Generalizing weak instrument robust IV statistics towards multiple parameters,unrestricted covariance matrices and identification statistics

We generalize the weak instrument robust score or Lagrange multiplier and likelihood ratio instrumental variables (IV) statistics towards multiple parameters and a general covariance matrix so they can be used in the generalized method of moments (GMM). The GMM extension of Moreira's [2003. A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048] conditional likelihood ratio statistic towards GMM preserves its expression except that it becomes conditional on a statistic that tests the rank of a matrix. We analyze the spurious power decline of Kleibergen's [2002. Pivotal statistics for testing structural parameters in instrumental variables regression. Econometrica 70, 1781–1803, 2005. Testing parameters in GMM without assuming that they are identified. Econometrica 73, 1103–1124] score statistic and show that an independent misspecification pre-test overcomes it. We construct identification statistics that reflect if the confidence sets of the parameters are bounded. A power study and the possible shapes of confidence sets illustrate the analysis.     

Performance of conditional Wald tests in IV regression with weak instruments

We compare the powers of five tests of the coefficient on a single endogenous regressor in instrumental variables regression. Following Moreira [2003, A conditional likelihood ratio test for structural models. Econometrica 71, 1027–1048], all tests are implemented using critical values that depend on a statistic which is sufficient under the null hypothesis for the (unknown) concentration parameter, so these conditional tests are asymptotically valid under weak instrument asymptotics. Four of the tests are based on k-class Wald statistics (two-stage least squares, LIML, Fuller's [Some properties of a modification of the limited information estimator. Econometrica 45, 939–953], and bias-adjusted TSLS); the fifth is Moreira's (2003) conditional likelihood ratio (CLR) test. The heretofore unstudied conditional Wald (CW) tests are found to perform poorly, compared to the CLR test: in many cases, the CW tests have almost no power against a wide range of alternatives. Our analysis is facilitated by a new algorithm, presented here, for the computation of the asymptotic conditional p-value of the CLR test.     

GEL statistics under weak identification

The central concern of this paper is the provision in a time series moment condition framework of practical recommendations of confidence regions for parameters whose coverage probabilities are robust to the strength or weakness of identification. To this end we develop Pearson-type test statistics based on GEL implied probabilities formed from general kernel smoothed versions of the moment indicators. We also modify the statistics suggested in Guggenberger and Smith (2008) for a general kernel smoothing function. Importantly for our conclusions, we provide GEL time series counterparts to GMM and GEL conditional likelihood ratio statistics given in Kleibergen (2005) and Smith (2007). Our analysis not only demonstrates that these statistics are asymptotically (conditionally) pivotal under both classical asymptotic theory and weak instrument asymptotics of Stock and Wright (2000) but also provides asymptotic power results in the weakly identified time series context. Consequently, the empirical null rejection probabilities of the associated tests and, thereby, the coverage probabilities of the corresponding confidence regions, should not be affected greatly by the strength or otherwise of identification. A comprehensive Monte Carlo study indicates that a number of the tests proposed here represent very competitive choices in comparison with those suggested elsewhere in the literature.     

Generalized reduced rank tests using the singular value decomposition

We propose a novel statistic to test the rank of a matrix. The rank statistic overcomes deficiencies of existing rank statistics, like: a Kronecker covariance matrix for the canonical correlation rank statistic of Anderson [Annals of Mathematical Statistics (1951), 22, 327–351] sensitivity to the ordering of the variables for the LDU rank statistic of Cragg and Donald [Journal of the American Statistical Association (1996), 91, 1301–1309] and Gill and Lewbel [Journal of the American Statistical Association (1992), 87, 766–776] a limiting distribution that is not a standard chi-squared distribution for the rank statistic of Robin and Smith [Econometric Theory (2000), 16, 151–175] usage of numerical optimization for the objective function statistic of Cragg and Donald [Journal of Econometrics (1997), 76, 223–250] and ignoring the non-negativity restriction on the singular values in Ratsimalahelo [2002, Rank test based on matrix perturbation theory. Unpublished working paper, U.F.R. Science Economique, University de Franche-Comté]. In the non-stationary cointegration case, the limiting distribution of the new rank statistic is identical to that of the Johansen trace statistic.     

Efficient information theoretic inference for conditional moment restrictions

The generalised method of moments estimator may be substantially biased in finite samples, especially so when there are large numbers of unconditional moment conditions. This paper develops a class of first-order equivalent semi-parametric efficient estimators and tests for conditional moment restrictions models based on a local or kernel-weighted version of the Cressie–Read power divergence family of discrepancies. This approach is similar in spirit to the empirical likelihood methods of Kitamura et al. [2004. Empirical likelihood-based inference in conditional moment restrictions models. Econometrica 72, 1667–1714] and Tripathi and Kitamura [2003. Testing conditional moment restrictions. Annals of Statistics 31, 2059–2095]. These efficient local methods avoid the necessity of explicit estimation of the conditional Jacobian and variance matrices of the conditional moment restrictions and provide empirical conditional probabilities for the observations.     

