Showing papers in "Econometric Theory in 1991"

Asymptotically Efficient Estimation of Cointegration Regressions

Abstract: An asymptotic optimality theory for the estimation of cointegration regressions is developed in this paper. The theory applies to a reasonably wide class of estimators without making any specific assumptions about the probability distribution or short-run dynamics of the data-generating process. Due to the nonstandard nature of the estimation problem, the conventional minimum variance criterion does not provide a convenient measure of asymptotic efficiency. An alternative criterion, based on the concentration or peakedness of the limiting distribution of an estimator, is therefore adopted. The limiting distribution of estimators with maximum asymptotic efficiency is characterized in the paper and used to discuss the optimality of some known estimators. A new asymptotically efficient estimator is also introduced. This estimator is obtained from the ordinary least-squares estimator by a time domain correction which is nonparametric in the sense that no assumption of a finite parameter model is required. The estimator can be computed with least squares without any initial estimations.

Asymptotics for Least Absolute Deviation Regression Estimators

Abstract: The LAD estimator of the vector parameter in a linear regression is defined by minimizing the sum of the absolute values of the residuals. This paper provides a direct proof of asymptotic normality for the LAD estimator. The main theorem assumes deterministic carriers. The extension to random carriers includes the case of autoregressions whose error terms have finite second moments. For a first-order autoregression with Cauchy errors the LAD estimator is shown to converge at a 1/n rate.

Effects of Model Selection on Inference

Abstract: The asymptotic properties of parameter estimators which are based on a model that has been selected by a model selection procedure are investigated. In particular, the asymptotic distribution is derived and the effects of the model selection process on subsequent inference are illustrated.

From characteristic function to distribution function: a simple framework for the theory

Abstract: A unified framework is established for the study of the computation of the distribution function from the characteristic function. A new approach to the proof of Gurland's and Gil-Pelaez's univariate inversion theorem is suggested. A multivariate inversion theorem is then derived using this technique.

Estimating Nonlinear Dynamic Models Using Least Absolute Error Estimation

Abstract: We consider least absolute error estimation in a dynamic nonlinear model with neither independent nor identically distributed errors. The estimator is shown to be consistent and asymptotically normal, with asymptotic covariance matrix depending on the errors through the heights of their density functions at their medians (zero). A consistent estimator of the asymptotic covariance matrix of the estimator is given, and the Wald, Lagrange multiplier, and likelihood ratio tests for linear restrictions on the parameters are discussed. A Lagrange multiplier test for heteroscedasticity based upon the absolute residuals is analyzed. This will be useful whenever the heights of the density functions are related to the dispersions.

Adaptive Rules for Seminonparametric Estimators That Achieve Asymptotic Normality

Abstract: In econometrics, seminonparametric (SNP) estimators originated in the consumer demand literature. The Fourier flexible form is a well-known example. The idea is to replace the consumer's indirect utility function with a truncated series expansion and then use a parametric procedure, such as nonlinear multivariate regression, to set a confidence interval on an elasticity. More recently, SNP estimators have been used in nonlinear time series analysis. A truncated Hermite expansion with an ARCH leading term is used as the conditional density of the process. The method of maximum likelihood is used to fit it to data.

A Shortcut to LAD Estimator Asymptotics

Abstract: Using generalized functions of random variables and generalized Taylor series expansions, we provide quick demonstrations of the asymptotic theory for the LAD estimator in a regression model setting. The approach is justified by the smoothing that is delivered in the limit by the asymptotics, whereby the generalized functions are forced to appear as linear functionals wherein they become real valued. Models with fixed and random regressors, and autoregressions with infinite variance errors are studied. Some new analytic results are obtained including an asymptotic expansion of the distribution of the LAD estimator.

