Least squares, preliminary test and Stein-type estimation in general vector AR(p) models

In the general vector autoregressive process AR ( p ), multivariate least square estimation (LSE)/maximum likelihood estimation (MLE) of a subset of the parameters is considered when the complementary subset is suspected to be redundant. This may be viewed as a special case of linear constraints of autoregressive parameters. We incorporate this nonsample information in the estimation process and propose preliminary test and Stein-type estimators for the target subset of parameters. Under local alternatives their asymptotic properties are investigated and compared with those of unrestricted and restricted LSE. The dominance picture of the estimators is presented.     

The method of elimination and substitution in the GMM estimation of mixed regressive,spatial autoregressive models

This paper proposes a computationally simple GMM for the estimation of mixed regressive spatial autoregressive models. The proposed method explores the advantage of the method of elimination and substitution in linear algebra. The modified GMM approach reduces the joint (nonlinear) estimation of a complete vector of parameters into estimation of separate components. For the mixed regressive spatial autoregressive model, the nonlinear estimation is reduced to the estimation of the (single) spatial effect parameter. We identify situations under which the resulting estimator can be efficient relative to the joint GMM estimator where all the parameters are jointly estimated.     

Een berekeningsmethode voor de logistische curve

Summary  " Simple estimation of the parameters of the logistic curve ."
By means of an appropriate transformation of the dependent variable Y _t into a simple function of Y _t, which depends only linearly on time, one of the parameters can be estimated in the usual way. A second transformation leads to the estimation of the two remaining parameters. As a numerical illustration two examples are presented.     

A Comparison of two Robust Estimation Methods for Business Surveys

Two alternative robust estimation methods often employed by National Statistical Institutes in business surveys are two‐sided M‐estimation and one‐sided Winsorisation, which can be regarded as an approximate implementation of one‐sided M‐estimation. We review these methods and evaluate their performance in a simulation of a repeated rotating business survey based on data from the Retail Sales Inquiry conducted by the UK Office for National Statistics. One‐sided and two‐sided M‐estimation are found to have very similar performance, with a slight edge for the former for positive variables. Both methods considerably improve both level and movement estimators. Approaches for setting tuning parameters are evaluated for both methods, and this is a more important issue than the difference between the two approaches. M‐estimation works best when tuning parameters are estimated using historical data but is serviceable even when only live data is available. Confidence interval coverage is much improved by the use of a bootstrap percentile confidence interval.     

ECONOMETRIC ESTIMATION OF ARMINGTON IMPORT ELASTICITIES FOR A REGIONAL CGE MODEL OF THE ILLINOIS ECONOMY

One of the main concerns associated with the development and use of regional CGE models is the determination of key parameter values, particularly substitution and other price elasticities. A common problem is the lack of appropriate regional data for econometric estimation. Consequently, it is important to identify key parameters that are likely to be important in determining quantitative results and then to prioritize these for estimation where appropriate data are available. In this paper, the focus is on the estimation of the regional trade (import) substitution parameters, which tend to be important in analysis for regional economies (given their openness to trade). Here, commodity import elasticities for the Illinois economy are estimated and tested in a single region CGE model of the Illinois economy. In our econometric estimation, we apply a model that takes account of market size and distance in estimating the substitutability between commodities produced in Illinois and other US states.     

Estimation and testing in time-series regression models with heteroscedastic disturbances

We consider ARMAX models with heteroscedastic residuals. Consistent estimation of the regression coefficient allows the Bicker-White approach to heteroscedasticity to be extended to moving averages of heteroscedastic disturbances. Tests for the presence of a moving-average or of heteroscedasticity are developed and estimation of the moving-average parameters considered.     

Approximate inference in multivariate nonlinear functional relationships

An estimation procedure based on estimating equations is presented for the parameters in a multivariate functional relationship model, where all observations are subject to error. The covariance matrix of the observational errors may be parametrized and is allowed to be different for different sets of observations. Estimators are defined for the unknown relation parameters and error parameters.
For linear models (i.e. where the model function is linear in the incidental parameters) the estimators are consistent and asymptotically normal. A consistent expression for the covariance matrix of the estimators is derived. The results are valid for general error distributions.
For nonlinear models the estimators are based on locally linear approximations to the model function. The afore mentioned properties of the estimators are now only approximately valid. The adequacy of the approximate inference, based on asymptotic theory for the linearized model, needs at least informal check. Some examples are given to illustrate the estimation procedure.     

