Estimating predictions,prediction errors and their standard deviations using constructed variables

Increasing use has been made of predictive tests for assessing model adequacy, but it is sometimes difficult to generate predictions and their standard errors in dynamic or simultaneous equation models. Following earlier suggestions by Salkever and Fuller, this paper shows how the requisite information may be obtained by the use of specially constructed variables in a regression framework. The main use of the method will be in those situations where prediction information is not available as a standard option in econometric packages.     

Estimating nonlinear time-series models using simulated vector autoregressions

This paper develops two new methods for conducting formal statistical inference in nonlinear dynamic economic models. The two methods require very little analytical tractability, relying instead on numerical simulation of the model's dynamic behaviour. Although one of the estimators is asymptotically more efficient than the other, a Monte Carlo study shows that, for a specific application, the less efficient estimator has smaller mean squared error in samples of the size typically encountered in macroeconomics. The estimator with superior small sample performance is used to estimate the parameters of a real business cycle model using observed US time-series data.     

Polynomial operators and the asymptotic distribution of dynamic multipliers

This paper employs formal manipulations of polynomial operators with matrix coefficients to derive dynamic and long-run multipliers. The same technique is employed to derive the asymptotic distribution of these quantities in a very simple manner. Some recursion formulae are given which make the computation of the covariance matrix of the limiting distribution a relatively simple process. The Klein Model I is used to provide a numerical illustration.     

Indirect inference in structural econometric models

This paper considers parametric inference in a wide range of structural econometric models. It illustrates how the indirect inference principle can be used in the inference of these models. Specifically, we show that an ordinary least squares (OLS) estimation can be used as an auxiliary model, which leads to a method that is similar in spirit to a two-stage least squares (2SLS) estimator. Monte Carlo studies and an empirical analysis of timber sale auctions held in Oregon illustrate the usefulness and feasibility of our approach.     

An algorithm for the solution of stochastic optimal control problems for large nonlinear econometric models

This paper considers the problem of solving an optimal control problem for large dynamic economic models which are both nonlinear and stochastic. It proposes a technique which combines conventional deterministic optimal control algorithms with the procedure of stochastic simulation, which calculates a numerical approximation to the distribution of the models endogenous variables. The new technique is computationally feasible for even large nonlinear models and, as an illustration of this, the Bank of England's large quarterly forecasting model is used in an example.     

Monitoring structural change in dynamic econometric models

The classical approach to testing for structural change employs retrospective tests using a historical data set of a given length. Here we consider a wide array of fluctuation‐type tests in a monitoring situation—given a history period for which a regression relationship is known to be stable, we test whether incoming data are consistent with the previously established relationship. Procedures based on estimates of the regression coefficients are extended in three directions: we introduce (a) procedures based on OLS residuals, (b) rescaled statistics and (c) alternative asymptotic boundaries. Compared to the existing tests our extensions offer ease of computation, improved size in finite samples for dynamic models and better power against certain alternatives, respectively. We apply our methods to three data sets, German M1 money demand, US labour productivity and S&P 500 stock returns. Copyright © 2005 John Wiley & Sons, Ltd.     

Estimation of econometric models of some discrete games

This paper discusses, estimates and compares some microeconometric models for simultaneous discrete endogenous variables. The models are based on the assumption that observed endogenous variables represent the outcome of a static discrete game. I discuss models based on non-cooperative equilibrium concepts (Nash, Stackelberg), as well as models which presume Pareto optimality of observed outcomes. The models are estimated using data on the joint labor force participation decisions of husbands and wives in a sample of Dutch households.     

The econometric analysis of models with risk terms

In this paper we have attempted to provide an integrated approach to the estimation of models with risk terms. It was argued that there exist orthogonality conditions between variables in the information set and higher-order moments of the unanticipated variable density. These could be exploited to provide consistent estimators of the parameters associated with the risk term. Specifically, it was recommended that an IV estimator should be applied, with instruments constructed from the information set. Four existing methods commonly used to estimate models with risk terms are examined, and applications of the techniques are made to the estimation of the risk term in the $US/$C exchange market, and the effects of price uncertainty upon production.     

