OHCh1-B

OHCh1-B - Chapter 1 An Overview of Regression Analysis...

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Chapter 1: An Overview of Regression Analysis Lecture 2 Revised on September 25, 2007

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1 The Population Regression 1.1 What is a regression? Example (Costs of an ice-cream stand): Imagine that you are thinking of starting an ice-cream stand. You are interested in the expected cost. Even if you do not sell any ice cream, you need to pay the rent for the stand. When you sell an ice cream bar, it cost you the wholesale price of the bar. There are other costs for electricity that vary with the temperature. The expected cost increases as the quantity of ice cream bars you sell. Example(Demand for a used CD): Imagine that you own a used music CD store and just got 24 copies of a CD by the Beatles. You are thinking
about setting the price for it. The expected value of demand for the music CD is likely to fall as you set its price higher. These examples show that the expected value of a random variable Y may depend on the value of another variable X . In that case, we write the expected value of Y given X as E ( Y j X ). E ( Y j X ) is a function of X . E ( Y j X ) = f ( X ) (1) This is called the population regression . Y : The dependent variable X : The independent (or explanatory) variable

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In this course, we assume that f ( X ) is a linear function of X : E ( Y j X ) = & 0 + 1 X (2) This is a strong assumption. However, we can often make a nonlinear Example: E ( Y j X ) = & 0 + 1 X 2 (3)
Then create a new variable Z = X 2 (4) then E ( Y j X ) = & 0 + 1 Z (5)

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1.2 In Equation (2), & 0 and 1 are called the , or regression coef- ±cients . & 0 : the constant or intercept term 1 : the . Example (Costs of an ice-cream stand-continued): Imagine that the cost is given by the rent for the stand of \$30 and the wholesale price of each bar of \$0.50. Let Y be the cost and X be the number of ice cream bars sold. Then
E ( Y j X ) = 30 + 0 : 5 X (6) The constant term is 30. This means that the cost is \$30 if the number of the cost increase by \$0.5 when the number of ice cream bars sold increases by one. In Economics, \$0.5 in this example is called marginal cost. On

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This note was uploaded on 07/17/2008 for the course ECON 444 taught by Professor Ogaki during the Fall '07 term at Ohio State.

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OHCh1-B - Chapter 1 An Overview of Regression Analysis...

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