Een berekeningsmethode voor de logistische curve

Summary “Simple estimation of the parameters of the logistic curve.” By means of an appropriate transformation of the dependent variable Y_t into a simple function of Y_t, which depends only linearly on time, one of the parameters can be estimated in the usual way. A second transformation leads to the estimation of the two remaining parameters. As a numerical illustration two examples are presented.     

Errors-in-variables in demand systems

The rejection of symmetry and other restrictions in demand systems may be due to measurement errors in the exogenous variables. It is shown that symmetry conditions can be used to identify and consistently estimate a linear model's parameters when measurement error exists. Several identification rules are derived and estimation of identified models is considered. Results are applied to estimation of the Almost Ideal Demand System for the United Kingdom.     

Estimation and model selection based inference in single and multiple threshold models

This paper evaluates the properties of a joint and sequential estimation procedure for estimating the parameters of single and multiple threshold models. We initially proceed under the assumption that the number of regimes is known á priori but subsequently relax this assumption via the introduction of a model selection based procedure that allows the estimation of both the unknown parameters and their number to be performed jointly. Theoretical properties of the resulting estimators are derived and their finite sample properties investigated.     

On new variance approximations for linear models with inequality constraints

In this paper, we examine the estimation of linear models subject to inequality constraints with a special focus on new variance approximations for the estimated parameters. For models with one inequality restriction, the proposed variance formulas are exact. The variance approximations proposed in this paper can be used in regression analysis, Kalman filtering, and balancing national accounts, when inequality constraints are to be incorporated in the estimation procedure.     

On the initialization of adaptive learning in macroeconomic models

We review and evaluate methods previously adopted in the applied literature of adaptive learning in order to initialize agents’ beliefs. Previous methods are classified into three broad classes: equilibrium-related, training sample-based, and estimation-based. We conduct several simulations comparing the accuracy of the initial estimates provided by these methods and how they affect the accuracy of other estimated model parameters. We find evidence against their joint estimation with standard moment conditions: as the accuracy of estimated initials tends to deteriorate with the sample size, spillover effects also deteriorate the accuracy of the estimates of the model’s structural parameters. We show how this problem can be attenuated by penalizing the variance of estimation errors. Even so, the joint estimation of learning initials with other model parameters is still subject to severe distortions in small samples. We find that equilibrium-related and training sample-based initials are less prone to these issues. We also demonstrate the empirical relevance of our results by estimating a New Keynesian Phillips curve with learning, where we find that our estimation approach provides robustness to the initialization of learning. That allows us to conclude that under adaptive learning the degree of price stickiness is lower compared to inferences under rational expectations.     

Posterior predictive p-values for classical null hypotheses

We show that for the purpose of testing a classical null hypothesis the posterior predictive check of Rubin (1984) may be inadequate. This inadequacy is caused by the estimation of the nuisance parameters under the null hypothesis. We show that this problem can be solved if the parameters are estimated under the encompassing model.     

Bayesian estimation of agent-based models

We consider Bayesian inference techniques for agent-based (AB) models, as an alternative to simulated minimum distance (SMD). Three computationally heavy steps are involved: (i) simulating the model, (ii) estimating the likelihood and (iii) sampling from the posterior distribution of the parameters. Computational complexity of AB models implies that efficient techniques have to be used with respect to points (ii) and (iii), possibly involving approximations. We first discuss non-parametric (kernel density) estimation of the likelihood, coupled with Markov chain Monte Carlo sampling schemes. We then turn to parametric approximations of the likelihood, which can be derived by observing the distribution of the simulation outcomes around the statistical equilibria, or by assuming a specific form for the distribution of external deviations in the data. Finally, we introduce Approximate Bayesian Computation techniques for likelihood-free estimation. These allow embedding SMD methods in a Bayesian framework, and are particularly suited when robust estimation is needed. These techniques are first tested in a simple price discovery model with one parameter, and then employed to estimate the behavioural macroeconomic model of De Grauwe (2012), with nine unknown parameters.     