Test Consistency with Varying Sampling Frequency

Abstract: This paper considers the consistency property of some test statistics based on a time series of data. While the usual consistency criterion is based on keeping the sampling interval fixed, we let the sampling interval take any equispaced path as the sample size increases to infinity. We consider tests of the null hypotheses of the random walk and randomness against positive autocorrelation (stationary or explosive). We show that tests of the unit root hypothesis based on the first-order correlation coefficient of the original data are consistent as long as the span of the data is increasing. Tests of the same hypothesis based on the first-order correlation coefficient of the first-differenced data are consistent against stationary alternatives only if the span is increasing at a rate greater than T ½ , where T is the sample size. On the other hand, tests of the randomness hypothesis based on the first-order correlation coefficient applied to the original data are consistent as long as the span is not increasing too fast. We provide Monte Carlo evidence on the power, in finite samples, of the tests Studied allowing various combinations of span and sampling frequencies. It is found that the consistency properties summarize well the behavior of the power in finite samples. The power of tests for a unit root is more influenced by the span than the number of observations while tests of randomness are more powerful when a small sampling frequency is available.

Strong Laws for Dependent Heterogeneous Processes

Abstract: This paper presents maximal inequalities and strong law of large numbers for weakly dependent heterogeneous random variables. Specifically considered are Lr mixingales for r > 1, strong mixing sequences, and near epoch dependent (NED) sequences. We provide the first strong law for Lr-bounded Lr mixingales and NED sequences for 1 > r > 2. The strong laws presented for α-mixing sequences are less restrictive than the laws of McLeish [8].

On the Asymptotic Behavior of Least-Squares Estimators in AR Time Series with Roots Near the Unit Circle

Abstract: Some asymptotic properties of the least-squares estimator of the parameters of an AR model of order p, p ? 1, are studied when the roots of the characteristic polynomial of the given AR model are on or near the unit circle. Specifically, the convergence in distribution is established and the corresponding limiting random variables are represented in terms of functionals of suitable Brownian motions. Further, the preceding convergence in distribution is strengthened to that of convergence uniformly over all Borel subsets. It is indicated that the method employed for this purpose has the potential of being applicable in the wider context of obtaining suitable asymptotic expansions of the distributions of leastsquares estimators.

Limit Theory for M-Estimates in an Integrated Infinite Variance

Abstract: We consider the limiting distributions of M-estimates of an “autoregressive” parameter when the observations come from an integrated linear process with infinite variance innovations. It is shown that M-estimates are, asymptotically, infinitely more efficient than the least-squares estimator (in the sense that they have a faster rate of convergence) and are conditionally asymptotically normal.

Estimating Orthogonal Impulse Responses via Vector Autoregressive Models

Abstract: Impulse response functions from time series models are standard tools for analyzing the relationship between economic variables. The asymptotic distribution of orthogonalized impulse responses is derived under the assumption that finite order vector autoregressive (VAR) models are fitted to time series generated by possibly infinite order processes. The resulting asymptotic distributions of forecast error variance decompositions are also given.

The Exact Likelihood Function for an Empirical Job Search Model

Abstract: The exact likelihood function for a prototypal job search model is analyzed. The optimality condition implied by the dynamic programming framework is fully imposed. Using the optimality condition allows identification of an offer arrival probability separately from an offer acceptance probability. The estimation problem is nonstandard. The geometry of the likelihood function in finite samples is considered, along with asymptotic properties of the maximum likelihood estimator.

Noninvertibility and Pseudo-Maximum Likelihood Estimation of Misspecified ARMA Models

Abstract: Recently Tanaka and Satchell [11] investigated the limiting properties of local maximizers of the Gaussian pseudo-likelihood function of a misspecified moving average model of order one in case the spectral density of the data process has a zero at frequency zero. We show that pseudo-maximum likelihood estimators in the narrower sense, that is, global maximizers of the Gaussian pseudo-likelihood function, may exhibit behavior drastically different from that of the local maximizers. Some general results on the limiting behavior of pseudo-maximum likelihood estimators in potentially misspecified ARMA models are also presented.

Estimation of the Covariance Matrix of the Least-Squares Regression Coefficients When the Disturbance Covariance Matrix Is of Unknown Form

Abstract: This paper deals with the problem of estimating the covariance matrix of the least-squares regression coefficients under heteroskedasticity and/or autocorrelation of unknown form. We consider an estimator proposed by White [17] and give a relatively simple proof of its consistency. Our proof is based on more easily verifiable conditions than those of White. An alternative estimator with improved small sample properties is also presented.