Bootstrapping cointegrating regressions

In this paper, we consider bootstrapping cointegrating regressions. It is shown that the method of bootstrap, if properly implemented, generally yields consistent estimators and test statistics for cointegrating regressions. For the cointegrating regression models driven by general linear processes, we employ the sieve bootstrap based on the approximated finite-order vector autoregressions for the regression errors and the first differences of the regressors. In particular, we establish the bootstrap consistency for OLS method. The bootstrap method can thus be used to correct for the finite sample bias of the OLS estimator and to approximate the asymptotic critical values of the OLS-based test statistics in general cointegrating regressions. The bootstrap OLS procedure, however, is not efficient. For the efficient estimation and hypothesis testing, we consider the procedure proposed by Saikkonen [1991. Asymptotically efficient estimation of cointegration regressions. Econometric Theory 7, 1–21] and Stock and Watson [1993. A simple estimator of cointegrating vectors in higher order integrating systems. Econometrica 61, 783–820] relying on the regression augmented with the leads and lags of differenced regressors. The bootstrap versions of their procedures are shown to be consistent, and can be used to do asymptotically valid inferences. A Monte Carlo study is conducted to investigate the finite sample performances of the proposed bootstrap methods.     

Short and long run causality measures: Theory and inference

The concept of causality introduced by Wiener [Wiener, N., 1956. The theory of prediction, In: E.F. Beckenback, ed., The Theory of Prediction, McGraw-Hill, New York (Chapter 8)] and Granger [Granger, C. W.J., 1969. Investigating causal relations by econometric models and cross-spectral methods, Econometrica 37, 424–459] is defined in terms of predictability one period ahead. This concept can be generalized by considering causality at any given horizon h$h$ as well as tests for the corresponding non-causality [Dufour, J.-M., Renault, E., 1998. Short-run and long-run causality in time series: Theory. Econometrica 66, 1099–1125; Dufour, J.-M., Pelletier, D., Renault, É., 2006. Short run and long run causality in time series: Inference, Journal of Econometrics 132 (2), 337–362]. Instead of tests for non-causality at a given horizon, we study the problem of measuring causality between two vector processes. Existing causality measures have been defined only for the horizon 1, and they fail to capture indirect causality. We propose generalizations to any horizon h$h$ of the measures introduced by Geweke [Geweke, J., 1982. Measurement of linear dependence and feedback between multiple time series. Journal of the American Statistical Association 77, 304–313]. Nonparametric and parametric measures of unidirectional causality and instantaneous effects are considered. On noting that the causality measures typically involve complex functions of model parameters in VAR and VARMA models, we propose a simple simulation-based method to evaluate these measures for any VARMA model. We also describe asymptotically valid nonparametric confidence intervals, based on a bootstrap technique. Finally, the proposed measures are applied to study causality relations at different horizons between macroeconomic, monetary and financial variables in the US.     

Local polynomial estimation of nonparametric simultaneous equations models

We define a new procedure for consistent estimation of nonparametric simultaneous equations models under the conditional mean independence restriction of Newey et al. [1999. Nonparametric estimation of triangular simultaneous equation models. Econometrica 67, 565–603]. It is based upon local polynomial regression and marginal integration techniques. We establish the asymptotic distribution of our estimator under weak data dependence conditions. Simulation evidence suggests that our estimator may significantly outperform the estimators of Pinkse [2000. Nonparametric two-step regression estimation when regressors and errors are dependent. Canadian Journal of Statistics 28, 289–300] and Newey and Powell [2003. Instrumental variable estimation of nonparametric models. Econometrica 71, 1565–1578].     

No unbounded arbitrage,weak no market arbitrage and no arbitrage price system conditions; Equivalent conditions

Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162] prove that the no unbounded arbitrage (NUBA), a special case of a condition in Page [Page, F.H., 1987. On equilibrium in Hart’s securities exchange model. Journal of Economic Theory 41, 392–404], is equivalent to the existence of a no arbitrage price system (NAPS) when no agent has non-null useless vectors. Allouch et al. [Allouch, N., Le Van, C., Page F.H., 2002. The geometry of arbitrage and the existence of competitive equilibrium. Journal of Mathematical Economics 38, 373–391] extend the NAPS introduced by Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418] and show that this condition is equivalent to the weak no market arbitrage (WNMA) of Hart [Hart, O., 1974. On the existence of an equilibrium in a securities model. Journal of Economic Theory 9, 293–311]. They mention that this result implies the one given by Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162]. In this note, we show that all these conditions are equivalent.     

