Jeffreys Prior Analysis of the Simultaneous Equations Model in the Case with n+1 Endogenous Variables

This paper analyzes the behavior of posterior distributions under the Jeffreys prior in a simultaneous equations model. The case under study is that of a general limited information setup with n + 1 endogenous variables. The Jeffreys prior is shown to give rise to a marginal posterior density which has Cauchy-like tails similar to that exhibited by the exact finite sample distribution of the corresponding LIML estimator. A stronger correspondence is established in the special case of a just-identified orthonormal canonical model, where the posterior density under the Jeffreys prior is shown to have the same functional form as the density of the finite sample distribution of the LIML estimator. The work here generalizes that of Chao and Phillips (1997), which gives analogous results for the special case of two endogenous variables.Cauchy tails, exact finite sample distributions, Jeffreys prior, just identification, limited information, posterior density, simultaneous equations model

Data Transformation and Forecasting in Models with Unit Roots and Cointegration

We perform a series of Monte Carlo experiments in order to evaluate the impact of data transformation on forecasting models, and find that vector error-corrections dominate differenced data vector autoregressions when the correct data transformation is used, but not when data are incorrectly tansformed, even if the true model contains cointegrating restrictions. We argue that one reason for this is the failure of standard unit root and cointegration tests under incorrect data transformation.Integratedness, Cointegratedness, Nonlinear transformation

Model Selection in Partially Nonstationary Vector Autoregressive Processes with Reduced Rank Structure

The current practice for determining the number of cointegrating vectors, or the cointegrating rank, in a vector autoregression (VAR) requires the investigator to perform a sequence of cointegration tests. However, as was shown in Johansen (1992), this type of sequential procedure does not lead to consistent estimation of the cointegrating rank. Moreover, these methods take as given the correct specification of the lag order of the VAR, though in actual applications the true lag length is rarely known, Simulation studies by Toda and Phillips (1994) and Chao (1993), on the other hand, have shown that test performance of these procedures can be adversely affected by lag misspecification. This paper addresses these issues by extending the analysis of Phillips and Ploberger (1996) on the Posterior Information Criterion (PIC) to a partially nonstationary vector autoregressive process with reduced rank structure. This extension allows lag length and cointegrating rank to be jointly selected by the criterion, and it leads to the consistent estimation of both. In addition, we also evaluate the finite sample performance of PIC relative to existing model selection procedures, BIC and AIC, through a Monte Carlo study. Results here show PIC to perform at least as well and sometimes better than the other two methods in all the cases examined

Uniform Inference in Panel Autoregression

This paper considers estimation and inference concerning the autoregressive coefficient (ρ) in a panel autoregression for which the degree of persistence in the time dimension is unknown. The main objective is to construct confidence intervals for ρ that are asymptotically valid, having asymptotic coverage probability at least that of the nominal level uniformly over the parameter space. It is shown that a properly normalized statistic based on the Anderson-Hsiao IV procedure, which we call the M statistic, is uniformly convergent and can be inverted to obtain asymptotically valid interval estimates. In the unit root case confidence intervals based on this procedure are unsatisfactorily wide and uninformative. To sharpen the intervals a new procedure is developed using information from unit root pretests to select alternative confidence intervals. Two sequential tests are used to assess how close ρ is to unity and to correspondingly tailor intervals near the unit root region. When ρ is close to unity, the width of these intervals shrinks to zero at a faster rate than that of the confidence interval based on the M statistic. Only when both tests reject the unit root hypothesis does the construction revert to the M statistic intervals, whose width has the optimal N^{-1/2}T^{-1/2} rate of shrinkage when the underlying process is stable. The asymptotic properties of this pretest-based procedure show that it produces confidence intervals with at least the prescribed coverage probability in large samples. Simulations confirm that the proposed interval estimation methods perform well in finite samples and are easy to implement in practice. A supplement to the paper provides an extensive set of new results on the asymptotic behavior of panel IV estimators in weak instrument settings

Consistent Estimation with a Large Number of Weak Instruments

This paper conducts a general analysis of the conditions under which consistent estimation can be achieved in instrumental variables regression when the available instruments are weak in the local-to-zero sense. More precisely, the approach adopted in this paper combines key features of the local-to-zero framework of Staiger and Stock (1997) and the many-instrument framework of Morimune (1983) and Bekker (1994) and generalizes both of these frameworks in the following ways. First, we consider a general local-to-zero framework which allows for an arbitrary degree of instrument weakness by modeling the first-stage coefficients as shrinking toward zero at an unspecified rate, say b n -1 . Our local-to-zero setup, in fact, reduces to that of Staiger and Stock (1997) in the case where b n = / n . In addition, we examine a broad class of single-equation estimators which extends the well-known k -class to include, amongst others, the Jackknife Instrumental Variables Estimator (JIVE) of Angrist, Imbens, and Krueger (1999). Analysis of estimators within this extended class based on a pathwise asymptotic scheme, where the number of instruments K n is allowed to grow as a function of the sample size, reveals that consistent estimation depends importantly on the relative magnitudes of r n , the growth rate of the concentration parameter, and K n . In particular, it is shown that members of the extended class which satisfy certain general condtions, such as LIML and JIVE, are consistent provided that sqrt{ K n / r n } → 0, as n → ∞. On the other hand, the two-stage least squares (2SLS) estimator is shown not to satisfy the needed conditions and is found to be consistent only if K n / r n → 0, as n → ∞. A main point of our paper is that the use of many instruments may be beneficial from a point estimation standpoint in empirical applications where the available instruments are weak but abundant, as it provides an extra source, by which the concentration parameter can grow, thus, allowing consistent estimation to be achievable, in certain cases, even in the presence of weak instruments. Our results, thus, add to the findings of Staiger and Stock (1997) who study a local-to-zero framework where K n is held fixed and the concentration parameter does not diverge as sample size grows; in consequence, no single-equation estimator is found to be consistent under their setup

