Specification Sensitivity in Right‐Tailed Unit Root Testing for Explosive Behaviour

This article aims to provide some empirical guidelines for the practical implementation of right‐tailed unit root tests, focusing on the recursive right‐tailed ADF test of Phillips et al. (2011b). We analyze and compare the limit theory of the recursive test under different hypotheses and model specifications. The size and power properties of the test under various scenarios are examined and some recommendations for empirical practice are given. Some new results on the consistent estimation of localizing drift exponents are obtained, which are useful in assessing model specification. Empirical applications to stock markets illustrate these specification issues and reveal their practical importance in testing.     

Regression‐based Tests for a Change in Persistence*

We show that the minimal forward (reverse) recursive unit tests of Banerjee, Lumsdaine and Stock [Journal of Business and Economics Statistics (1992) Vol. 10, pp. 271–288] are consistent against the alternative of a change in persistence from I(0) to I(1) [I(1) to I(0)]. However, these statistics are also shown to diverge for series which are I(0) throughout. Consequently, a rejection by these tests does not necessarily imply a change in persistence. We propose a further test, based on the ratio of these statistics, which is consistent against changes either from I(0) to I(1), or vice versa, yet does not over‐reject against constant I(0) series. Consistent breakpoint estimators are proposed.     

Bayesian VARs: Specification Choices and Forecast Accuracy

In this paper we discuss how the point and density forecasting performance of Bayesian vector autoregressions (BVARs) is affected by a number of specification choices. We adopt as a benchmark a common specification in the literature, a BVAR with variables entering in levels and a prior modeled along the lines of Sims and Zha (International Economic Review 1998; 39 : 949–968). We then consider optimal choice of the tightness, of the lag length and of both; evaluate the relative merits of modeling in levels or growth rates; compare alternative approaches to h‐step‐ahead forecasting (direct, iterated and pseudo‐iterated); discuss the treatment of the error variance and of cross‐variable shrinkage; and assess rolling versus recursive estimation. Finally, we analyze the robustness of the results to the VAR size and composition (using also data for France, Canada and the UK, while the main analysis is for the USA). We obtain a large set of empirical results, but the overall message is that we find very small losses (and sometimes even gains) from the adoption of specification choices that make BVAR modeling quick and easy, in particular for point forecasting. This finding could therefore further enhance the diffusion of the BVAR as an econometric tool for a vast range of applications. Copyright © 2013 John Wiley & Sons, Ltd.     

The stability of macroeconomic systems with Bayesian learners

We study abstract macroeconomic systems in which expectations play an important role. Consistent with the recent literature on recursive learning and expectations, we replace the agents in the economy with econometricians. Unlike the recursive learning literature, however, the econometricians in the analysis here are Bayesian learners. We are interested in the extent to which expectational stability remains the key concept in the Bayesian environment. We isolate conditions under which versions of expectational stability conditions govern the stability of these systems just as in the standard case of recursive learning. We conclude that Bayesian learning schemes, while they are more sophisticated, do not alter the essential expectational stability findings in the literature.     

Marsaglia’s lattice test and non-linear congruential pseudo random number generators

Sequences of integers defined by a non-linear recursive congruential pseudo random number generator with prime modulusp and maximal period length are divided into vectors ofd consecutive numbers. The lattice spanned by these vectors is studied. i.e. the minimal lattice which contains all these vectors. It is shown that this lattice coincides with the full integer lattice, at least ford=2 andd=3, i.e. non-linear generators pass Marsaglia’s lattice test at least ford ⩽ 3. For a special class of non-linear generators introduced in Eichenauer and Lehn (1986) it is proved that these generators pass the test even for dimensionsd ⩽ (p − 1)/2. Supported by Deutsche Forschungsgemeinschaft.     

Distribution theory for the studentized mean for long,short, and negative memory time series

We consider the problem of estimating the variance of the partial sums of a stationary time series that has either long memory, short memory, negative/intermediate memory, or is the first-difference of such a process. The rate of growth of this variance depends crucially on the type of memory, and we present results on the behavior of tapered sums of sample autocovariances in this context when the bandwidth vanishes asymptotically. We also present asymptotic results for the case that the bandwidth is a fixed proportion of sample size, extending known results to the case of flat-top tapers. We adopt the fixed proportion bandwidth perspective in our empirical section, presenting two methods for estimating the limiting critical values—both the subsampling method and a plug-in approach. Simulation studies compare the size and power of both approaches as applied to hypothesis testing for the mean. Both methods perform well–although the subsampling method appears to be better sized–and provide a viable framework for conducting inference for the mean. In summary, we supply a unified asymptotic theory that covers all different types of memory under a single umbrella.     

