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Lecture 5 Prof. Arkonac's slides (Ch 4.4 to 5.4) for ECO 4000

Lecture 5 Prof. Arkonac's slides (Ch 4.4 to 5.4) for ECO 4000

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Linear Regression with One Regressor ECO 4000, Statistical Analysis for Economics and Finance Fall 2010 Lecture 5 Prof: Seyhan Arkonac, PhD 1

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2 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.
3 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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4 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 :
5 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. Least squares assumption #1, ctd.

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6 Least squares assumption #2: ( X i , Y i ), i = 1,…, n are i.i.d. This arises automatically if the entity (individual, district) is sampled by simple random sampling: the entity is selected then, for that entity, X and Y are observed (recorded). The main place we will encounter non-i.i.d. sampling is when data are recorded over time (“time series data”) – this will introduce some extra complications.
7 Least squares assumption #3: Large outliers are rare Technical statement: E ( X 4) < and E ( Y 4) < A large outlier is an extreme value of X or Y On a technical level, if X and Y are bounded, then they have finite fourth moments. (Standardized test scores automatically satisfy this; STR , family income, etc. satisfy this too).

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Lecture 5 Prof. Arkonac's slides (Ch 4.4 to 5.4) for ECO 4000

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