Bayesian analysis of random coefficient logit models using aggregate data

We present a Bayesian approach for analyzing aggregate level sales data in a market with differentiated products. We consider the aggregate share model proposed by Berry et al. [Berry, Steven, Levinsohn, James, Pakes, Ariel, 1995. Automobile prices in market equilibrium. Econometrica. 63 (4), 841–890], which introduces a common demand shock into an aggregated random coefficient logit model. A full likelihood approach is possible with a specification of the distribution of the common demand shock. We introduce a reparameterization of the covariance matrix to improve the performance of the random walk Metropolis for covariance parameters. We illustrate the usefulness of our approach with both actual and simulated data. Sampling experiments show that our approach performs well relative to the GMM estimator even in the presence of a mis-specified shock distribution. We view our approach as useful for those who are willing to trade off one additional distributional assumption for increased efficiency in estimation.     

Approximately normal tests for equal predictive accuracy in nested models

Forecast evaluation often compares a parsimonious null model to a larger model that nests the null model. Under the null that the parsimonious model generates the data, the larger model introduces noise into its forecasts by estimating parameters whose population values are zero. We observe that the mean squared prediction error (MSPE) from the parsimonious model is therefore expected to be smaller than that of the larger model. We describe how to adjust MSPEs to account for this noise. We propose applying standard methods [West, K.D., 1996. Asymptotic inference about predictive ability. Econometrica 64, 1067–1084] to test whether the adjusted mean squared error difference is zero. We refer to nonstandard limiting distributions derived in Clark and McCracken [2001. Tests of equal forecast accuracy and encompassing for nested models. Journal of Econometrics 105, 85–110; 2005a. Evaluating direct multistep forecasts. Econometric Reviews 24, 369–404] to argue that use of standard normal critical values will yield actual sizes close to, but a little less than, nominal size. Simulation evidence supports our recommended procedure.     

Efficient tests for the presence of a pair of complex conjugate unit roots in real time series

We introduce a framework which allows us to draw a clear parallel between the test for the presence of seasonal unit roots and that for unit root at frequency 0 (or π$\pi$). It relies on the properties of the complex conjugate integrated of order one processes which are implicitly at work in the real time series. In the same framework as that of Phillips and Perron (Biometrica 75 (1988) 335), we derive tests for the presence of a pair of conjugate complex unit roots. The asymptotic distribution we obtain are formally close to those derived by these authors but expressed with complex Wiener processes. We then introduce sequences of near-integrated processes which allow us to study the local-to-unity asymptotic of the above test statistics. We state a result on the weak convergence of the partial sum of complex near-random walks which leads to complex Orstein–Uhlenbeck processes. Drawing on Elliott et al. (Econometrica 64 (1996) 813) we then study the design of point-optimal invariant test procedures and compute their envelope employing local-to-unity asymptotic approximations. This leads us to introduce new feasible and near efficient seasonal unit root tests. Their finite sample properties are investigated and compared with the different test procedures already available (J. Econometrics 44 (1991) 215; 62 (1994) 415; 85 (1998) 269) and those introduced in the first part of the paper.     

Estimation of Markov regime-switching regression models with endogenous switching

Following Hamilton [1989. A new approach to the economic analysis of nonstationary time series and the business cycle. Econometrica 57, 357–384], estimation of Markov regime-switching regressions typically relies on the assumption that the latent state variable controlling regime change is exogenous. We relax this assumption and develop a parsimonious model of endogenous Markov regime-switching. Inference via maximum likelihood estimation is possible with relatively minor modifications to existing recursive filters. The model nests the exogenous switching model, yielding straightforward tests for endogeneity. In Monte Carlo experiments, maximum likelihood estimates of the endogenous switching model parameters were quite accurate, even in the presence of certain model misspecifications. As an application, we extend the volatility feedback model of equity returns given in Turner et al. [1989. A Markov model of heteroskedasticity, risk, and learning in the stock market. Journal of Financial Economics 25, 3–22] to allow for endogenous switching.     

Pseudo-likelihood estimation and bootstrap inference for structural discrete Markov decision models

This paper analyzes the higher-order properties of the estimators based on the nested pseudo-likelihood (NPL) algorithm and the practical implementation of such estimators for parametric discrete Markov decision models. We derive the rate at which the NPL algorithm converges to the MLE and provide a theoretical explanation for the simulation results in Aguirregabiria and Mira [Aguirregabiria, V., Mira, P., 2002. Swapping the nested fixed point algorithm: A class of estimators for discrete Markov decision models. Econometrica 70, 1519–1543], in which iterating the NPL algorithm improves the accuracy of the estimator. We then propose a new NPL algorithm that can achieve quadratic convergence without fully solving the fixed point problem in every iteration and apply our estimation procedure to a finite mixture model. We also develop one-step NPL bootstrap procedures for discrete Markov decision models. The Monte Carlo simulation evidence based on a machine replacement model of Rust [Rust, J., 1987. Optimal replacement of GMC bus engines: An empirical model of Harold Zurcher. Econometrica 55, 999–1033] shows that the proposed one-step bootstrap test statistics and confidence intervals improve upon the first order asymptotics even with a relatively small number of iterations.     

