Confidence Intervals for Partially Identified Parameters

Recently a growing body of research has studied inference in settings where parameters of interest are partially identified. In many cases the parameter is real‐valued and the identification region is an interval whose lower and upper bounds may be estimated from sample data. For this case confidence intervals (CIs) have been proposed that cover the entire identification region with fixed probability. Here, we introduce a conceptually different type of confidence interval. Rather than cover the entire identification region with fixed probability, we propose CIs that asymptotically cover the true value of the parameter with this probability. However, the exact coverage probabilities of the simplest version of our new CIs do not converge to their nominal values uniformly across different values for the width of the identification region. To avoid the problems associated with this, we modify the proposed CI to ensure that its exact coverage probabilities do converge uniformly to their nominal values. We motivate this modified CI through exact results for the Gaussian case.     

Confidence Intervals for Diffusion Index Forecasts and Inference for Factor‐Augmented Regressions

We consider the situation when there is a large number of series, N, each with T observations, and each series has some predictive ability for some variable of interest. A methodology of growing interest is first to estimate common factors from the panel of data by the method of principal components and then to augment an otherwise standard regression with the estimated factors. In this paper, we show that the least squares estimates obtained from these factor‐augmented regressions are consistent and asymptotically normal if . The conditional mean predicted by the estimated factors is consistent and asymptotically normal. Except when T/N goes to zero, inference should take into account the effect of “estimated regressors” on the estimated conditional mean. We present analytical formulas for prediction intervals that are valid regardless of the magnitude of N/T and that can also be used when the factors are nonstationary.     

Inference for the Identified Set in Partially Identified Econometric Models

This paper provides computationally intensive, yet feasible methods for inference in a very general class of partially identified econometric models. Let P denote the distribution of the observed data. The class of models we consider is defined by a population objective function Q(θ, P) for θ∈Θ. The point of departure from the classical extremum estimation framework is that it is not assumed that Q(θ, P) has a unique minimizer in the parameter space Θ. The goal may be either to draw inferences about some unknown point in the set of minimizers of the population objective function or to draw inferences about the set of minimizers itself. In this paper, the object of interest is Θ₀(P)=argmin_θ∈ΘQ(θ, P), and so we seek random sets that contain this set with at least some prespecified probability asymptotically. We also consider situations where the object of interest is the image of Θ₀(P) under a known function. Random sets that satisfy the desired coverage property are constructed under weak assumptions. Conditions are provided under which the confidence regions are asymptotically valid not only pointwise in P, but also uniformly in P. We illustrate the use of our methods with an empirical study of the impact of top‐coding outcomes on inferences about the parameters of a linear regression. Finally, a modest simulation study sheds some light on the finite‐sample behavior of our procedure.     

Uniform Inference in Autoregressive Models

The purpose of this paper is to provide theoretical justification for some existing methods for constructing confidence intervals for the sum of coefficients in autoregressive models. We show that the methods of Stock (1991), Andrews (1993), and Hansen (1999) provide asymptotically valid confidence intervals, whereas the subsampling method of Romano and Wolf (2001) does not. In addition, we generalize the three valid methods to a larger class of statistics. We also clarify the difference between uniform and pointwise asymptotic approximations, and show that a pointwise convergence of coverage probabilities for all values of the parameter does not guarantee the validity of the confidence set.     

Hybrid and Size‐Corrected Subsampling Methods

This paper considers inference in a broad class of nonregular models. The models considered are nonregular in the sense that standard test statistics have asymptotic distributions that are discontinuous in some parameters. It is shown in Andrews and Guggenberger (2009a) that standard fixed critical value, subsampling, and m out of n bootstrap methods often have incorrect asymptotic size in such models. This paper introduces general methods of constructing tests and confidence intervals that have correct asymptotic size. In particular, we consider a hybrid subsampling/fixed‐critical‐value method and size‐correction methods. The paper discusses two examples in detail. They are (i) confidence intervals in an autoregressive model with a root that may be close to unity and conditional heteroskedasticity of unknown form and (ii) tests and confidence intervals based on a post‐conservative model selection estimator.     