The numerical values of some key parameters in econometric models

In the case of two endogenous variables, exogenous predetermined variables, and normally distributed disturbances, the distributions of the Two-Stage Least Squares (TSLS) and Limited Information Maximum Likelihood (LIML) estimators can be compared on the basis of three key parameters: the non-centrality parameter, a standardization of the structural coefficient, and the number of excluded exogenous variables. In this paper the values of these parameters are estimated in eleven structural equations from various actual econometric models. The distribution functions of the normalized TSLS and LIML estimators are given for the first two key parameters set at approximately their trimmed means, and the third at its median.     

Some applications for Basil's independence theorem in testing econometric models

Basu's Independence Theorem is used to show how tests of compound econometric hypotheses, involving two or more naturally ordered hypotheses, may be based upon mutually independent test statistics.     

Linear and nonlinear regression with stable errors

In this paper we describe methods and evaluate programs for linear regression by maximum likelihood when the errors have a heavy tailed stable distribution. The asymptotic Fisher information matrix for both the regression coefficients and the error distribution parameters are derived, giving large sample confidence intervals for all parameters. Simulated examples are shown where the errors are stably distributed and also where the errors are heavy tailed but are not stable, as well as a real example using financial data. The results are then extended to nonlinear models and to non-homogeneous error terms.     

Identification in models with autoregressive errors

Consider the model

$\text{A(L)x}{}_{\mathrm{t}}\text{=B(L)y}{}_{\mathrm{t}}\text{+C(L)z}{}_{\mathrm{t}}\text{=u}{}_{\mathrm{t}}\text{, t=1,…,T}$

, where

$\text{A(L)=(B(L):C(L))}$

is a matrix of polynomials in the lag operator so that $\text{L}{}^{\mathrm{r}}\text{x}{}_{\mathrm{t}}\text{=x}{}_{\mathrm{t?r}}$, and $\text{y}{}_{\mathrm{t}}$ is a vector of n endogenous variables,

$\text{B(L)=}\text{∑}\text{s=0}\text{k}\text{B}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

$\text{B}{}_0\text{I}{}_{\mathrm{n}}$, and the remaining $\text{B}{}_{\mathrm{s}}$ are $\text{n × n}$ square matrices,

$\text{C(L)=}\text{∑}\text{s=0}\text{k}\text{C}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

, and $\text{C}{}_{\mathrm{s}}$ is $\text{n × m}$.Suppose that $\text{u}{}_{\mathrm{t}}$ satisfies

$\text{R(L)u}{}_{\mathrm{t}}\text{=e}{}_{\mathrm{t}}$

, where

$\text{R(L)=}\text{∑}\text{s=0}\text{r}\text{R}{}_{\mathrm{s}}\text{L}{}^{\mathrm{s}}$

, $\text{R}{}_0\text{=I}{}_{\mathrm{n}}$, and $\text{R}{}_{\mathrm{s}}$ is a $\text{n × n}$ square matrix. $\text{e}{}_{\mathrm{t}}$ may be white noise, or generated by a vector moving average stochastic process.Now writing

$\text{Ψ(L)=R(L)A(L)}$

, it is assumed that ignoring the implicit restrictions which follow from eq. (1), $\text{Ψ(L)}$ can be consistently estimated, so that if the equation

$\text{Ψ(L)x}{}_{\mathrm{t}}\text{=e}{}_{\mathrm{t}}$

has a moving average error stochastic process, suitable conditions [see E.J. Hannan] for the identification of the unconstrained model are satisfied, and that the appropriate conditions (lack of multicollinearity) on the data second moments matrices discussed by Hannan are also satisfied. Then the essential conditions for identification of the A(L) and R(L) can be considered by requiring that for the true $\text{Ψ(L)}$ eq. (1) has a unique solution for $\text{A(L)}$ and $\text{R(L)}$.There are three types of lack of identification to be distinguished. In the first there are a finite number of alternative factorisations. Apart from a factorisation condition which will be satisfied with probability one a necessary and sufficient condition for lack of identification is that $\text{A(L)}$ has a latent root λ in the sense that for some non-zero vector β,

