C7D6 - 1 We will illustrate the heteroscedasticity theory...

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Unformatted text preview: 1 We will illustrate the heteroscedasticity theory with a Monte Carlo simulation in which Y = 10 + 2 X , the data for X are the integers from 5 to 54, and u = X ε , where ε is iid N(0,1) (identically and independently distributed, drawn from a normal distribution with zero mean and unit variance). HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION Y i = 10 + 2.0 X i + u i X i = {5,6, ..., 54} u i = X i ε i ε i ~ N(0,1) 20 40 60 80 100 120 140 160 180 10 20 30 40 50 1 standard deviation of u 2 The blue circles give the nonstochastic component of Y in the observations. The lines give the points one standard deviation of u above and below the nonstochastic component of Y and show how the distribution of Y spreads in the vertical dimension as X increases. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION Y i = 10 + 2.0 X i + u i X i = {5,6, ..., 54} u i = X i ε i ε i ~ N(0,1) 20 40 60 80 100 120 140 160 180 10 20 30 40 50 1 standard deviation of u 3 The standard deviation of u i is equal to X i . The heteroscedasticity is thus of the type detected by the Goldfeld–Quandt test and it may be eliminated using weighted least squares (WLS). We scale through by X , weighting observation i by multiplying it by 1/ X i . HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION Y i = 10 + 2.0 X i + u i X i = {5,6, ..., 54} u i = X i ε i ε i ~ N(0,1) 20 40 60 80 100 120 140 160 180 10 20 30 40 50 1 standard deviation of u 4 The diagram shows the results of estimating the slope coefficient using WLS and OLS for one million samples. HETEROSCEDASTICITY: MONTE CARLO ILLUSTRATION 0.5 1 1.5 2 0.5 1 1.5 2 2.5 3 3.5 4 Y i = 10 + 2.0 X i + u i X i = {5,6, ..., 54} u i = X i ε i ε i ~ N(0,1) Slope coefficient estimated using WLS...
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This note was uploaded on 05/26/2010 for the course ECON 301 taught by Professor Öcal during the Spring '10 term at Middle East Technical University.

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C7D6 - 1 We will illustrate the heteroscedasticity theory...

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