The exact estimation of logit models by non-linear least squares

The purpose of this paper is to examine the relationship between least squares and maximum likelihood estimation, where the likelihood function is the product of two explicit functions. We illustrate the correspondence for the particular case of the logit model, and show that this can be estimated by commonly accessible non-linear least squares estimation packages. Unlike the conventional non-linear least squares approach the estimates obtained following the proposed method are maximum likelihood for all sample sizes.     

Forecasting technology diffusion with the Richards model

The Richards model has a shape parameter m that allows it to fit any sigmoidal curve. This article demonstrates the ability of a modified Richards model to fit a variety of technology diffusion curvilinear data that would otherwise be fit by Bass, Gompertz, Logistic, and other models. The performance of the Richards model in forecasting was examined by analyzing fragments of data computed from the model itself, where the fragments simulated either an entire diffusion curve but with sparse data points, or only the initial trajectory of a diffusion curve but with dense data points. It was determined that accurate parameter estimates could be obtained when the data was sparse but traced out the curve at least up to the third inflection point (concave down), and when the data was dense and traced out the curve up to the first inflection point (concave up). Rogers' Innovation I, II and III are discussed in the context of the Richards model. Since m is scale independent, the model allows for a typology of diffusion curves and may provide an alternative to Rogers' typology.     

A note on estimating disequilibrium models with aggregation

When an aggregate disequilibrium is the result of disequilibrium in several submarkets, the usual maximum likelihood estimation, which is based on the min of aggregate demand and supply, represents a misspecification. The present paper compares ML with several nonlinear least squares methods that are appropriate for this situation. Monte Carlo experiments suggest that ML is robust with respect to the misspecification and may be preferable to the nonlinear least suqares methods in some situations.     

Detecting intra-national PPP model in China: A median-unbiased estimation approach

This paper studies the speed of price adjustments toward the intra-national PPP level within China based on half-life measurement. We calculate the point estimates of half-lives and the corresponding confidence intervals using median-unbiased estimation methods to correct for downward bias in least squares estimates. The point estimates of half-lives indicate that the speed of mean reversion toward convergence between region pairs is fast. A strong conclusion about the high speed of adjustments is, however, unwarranted in some regions where the confidence intervals for half-life estimates are much wider than others.     

A conditional least squares estimation procedure for a disequilibrium market model with autocorrelated errors

Maximum likelihood estimation of parameters in a market disequilibrium model with unknown sample separation and autocorrelated errors is computationally intractable [Quandt (1981) and Lee (1984)]. This paper presents a computationally convenient conditional least squares procedure for estimation of parameters in this model.     

Maximum likelihood estimation for vector autoregressions with multivariate stochastic volatility

This paper analyzes the maximum likelihood estimation for vector autoregressions with stochastic volatility. The stochastic volatility is modeled following Uhlig (1997). The asymptotic distribution of the maximum likelihood estimate is discussed under mild regularity conditions. The maximum likelihood estimate can be obtained via an iterative method. In that case, the maximum likelihood estimate becomes the iteratively reweighted least squares estimate analyzed in Rubin (1983). The iteratively reweighted least squares estimate is computationally much simpler than the Bayesian method offered by Uhlig (1997).     

Estimating Costs of Children from Micro-Unit Records: A New Procedure Applied to Australian Data

Measuring the costs of children is of immense practical importance in a range of economic and social policy areas. In this paper, we introduce a new econometric procedure that improves on existing methods for obtaining estimates of such costs from a demand system. We develop, using an extended linear expenditure system, an iterative maximum likelihood estimator that overcomes possible estimation problems that arise from the 2-step estimation procedures employed by earlier authors. We also allow for a more general assumption about the equation “errors”, that of non-zero correlation between the errors for different commodities in the same household. Another important contribution is the development of an estimation procedure for sets of seemingly unrelated regressions where the different sets of equations are linked by some common parameters. The proposed procedure is applied to the 1988–89 and 1993–94 Australian Household Expenditure Surveys and results obtained update estimates of both the commodity-specific and general scales previously obtained for Australia.

