Assumption MLR 3 Notes No Perfect Collinearity Perfect

Assumption MLR. 3 Notes (No Perfect Collinearity) Perfect Collinearity can exist if: 1) One variable is a constant multiple of another 2) Logs are used inappropriately 3) One variable is a linear function of two or more other variables In general, all of these issues are easy to fix, once they are identified.

Assumption MLR. 3 Notes (No Perfect Collinearity) 1) One variable is a constant multiple of another -ie: Assume that Joe only drinks coffee at work, and drinks exactly 3 cups of coffee every day he works. Therefore: -including both coffee and work in the regression would cause perfect collinearity; the regression would fail

Assumption MLR. 3 Notes (No Perfect Collinearity) 2) Logs are used inappropriately -Consider the following equation and apply log rules: -a variable is included twice, causing an inability to estimate B 1 and B 2 separately -note that geek and geek 2 could both have been used, as they are not linearly related

Assumption MLR. 3 Notes (No Perfect Collinearity) 3) One variable is a linear function of two or more other variables -Consider a teenager who spends all their income on movies and clothes: -if income and expenditures on both movies and clothes are in the regression, perfect collinearity exists and the regression fails

3. 3 Fixing Multicollinearity Drop a variable from the model. 1. If one variable is a multiple of another, it adds nothing to consider it twice 2. Ditto for logs 3. If the elements of a sum are in a regression, the sum itself is redundant. (Alternately one of the elements can be omitted).

3. 3 Multicollinearity and N -Assumption MLR. 3 can also fail is N is too small -in general, MLR. 3 will always fail if n<k+1 (the number of parameters) -even if n>k+1, MLR. 3 may fail due to a bad sample Next we have the most important assumption for proving OLS’s unbiasedness:

Assumption MLR. 4 (Zero Conditional Mean) The error u has an expected value of zero given any values of the independent variables. In other words,

Assumption MLR. 4 Notes (Zero Conditional Mean) MLR. 4 fails if the functional relationship is misspecified: 1) A variable is not included the correct way -ie: consumption is included in the regression but not consumption 2 and the true relationship is quadratic 2) A variable is included the incorrect way -ie: log(consumption) is included in the regression but consumption is the true relationship -In these cases, the estimators are biased

Assumption MLR. 4 Notes (Zero Conditional Mean) MLR. 4 also fails if one omits an important factor correlated with any x -this can be due to ignorance or data restrictions MLR. 4 also fails due to: 1) Measurement error (ch. 15) 2) An independent variable is jointly determined with y (ch. 16)

Assumption MLR. 4 Notes (Zero Conditional Mean) When MLR. 4 holds, we have EXOGENOUS EXPLANATORY VARIABLES When MLR. 4 does not hold (xj is correlated with u), xj is an ENDOGENOUS EXPLANATORY VARIABLE

MLR. 3 vs. MLR. 4 (Cage Match) MRL. 3 deals with relationships among independent variables -if it fails, OLS cannot run MLR. 4 deals with relationships between u and independent variables -it is easier to miss -it is more important

Theorem 3. 1 (Unbiasedness of OLS) Under assumptions MLR. 1 through MLR. 4, For any values of the population parameter Bj. In other words, the OLS estimators are unbiased estimators of the population parameters.

3. 3 Is OLS valid? • IF OLS runs (B estimates are found) -MLR. 3 is satisfied • IF the sample is random -MLR. 2 is satisfied • IF we have some reason to suspect a true relationship -MLR. 1 is valid • Therefore if we believe MLR. 4 holds true -OLS is valid!

3. 3 What is unbiasedness? • Our estimates of Bhat are all numbers -numbers are fixed, and cannot be biased or unbiased • MLR. 1 through MLR. 4 comments on the OLS PROCEEDURE -Is our assumptions hold true, our OLS PROCEEDURE is unbiased. • In other words: “we have no reason to believe our estimate is more likely to be too big or more likely to be too small. ”

3. 3 Irrelevant Variables in a Regression Model Including independent variables that do not actually affect y (irrelevant variables) is also called OVERSPECIFYING THE MODEL -Consider the model: -where x 3 has no impact on y; B 3=0 -x 3 may or may not be correlated with x 2 and x 1 -in terms of expectations:

3. 3 Irrelevant Variables in a Regression Model From theorem 3. 1, B 1 hat and B 2 hat are unbiased since MLR. 1 to MLR. 4 still hold We furthermore expect that: -even though B 3 hat may not be zero, it will average out to zero across samples -Including irrelevant variables doesn’t affect OLS unbiasedness, but we will see it affect OLS variance