Bootstrap Determination of the Co‐Integration Rank in Vector Autoregressive Models

This paper discusses a consistent bootstrap implementation of the likelihood ratio (LR) co‐integration rank test and associated sequential rank determination procedure of Johansen (1996). The bootstrap samples are constructed using the restricted parameter estimates of the underlying vector autoregressive (VAR) model that obtain under the reduced rank null hypothesis. A full asymptotic theory is provided that shows that, unlike the bootstrap procedure in Swensen (2006) where a combination of unrestricted and restricted estimates from the VAR model is used, the resulting bootstrap data are I(1) and satisfy the null co‐integration rank, regardless of the true rank. This ensures that the bootstrap LR test is asymptotically correctly sized and that the probability that the bootstrap sequential procedure selects a rank smaller than the true rank converges to zero. Monte Carlo evidence suggests that our bootstrap procedures work very well in practice.     

Folklore Theorems,Implicit Maps,and Indirect Inference

The delta method and continuous mapping theorem are among the most extensively used tools in asymptotic derivations in econometrics. Extensions of these methods are provided for sequences of functions that are commonly encountered in applications and where the usual methods sometimes fail. Important examples of failure arise in the use of simulation‐based estimation methods such as indirect inference. The paper explores the application of these methods to the indirect inference estimator (IIE) in first order autoregressive estimation. The IIE uses a binding function that is sample size dependent. Its limit theory relies on a sequence‐based delta method in the stationary case and a sequence‐based implicit continuous mapping theorem in unit root and local to unity cases. The new limit theory shows that the IIE achieves much more than (partial) bias correction. It changes the limit theory of the maximum likelihood estimator (MLE) when the autoregressive coefficient is in the locality of unity, reducing the bias and the variance of the MLE without affecting the limit theory of the MLE in the stationary case. Thus, in spite of the fact that the IIE is a continuously differentiable function of the MLE, the limit distribution of the IIE is not simply a scale multiple of the MLE, but depends implicitly on the full binding function mapping. The unit root case therefore represents an important example of the failure of the delta method and shows the need for an implicit mapping extension of the continuous mapping theorem.     

On the Failure of the Bootstrap for Matching Estimators

Matching estimators are widely used in empirical economics for the evaluation of programs or treatments. Researchers using matching methods often apply the bootstrap to calculate the standard errors. However, no formal justification has been provided for the use of the bootstrap in this setting. In this article, we show that the standard bootstrap is, in general, not valid for matching estimators, even in the simple case with a single continuous covariate where the estimator is root‐N consistent and asymptotically normally distributed with zero asymptotic bias. Valid inferential methods in this setting are the analytic asymptotic variance estimator of Abadie and Imbens (2006a) as well as certain modifications of the standard bootstrap, like the subsampling methods in Politis and Romano (1994).     

Nonparametric Test for Causality with Long‐range Dependence

This paper introduces a nonparametric Granger‐causality test for covariance stationary linear processes under, possibly, the presence of long‐range dependence. We show that the test is consistent and has power against contiguous alternatives converging to the parametric rate T^−1/2. Since the test is based on estimates of the parameters of the representation of a VAR model as a, possibly, two‐sided infinite distributed lag model, we first show that a modification of Hannan's (1963, 1967) estimator is root‐ T consistent and asymptotically normal for the coefficients of such a representation. When the data are long‐range dependent, this method of estimation becomes more attractive than least squares, since the latter can be neither root‐ T consistent nor asymptotically normal as is the case with short‐range dependent data.     

