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Unformatted text preview: Stat 104: Quantitative Methods for Economists Class 28: Regression Output and Hypothesis Testing 1 All we have done so far is learned how to fit a line to some (X,Y) data. There really hasnt been very much statistics involved at all. Introduction In this section we detail some statistical theory necessary to discuss how good a line we have and how accurate predictions using the fitted line will be. 2 Many relationships in science are exact: distance = rate x time E = mc 2 Exact versus Inexact Relationships: Force = mass x acceleration But most relationships in business are inexact: How do we express uncertainty in our relationships ? 3 Here is the Accord regression output again: 4 For example, we know that there isnt an exact relationship between mileage of a car and its price (how do we know this, by the way?) price = $17067$.06(odometer) That is, not every Accord with 30000 miles will sell for $15267. Some will sell for more, and some houses will sell for less. A more realistic statement is that Average Car Price = $17067$.06(odometer) This is a main point about regression: we model the average of something rather than the something itself. 5 The Average Line s The regression line should be viewed as the average value of Y for a given X, or in symbols E(YX). 15500 16000 6 13500 14000 14500 15000 20000 30000 40000 50000 Odometer Fitted values Price In symbols, this line of averages can be expressed as: E Y X X (  ) = + 1 where E(YX) is the expected value (average) of Y r a given X value. for a given X value. In our housing example, we might think of as the average price of cars with mileage X. (  ) E Y X 7 Another way of explaining this concept is to write the model as Y X or = + + 1 Y E Y X = + (  ) where refers to some random noise if there were no noise in the system, what would be the relationship between X and Y ? 8 To describe what the likely value of may be, we let it be a normally distributed random variable: ~ ( , ) N 2 ean of 0. ariance of 2 mean of is 0. Sometimes Y will be above the line, sometimes below. variance of is If is small, will tend to be small (close to 0). If is big, could be big (far from 0). 9 ASSUMPTIONS of the Simple Linear Regression Model Y X = + + 1 1 + X the part of Y related to X the part of Y unrelated to X: ~ ( , ) N 2 Note: the distribution of does not depend on X is independent of X ....
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This note was uploaded on 03/27/2012 for the course STATS 104 taught by Professor Michaelparzen during the Fall '11 term at Harvard.
 Fall '11
 MichaelParzen
 Statistics

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