results.     

Testing for Flexible Nonlinear Trends with an Integrated or Stationary Noise Component

This paper proposes a new test for the presence of a nonlinear deterministic trend approximated by a Fourier expansion in a univariate time series for which there is no prior knowledge as to whether the noise component is stationary or contains an autoregressive unit root. Our approach builds on the work of Perron and Yabu ( 2009a ) and is based on a Feasible Generalized Least Squares procedure that uses a super‐efficient estimator of the sum of the autoregressive coefficients α when α = 1. The resulting Wald test statistic asymptotically follows a chi‐square distribution in both the I(0) and I(1) cases. To improve the finite sample properties of the test, we use a bias‐corrected version of the OLS estimator of α proposed by Roy and Fuller ( 2001 ). We show that our procedure is substantially more powerful than currently available alternatives. We illustrate the usefulness of our method via an application to modelling the trend of global and hemispheric temperatures.     

Econometric Methods for Modelling Systems With a Mixture of i(1) and i(0) Variables

This paper considers structural models with both I(1) and I(0) variables. The structural shocks associated with either set of variables could be permanent or transitory. We classify the shocks as (P1,P0) and (T1,T0), where P/T distinguishes permanent/transitory, while 1/0 means they are attached to structural equations with either I(1) or I(0) variables as their ‘dependent’ variable. We show that P0 shocks can affect cointegration analysis and provide a formula to compute the permanent component if they are present. Finally, we reformulate a well‐known empirical structural vector autoregression showing the impact of P0 shocks when there are just long‐run parametric and sign restrictions. Copyright © 2015 John Wiley & Sons, Ltd.     

Pyrrho's lemma,or have it your way

A simple lemma is derived to support the claim that regression models can be manipulated to a very large extent: by simply adding one regressor one can obtain essentially every set of desired regression coefficients and predictions as well ast-values and standard errors. Consequently, if the product of a specification search is not shown to be generalizable, a sceptical attitude towards its validity is well-founded.     

Bias Reduction in Dynamic Panel Data Models by Common Recursive Mean Adjustment*

The within‐group estimator (same as the least squares dummy variable estimator) of the dominant root in dynamic panel regression is known to be biased downwards. This article studies recursive mean adjustment (RMA) as a strategy to reduce this bias for AR(p) processes that may exhibit cross‐sectional dependence. Asymptotic properties for N,T→∞ jointly are developed. When ( log ²T)(N/T)→ζ, where ζ is a non‐zero constant, the estimator exhibits nearly negligible inconsistency. Simulation experiments demonstrate that the RMA estimator performs well in terms of reducing bias, variance and mean square error both when error terms are cross‐sectionally independent and when they are not. RMA dominates comparable estimators when T is small and/or when the underlying process is persistent.     

CCE in fixed‐T panels

The presence of unobserved heterogeneity and its likely detrimental effect on inference has recently motivated the use of factor‐augmented panel regression models. The workhorse of this literature is based on first estimating the unknown factors using the cross‐section averages of the observables, and then applying ordinary least squares conditional on the first‐step factor estimates. This is the common correlated effects (CCE) approach, the existing asymptotic theory for which is based on the requirement that both the number of time series observations, T, and the number of cross‐section units, N, tend to infinity. The obvious implication of this theory for empirical work is that both N and T should be large, which means that CCE is impossible for the typical micro panel where only N is large. In the current paper, we put the existing CCE theory and its implications to a test. This is done by developing a new theory that enables T to be fixed. The results show that many of the previously derived large‐T results hold even if T is fixed. In particular, the pooled CCE estimator is still consistent and asymptotically normal, which means that CCE is more applicable than previously thought. In fact, not only do we allow T to be fixed, but the conditions placed on the time series properties of the factors and idiosyncratic errors are also much more general than those considered previously.     

Combining Significance of Correlated Statistics with Application to Panel Data*

The inverse normal method, which is used to combine P‐values from a series of statistical tests, requires independence of single test statistics in order to obtain asymptotic normality of the joint test statistic. The paper discusses the modification by Hartung (1999, Biometrical Journal, Vol. 41, pp. 849–855) , which is designed to allow for a certain correlation matrix of the transformed P‐values. First, the modified inverse normal method is shown here to be valid with more general correlation matrices. Secondly, a necessary and sufficient condition for (asymptotic) normality is provided, using the copula approach. Thirdly, applications to panels of cross‐correlated time series, stationary as well as integrated, are considered. The behaviour of the modified inverse normal method is quantified by means of Monte Carlo experiments.     

