Squares Estimation Of Models Research Articles

Bootstrapping periodically autoregressive models

The main objective of this paper is to establish the residual and the wild bootstrap procedures for periodically autoregressive models. We use the least squares estimators of model’s parameters and generate their bootstrap equivalents. We prove that the bootstrap procedures for causal periodic autoregressive time series with finite fourth moments are weakly consistent. Finally, we confirm our theoretical considerations by simulations.

A Cointegration Model of Age-Specific Fertility and Female Labor Supply in the United States

1. IntroductionModels of fertility based on economic theories of behavior have been subjected to rigorous conceptual and empirical scrutiny (for surveys, see Olsen 1994 and Macunovich 1996a; for critical reviews, see Murphy 1992 and Smith 1981). However, much empirical analysis of aggregate fertility patterns in the United States has relied on traditional regression methods, with little influence from recent developments in multiple time-series methods appropriate for nonstationary variables.Although important theoretical propositions are testable with individual data, understanding of trends and patterns in fertility behavior at the societal level requires aggregate analysis (Ryder 1980). The aggregation of individual effects to make statements about total fertility is problematic, as the composition of the population changes over time. Some effects that are measured at the individual level may reflect changes in individuals' positions within a society, and these effects will not be present at the societal level. Alternatively, social interaction may induce behavioral changes across a population that are not reflected in individual responses. As described by Kohler (2001), changes in societal norms and institutions can create feedback loops between aggregate variables and individual incentives toward childbearing, so that effects between aggregates may exceed substantially the responses measured at the level of the household.1Analysis of aggregate time-series data has its own considerable challenges. Fertility and its determinants are most likely nonstationary time series that trend or drift persistently away from their initial values. Such nonstationarity may undermine classical estimation and inference with traditional regression procedures, leading to spurious inferences about relations among variables. Furthermore, the principal determinants of fertility, for example, women's wages and education levels, female labor force participation, unemployment rates, and husband's incomes, are quite possibly endogenously determined in conjunction with fertility decisions. This problem of endogenous regressons can undermine the identifiability of the fertility model, rendering the relations unestimable. Even if the relations are identified, the problem of endogenous regressors leads to inconsistent least squares estimators of model parameters.The objective of this paper is to revisit a simple economic model of fertility, employing the cointegration model of Johansen (1995) that is appropriate for analyzing relations between nonstationary time series. Johansen's procedure allows the empirical determination of the number of stationary relations and produces maximum likelihood estimators of the parameters of these relations that are consistent and normally distributed, even in the presence of endogenous explanatory variables. Furthermore, to capture information on both the level and the timing of fertility, the analysis is applied to two age-specific fertility rates covering the prime childbearing years of U.S. women.2. Empirical Studies of Fertility with Aggregate DataEconomic models of fertility are grounded in either Easterlin's (1980a) relative income hypothesis or the New Home Economics (NHE) of Willis (1973) and Becker (1981). The former theory emphasizes the role of male incomes, relative to economic aspirations, as the driving force behind fertility and female labor force participation. Economic aspirations of young adults are determined by material conditions prevailing in their parental homes during their teenage years, when their parents would be close to their prime in earnings capacity. An increase in relative income shifts preferences in favor of childbearing and away from labor force activity by young adult women.In the full Easterlin model, relative income is determined by the size of the young adult cohort relative to that of prime aged adults, both measured contemporaneously (Easterlin 1980b). …

A SIMPLE VARIABLE SELECTION TECHNIQUE FOR NONLINEAR MODELS

Applying nonparametric variable selection criteria in nonlinear regression models generally requires a substantial computational effort if the data set is large. In this paper we present a selection technique that is computationally much less demanding and performs well in comparison with methods currently available. It is based on a polynomial approximation of the nonlinear model. Performing the selection only requires repeated least squares estimation of models that are linear in parameters. The main limitation of the method is that the number of variables among which to select cannot be very large if the sample is small and the order of an adequate polynomial at the same time is high. Large samples can be handled without problems.

Exact finite-sample relative efficiency of suboptimally weighted least squares estimators in models with ordered heteroscedasticity

This paper is concerned with regression models having heteroscedastic errors whose variances are not precisely known but can be ordered by size. The object is to use this limited information to devise regression coefficient estimators which are more efficient in finite samples than the ordinary least squares (OLS) estimator. Three common relative efficiency measures (determinant, trace, and extreme eigenvalue) are considered. A class of weighted least squares (WLS) estimators is proved to be efficient relative to the OLS estimator on the determinant (generalized variance) measure. Only a partial result can be obtained for the trace measure. A counterexample shows that no similar results are possible for the eigenvalue measure under the weak conditions of this paper. The efficiency value of additional prior information on variance is also investigated.

Two-step two-stage least squares estimation in models with rational expectations

The paper develops a two-step estimator for use in rational-expectations models with autocorrelated residuals and predetermined, but not strictly exogenous, instruments. The estimator extends the applicability of McCallum's (1976) error-in-variablesapproach to estimating such models, and is asymptotically efficient in a class of intrumental-variables estimators. As an application we use instrumental-variables techniques to estimate Taylor's (1979) rational-expectations macroeconomic model of the United States.