Lecture_5__Prof._Arkonac's__slides_(Ch_4.4_to_5.4)

Lecture_5__Prof._Arkonac's__slides_(Ch_4.4_to_5.4) -...

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Regression with a single Regressor: Hypothesis Testing and Confidence Intevals Lecture 5 Prof: Seyhan Erden Arkonac, PhD Problem Set 1 is due NOW! Problem Set 2 is due on Sept 28 th . 1
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TA Information: Naihobe Gonzalez E-mail: ndg2109@columbia.edu Office Hours: Thurs 12-1 (Uris Library), Recitation: Thurs 11-11:50 (PUP 424) TA Information: Ju Hyun Kim E-mail: jk3201@columbia.edu Office Hours: Recitation: Fri 2:10-3PM(PUP 424), Office Hours: Fri 3:10- 4:10PM(Lehman) TA Information: WooRam Park E-mail: wp2135@columbia.edu Office Hours: Office hours : Thurs 2:00~3:00 IAB 1006A Recitation Thurs 3:10~4:00 IAB 403 TA Information: Ran Huo E-mail: rh2346@columbia.edu Office Hours: Recitation: Thursday 12:00-12:50 404IAB; Office Hour: Wednesday 1-2 1006A IAB TA Information: Shreya Agarwal E-mail: sa2628@columbia.edu Office Hours: Mon 12:30pm - 1:30 pm (Uris Library Common Area) Recitation: Fri 11:00am - 11:50am 2
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Stata sessions: Thursday 9/23, 11-11:50am, SCH 558 (Naihobe) Thursday 9/23, 12-12:50pm, SCH 558 (Ran) Thursday 9/23, 4:10-6pm, SCH 558 (WooRam) Friday 9/24, 2:10-3pm, SCH 558 (Ju Hyun) Every week starting 9/24 11-1150am SCH 558 (Shreya)
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4 The Least Squares Assumptions (SW Section 4.4) What, in a precise sense, are the properties of the OLS estimator? We would like it to be unbiased, and to have a small variance. Does it? Under what conditions is it an unbiased estimator of the true population parameters? To answer these questions, we need to make some assumptions about how Y and X are related to each other, and about how they are collected (the sampling scheme) These assumptions – there are three – are known as the Least Squares Assumptions.
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5 The Least Squares Assumptions Y i = 0 + 1 X i + u i , i = 1,…, n 1. The conditional distribution of u given X has mean zero, that is, E ( u | X = x ) = 0. This implies that 1 ˆ is unbiased 2. ( X i ,Y i ), i =1,…, n , are i.i.d. This is true if X, Y are collected by simple random sampling This delivers the sampling distribution of 0 ˆ and 1 ˆ 3. Large outliers in X and/or Y are rare. Technically, X and Y have finite fourth moments Outliers can result in meaningless values of 1 ˆ
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6 Least squares assumption #1: E ( u | X = x ) = 0. Example: Test Score i = 0 + 1 STR i + u i , u i = other factors What are some of these “other factors”? Is E ( u | X = x ) = 0 plausible for these other factors? For any given value of X, the mean of u is zero :
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7 A benchmark for thinking about this assumption is to consider an ideal randomized controlled experiment : X is randomly assigned to people (students randomly assigned to different size classes; patients randomly assigned to medical treatments). Randomization is done by computer – using no information about the individual. Because X is assigned randomly, all other individual characteristics – the things that make up u – are independently distributed of X Thus, in an ideal randomized controlled experiment, E ( u | X = x ) = 0 (that is, LSA #1 holds) In actual experiments, or with observational data, we will need to think hard about whether E ( u | X = x ) = 0 holds.
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This note was uploaded on 11/10/2011 for the course ECON 3142 taught by Professor Arkonac during the Spring '11 term at Columbia.

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Lecture_5__Prof._Arkonac's__slides_(Ch_4.4_to_5.4) -...

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