An analysis of stock market transactions data

Many studies have argued against the strict form of the efficient markets hypothesis (EMH) by concluding that a lagged relationship exists between volume and the absolute value of a price change. These studies have denied a priori the possibility of a contemporaneous relationship. If a simultaneous relationship exists then least squares with only lagged variables suffers from omitted variable bias, and least squares with contemporaneous variables may suffer from simultaneous equations bias. Investigating these possibilities, this study demonstrates that simultaneity exists and that previous findings of a lagged relationship between the variables are therefore due to specification error. System estimation techniques suggest that the price-volume relationship is recursive, with the absolute value of a price change causing volume contemporaneously, but not conversely.     

Solving the multi-country real business cycle model using a Smolyak-collocation method

We describe a sparse-grid collocation method to compute recursive solutions of dynamic economies with a sizable number of state variables. We show how powerful this method can be in applications by computing the non-linear recursive solution of an international real business cycle model with a substantial number of countries, complete insurance markets and frictions that impede frictionless international capital flows. In this economy, the aggregate state vector includes the distribution of world capital across different countries as well as the exogenous country-specific technology shocks. We use the algorithm to efficiently solve models with up to 10 countries (i.e., up to 20 continuous-valued state variables).     

Smooth minimum distance estimation and testing with conditional estimating equations: Uniform in bandwidth theory

To study the influence of a bandwidth parameter in inference with conditional moments, we propose a new class of estimators and establish an asymptotic representation of our estimator as a process indexed by a bandwidth, which can vary within a wide range including bandwidths independent of the sample size. We study its behavior under misspecification. We also propose an efficient version of our estimator. We develop a procedure based on a distance metric statistic for testing restrictions on parameters as well as a bootstrap technique to account for the bandwidth’s influence. Our new methods are simple to implement, apply to non-smooth problems, and perform well in our simulations.     

Prediction from ARFIMA models: Comparisons between MLE and semiparametric estimation procedures

This paper considers the effects on multi-step prediction of using semiparametric local Whittle estimators rather than MLE for long memory ARFIMA models. We consider various representations of the minimum MSE predictor with known parameters. We then conduct a detailed simulation study for when the true parameters are replaced with estimates. The predictor based on MLE is found to be superior, in the MSE sense, to the predictor based on the two-step local Whittle estimation. The “optimal” bandwidth local Whittle estimator produces worse predictions than the local Whittle using an agnostic bandwidth of the square root of the sample size.     

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.     

Theoretical approaches to lowest unique bid auctions

In this paper, we analyze a game called the lowest unique bid auction (LUBA). It is known that there is no closed form solution for the symmetric mixed strategy equilibria in LUBA. We propose two alternative approaches. We use weak dominance to identify upper bounds for players’ bids. Under symmetry, we provide a method to compute a mixed strategy equilibrium by utilizing the recursive structure of the winning chances of the bids under one assumption.     

Prewhitening Bias in HAC Estimation*

Heteroskedasticity and autocorrelation consistent (HAC) estimation commonly involves the use of prewhitening filters based on simple autoregressive models. In such applications, small sample bias in the estimation of autoregressive coefficients is transmitted to the recolouring filter, leading to HAC variance estimates that can be badly biased. The present paper provides an analysis of these issues using asymptotic expansions and simulations. The approach we recommend involves the use of recursive demeaning procedures that mitigate the effects of small‐sample autoregressive bias. Moreover, a commonly used restriction rule on the prewhitening estimates (that first‐order autoregressive coefficient estimates, or largest eigenvalues, >0.97 be replaced by 0.97) adversely interferes with the power of unit‐root and [ Kwiatkowski, Phillips, Schmidt and Shin (1992) Journal of Econometrics, Vol. 54, pp. 159–178] (KPSS) tests. We provide a new boundary condition rule that improves the size and power properties of these tests. Some illustrations of the effects of these adjustments on the size and power of KPSS testing are given. Using prewhitened HAC estimates and the new boundary condition rule, the KPSS test is consistent, in contrast to KPSS testing that uses conventional prewhitened HAC estimates [ Lee, J. S. (1996) Economics Letters, Vol. 51, pp. 131–137].     

