Random Sampling

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Random Sampling

IV. Random Sampling V. Estimators VI. Large Sample Estimators

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Lecture Note
Cornell University
Econ 3120 - Applied Econometrics

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Lecture 4 Outline of Last Lecture I. Covariance II. Conditional Expectation III. Normal Distribution Outline of Current Lecture IV. Random Sampling V. Estimators VI. Large Sample Estimators Current Lecture Random sampling This lecture discusses how to estimate the mean from a population of interest. Normally it is impractical or impossible to examine the whole population. If we could do this, we could simply take the mean of the population and we’d be done. Given that we can only examine a sample, we have to use statistical inference to 1) estimate the parameters that we care about (in this case it’s the mean) and 2) test hypotheses about these parameters. To estimate the mean (or any other parameter of interest), we’ll focus on random samples from the population. Formally, a random sample is a set of independent and identically distributed (i.i.d) random variables {Y1,Y2,...,Yn} that share a probability density function f(y;θ). For the first part of the lecture, we’re going to assume that the population is distributed Normal(µ,σ 2 ), an assumption we will relax when we discuss large sample properties later in the lecture. 2 Estimators Once we have our random sample, we can use it to estimate the mean (or any other parameter of interest). An estimator is a rule (or a function) that uses the outcome of random sampling to assign a value to the parameter based on the sample. θˆ = h(Y1,Y2,...,Yn) For the mean, the most obvious estimator is the sample mean or sample average: Y¯ = 1 n n ∑ i=1 Yi We often write estimates by putting a “hat” on top of the parameter, i.e., µˆ1 = Y¯. Note that this is just one possible way to estimate the mean. We could estimate the mean by simply looking at the first observation, i.e., µˆ2 = Y1. In fact, an estimator doesn’t even need to depend on the random sample at all. µˆ3 = 4 is a perfectly valid estimator. We’ll see, though, that the sample mean is preferable because it is unbiased and efficient. Estimators, like the samples they come from, have their own distributions, called sampling distributions. Estimators have distributions since they are simply functions of realizations of random variables, and we have seen that functions of random variables have distributions. 2.1 Unbiasedness An estimator is unbiased if its expectation equals the parameter of interest, that is, E(θˆ) = θ The bias of an estimator is computed as the difference between the expectation and the true parameter: Bias(θˆ) = E(θˆ)−θ Example: Show that µˆ1 = Y¯ and µˆ2 = Y1 are unbiased estimates of the mean, and that µˆ3 = 4 is biased. 2.2 Sample variance and sampling variance of estimators In order to conduct inference (that is, say something about how accurate our estimator is), we need to be able to estimate the variance of a Econ 3120 1st Edition

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