GARCH models without positivity constraints: Exponential or log GARCH?

This paper provides a probabilistic and statistical comparison of the log-GARCH and EGARCH models, which both rely on multiplicative volatility dynamics without positivity constraints. We compare the main probabilistic properties (strict stationarity, existence of moments, tails) of the EGARCH model, which are already known, with those of an asymmetric version of the log-GARCH. The quasi-maximum likelihood estimation of the log-GARCH parameters is shown to be strongly consistent and asymptotically normal. Similar estimation results are only available for the EGARCH (1,1) model, and under much stronger assumptions. The comparison is pursued via simulation experiments and estimation on real data.     

Seemingly unrelated nonlinear regressions

The article considers the estimation of the parameters of a set of nonlinear regression equations when the responses are contemporaneously but not serially correlated. Conditions are set forth such that the estimator obtained is strongly consistent, asymptotically normally distributed, and asymptotically more efficient than the single-equation least squares estimator. The methods presented allow estimation of the parameters subject to nonlinear restrictions across equations. The article includes a discussion of methods to perform the computations and a Monte Carlo simulation.     

Production risk and the estimation of ex-ante cost functions

Cost function estimation under production uncertainty is problematic because the relevant cost is conditional on unobservable expected output. If input demand functions are also stochastic, then a nonlinear errors-in-variables model is obtained and standard estimation procedures typically fail to attain consistency. But by exploiting the full implications of the expected profit maximization hypothesis that gives rise to ex-ante cost functions, it is shown that the errors-in-variables problem can be effectively removed, and consistent estimation of the parameters of interest achieved. A Monte Carlo experiment illustrates the advantages of the proposed procedure as well as the pitfalls of other existing estimators.     

A minimum chi-squared method for indirect parameter estimation from Poisson data

In this paper, we introduce a new algorithm for estimating non-negative parameters from Poisson observations of a linear transformation of the parameters. The proposed objective function fits both a weighted least squares (WLS) and a minimum χ² estimation framework, and results in a convex optimization problem. Unlike conventional WLS methods, the weights do not need to be estimated from the datas, but are incorporated in the objective function. The iterative algorithm is derived from an alternating projection procedure in which "distance" is determined by the chi-squared test statistic, which is interpreted as a measure of the discrepancy between two distributions. This may be viewed as an alternative to the Kullback-Leibler divergence which corresponds to the maximum likelihood (ML) estimation. The algorithm is similar in form to, and shares many properties with, the expectation maximization algorithm for ML estimation. In particular, we show that every limit point of the algorithm is an estimator, and the sequence of projected (by the linear transformation into the data space) means converge. Despite the similarities, we show that the new estimators are quite distinct from ML estimators, and obtain conditions under which they are identical.     

Nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters

We consider nonparametric/semiparametric estimation and testing of econometric models with data dependent smoothing parameters. Most of the existing works on asymptotic distributions of a nonparametric/semiparametric estimator or a test statistic are based on some deterministic smoothing parameters, while in practice it is important to use data-driven methods to select the smoothing parameters. In this paper we give a simple sufficient condition that can be used to establish the first order asymptotic equivalence of a nonparametric estimator or a test statistic with stochastic smoothing parameters to those using deterministic smoothing parameters. We also allow for general weakly dependent data.     

CHOOSING THE VARIABLES TO ESTIMATE SINGULAR DSGE MODELS

We propose two methods to choose the variables to be used in the estimation of the structural parameters of a singular DSGE model. The first selects the vector of observables that optimizes parameter identification; the second selects the vector that minimizes the informational discrepancy between the singular and non‐singular model. An application to a standard model is discussed and the estimation properties of different setups compared. Practical suggestions for applied researchers are provided. Copyright © 2014 John Wiley & Sons, Ltd.