Who Invented Local Power Analysis

Abstract: Asymptotic local power analysis has become an important and increasingly used technique in econometrics. This paper reviews the history of local power analysis and delineates the contribution of J.Neyman, E.J.G. Pitman, and G. Noether.

The Et Interview: Professor Sir Richard Stone

Abstract: Sir Richard Stone, knighted in 1978 and Nobel Laureate in Economics in 1984, is one of the pioneering architects of national income and social accounts, and is one of the few economists of his generation to have faced the challenge of economics as a science by combining theory and measurement within a cohesive framework. He was awarded the Nobel Prize in Economics for his “fundamental contributions to the development of national accounts,” but he has made equally significant contributions to the empirical analysis of consumer behavior. His work on the “Growth Project” has also been instrumental in the development of appropriate econometric methodology for the construction and the analysis of large disaggregated macroeconometric models. Throughout his long and productive career, stretching over more than half a century, Stone has been an inspiration to applied econometricians all over the world. His influence goes well beyond his written work. He has made a lasting impact on the large number of (now prominent) economists and statisticians who visited the Department of Applied Economics when he was its Director. He is a scientist, a scholar, and above all, a gentleman. He gives generously of himself and is always willing to help the cause of applied econometrics. He has been a Fellow of King's College, Cambridge since 1945 and has served as the President of the Econometric Society (in 1955) and the President of the Royal Economic Society (during 1978–1980). In the interview that follows, Richard Stone gives us a delightful account of his time as a student at Westminster School, his early introduction to economics at Cambridge University, and he shares with us his memories and thoughts on a long and productive career. The interview was conducted in Stones' magnificent private library in Cambridge, and I hope that readers enjoy reading the interview as much as I enjoyed recording it. Further details of Richard Stone's biography and research activities can be found in: Deaton, A. Stone, John Richard Nicholas. In J. Eatwell, M. Milgate and P. Newman (eds.), The New Palgrave: A Dictionary of Economics , Vol. 4, pp. 509–512. London: Macmillan, 1987. Stone, J.R.N. An autobiographical sketch. In Les Prix Nobel 1984 . Stockholm: Almquist and Wicksell International, 1985.

Robust m-tests

Abstract: This paper investigates the local robustness properties of a general class of multidimensional tests based on M-estimators. These tests are shown to inherit the efficiency and robustness properties of the estimators on which they are based. In particular, it is shown that small perturbations of the distribution of the observations can have arbitrarily large effects on the asymptotic level and power of tests based on estimators that do not possess a bounded influence function. An asymptotic 'admissibility' result is also presented, which provides a justification for tests based on optimal bounded-influence estimators. The problem of the robustness of a test, that is, the stability of its level and power under small changes in the underlying probability distribution of the observations, has received considerable attention in the statistical literature, mainly with reference to one-dimensional tests or the linear model (Rieder [18], Schrader and Hettmansperger [22], Lambert [14], Rousseeuw and Ronchetti [21], Kent [12], Ronchetti [19], Wang [25], and Hampel et al. [7]), but has been largely ignored by econometricians. This paper investigates the local robustness properties of a broad class of multidimensional tests, called M-tests because they are based on M-estimators. This class of tests includes most common tests in econometrics, such as Wald, score and Hausman tests. We study the asymptotic properties of M-tests under small perturbations of the assumed probability distribution of the observations. The particular kind of perturbations that we consider are "contamination models" where the assumed distribution is contaminated, with small but positive probability, by some extraneous distribution. This is a convenient way of representing the fact that an econometric model is at best an approximation to the true data-generation process, and this approximation may be adequate for the majority but not all the observations. Our approach builds on earlier work of Rieder [18], Rousseeuw and Ronchetti [21], Ronchetti [19], and Wang [25] for one-dimensional, one-sided tests, and on results of Hampel et al. [7] for the linear model. We show that contamination of the assumed model can

The Bias of Forecasts from a First-Order Autoregression

Abstract: The exact finite sample behavior is investigated on the bias of multiperiod leastsquares forecasts in the normal autoregressive model y t = α + β y t –1 + u t . Necessary and sufficient conditions are given for the existence of the bias and an expression is presented which we use to obtain exact numerical results for finite samples. The unit root and near unit root behavior is studied in detail and some popular preconceptions about the behavior of the bias are shown to be false.