$\text{β′A(λ)=0}$

.The second concept of lack of identification corresponds to the Fisher conditions for local identifiability on the derivatives of the constraints. It is shown that a necessary and sufficient condition that the model is locally unidentified in this sense is that R(L) and A(L) have a common latent root, i.e., that for some vectors δ and β,

$\text{R(λ)δ=0 and β′A(λ)=0}$

.Firstly it is shown that only if further conditions are satisfied will this lead to local unidentifiability in the sense that there are solutions of the equation

$\text{Ψ(z)=R(z)A(z)}$

in any neighbourhood of the true values.     

Multisectoral econometric models of the centrally planned economies and the disequilibria

The paper provides a systematic analysis of the properties of large econometric macro models built for the Centrally Planned economies of the '70s and '80s. A specific typology of macro models is introduced, distinguishing between the demand and quasi-demand determined systems on the one hand and the supply and quasi-supply determined systems on the other, and also between complete and incomplete models specifying only one side of economic activities (i.e. generating either demand or supply). It is shown that, under the command type economy, the incomplete, fully supply determined systems prevailed, generating production and showing its allocation. The economic reforms of the '80s, which aimed at a slow transition towards market economies, brought about a tendency towards constructing complete quasi-supply determined systems (especially for Poland and the CSRS). Since, in principle, they were built for shortage economies, the model builders had to allow for unobservables (final and intermediate demand, capacity utilization) and, on the other hand, for an increasing role of the financial phenomena (including prices) and financial policy instruments. A summary of the applications of macro models in forecasting and policy simulations is provided and new tendencies associated with regaining economic equilibria and approaching the market mechanisms emerging in the period of transition are shown.This is a revised and extended version of our paper delivered at the Project LINK Meeting in Paris, 1989 (see W. Welfe 1989b), which made use of our earlier publication (W. Welfe 1988).     

Structural measurement errors in nonseparable models

This paper considers measurement error from a new perspective. In surveys, response errors are often caused by the fact that respondents recall past events and quantities imperfectly. We explore the consequences of limited recall for the identification of marginal effects. Our identification approach is entirely nonparametric, using Matzkin-type nonseparable models that nest a large class of potential structural models. We show that measurement error due to limited recall will generally exhibit nonstandard behavior, in particular be nonclassical and differential, even for left-hand side variables in linear models. We establish that information reduction by individuals is the critical issue for the severity of recall measurement error. In order to detect information reduction, we propose a nonparametric test statistic. Finally, we propose bounds to address identification problems resulting from recall errors. We illustrate our theoretical findings using real-world data on food consumption.     

Estimating moments in ANOVA-type mixed models

In the paper, a simple projection-based method is systematically developed to estimate the qth (\(q\ge 2\)) order moments of random effects and errors in the ANOVA type mixed model (ANOVAMM), where the response may not be divided into independent sub-vectors. All the estimates are weakly consistent and the second-order moment estimates are strongly consistent. Besides, the derived estimates are different from those in mixed models with cluster design. Simulation studies are conducted to examine the finite sample performance of the estimates and two real data examples are analyzed for illustration.     

Specification diagnostics based on Laguerre alternatives for econometric models of duration

Score tests of the null hypothesis of exponentially distributed durations (conditional on regressors) against alternatives in a family of approximations to arbitrary distributions for non-negative random variables are developed. The test statistics take a simple, easily calculated and interpreted form. The alternatives considered are expansions of various orders in Laguerre polynomials. An economic model generating exponential unemployment duration is presented. The statistics are applied in a look at unemployment durations in the Denver Income Maintenance Experiment data.