Likelihood Inference on Semiparametric Models: Average Derivative and Treatment Effect

Over the past few decades, much progress has been made in semiparametric modelling and estimation methods for econometric analysis. This paper is concerned with inference (i.e. confidence intervals and hypothesis testing) in semiparametric models. In contrast to the conventional approach based on t‐ratios, we advocate likelihood‐based inference. In particular, we study two widely applied semiparametric problems, weighted average derivatives and treatment effects, and propose semiparametric empirical likelihood and jackknife empirical likelihood methods. We derive the limiting behaviour of these empirical likelihood statistics and investigate their finite sample performance through Monte Carlo simulation. Furthermore, we extend the (delete‐1) jackknife empirical likelihood toward the delete‐d version with growing d and establish general asymptotic theory. This extension is crucial to deal with non‐smooth objects, such as quantiles and quantile average derivatives or treatment effects, due to the well‐known inconsistency phenomena of the jackknife under non‐smoothness.     

The asymptotic efficiency of semiparametric estimators for censored linear regression models

This paper presents numerical comparisons of the asymptotic mean square estimation errors of semiparametric generalized least squares (SGLS), quantite, symmetrically censored least squares (SCLS), and tobit maximum likelihood estimators of the slope parameters of censored linear regression models with one explanatory variable. The results indicate that the SCLS estimator is less efficient than the other two semiparametric estimators. The SGLS estimator is more efficient than quantile estimators when the tails of the distribution of the random component of the model are not too thick and the probability of censoring is not too large. The most efficient semiparametric estimators usually have smaller mean square estimation errors than does the tobit estimator when the random component of the model is not normally distributed and the sample size is 500–1,000 or more.     

ESTIMATION WITH CENSORED REGRESSORS: BASIC ISSUES*

We study issues that arise for estimation of a linear model when a regressor is censored. We discuss the efficiency losses from dropping censored observations, and illustrate the losses for bound censoring. We show that the common practice of introducing a dummy variable to “correct for” censoring does not correct bias or improve estimation. We show how censored observations generally have zero semiparametric information, and we discuss implications for estimation. We derive the likelihood function for a parametric model of mixed bound‐independent censoring, and apply that model to the estimation of wealth effects on consumption.     

An analysis of housing expenditure using semiparametric cross-section models

In this paper we model expenditure on housing for owners and renters by means of endogenous switching regression models using cross-section data. We explain the share of housing in total expenditure from family characteristics and total expenditure, where the latter is allowed to be endogenous. We apply various existing parametric and semiparametric techniques for cross-section data. Exogeneity of total expenditure is rejected for the parametric models but not for most semiparametric models. The results are compared on the basis of budget elasticities and graphs of the estimated relationship between the budget share spent on housing and the logarithm of total expenditure. First version received: November 1997 / Final version received: January 2000     

Nonlinearity in medical expenditures: a new semiparametric approach

Research in empirical health economics has found that the relationship between medical expenditures and age, income and other variables can be highly nonlinear. Moreover, men and women can have quite different medical expenditure patterns due to their differences in life expectancy. Thus it may be difficult to find an appropriate parametric model to capture the highly complicated nonlinearity in medical expenditures, and introducing a simple gender dummy variable in a parametric model may not reveal all the medical expenditure differences between men and women. This study takes a semiparametric approach. In particular, an additive partially linear specification is employed to study the relationship between medical expenditures and age, income, gender and other individual characteristics. Using data from the National Medical Expenditure Survey, the results indicate that the semiparametric approach taken in this study is very promising. They confirm that medical expenditures are nonlinear in income and age, and that men and women have quite different medical expenditure patterns.     

Food quality,calories and household income

We investigate the relationship between calories, food quality and household per capita expenditure using regression and semiparametric methods on a sample of poor households from rural Mexico, where Programa de Apoyo Alimentario (PAL), a targeted nutritional programme, is operating. The semiparametric method yields an estimate of the elasticity between calories and expenditure of 0.39 below the median and 0.28 above. The corresponding estimates of the elasticity of the calorie price are 0.48 below the median and 0.45 above. We also examine the extent to which the expenditure elasticity of the calorie price is explained by substitution between and within major food groups. We find that there is a very high incidence of substitution within cereals (especially for poor households) and that between group substitution explains at most 59% of the income elasticity for food quality. These estimates suggest that the potential of a cash transfer programme to have a positive impact on the food diversity and the nutritional status of households is quite limited.     