Discrete Models for Estimating General Linear Continuous Time Systems

Abstract: This paper derives discrete models for estimating systems of both first- and second-order linear differential equations in which derivatives of the exogenous variables appear in addition to their levels.

A Continuous Time Approximation to the Stationary First-Order Autoregressive Model

Abstract: We consider the least-squares estimator in a strictly stationary first-order autoregression without an estimated intercept. We study its continuous time asymptotic distribution based on an asymptotic framework where the sampling interval converges to zero as the sample size increases. We derive a momentgenerating function which permits the calculation of percentage points and moments of this asymptotic distribution and assess the adequacy of the approximation to the finite sample distribution. In general, the approximation is excellent for values of the autoregressive parameter near one. We also consider the behavior of the power function of tests based on the normalized leastsquares estimator. Interesting nonmonotonic properties are uncovered. This analysis extends the study of Perron [15] and helps to provide explanations for the finite sample results established by Nankervis and Savin [13].

Open Higher Order Continuous-Time Dynamic Model with Mixed Stock and Flow Data and Derivatives of Exogenous Variables

Abstract: This paper is concerned with deriving formulae for higher order derivatives of exogenous variables for use in estimating the parameters of an open secondorder continuous time model with mixed stock and flow data and first and second order derivatives of exogenous variables which are not observable. This should provide the basis for the future estimation of continuous time models in a range of applied areas using the new Gaussian estimation computer program developed by Nowman [4].

NONUNIFORM BOUNDS FOR NONPARAMETRIC t-TESTS

Abstract: This paper gives simple nonuniform bounds on the tail areas of the permutation distribution of the usual Student's t-statistic when the observations are independent with symmetric distributions. As opposed to uniform bounds, nonuniform bounds depend on the observed sample. It is shown that the nonuniform bounds proposed are always tighter than uniform exponential bounds previously suggested. The use of the bounds to perform nonparametric t-tests is discussed and numerical examples are presented. Further, the bounds are extended to t-tests in the context of a simple linear regression.

On the Estimated Variances of Regression Coefficients in Misspecified Error Components Models

Abstract: In a regression model with an arbitrary number of error components, the covariance matrix of the disturbances has three equivalent representations as linear combinations of matrices. Furthermore, this property is invariant with respect to powers, matrix addition, and matrix multiplication. This result is applied to the derivation and interpretation of the inconsistency of the estimated coefficient variances when the error components structure is improperly restricted. This inconsistency is defined as the difference between the asymptotic variance obtained when the restricted model is correctly specified, and the asymptotic variance obtained when the restricted model is incorrectly specified; when some error components are improperly omitted, and the remaining variance components are consistently estimated, it is always negative. In the case where the time component is improperly omitted from the two-way model, we show that the difference between the true and estimated coefficient variances is of order greater than N–1 in probability.

On Limited Dependent Variable Models: Maximum Likelihood Estimation and Test of One-sided Hypothesis

Abstract: The limited dependent variable models with errors having log-concave density functions are studied here. For such models with normal errors, the asymptotic normality of the maximum likelihood estimator was established by Amemiya [1]. We show, when the density of the error distribution is log-concave, that the maximum likelihood estimator exists with arbitrarily large probability for large sample sizes, and is asymptotically normal. The general theory presented here includes the important special cases of normal, logistic, and extreme value error distributions. The main results are established under rather weak conditions. It is also shown that, under the null hypothesis, the asymptotic distribution of the likelihood ratio statistic for testing a one-sided alternative hypothesis is a weighted sum of chi-squares.