Analysis of Sampling Distributions and Fixed Effects Methods

This assignment solution delves into key statistical concepts, beginning with an explanation of sampling distributions, emphasizing their theoretical nature and the process of obtaining them through random sampling from a population. It highlights the desired characteristics of an estimator, focusing on unbiasedness and consideration of sampling variability. The solution then explores the fixed effects method in detail, outlining its role in reducing bias by controlling stable characteristics and within-person variation, while acknowledging its limitations regarding between-person variation. Finally, it examines the method of instrumental variables, explaining its use in estimating causal relationships when controlled experiments are not feasible, and discussing its application in addressing issues like variable omission and measurement error in OLS regression, providing an example of its use in addressing ability bias in education return analysis.

Part 1: Explain the concept of a sampling distribution and what sampling distribution
characteristics are desired in an estimator.
Answer: Sampling Distribution is concerned with the probability distribution. The sampling
distribution is theoretical in nature. It is obtained with the help of substantial size of the
population. The samples are collected on random basis (Wilks, 1938). The sampling distributions
are achieved with the help of certain set of data that are further processed using various statistical
tools such as mean and mode (Hirose et al, 2015).
Three balls have been considered with Number 1, 2, and 3 written on them. Selection of two
balls has been done at random and below table developed showing entire possible outcomes:
The frequencies of the mean are as:

The figure below has been developed based on the table above showing the distribution. This
figure shows the sampling distribution of the mean:
It is expected that the estimator is not biased, that is the estimator should not undermine the
parameter and should not also over-estimate the same. The sampling variability should be under
consideration.

Part 2: In detail, explain the method of fixed effects and why it is used.
Answer: The fixed effects method considers the almost entire data provision and the use of fixed
effect method leads to the reduction in the bias. It helps in controlling the stable characteristics
which leads to the elimination of the substantial bias. Fixed method considers within-person
variation and tries to eliminate it (Bai, 2013). However, it rarely considers the between-person
variation. Still it is beneficial to use it as it has been found that the major identification of
variation has been done in within-person than between-person. It does negatively impact the
sampling variability and is a kind of trade-off (Gunasekara et a, 2014). Fixed methods are largely
suitable for non-experimental data as it helps in substantial reduction of bias.
Let us consider the regression analysis of the income of a household and the satisfaction level of
the parents of those households regarding the education of the child. In this, if the simple
regression result is considered then it can be found that the parents are either happy or not happy
but the hidden variables will not be there for assistance to the researchers. This issue can be
handled by the consideration of within-person factors.
Part 3: Explain in detail the method of instrumental variables, the reasons for using
instrumental variables and the potential problems of using this method.
Answers: The method of instrumental variables is concerned with the estimation of the causal
relationships whenever the controlled experiments are not possible. The instrumental variables
estimator helps in obtaining consistent estimation of the parameter (Cawley and Meyerhoefer,
2012). The use of instrumental variables is done for the solution of the problems that are
identified in the OLS regression, such as variable bias omission, and error in measurement.

Example of this can be, “Ability Bias in the Returns to Education” and it with the usual
regression method might result into biased estimation. However, the unbiased estimates can be
achieved using the instrumental variables.
References
Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite
hypotheses. The Annals of Mathematical Statistics, 9(1), 60-62.
Hirose, Y., Preedalikit, K., Liu, I., Sibanda, N., & Fernández, D. (2015). Large sample
distribution of estimators in a mixture of semiparametric models.
Bai, J. (2013). Fixed‐Effects Dynamic Panel Models, a Factor Analytical Method. Econometrica,
81(1), 285-314.
Gunasekara, F. I., Richardson, K., Carter, K., & Blakely, T. (2014). Fixed effects analysis of
repeated measures data. International journal of epidemiology, 43(1), 264-269.
Cawley, J., & Meyerhoefer, C. (2012). The medical care costs of obesity: an instrumental
variables approach. Journal of health economics, 31(1), 219-230.