Stochastic cointegration: estimation and inference

This paper considers the estimation of a stochastically cointegrating regression within the stochastic cointegration modelling framework introduced in McCabe et al. (Stochastic cointegration: testing, 2001). A stochastic cointegrating regression allows some or all of the variables to be conventionally or heteroscedastically integrated. This generalizes Hansen's (J. Econom. 54 (1992) 139) heteroscedastic cointegrating regression model, where the dependent variable is heteroscedastically integrated, but all the regressor variables are restricted to being conventionally integrated. In contrast to conventional and heteroscedastic cointegrating regression, ordinary least-squares (OLS) estimation is shown to be inconsistent, in general, in a stochastically cointegrating regression. As a solution, a new instrumental variables (IVs) estimator is proposed and is shown to be consistent. Under a suitable exogeneity assumption, standard asymptotic inference on the stochastic cointegrating vector can be carried out based on the IV estimator. The finite sample properties of the test statistics, including their robustness to the exogeneity assumption, are examined by simulation.     

Robustness of Logit Analysis: Unobserved Heterogeneity and Mis‐specified Disturbances*

In probit and logit models, the β coefficients vary inversely with the variance of the disturbances. The omission of a relevant orthogonal regressor leads to increased unobserved heterogeneity, and this depresses the β coefficients of the remaining regressors towards zero. For the probit model, Wooldridge (Econometric Analysis of Cross Section and Panel Data, MIT Press, Cambridge, MA, 2002) has shown that this bias does not carry over to the effect of these regressors on the outcome. We find by simulations that this also holds for the logit model, even when omitting a variable leads to severe mis‐specification of the disturbance. More simulations show that logit analysis is quite insensitive to pure mis‐specification of the disturbance as such.     

Robust Non‐nested Testing for Ordinary Least Squares Regression when Some of the Regressors are Lagged Dependent Variables*

The problem of testing non‐nested regression models that include lagged values of the dependent variable as regressors is discussed. It is argued that it is essential to test for error autocorrelation if ordinary least squares and the associated J and F tests are to be used. A heteroskedasticity–robust joint test against a combination of the artificial alternatives used for autocorrelation and non‐nested hypothesis tests is proposed. Monte Carlo results indicate that implementing this joint test using a wild bootstrap method leads to a well‐behaved procedure and gives better control of finite sample significance levels than asymptotic critical values.     

Diagnostic Tests of Cross‐section Independence for Limited Dependent Variable Panel Data Models*

This article considers the problem of testing for cross‐section independence in limited dependent variable panel data models. It derives a Lagrangian multiplier (LM) test and shows that in terms of generalized residuals of Gourieroux et al. (1987) it reduces to the LM test of Breusch and Pagan (1980) . Because of the tendency of the LM test to over‐reject in panels with large N (cross‐section dimension), we also consider the application of the cross‐section dependence test (CD) proposed by Pesaran (2004) . In Monte Carlo experiments it emerges that for most combinations of N and T the CD test is correctly sized, whereas the validity of the LM test requires T (time series dimension) to be quite large relative to N. We illustrate the cross‐sectional independence tests with an application to a probit panel data model of roll‐call votes in the US Congress and find that the votes display a significant degree of cross‐section dependence.     

Estimating the Kullback–Liebler risk based on multifold cross‐validation

This paper concerns a class of model selection criteria based on cross‐validation techniques and estimative predictive densities. Both the simple or leave‐one‐out and the multifold or leave‐m‐out cross‐validation procedures are considered. These cross‐validation criteria define suitable estimators for the expected Kullback–Liebler risk, which measures the expected discrepancy between the fitted candidate model and the true one. In particular, we shall investigate the potential bias of these estimators, under alternative asymptotic regimes for m. The results are obtained within the general context of independent, but not necessarily identically distributed, observations and by assuming that the candidate model may not contain the true distribution. An application to the class of normal regression models is also presented, and simulation results are obtained in order to gain some further understanding on the behavior of the estimators.