It is discarded in calculating the coefficients.26
An illustrative example using Venn diagramsYXBlueZRedGreenOrangeBecause the contribution in red is not discernible whether it comes from Z or X it is not used and only unique variation in blue and greenare used leading tounbiased estimates.In this example are the SLR models of Y on Z and Y on X producing unbiased estimates?No, both produce biased estimates. Here including the variables in the MLR leads to unbiased estimates at the cost of higher variances which is almost always the case.27
An illustrative example using Venn diagramsYXBlue00000ZGreenHow about now?Are the SLR models producing unbiased estimates?Yes. There is no correlation in the independent variables now. These variables are said to be orthogonal.28
An illustrative example using Venn diagramsYXBlue00000ZGreenWhat is this illustrating?There is now a great deal correlation in the independent variables.RedThis is foreshadowing the MLR assumptions but you can easily see that little unique variation is left for the estimation of the coefficients on X and Z.And using similar logic as before the variances on these estimators will be quite high.29
An illustrative example using Venn diagramsYXYellowBlueZRedGreenOrangeConceptually speaking what do you think R2looks like here?Here the ratio of the (Blue + Red + Green) to the ratio (Blue + Red + Green + Yellow) is R2.30
An illustrative example using Venn diagramsYXYellowBlueZRedGreenOrangeFinally, what is Yellow representing?31
An illustrative example using Venn diagramsYXYellowBlueZRedGreenOrangeWhich implies what if we remove an independent variable?That’s right the error variance is larger.32
You've reached the end of your free preview.
Want to read all 32 pages?