Bootstrap Testing of Hypotheses on Co‐Integration Relations in Vector Autoregressive Models

It is well known that the finite‐sample properties of tests of hypotheses on the co‐integrating vectors in vector autoregressive models can be quite poor, and that current solutions based on Bartlett‐type corrections or bootstrap based on unrestricted parameter estimators are unsatisfactory, in particular in those cases where also asymptotic χ² tests fail most severely. In this paper, we solve this inference problem by showing the novel result that a bootstrap test where the null hypothesis is imposed on the bootstrap sample is asymptotically valid. That is, not only does it have asymptotically correct size, but, in contrast to what is claimed in existing literature, it is consistent under the alternative. Compared to the theory for bootstrap tests on the co‐integration rank (Cavaliere, Rahbek, and Taylor, 2012), establishing the validity of the bootstrap in the framework of hypotheses on the co‐integrating vectors requires new theoretical developments, including the introduction of multivariate Ornstein–Uhlenbeck processes with random (reduced rank) drift parameters. Finally, as documented by Monte Carlo simulations, the bootstrap test outperforms existing methods.     

Likelihood Inference for a Fractionally Cointegrated Vector Autoregressive Model

We consider model based inference in a fractionally cointegrated (or cofractional) vector autoregressive model, based on the Gaussian likelihood conditional on initial values. We give conditions on the parameters such that the process X_t is fractional of order d and cofractional of order d−b; that is, there exist vectors β for which β^′X_t is fractional of order d−b and no other fractionality order is possible. For b=1, the model nests the I(d−1) vector autoregressive model. We define the statistical model by 0 < bd, but conduct inference when the true values satisfy 0d₀−b₀<1/2 and b₀≠1/2, for which β₀^′X_t is (asymptotically) a stationary process. Our main technical contribution is the proof of consistency of the maximum likelihood estimators. To this end, we prove weak convergence of the conditional likelihood as a continuous stochastic process in the parameters when errors are independent and identically distributed with suitable moment conditions and initial values are bounded. Because the limit is deterministic, this implies uniform convergence in probability of the conditional likelihood function. If the true value b₀>1/2, we prove that the limit distribution of is mixed Gaussian, while for the remaining parameters it is Gaussian. The limit distribution of the likelihood ratio test for cointegration rank is a functional of fractional Brownian motion of type II. If b₀<1/2, all limit distributions are Gaussian or chi‐squared. We derive similar results for the model with d = b, allowing for a constant term.     

Sieve Wald and QLR Inferences on Semi/Nonparametric Conditional Moment Models

This paper considers inference on functionals of semi/nonparametric conditional moment restrictions with possibly nonsmooth generalized residuals, which include all of the (nonlinear) nonparametric instrumental variables (IV) as special cases. These models are often ill‐posed and hence it is difficult to verify whether a (possibly nonlinear) functional is root‐n estimable or not. We provide computationally simple, unified inference procedures that are asymptotically valid regardless of whether a functional is root‐n estimable or not. We establish the following new useful results: (1) the asymptotic normality of a plug‐in penalized sieve minimum distance (PSMD) estimator of a (possibly nonlinear) functional; (2) the consistency of simple sieve variance estimators for the plug‐in PSMD estimator, and hence the asymptotic chi‐square distribution of the sieve Wald statistic; (3) the asymptotic chi‐square distribution of an optimally weighted sieve quasi likelihood ratio (QLR) test under the null hypothesis; (4) the asymptotic tight distribution of a non‐optimally weighted sieve QLR statistic under the null; (5) the consistency of generalized residual bootstrap sieve Wald and QLR tests; (6) local power properties of sieve Wald and QLR tests and of their bootstrap versions; (7) asymptotic properties of sieve Wald and SQLR for functionals of increasing dimension. Simulation studies and an empirical illustration of a nonparametric quantile IV regression are presented.     