     

Estimation of Structural Nonlinear Errors‐in‐Varibles Models by Simulated Least‐Squares Method

This article proposes a simulation approach to obtain least‐squares or generalized least‐squares estimators of structural nonlinear errors‐in‐variables models. The proposed estimators are computationally attractive because they do not need numerical integration nor huge numbers of simulations per observable. In addition, the asymptotic covariance matrix of the estimator has a simple decomposition that may be used to guide selection of appropriate simulation sizes. The method is also useful for models with missing data or imperfect surrogate covariates, where application of conventional least‐squares and maximum‐likelihood methods is restricted by numerical multidimensional integrations.     

Estimation and testing of alternative production function models

The paper deals with four different decision models for a firm operating with a constant elasticity of substitution production function. The assumptions underlying the different models and their consequences for estimation are carefully specified. Maximum likelihood estimates of the parameters are obtained of the full four-equation models, using data pertaining to the Dutch manufacturing sector. Finally likelihood ratio test procedures are developed in order to determine whether one or more of the decision models can be rejected on the basis of this data.     

Empirical implications of mathematical functions used to analyze market penetration of new products—Cigarettes case study

Three types of S-shaped growth curves—the logistic, the lognormal, and the Gompertz—are widespread in the literature on analysis of market penetration of new products/processes. This article discusses the mathematical properties of these function in the light of their empirical implications, such as location of the point of maximum growth rate, the symmetry/asymmetry of the growth rates around that inflection point, and the ease of estimation (linear and nonlinear regressions). The empirical economic expectations cannot specify these phenomena a priori so that a less restrictive function should be preferred. Such a function is the Gompertz function. It is, therefore, applied to the study of market penetration of filtered and menthol cigarettes and shown that the empirical verification vindicates the theoretical postulates of that function. The empirical estimates are also closer to actual estimates of growth rates of adoption.     

TRADE,STRUCTURAL REFORM,AND INSTITUTIONS IN SUB‐SAHARAN AFRICA

Sub‐Saharan African countries have traditionally lagged the rest of the developing world in terms of overall trade relative to gross domestic product. But, there is growing interest among these countries to initiate trade policies and improve quality of institutions as a way to promote trade and boost foreign direct investment. This article extends the gravity model of trade to include proxies for trade reform policy and institutional quality among the 15 countries of the Economic Community of West African States (ECOWAS) for data spanning 1984–2006. Alternative methods of estimation based on ordinary least squares, Heckman two‐step procedure, and Poisson pseudo‐maximum likelihood produce predictions that are consistent with the standard gravity model. They further highlight the evidence of restrictive trade policies and weak institutions that contribute to the failure of ECOWAS countries to boost bilateral trade. (JEL F13, F15, O19, O55)     

Revisiting the Olympic Effect

By utilizing the log‐linear gravity model, other authors have found statistically robust, permanent and large effects of hosting mega‐events (e.g. Olympics) on international exports. Surprisingly, they found that the unsuccessful bidders to host the Olympics experienced a similar impact on exports. Utilizing alternate specification such as the Poisson pseudo‐maximum likelihood (PPML) estimation that allows for heteroskedasticity prevalent in trade data, the this paper fails to find a robust positive effect of hosting and bidding for a mega‐event on total aggregate exports. Under heteroskedasticity, the parameters of log‐linearized models estimated by ordinary least squares (OLS) lead to biased estimates of the true elasticities.     

Measuring the Benefits of Neighbourhood Park Amenities: Application and Comparison of Spatial Hedonic Approaches

The hedonic price method was used to estimate the influence of parks on the rental prices of single-room dwellings in Setagaya Ward, Tokyo, Japan. A simple least squares method is not optimal when the data set contains spatial autocorrelation. To improve the accuracy of estimates, we employed spatial autoregression and kriging models, resulting in a higher validity of the spatial models compared to the least squares model. Kriging models were superior to others particularly in terms of prediction accuracy, indicating that these models should be employed if the objective is superior prediction rather than estimation. The results showed that the effect of parks on property values varied with the buffer distance and park size.     

On learning equilibria

Bullard (1994) and Schönhofer (1999) show that endogenous business cycles may emerge in an inflationary overlapping generations model where households predict future inflation rates by running a least squares regression on prices. We show that given the same beliefs but under an alternative, more natural, estimation procedure based upon inflation rates the monetary steady state will be globally stable for a large set of savings functions. We also study an evolutionary competition between the two estimation procedures. Although the dynamics are stabilized for a large set of parameter values, endogenous business cycles may still emerge in this heterogeneous beliefs framework.     