Bayesian Posterior Distributions in Limited Information Analysis of the Simultaneous Equations Model Using the Jeffreys’ Prior

This paper studies the use of the Jeffreys’ prior in Bayesian analysis of the simultaneous equations model (SEM). Exact representations are obtained for the posterior density of the structural coefficient beta in canonical SEM’s with two endogenous variables. For the general case with m endogenous variables and an unknown covariance matrix, the Laplace approximation is used to derive an analytic formula for the same posterior density. Both the exact and the approximate formulas we derive are found to exhibit Cauchy-like tails analogous to comparable results in the classical literature on LIML estimation. Moreover, in the special case of a two-equation, just-identified SEM in canonical form, the posterior density of beta is shown to have the same infinite series representation as the density of the finite sample distribution of the corresponding LIML estimator. This paper also examines the occurrence of a nonintegrable asymptotic cusp in the posterior distribution of the reduced form parameter Pi, first documented in Kleibergen and van Dijk (1994). This phenomenon is explained in terms of the jacobian of the mapping from the structural model to the reduced form. This interpretation assists in understanding the success of the Jeffreys’ prior in resolving this proble

Alternative Approximations of the Bias and MSE of the IV Estimator under Weak Identification with an Application to Bias Correction

We provide analytical formulae for the asymptotic bias (ABIAS) and mean squared error (AMSE) of the IV estimator, and obtain approximations thereof based on an asymptotic scheme which essentially requires the expectation of the first stage F -statistic to converge to a finite (possibly small) positive limit as the number of instruments approaches infinity. The approximations so obtained are shown, via regression analysis, to yield good approximations for ABIAS and AMSE functions, and the AMSE approximation is shown to perform well relative to the approximation of Donald and Newey (2001). Additionally, the manner in which our framework generalizes that of Richardson and Wu (1971) is discussed. One consequence of the asymptotic framework adopted here is that consistent estimators for the ABIAS and AMSE can be obtained. As a result, we are able to construct a number of bias corrected OLS and IV estimators, which we show to be consistent under a sequential asymptotic scheme. These bias-corrected estimators are also robust, in the sense that they remain consistent in a conventional asymptotic setup, where the model is fully identified. A small Monte Carlo experiment documents the relative performance of our bias adjusted estimators versus standard IV, OLS, LIML estimators, and it is shown that our estimators have lower bias than LIML for various levels of endogeneity and instrument relevance

Computational Aspects of Optional P\'{o}lya Tree

Optional P\'{o}lya Tree (OPT) is a flexible non-parametric Bayesian model for density estimation. Despite its merits, the computation for OPT inference is challenging. In this paper we present time complexity analysis for OPT inference and propose two algorithmic improvements. The first improvement, named Limited-Lookahead Optional P\'{o}lya Tree (LL-OPT), aims at greatly accelerate the computation for OPT inference. The second improvement modifies the output of OPT or LL-OPT and produces a continuous piecewise linear density estimate. We demonstrate the performance of these two improvements using simulations

Model Selection in Partially Nonstationary Vector Autoregressive Processes with Reduced Rank Structure

The current practice for determining the number of cointegrating vectors, or the cointegrating rank, in a vector autoregression (VAR) requires the investigator to perform a sequence of cointegration tests. However, as was shown in Johansen (1992), this type of sequential procedure does not lead to consistent estimation of the cointegrating rank. Moreover, these methods take as given the correct specification of the lag order of the VAR, though in actual applications the true lag length is rarely known, Simulation studies by Toda and Phillips (1994) and Chao (1993), on the other hand, have shown that test performance of these procedures can be adversely affected by lag misspecification. This paper addresses these issues by extending the analysis of Phillips and Ploberger (1996) on the Posterior Information Criterion (PIC) to a partially nonstationary vector autoregressive process with reduced rank structure. This extension allows lag length and cointegrating rank to be jointly selected by the criterion, and it leads to the consistent estimation of both. In addition, we also evaluate the finite sample performance of PIC relative to existing model selection procedures, BIC and AIC, through a Monte Carlo study. Results here show PIC to perform at least as well and sometimes better than the other two methods in all the cases examined.Cointegrating rank, information criterion, order selection, PIC, reduced rank regression, vector autoregression