The Time Series and Cross‐Section Asymptotics of Dynamic Panel Data Estimators

In this paper we derive the asymptotic properties of within groups (WG), GMM, and LIML estimators for an autoregressive model with random effects when both T and N tend to infinity. GMM and LIML are consistent and asymptotically equivalent to the WG estimator. When T/N→ 0 the fixed T results for GMM and LIML remain valid, but WG, although consistent, has an asymptotic bias in its asymptotic distribution. When T/N tends to a positive constant, the WG, GMM, and LIML estimators exhibit negative asymptotic biases of order 1/T, 1/N, and 1/(2N−T), respectively. In addition, the crude GMM estimator that neglects the autocorrelation in first differenced errors is inconsistent as T/N→c>0, despite being consistent for fixed T. Finally, we discuss the properties of a random effects pseudo MLE with unrestricted initial conditions when both T and N tend to infinity.     

Nonlinear Regressions with Integrated Time Series

An asymptotic theory is developed for nonlinear regression with integrated processes. The models allow for nonlinear effects from unit root time series and therefore deal with the case of parametric nonlinear cointegration. The theory covers integrable and asymptotically homogeneous functions. Sufficient conditions for weak consistency are given and a limit distribution theory is provided. The rates of convergence depend on the properties of the nonlinear regression function, and are shown to be as slow as n^1/4 for integrable functions, and to be generally polynomial in n^1/2 for homogeneous functions. For regressions with integrable functions, the limiting distribution theory is mixed normal with mixing variates that depend on the sojourn time of the limiting Brownian motion of the integrated process.     

One‐Dimensional Inference in Autoregressive Models With the Potential Presence of a Unit Root

This paper examines the problem of testing and confidence set construction for one‐dimensional functions of the coefficients in autoregressive (AR(p)) models with potentially persistent time series. The primary example concerns inference on impulse responses. A new asymptotic framework is suggested and some new theoretical properties of known procedures are demonstrated. I show that the likelihood ratio (LR) and LR^± statistics for a linear hypothesis in an AR(p) can be uniformly approximated by a weighted average of local‐to‐unity and normal distributions. The corresponding weights depend on the weight placed on the largest root in the null hypothesis. The suggested approximation is uniform over the set of all linear hypotheses. The same family of distributions approximates the LR and LR^± statistics for tests about impulse responses, and the approximation is uniform over the horizon of the impulse response. I establish the size properties of tests about impulse responses proposed by Inoue and Kilian (2002) and Gospodinov (2004), and theoretically explain some of the empirical findings of Pesavento and Rossi (2007). An adaptation of the grid bootstrap for impulse response functions is suggested and its properties are examined.     

Instrumental Variable Estimation of Nonlinear Errors‐in‐Variables Models

This paper establishes that instruments enable the identification of nonparametric regression models in the presence of measurement error by providing a closed form solution for the regression function in terms of Fourier transforms of conditional expectations of observable variables. For parametrically specified regression functions, we propose a root n consistent and asymptotically normal estimator that takes the familiar form of a generalized method of moments estimator with a plugged‐in nonparametric kernel density estimate. Both the identification and the estimation methodologies rely on Fourier analysis and on the theory of generalized functions. The finite‐sample properties of the estimator are investigated through Monte Carlo simulations.     

A Bias–Reduced Log–Periodogram Regression Estimator for the Long–Memory Parameter

In this paper, we propose a simple bias–reduced log–periodogram regression estimator, ^d_r, of the long–memory parameter, d, that eliminates the first– and higher–order biases of the Geweke and Porter–Hudak (1983) (GPH) estimator. The bias–reduced estimator is the same as the GPH estimator except that one includes frequencies to the power 2k for k=1,…,r, for some positive integer r, as additional regressors in the pseudo–regression model that yields the GPH estimator. The reduction in bias is obtained using assumptions on the spectrum only in a neighborhood of the zero frequency. Following the work of Robinson (1995b) and Hurvich, Deo, and Brodsky (1998), we establish the asymptotic bias, variance, and mean–squared error (MSE) of ^d_r, determine the asymptotic MSE optimal choice of the number of frequencies, m, to include in the regression, and establish the asymptotic normality of ^d_r. These results show that the bias of ^d_r goes to zero at a faster rate than that of the GPH estimator when the normalized spectrum at zero is sufficiently smooth, but that its variance only is increased by a multiplicative constant. We show that the bias–reduced estimator ^d_r attains the optimal rate of convergence for a class of spectral densities that includes those that are smooth of order s≥1 at zero when r≥(s−2)/2 and m is chosen appropriately. For s>2, the GPH estimator does not attain this rate. The proof uses results of Giraitis, Robinson, and Samarov (1997). We specify a data–dependent plug–in method for selecting the number of frequencies m to minimize asymptotic MSE for a given value of r. Some Monte Carlo simulation results for stationary Gaussian ARFIMA (1, d, 1) and (2, d, 0) models show that the bias–reduced estimators perform well relative to the standard log–periodogram regression estimator.     

