Econ 399 Chapter6b - 6.4Prediction...

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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 = = = = + + + + = θ β -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 θ 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 β - - - - = ... 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 β θ -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
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Econ 399 Chapter6b - 6.4Prediction...

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