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In this paper, we derive the finite sample bias and MSE of the fully aggregated estimator (FAE) for the stationary AR(1) model with intercept, which is proposed by Han, Phillips, and Sul (2011). Our analytical results show why the FAE is less biased than the ordinary least square estimator in finite sample case and is not biased by non-normality of error distribution and by intercept term at least O(1/T ), where T is sample size. We also propose a second order unbiased FAE using the analytical result. Finally, we examine the Monte Carlo simulation and show that it is consistent with the theoretical results.
The proposed paper studies the bias in the two-stage least squares, or 2SLS, estimator that is caused by the compliance-effect covariance (hereafter, the compliance-effect bias). It starts by deriving the formula for the bias in an infinite sample (i.e., in the absence of finite sample bias) under different circumstances. Specifically, it considers the following cases: (1) A single site study with one mediator; (2) A multiple site study with one mediator; and (3) A multiple site study with multiple mediators. The formulas demonstrate how the magnitude of the compliance-effect bias varies with different parameters (e.g., compliance-effect correlation, mean and variance of the compliance and effect) in infinite samples. However, as the situation under consideration gets more complicated, the bias formula quickly becomes intractable. The second part of the paper, therefore, uses simulations to demonstrate the relationship between the compliance-effect bias and various parameters, as well as the behavior of the estimated 2SLS standard errors. Furthermore, the simulation exercise assesses how the compliance-effect bias interacts with the finite sample bias when the analysis sample is small or when the instrument is weak. The paper also uses simulations to compare the properties of the 2SLS estimator with those of the ordinary least squares (OLS) estimator in the presence of the compliance-effect bias, the finite sample bias, and the omitted variable bias. (Contains 6 footnotes.).
This paper investigates the finite sample distribution of the least squares estimator of the autoregressive parameter in a first-order autoregressive model. Uniform asymptotic expansion for the distribution applicable to both stationary and nonstationary cases is obtained. Accuracy of the approximation to the distribution by a first few terms of this expansion is then investigated. It is found that the leading term of this expansion approximates well the distribution. The approximation is, in almost all cases, accurate to the second decimal place throughout the distribution. In the literature, there exists a number of approximations to this distribution which are specifically designed to apply in some special cases of this model. The present approximation compares favorably with those approximations and in fact, its accuracy is, with almost no exception, as good as or better than these other approximations. Convenience of numerical computations seems also to favor the present approximations over the others. An application of the finding is illustrated with examples.
The least squares estimator of the autoregressive parameter, LS((gamma)), in a first-order stochastic difference equation with independent, identically distributed random innovations is known to be asymptotically unbiased, efficient and consistent (as T ( -->) (INFIN) or (sigma) ( -->) 0) under the proper model specification. Further, LS((gamma)) has a limiting normal distribution around the true parameter, (gamma), if the random innovations are drawn from a normal population. These properties are not observed, however, in sample sizes that are typical of economic time series.
We study the consistency, robustness and efficiency of parameter estimation in different but related models via semiparametric approach. First, we revisit the second-order least squares estimator proposed in Wang and Leblanc (2008) and show that the estimator reaches the semiparametric efficiency. We further extend the method to the heteroscedastic error models and propose a semiparametric efficient estimator in this more general setting. Second, we study a class of semiparametric skewed distributions arising when the sample selection process causes sampling bias for the observations. We begin by assuming the anti-symmetric property to the skewing function. Taking into account the symmetric nature of the population distribution, we propose consistent estimators for the center of the symmetric population. These estimators are robust to model misspecification and reach the minimum possible estimation variance. Next, we extend the model to permit a more flexible skewing structure. Without assuming a particular form of the skewing function, we propose both consistent and efficient estimators for the center of the symmetric population using a semiparametric method. We also analyze the asymptotic properties and derive the corresponding inference procedures. Numerical results are provided to support the results and illustrate the finite sample performance of the proposed estimators.
I derive the approximate bias and mean squared error of the least squares estimator of the autoregressive coefficient in a stationary first-order dynamic regression model, with or without an intercept, under a general error distribution. It is shown that the effects of nonnormality on the approximate moments of the least squares estimator come into play through the skewness and kurtosis coefficients of the nonnormal error distribution.