Gujarati 7.15 d

[ArpitB]

Hi David,

This is from Gujarati Q.07.15 d
I am having little difficulty in understanding this solution.


What does Adjustment mean here ? and how did we calculated other things? Sorry it might be very trivial question, but i am unable to infer it properly.

Thank you,

Regards,
Arpit

 

[David Harper CFA FRM]

Hi @ArpitB

Please note it this is well beyond exam scope, although a great question. (I tried to google to find a simple reference but cannot find...). The confidence interval around the predicted Y value in an OLS regression is not a set of parallel lines; i.e., the confidence interval is not a constant width. The interval is narrow at the unconditional mean of the independent (X) value but wider as X value varies from the mean. In this way, as you would expect, the key determinant (as reflected in the XLS) of the confidence interval is the variance of the error, or the square of the familiar standard error. A common mistake is to use this standard error unconditionally to inform a confidence interval, regardless of the X value upon which the prediction is based.

The unconditional mean of X, above, is 453.67; i.e., the average of the X values. The predicted Y given by an X of 460, which is above the mean X, returns a confidence interval of width 2.07 (505.56 - 503.49). If I increase the X to 470 (i.e., further from the mean X), the confidence interval shifts to 509.91 (lower) and 513.42), which is an increase in the width, up to 3.51. The regression's standard error remains the key informant, but the adjustment is widening the confidence interval as the X moves away from its mean value (in a sense, as the X moves toward its sample boundry, the prediction is becoming progressively less certain. The confidence interval band is a set of concave/convext curves). I hope that explains!