1. 2.4 Units of Measurement and Functional Form -Two important econometric issues are: 1) Changing measurement -When does scaling variables have an effect on OLS estimates? -When does scaling variables have no effect on OLS estimates? 2) Functional Forms -How do natural logs affect linear regressions? -How do functional forms impact elasticity?

2. 2.4 Units of Measurement and Functional Form Consider the following model: -where the number of squirrels and trees are in single units -if trees=2000 then the predicted number of squirrels becomes 1500 -if there are no trees, there are 500 squirrels -how does this change if squirrels are measured in hundreds (ie: divided by 100)?

3. 2.4 Units of Measurement and Functional Form -If squirrels are measured in hundreds, then zero trees would produce 500 squirrels, so B 0 hat=5 (divided by 100) -If there are 2,000 trees, then B 1 hat*2000 must equal 10 (15-5, or 1,000 squirrels) -Therefore B 1 hat=0.005 (divided by 100)

4. 2.4 Acorns are for the squirrels Therefore When the dependent variable is multiplied (divided) by a constant… Multiply (divide) the OLS intercept and slope by the same constant. IE: Change the y – change OLS the same way.

5. 2.4 Units of Measurement and Functional Form Consider the following model: -where the number of treehuggers and trees are in single units -if trees=800 then the predicted number of treehuggers becomes 900 -if there are no trees, there are 500 treehuggers -how does this change if trees are measured in hundreds (ie: divided by 100)?

6. 2.4 Units of Measurement and Functional Form -If trees are measured in hundreds, then zero trees would produce 500 treehuggers, so B 0 hat=500 (nothing has changed) -If there are 800 trees, then trees=8 and 8(B 1 hat) must equal 400 (900-500) -Therefore B 1 hat=50 (multiplied by 100)

7. 2.4 Acorns are for the squirrels Therefore When the independent variable is multiplied (divided) by a constant… Divide (multiply) the OLS slope (not intercept) by the same constant. IE: Change the x – change OLS slope the opposite way.

8. 2.4 Units of Measurement and Functional Form How does R 2 (goodness of fit) change when a variable is scaled? -It doesn’t -R 2 calculates how much of the variation in y is explained by x -this doesn’t depend on scaling -a similar “best fit line” is drawn through data points, regardless of scaling

9. 2.4 Functional Form Thus far, we have focused on LINEAR relationships -Linear relationships don’t capture all of the possible interaction between variables -Linear relationships assume that the first x has the same impact on y as the last x -changing impacts can be captured through the use of NATURAL LOGARITHMS

10. 2.4 Log-Lin Model When a variable has an increasing (percentage) impact on y, the log-lin model is appropriate: -note that log(y) indicates the natural log of y -if we assume that u doesn’t change, -note as x increases, y increases, therefore this equation expresses INCREASING return

11. 2.4 Log-Lin Model Assume that absence does make the heart grow fonder: -assuming (for simplicity) that u=0, 2 days absence causes fondness of e 4 (54.6) while 10 day’s absence causes fondness of e 8 (2,981) -therefore, given another day of absence:

12. 2.4 Lin-Log Model Therefore: -the 3 rd day of absence increases fondness by 27.3 -the 11 th day of absence increases fondness by 1,490.5 -we have INCREASING RETURNS -Note: a Log-lin model can also be expressed:

13. 2.4 Log-Log Model Recall from Econ 299 that elasticity is calculated as: -if constant elasticity is theoretically important to a model, a log-log functional form ensures that elasticity is constant as B 1 :

14. 2.4 Scaling and Dependent Logs Consider a Log-Lin model where the y value is multiplied by c: -scaling a dependent variable in log form changes the intercept but does not affect the slope

15. 2.4 Units of Measurement and Functional Form Different Functional Forms are Summarized as Follows: ModelFunctionInterpretation of B 1 Lin-Liny=f(x)∆y= B 1 ∆x Lin-Logy=f(log(x))∆y= (B 1 /100)%∆x Log-LinLog(y)=f(x)%∆y= 100B 1 ∆x *also call semi-elasticity Log-LogLog(y)=f(log(x))%∆y= B 1 %∆x

16. 2.4 Units of Measurement and Functional Form Notes: 1)Even though non-linear variables are included in models (ie: log(y) or y 2 ), the models are still considered “Linear Regressions” as they are linear in the parameters B 1 and B 2 2)Non-linear variables make interpreting B 1 and B 2 more complicated 3)Some estimated models are NOT linear regression models

17. 2.5 Expected Values and Variances of the OLS Estimators This section will, using classical Gauss-Markov Assumptions, find 3 OLS properties: 1) OLS is unbiased 2) Sample Variance of OLS Estimators 3) Estimated Error Variance This will be done viewing B 0 hat and B 1 hat as estimators of the population model:

18. Gauss-Markov Assumption SLR.1 (Linear in Parameters) In the population model, the dependent variable, y, is related to the independent variable, x, and the error (or disturbance), u, as Where B 0 and B 1 are the population intercept and slope parameters, respectively.

