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stat104lec28v2_1up

# stat104lec28v2_1up - Stat 104 Quantitative Methods for...

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Stat 104: Quantitative Methods for Economists Class 28: Regression Output and Hypothesis Testing 1

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All we have done so far is learned how to fit a line to some (X,Y) data. There really hasn’t been very much statistics involved at all. In this section we detail some statistical theory Introduction 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 Force = mass x acceleration Exact versus Inexact Relationships: But most relationships in business are inexact: How do we express uncertainty in our relationships ? 3

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Here is the Accord regression output again: 4
For example, we know that there isn’t 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

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The Average Line square6 The regression line should be viewed as the average value of Y for a given X, or in symbols E(Y|X). 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 ( | ) = + β β 0 1 where E(Y|X) is the expected value (average) of Y 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

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Another way of explaining this concept is to write the model as Y X or = + + β β ε 0 1 Y E Y X = + ε ( | ) where ε 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 ε may be, we let it be a normally distributed random variable: ε σ ~ ( , ) N 0 2 mean of ε is 0. variance of ε is σ 2 Sometimes Y will be above the line, sometimes below. If σ is small, ε will tend to be small (close to 0). If σ is big, ε could be big (far from 0). 9

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ASSUMPTIONS of the Simple Linear Regression Model Y X = + + β β ε 0 1 β β 0 1 + X the part of Y related to X ε the part of Y unrelated to X: ε σ ~ ( , ) N 0 2 Note: the distribution of ε does not depend on X ε is independent of X .
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stat104lec28v2_1up - Stat 104 Quantitative Methods for...

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