Bootstrap Algorithms for Testing and Determining the Cointegration Rank in VAR Models1

In this paper a bootstrap algorithm for a reduced rank vector autoregressive model with a restricted linear trend and independent, identically distributed errors is analyzed. For testing the cointegration rank, the asymptotic distribution under the hypothesis is the same as for the usual likelihood ratio test, so that the bootstrap is consistent. It is furthermore shown that a bootstrap procedure for determining the rank is asymptotically consistent in the sense that the probability of choosing the rank smaller than the true one converges to zero.     

Regression Versus Causation,Revisited

I recently discussed pitfalls in attempted causal inference based on reduced‐form regression models. I used as motivation a real‐world example from a paper by Dr. Sneeringer, which interpreted a reduced‐form regression analysis as implying the startling causal conclusion that “doubling of [livestock] production leads to a 7.4% increase in infant mortality.” This conclusion is based on: (A) fitting a reduced‐form regression model to aggregate (e.g., county‐level) data; and (B) (mis)interpreting a regression coefficient in this model as a causal coefficient, without performing any formal statistical tests for potential causation (such as conditional independence, Granger‐Sims, or path analysis tests). Dr. Sneeringer now adds comments that confirm and augment these deficiencies, while advocating methodological errors that, I believe, risk analysts should avoid if they want to reach logically sound, empirically valid, conclusions about cause and effect. She explains that, in addition to (A) and (B) above, she also performed other steps such as (C) manually selecting specific models and variables and (D) assuming (again, without testing) that hand‐picked surrogate variables are valid (e.g., that log‐transformed income is an adequate surrogate for poverty). In her view, these added steps imply that “critiques of A and B are not applicable” to her analysis and that therefore “a causal argument can be made” for “such a strong, robust correlation” as she believes her regression coefficient indicates. However, multiple wrongs do not create a right. Steps (C) and (D) exacerbate the problem of unjustified causal interpretation of regression coefficients, without rendering irrelevant the fact that (A) and (B) do not provide evidence of causality. This reply focuses on whether any statistical techniques can produce the silk purse of a valid causal inference from the sow's ear of a reduced‐form regression analysis of ecological data. We conclude that Dr. Sneeringer's analysis provides no valid indication that air pollution from livestock operations causes any increase in infant mortality rates. More generally, reduced‐form regression modeling of aggregate population data—no matter how it is augmented by fitting multiple models and hand‐selecting variables and transformations—is not adequate for valid causal inference about health effects caused by specific, but unmeasured, exposures.     

End‐of‐Sample Instability Tests

This paper considers tests for structural instability of short duration, such as at the end of the sample. The key feature of the testing problem is that the number, m, of observations in the period of potential change is relatively small—possibly as small as one. The well‐known F test of Chow (1960) for this problem only applies in a linear regression model with normally distributed iid errors and strictly exogenous regressors, even when the total number of observations, n+m, is large. We generalize the F test to cover regression models with much more general error processes, regressors that are not strictly exogenous, and estimation by instrumental variables as well as least squares. In addition, we extend the F test to nonlinear models estimated by generalized method of moments and maximum likelihood. Asymptotic critical values that are valid as n→∞ with m fixed are provided using a subsampling‐like method. The results apply quite generally to processes that are strictly stationary and ergodic under the null hypothesis of no structural instability.