Demographic versus expenditure flexibility in Engel curves

We consider the effects of demographic and expenditure variables on consumer demand in a system of Engel curves using a smooth coefficient semiparametric model where the expenditure effects on the budget shares vary nonparametrically with demographic variables such as the age of head and number of children in the household. Our findings, based on UK micro data, suggest that with a smooth coefficient semiparametric model there is no need for nonlinear logarithmic expenditure effects in the budget shares. Furthermore, we find evidence of a trade-off between demographic and expenditure effects in Engel curves and that a rank-2 system of Engel curves where the logarithmic expenditure effects are allowed to vary with demographic characteristics either nonparametrically or as a third degree polynomial function cannot be rejected against a rank-3 (quadratic logarithmic) model. The implications on household behavior and welfare are also examined. We would like to thank an anonymous referee and Baldev Raj, the editor, for useful comments and suggestions. We would also like to thank the University of Cyprus for financial support, Theofanis Mamuneas for stimulating discussions and the Office of National Statistics for making available the UK Family Expenditure Survey data through the ESRC Data Archive. The last author would also like to acknowledge the financial support from SSHRC of Canada.     

Cigarette demand and tax policy for race groups in South Africa

This paper calculates cigarette demand for race groups in South Africa. Elasticities are the most important information a tax policy analyst can have. Elasticities determine how the tax base will change with a change in the tax rate and thus how government revenues will respond to the tax. Elasticities also determine the excess burden that consumers will bear as a result of the tax. As such, own price, crossprice, and expenditure elasticities are calculated along with government revenue maximizing tax rates, and total and excess burdens. Parametric and semiparametric estimation techniques are used and compared. Results show that a tax on cigarettes will discourage nonsmokers from starting to smoke and mainly raise revenue from current smokers. Furthermore, it is found that consumption behaviours between groups are different implying different government revenue maximizing tax rates for each group affecting the distribution of income.     

Long memory in the interest rates in some Asian countries

In this paper, the stochastic behavior of short run interest rates in some Asian development countries is examined by means of using fractionally integrated semiparametric techniques. In doing so, a much richer flexibility is allowed in the dynamic behavior of the series not achieved by the classical representations based on I(0) or I(1) processes. The author uses a quasi-maximum likelihood estimation procedure of Robinson [QMLE, 1995a], which has some advantages with respect to other methods. The results show that the orders of integration of the short run interest rates in Singapore and Thailand are strictly below 1, implying mean reversion. On the contrary, the results for Malaysia, South Korea, and Philippines are less conclusive, the values of d oscillating around the unit root case.     

Endogeneity in Semiparametric Binary Response Models

This paper develops and implements semiparametric methods for estimating binary response (binary choice) models with continuous endogenous regressors. It extends existing results on semiparametric estimation in single-index binary response models to the case of endogenous regressors. It develops a control function approach to account for endogeneity in triangular and fully simultaneous binary response models. The proposed estimation method is applied to estimate the income effect in a labour market participation problem using a large micro data-set from the British Family Expenditure Survey. The semiparametric estimator is found to perform well, detecting a significant attenuation bias. The proposed estimator is contrasted to the corresponding probit and linear probability specifications.     

A NOTE ON COMPUTATION OF THE DOUBLE-HURDLE MODEL WITH DEPENDENCE WITH AN APPLICATION TO TOBACCO EXPENDITURE

This note shows how the sample likelihood of the full double-hurdle model with dependence can be reformulated to allow estimation with standard econometric software. An illustrative example is provided, using data on tobacco expenditure by households in the 1984 UK Family Expenditure Survey.     

Semiparametric Estimation of Simultaneous-Equation Microeconometric Models with Index Restrictions

This article introduces semiparametric methods for the estimation of simultaneous-equation microeconometric models with index restrictions. The methods are motivated by a semiparametric minimum-distance procedure, which unifies the estimation of both regression-type and linear or nonlinear simultaneous-equation models without emphasis on the construction of instrumental variables. Single-equation and systematic estimation methods and optimal weighting procedures are considered. The estimators are √ n -consistent and asymptotically normal. For the estimation of nonparametric regression and some sample selection models where the variances of disturbances are functions of the same indices, the optimal weighted estimator attains Chamberlain's efficient bound for models with conditional moment restrictions. The weighted estimator is shown to be optimal within a class of semiparametric instrumental variables estimators.
JEL classification numbers: C14, C24, C34.     

SEMINONPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATION OF CONDITIONAL MOMENT RESTRICTION MODELS*

This article studies estimation of a conditional moment restriction model with the seminonparametric maximum likelihood approach proposed by Gallant and Nychka (Econometrica 55 (March 1987), 363–90). Under some sufficient conditions, we show that the estimator of the finite dimensional parameter θ is asymptotically normally distributed and attains the semiparametric efficiency bound and that the estimator of the density function is consistent under L₂ norm. Some results on the convergence rate of the estimated density function are derived. An easy to compute covariance matrix for the asymptotic covariance of the θ estimator is presented.     

Parametric versus nonparametric methods in risk scoring: an application to microcredit

The importance of credit access to improve economic opportunities in developing markets is well established in the literature. However, there exists a strong need to mitigate adverse selection problems in microlending. A risk scoring model that more accurately predicts the likelihood of repayment of potential borrowers can help address this market imperfection and to benefit both lenders and borrowers. This paper compares the performance of nonparametric versus semiparametric and traditional parametric risk scoring models based on default probabilities. We show the advantages of relying on less structured, data-driven methods for risk scoring using both simulated data and data from credit loans granted to small and microenterprises in rural Peru. The estimation results indicate that nonparametric methods lead to a better evaluation of credit worthiness and can help prevent including potential “bad” borrowers and excluding “good” borrowers from sensitive microcredit markets.     

Testing for structural breaks in panel varying coefficient models: with an application to OECD health expenditure

In this article, we propose a model selection approach for testing structural breaks in a semiparametric panel varying coefficient model. Monte Carlo evidence shows that the proposed model selection approach performs well in finite sample settings. Applying the method to an empirical data, we find evidence of structural breaks in Organisation for Economic Co-operation and Development (OECD) health expenditure data by allowing for income elasticity to be state (income)-dependent. The relationship between health expenditure and income is subject to two types of structural changes: smooth changes over income and structural breaks in the time dimension. The findings hold for both foreign exchange rate-converted and Purchasing Power Parity-converted expenditure and GDP.     

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.

     

Efficient Estimation of Additive Partially Linear Models

I consider the problem of estimating an additive partially linear model using general series estimation methods with polynomial and splines as two leading cases. I show that the finite-dimensional parameter is identified under weak conditions. I establish the root-n-normality result for the finite-dimensional parameter in the linear part of the model and show that it is asymptotically more efficient than a semiparametric estimator that ignores the additive structure. When the error is conditional homoskedastic, my finite-dimensional parameter estimator reaches the semiparametric efficiency bound. Efficient estimation when the error is conditional heteroskedastic is also discussed.     

A NOTE ON SEMIPARAMETRIC ESTIMATION OF FINITE MIXTURES OF DISCRETE CHOICE MODELS WITH APPLICATION TO GAME THEORETIC MODELS*

We view a game abstractly as a semiparametric mixture distribution and study the semiparametric efficiency bound of this model. Our results suggest that a key issue for inference is the number of equilibria compared to the number of outcomes. If the number of equilibria is sufficiently large compared to the number of outcomes, root‐n consistent estimation of the model will not be possible. We also provide a simple estimator in the case when the efficiency bound is strictly above zero.     

Semiparametric Bayesian inference of the Kuznets hypothesis

In contrast to the parametric (quadratic) specification used for examining the Kuznets hypothesis, this study relies on the semiparametric Bayesian inference of the partially linear regression to re-assess the validity of an inverted-U shape of the Kuznets curve. The simple framework permits us to perform estimation and model comparison in a unified way. Empirical results using cross-sectional data on 75 countries indicate that there exists an (approximately) inverted-√, hence, asymmetric relation between inequality and per capita GDP. Moreover, judged by the Bayes factor, overwhelming evidence is found in favor of our semiparametric specification. Finally, robustness check using alternative measure of inequality and pool data confirms our findings.     

Matching using semiparametric propensity scores

This paper considers the application of semiparametric methods to estimate propensity scores or probabilities of program participation, which are central to certain program evaluation methods. To evaluate the practical benefits, we first conduct a Monte Carlo study. Second, we use data from the NSW experiment, CPS, and PSID. We compare treatment effect and evaluation bias estimates using propensity scores estimated from parametric logit, semiparametric single index, and semiparametric binary quantile regression models. Our results suggest that it is important to account for very general forms of heterogeneity in (semiparametric) estimation of the propensity score, particularly when the treatment effects vary in an unsystematic manner with the true propensity score.