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Example


Population {1, 2, 3, 4} Draw samples {Y1, Y2} with sample size n=2 each time. Total possible samples {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}
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The sampling distribution of
Y
Y is a random variable, and its properties are determined by the sampling distribution of Y
The individuals in the sample are drawn at random. Thus the values of (Y1,…, Yn) are random Thus functions of (Y1,…, Yn), such as Y , are random: had a different sample been drawn, they would have taken on a different value The distribution of Y over different possible samples of size n is called the sampling distribution of Y . The mean and variance of Y are the mean and variance of its sampling distribution, E(Y ) and var(Y ). The concept of the sampling distribution underpins all of econometrics.
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Estimators and Estimates

Typically, we can’t observe the whole population, so we must make inferences based on the estimate from a random sample An estimator is just a mathematical formula for estimating a population parameter from sample data An estimate is the actual value the formula produces from the sample data
Econometrics
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Econometrics
Instructor: Chui Chin Man (崔展文) Office: 511-2 (嘉庚二) E-mail: cmchui@
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Course Requirement
Lectures: Tuesday 2:30-5:30 p.m. (Room 501嘉庚二 )
We will assume simple random sampling Choose and individual (district, entity) at random from the population Randomness and data Prior to sample selection, the value of Y is random because the individual selected is random Once the individual is selected and the value of Y is observed then Y is just a number – not random The data set is (Y1, Y2,…, Yn), where Yi = value of Y for the ith individual (district, entity) sampled
S2
2 ( Y Y ) i 1 i n
n 1
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Estimation Y is the natural estimator of the mean. But: (a) What are the properties of Y ? (b) Why should we use Y rather than some other estimator? Y1 (the first observation) maybe unequal weights – not simple average median(Y1,…, Yn) The starting point is the sampling distribution of Y …
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Population and Sample



Population — a population is the group of all items of interest to a statistics practitioner. — frequently very large; sometimes infinite.
E.g. 13 billion people in China


Sample — A sample is a set of data drawn from the population. — Potentially very large, but less than the population.


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Commonly used Estimators

We use sample mean to estimate the population mean
Y i 1Yi
n

We use sample variance to estimate the population variance
Байду номын сангаас
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Previous example



Population {1, 2, 3, 4} population mean=2.5 Draw samples {Y1, Y2} with sample size n=2. {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} Sample means based on the 6 samples {1.5, 2, 2.5, 2,5, 3, 3.5} is the sample distribution of Y Expected value and variance of Y ?
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Lecture 1 Quick review of some important concepts in statistics
(Appendix C of Wooldridge)
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Outline

Sample distribution Estimation and estimator Properties of estimator
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Distribution of Y1,…, Yn under simple random sampling
Because individuals {Yi} are selected at random, we further make assumptions that {Yi}, i = 1,…, n, are independently distributed {Yi}, i = 1,…, n, come from the same distribution, that is, {Yi} are identically distributed That is, under simple random sampling, {Yi}, i = 1,…, n, are independently and identically distributed (i.i.d.) This framework allows rigorous statistical inferences about moments of population distributions using a sample of data from that population …
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The mean and variance of the sampling distribution of Y
General case – that is, for Yi i.i.d. from any distribution, not just Bernoulli: 1 n 1 n 1 n mean: E(Y ) = E( Yi ) = E (Yi ) = Y = Y n i 1 n i 1 n i 1
E.g. a sample of 4 million people from Xiamen
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Statistical Inference

Statistical inference is the process of making an estimate, prediction, or decision about a population parameter based on a sample statistic.

Grading

Assignments and computer exercises One mid-term test One final examination
10% 40% 50%
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Textbook

Wooldridge, J. “Introductory Econometrics: A modern approach” 3 edition