19. Gauss-Markov Assumption SLR.1 (Linear in Parameters) Notes: 1)In reality, x, y and u are all viewed as random variables 2)Since OLS needs only be linear in B 1 and B 2, SLR.1 is far from restrictive Given an equation, an assumption must now be made concerning data

20. Gauss-Markov Assumption SLR.2 (Random Sampling) We have a random sample of size n, {(x,y): i=1,2,…..n}, following the population model in equation (2.47).

21. Gauss-Markov Assumption SLR.2 (Random Sample) We will see in later chapters that random sampling can fail, especially in time series data but also in cross-sectional data -Now that we have a population equation and an assumption about data, (2.47) can be re- written as: Where u i captures all unobservables for observation I and differs from u i hat -to estimate B 0 and B 1 we need a 3 rd assumption:

22. Gauss-Markov Assumption SLR.3 (Sample Variation in the Explanatory Variable) Sample outcomes of x, namely, {x i, u=1,….,n}, are not all the same value.

23. Gauss-Markov Assumption SLR.3 (Sample Variation in the Explanatory Variable) -This assumption ensures that the denominator of B 1 hat is not zero -This assumption is violated if: -The variance of x is zero -The standard deviation of x is zero -The minimum value of x is equal to the maximum value -Although we can now obtain OLS estimates, we need one more assumption to ensure unbiasedness

24. Gauss-Markov Assumption SLR.4 (Zero Conditional Mean) The error u has an expected value of zero given any value of the explanatory variable. In other words,

25. Gauss-Markov Assumption SLR.4 (Zero Conditional Mean) Given our assumption about random sampling, we can further conclude: -this is read “for all 1=1, 2,….n” -given SLR.2 and SLR.4, we can derive the properties of OLS estimators as conditional on x i ’s values -given these 2 assumptions, nothing is lost in derivation by assuming x i is nonrandom

26. 2.4 OLS is Unbiased In order to prove OLS’s unbiasedness, B 1 hat must first be algebraically manipulated: -By a familiar mathematical property. -Substituting out y i and restating the denominator gives us: (note that SST x is not the same as SST)

27. 2.4 OLS is Unbiased Using summation properties, the numerator becomes: -Which is simplified using the properties:

28. 2.4 OLS is Unbiased Returning to our B 1 hat estimate, we now have: -Which indicates that the estimate of B 1 equals B 1 plus a term that is a linear combination of errors -Conditional on values of x, B 1 hat’s randomness is due solely to the errors -Note:

29. Theorem 2.1 (Unbiasedness of OLS) Using assumptions SLR.1 through SLR.4, for any values of B 0 and B 1. In other words, B 0 hat is unbiased for B 0 and B 1 hat is unbiased for B 1.

30. Theorem 2.1 Proof Since expected values are conditional on samples of x, and SST x and d i are functions only of x i, they are nonrandom in conditioning. Therefore:

31. Theorem 2.1 Proof -this is proved from the fact that each u i (conditional on sample x’s) is zero from SLR.2 and SLR.4 -”since unbiasedness holds for any outcome on {x 1, x 2,…,x n }, unbiasedness also holds without conditioning on {x 1, x 2,…,x n } -unbiasness of B 1 hat is now straightforward:

32. Theorem 2.1 Proof Since we already proved that

33. Theorem 2.1 Notes -Remember that unbiasedness is a feature of the sample distributions of B 1 hat and B 2 hat -if we have a poor sample, our OLS estimates would be far from the true values -if any of our 4 initial Gauss-Markov assumptions are not true, OLS’s unbiasedness fails

34. Assumption Failure -If SLR.1 fails (y and x are not linearly related), very advanced estimation methods are needed -Failure of SLR.2 (random sampling) is discussed in Chapters 9 and 17 -common in time series and possible in cross-sectional data -If SLR.3 fails (x’s are all the same), we cannot obtain OLS estimates -If SLR.4 fails, OLS estimators are biased, which can be corrected

35. SLR.4 Failure -If x is correlated with u, we have spurious correlation -the relationship between x and y is influenced by other factors connected with x -note that some vague connection is always possible but not statistically significant

36. SLR.4 Failure Example -Saskatchewan instituted a hypothetical drunk driving awareness (DDA) campaign as an alternative to jail time for DUI -It was found that the relationship between DUI’s and enrolment in the program is as follows: -Even though the program looks to have failed, it is due to a spurious correlation: -the existence of drunk drivers both increases the number of DUI’s and the enrolment in the program