Econ 399 Chapter6b - 6.4Prediction However,...

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6.4 Prediction -We have already seen how to make predictions about our dependent variable using our OLS estimates and values for our independent variables -However, these estimates are subject to sampling variation -a confidence interval is therefore often preferable to a point estimate
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6.4 Prediction -Assume we want to obtain the value of y (denoted by θ) given certain independent values x j =c j : ) ,..., , | ( (6.27) ... 2 2 1 1 2 2 1 1 0 k k k k c x c x c x y E c c c -since we don’t have actual values for B J , our estimator of θ becomes: (6.29) ˆ ... ˆ ˆ ˆ ˆ 2 2 1 1 0 k k c c c
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6.4 Prediction -However, due to uncertainty, we may prefer a confidence interval estimate for θ: ) ˆ ( * ˆ se t CI -Since we don’t know the above standard error, similar to F tests we construct a new regression using the calculation: k k c c c ... 2 2 1 1 0
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6.4 Prediction -substituting this calculation into the population regression function gives us: (6.30) u ) ( ... ) ( ) ( 2 2 2 1 1 1 k k k c x c x c x y -In this regression, the intercept gives us the needed predicted value and intercept to construct our confidence interval -note that all our slope coefficients and their standard errors must be the same as the original regression
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6.4 Prediction Notes 1) Note that the variance of this prediction will be smallest at the mean values of x j -This is due to the fact that we have
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