Finite sample multivariate structural change tests with application to energy demand models

This paper considers finite sample motivated structural change tests in the multivariate linear regression model with application to energy demand models, in which case commonly used structural change tests remain asymptotic. As in Dufour and Kiviet [1996. Exact tests for structural change in first-order dynamic models. Journal of Econometrics 70, 39–68], we account for intervening nuisance parameters through a two-stage maximized Monte Carlo test procedure. Our contributions can be classified into five categories: (i) we extend tests for which a finite-sample theory has been supplied for Gaussian distributions to the non-Gaussian context; (ii) we show that Bai et al. [1998. Testing and dating common breaks in multi-variate time series. The Review of Economic Studies 65 (3), 395–432] test severely over-rejects and propose exact variants of this test; (iii) we consider predictive break test approaches which generalize tests in Dufour [1980. Dummy variables and predictive tests for structural change. Economics Letters 6, 241–247] and Dufour and Kiviet [1996. Exact tests for structural change in first-order dynamic models. Journal of Econometrics 70, 39–68]; (iv) we propose exact (non-Bonferonni based) extensions of the multivariate outliers test from Wilks [1963. Multivariate statistical outliers. Sankhya Series A 25, 407–426] to models with covariates; (v) we apply these tests to the energy demand system analyzed by Arsenault et al. [1995. A total energy demand model of Québec: forecasting properties. Energy Economics 17 (2), 163–171]. For two out of the six industrial sectors analyzed over the 1962–2000 period, break and further goodness-of-fit and diagnostic tests allow to identify (and correct) specification problems arising from historical regulatory changes or (possibly random) industry-specific effects. The procedures we propose have potential useful applications in statistics, econometrics and finance (e.g. event studies).     

Hodges–Lehmann optimality for testing moment conditions

This paper studies the Hodges and Lehmann (1956) optimality of tests in a general setup. The tests are compared by the exponential rates of growth to one of the power functions evaluated at a fixed alternative while keeping the asymptotic sizes bounded by some constant. We present two sets of sufficient conditions for a test to be Hodges–Lehmann optimal. These new conditions extend the scope of the Hodges–Lehmann optimality analysis to setups that cannot be covered by other conditions in the literature. The general result is illustrated by our applications of interest: testing for moment conditions and overidentifying restrictions. In particular, we show that (i) the empirical likelihood test does not necessarily satisfy existing conditions for optimality but does satisfy our new conditions; and (ii) the generalized method of moments (GMM) test and the generalized empirical likelihood (GEL) tests are Hodges–Lehmann optimal under mild primitive conditions. These results support the belief that the Hodges–Lehmann optimality is a weak asymptotic requirement.     

Ordinal notions of submodularity

We consider several ordinal formulations of submodularity, defined for arbitrary binary relations on lattices. Two of these formulations are essentially due to Kreps [Kreps, D.M., 1979. A representation theorem for “Preference for Flexibility”. Econometrica 47 (3), 565–578] and one is a weakening of a notion due to Milgrom and Shannon [Milgrom, P., Shannon, C., 1994. Monotone comparative statics. Econometrica 62 (1), 157–180]. We show that any reflexive binary relation satisfying either of Kreps’s definitions also satisfies Milgrom and Shannon’s definition, and that any transitive and monotonic binary relation satisfying the Milgrom and Shannon’s condition satisfies both of Kreps’s conditions.     

Exact permutation tests for non-nested non-linear regression models

This paper proposes exact distribution-free permutation tests for the specification of a non-linear regression model against one or more possibly non-nested alternatives. The new tests may be validly applied to a wide class of models, including models with endogenous regressors and lag structures. These tests build on the well-known J test developed by Davidson and MacKinnon [1981. Several tests for model specification in the presence of alternative hypotheses. Econometrica 49, 781–793] and their exactness holds under broader assumptions than those underlying the conventional J test. The J-type test statistics are used with a randomization or Monte Carlo resampling technique which yields an exact and computationally inexpensive inference procedure. A simulation experiment confirms the theoretical results and also shows the performance of the new procedure under violations of the maintained assumptions. The test procedure developed is illustrated by an application to inflation dynamics.