Edgeworth approximations for semiparametric instrumental variable estimators and test statistics

We establish the validity of higher order asymptotic expansions to the distribution of a version of the nonlinear semiparametric instrumental variable estimator considered in Newey (Econometrica 58 (1990) 809) as well as to the distribution of a Wald statistic derived from it. We employ local polynomial smoothing with variable bandwidth, which includes local linear, kernel, and (a version of) nearest neighbor estimates as special cases. Our expansions are valid to order n^−2ε for some 0<ε<1/2, where ε depends on the smoothness and dimensionality of the data distribution and on the order of the polynomial chosen by the practitioner. We use the expansions to define optimal bandwidth selection methods for both estimation and testing problems and apply our methods to simulated data.     

General‐to‐Specific Model Selection Procedures for Structural Vector Autoregressions*

Structural vector autoregressive (SVAR) models have emerged as a dominant research strategy in empirical macroeconomics, but suffer from the large number of parameters employed and the resulting estimation uncertainty associated with their impulse responses. In this paper, we propose general‐to‐specific (Gets) model selection procedures to overcome these limitations. It is shown that single‐equation procedures are generally efficient for the reduction of recursive SVAR models. The small‐sample properties of the proposed reduction procedure (as implemented using PcGets) are evaluated in a realistic Monte Carlo experiment. The impulse responses generated by the selected SVAR are found to be more precise and accurate than those of the unrestricted VAR. The proposed reduction strategy is then applied to the US monetary system considered by Christiano, Eichenbaum and Evans (Review of Economics and Statistics, Vol. 78, pp. 16–34, 1996) . The results are consistent with the Monte Carlo and question the validity of the impulse responses generated by the full system.     

Two estimators of the long-run variance: Beyond short memory

This paper deals with the estimation of the long-run variance of a stationary sequence. We extend the usual Bartlett-kernel heteroskedasticity and autocorrelation consistent (HAC) estimator to deal with long memory and antipersistence. We then derive asymptotic expansions for this estimator and the memory and autocorrelation consistent (MAC) estimator introduced by Robinson [Robinson, P. M., 2005. Robust covariance matrix estimation: HAC estimates with long memory/antipersistence correction. Econometric Theory 21, 171–180]. We offer a theoretical explanation for the sensitivity of HAC to the bandwidth choice, a feature which has been observed in the special case of short memory. Using these analytical results, we determine the MSE-optimal bandwidth rates for each estimator. We analyze by simulations the finite-sample performance of HAC and MAC estimators, and the coverage probabilities for the studentized sample mean, giving practical recommendations for the choice of bandwidths.     

A remark on the recursive estimation of ARMA order

Summary It is well known how, for an ARMA process of order (p ₀,q ₀), max (p ₀,q ₀) may be recursively estimatedHannan/Rissanen. Assuming max (p ₀,q ₀) to be known and, in addition,p ₀q ₀, a simple procedure for the recursive estimation of (p ₀,q ₀) is presented.     

Recursive Nash bargaining over a productive asset

We propose an equilibrium concept (the recursive Nash bargaining solution) that describes the outcome of repeated negotiations between two rational agents under the assumptions that the state of the economic system under consideration changes according to the actions of the players and that neither party can make binding commitments to future behavior. This equilibrium is dynamically consistent but typically not Pareto-efficient. As an application, we compute the recursive Nash bargaining solution in a model of two heterogeneous agents bargaining over the use of a productive asset with constant gross rate of return and study how the time-preference rates and the elasticities of substitution affect the solution.     

Asset prices in an exchange economy when agents have heterogeneous homothetic recursive preferences and no risk free bond is available

We study a pure exchange economy under incomplete markets where households have heterogeneous homothetic recursive preferences and lending and borrowing are precluded. We fully characterize the properties of the efficient allocations and the equilibrium asset price. The ownership distribution dynamics reveal the emergence of a dominant agent, who after some finite time, remains the only investor that increases asset holdings until asymptotically owning the entire wealth. Investors can be ranked according to a unique parameter that aggregates agents’ preference characteristics and we show how time discount rate, attitude towards risk and intertemporal substitution contribute to capital accumulation.     

Pricing of the time-change risks

We develop an equilibrium endowment economy with Epstein-Zin recursive utility and a Lévy time-change subordinator, which represents a clock that connects business and calendar time. Our setup provides a tractable equilibrium framework for pricing non-Gaussian jump-like risks induced by the time-change, with closed-form solutions for asset prices. Persistence of the time-change shocks leads to predictability of consumption and dividends and time-variation in asset prices and risk premia in calendar time. In numerical calibrations, we show that the risk compensation for Lévy risks accounts for about one-third of the overall equity premium.