Testing whether waterfowl hunting benefits increase with greater water deliveries to wetlands

The change in waterfowl hunting benefits due to an increase in water deliveries to the levels required for biologically optimal wildlife refuge management at California's San Joaquin Valley National Wildlife Refuges are estimated with the Travel Cost Method, using both ordinary least squares and Poisson count data estimators. To test whether these increases were statistically significant, the Krinsky and Robb technique was used to find confidence intervals around the consumer surplus point estimates. The increases in consumer surplus were found to be statistically significant in 5 of the 6 refuges based on OLS regression estimates and in all 6 refuges using Poisson count data regression estimates. In addition, a comparison of the marginal value of an acre-foot of water in consumptive recreational use versus agriculture use is made, with the finding that the marginal value of water in waterfowl hunting was greater than the marginal value of water in agriculture for one of the six refuges.     

Bootstrap Testing in Nonlinear Models

Bootstrap testing of nonlinear models normally requires at least one nonlinear estimation for every bootstrap sample. We show how to reduce computational costs by performing only a fixed, small number of Newton or quasi-Newton steps for each bootstrap sample. The number of steps is smaller for likelihood ratio tests than for other types of classical tests and smaller for Newton's method than for quasi-Newton methods. The suggested procedures are applied to tests of slope coefficients in the tobit model and to tests of common factor restrictions. In both cases, bootstrap tests work well, and very few steps are needed.     

Interval regression models with endogenous explanatory variables

We consider the estimation of linear models where the dependent variable is observed by intervals and some continuous regressors may be endogenous. Our approach, an IV version of the technique devised by Stewart (Rev Econ Stud 50(3):737?C753, 1983), is fully parametric and two estimators are proposed: a two-step estimator and a limited-information maximum-likelihood estimator. The results can be summarized as follows: the two-step estimator has an intuitive appeal, and a Monte Carlo experiment suggests that its relative efficiency is rather satisfactory. The limited-information maximum-likelihood estimator, however, is probably simpler to implement and has the advantage of providing a framework in which several testing procedures are more straightforward to perform. The application of two-stage least squares to a proxy of the dependent variable built by taking midpoints, on the other hand, leads to inconsistent estimates; Monte Carlo evidence suggests that the bias arising from the ??midpoint?? technique is much worse than the effect of distributional misspecification. An example application is also included, which uses Australian data on migrants?? remittances; endogeneity effects are substantial and using conventional estimation methods leads to substantially misleading inference.     

The role of ad hoc factors in policy rate settings

Policymakers do not always follow a simple rule for setting policy interest rates for various reasons. Thus their behavior can be represented by a standard Taylor type policy rule amended with an additional variable representing an ad hoc factor. Consequently, ignoring the presence of the ad hoc factor causes bias in conventional policy rule estimators. I contrast the unbiased estimates of a procedure that accounts for the ad hoc factors and the bias of least squares on a unique data set of an unconditional inflation targeting episode.     

Mixed estimation as generated data

Mixed estimation with an informative prior can be regarded as a form of least squares estimation with an enlarged data set. The relationship between such additional ‘pseudo data’ and the sample data is studied.     

Generalized least squares and maximum likelihood estimation of the logistic function for technology diffusion

Given a simple stochastic model of technology adoption, we derive a function for technological diffusion that is logistic in the deterministic part and has an error term based on the binomial distribution. We derive two estimators—a generalized least squares (GLS) estimator and a maximum likelihood (ML) estimator—which should be more efficient than the ordinary least squares (OLS) estimators typically used to estimate technological diffusion functions. We compare the two new estimators with OLS using Monte-Carlo techniques and find that under perfect specification, GLS and ML are equally efficient and both are more efficient than OLS. There was no evidence of bias in any of the estimators. We used the estimators on some example data and found evidence suggesting that under conditions of misspecification, the estimated variance-covariance of the ML estimator is badly biased. We verified the existence of the bias with a second Monte-Carlo experiment performed with a known misspecification. In the second experiment, GLS was the most efficient estimator, followed by ML, and OLS was least efficient. We conclude that the GLS estimator of choice.