a425_form - Functional Form & Qualitative Variables APS...

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Functional Form & Qualitative Variables APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2001-2010 1 APS 425 – Winter 2010 Functional Form and Qualitative Variables Instructor: G William Schwer Instructor: G. William Schwert 275-2470 schwert@schwert.simon.rochester.edu Topics • Transformations to linearity • Dummy variables Interaction variable • Interaction variables
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Functional Form & Qualitative Variables APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2001-2010 2 Transformations to Linearity • Linear regression assumes that Y is linearly related to X 1 , X 2 , etc. with an additive error term • Scatter diagrams can help us understand whether the variables we are considering need to be transformed to create a linear relation – e.g., if we take logarithms of the variables (assuming that they are all positive numbers), this implies a constant elasticity relation between Y and X and log Y = 0 + 1 log X + => Y = 0 X exp( ) where log is the natural logarithm and exp(•) is the exponential function Transformations to Linearity Quantity Demanded at Different Prices $6 $8 $10 $12 rice per Unit Quantity Demanded at Different Prices (log transformations) 1.50 2.00 2.50 Price per Unit $0 $2 $4 0 200 400 600 800 1000 1200 Units Demanded at a Given Price Pr 0.00 0.50 1.00 012345678 Log of Units Demanded at a Given Price Log of
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Functional Form & Qualitative Variables APS 425 - Advanced Managerial Data Analysis (c) Prof. G. William Schwert, 2001-2010 3 Logs as Percent Changes • Suppose that log(Y) is linearly related to some variable X log Y = 0 + 1 X + Then the slope coefficient 1 measures the percent change in Y caused by a change in X Fact log (1 + r) r for |r|< 15 Fact: log (1 + r) r for | r | < .15 To the extent that the distribution of % changes is more likely to have a constant variance than the distribution of changes, the log transformation can help solve heteroskedasticity problems Logs as Percent Changes Logs as approximate percentage ln(1+r ln(1+r Excel function ln(•) is the r ln(1+r) r ln(1+r) -0.150 -0.163 0.150 0.140 -0.140 -0.151 0.140 0.131 -0.130 -0.139 0.130 0.122 -0.120 -0.128 0.120 0.113 -0.110 -0.117 0.110 0.104 -0.100 -0.105 0.100 0.095 -0.090 -0.094 0.090 0.086 -0.080 -0.083 0.080 0.077 Excel function ln () is the natural logarithm -0.070 -0.073 0.070 0.068 -0.060 -0.062 0.060 0.058 -0.050 -0.051 0.050 0.049 -0.040 -0.041 0.040
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a425_form - Functional Form &amp; Qualitative Variables APS...

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