1. 2.5 Variances of the OLS Estimators
-We have proven that the sampling distribution of OLS estimates (B0hat and B1hat) are centered around the true value
-How FAR are they distributed around the true values?
-The best estimator will be most narrowly distributed around the true values
-To determine the best estimator, we must calculate OLS variance or standard deviation
-recall that standard deviation is the square root of the variance.

2. Gauss-Markov Assumption SLR.5 (Homoskedasticity)
The error u has the same variance given any value of the explanatory variable. In other words,

3. Gauss-Markov Assumption SLR.5 (Homoskedasticity)
-While variance can be calculated via assumptions SLR.1 to SLR.4, this is very complicated
-The traditional simplifying assumption is homoskedasticity, or that the unobservable error, u, has a CONSTANT VARIANCE
-Note that SLR.5 has no impact on unbiasness
-SLR.5 simply simplifies variance calculations and gives OLS certain efficiency properties

4. Gauss-Markov Assumption SLR.5 (Homoskedasticity)
-While assuming x and u are independent will also simplify matters, independence is too strong an assumption
-Note that:

5. Gauss-Markov Assumption SLR.5 (Homoskedasticity)
-if the variance is constant given x, it is always constant
-Therefore:
-σ2 is also called the ERROR VARIANCE or DISTURBANCE VARIANCE

6. Gauss-Markov Assumption SLR.5 (Homoskedasticity)
-SLR.4 and SLR.5 can be rewritten as conditions of y (using the fact we expect the error to be zero and y only varies due to the error)

7. Heteroskedastistic Example
Consider the following model:
-here it is assumed that weight is a function of income
-SLR.5 requires that:
-but income affects weight range
-rich people can both afford fatty foods and expensive weight loss
-HETEROSKEDATICITY is present

8. Theorem 2.2 (Sampling Variances of the OLS Estimators)
Using assumptions SLR.1 through SLR.5,
Where these are conditional on sample values {x1,….,xn}

9. Theorem 2.2 Proof
From OLS’s unbiasedness we know,
-we know that B1 is a constant and SST and d are non-random, therefore:

10. Theorem 2.2 Notes
-(2.57) and (2.58) are “standard” ols variances which are NOT valid if heteroskedasticity is present
1) Larger error variances increase B1’s variance
-more variance in y’s determinants make B1 harder to estimate
2) Larger x variances decrease B1’s variance
-more variance in x makes OLS easier to accurately plot
3) Bigger sample size increases x’s variation
-therefore bigger sample size decreases B1’s variation

11. Theorem 2.2 Notes
-to get the best estimate of B1, we should choose x to be as spread out as possible
-this is most easily done by obtaining a large sample size
-large sample sizes aren’t always possible
-In order to conduct tests and create confidence intervals, we need the STANDARD DEVIATIONS of B1hat and B2hat
-these are simply the square roots of the variances

12. 2.5 Estimating the Error Variance
Unfortunately, we rarely know σ2, although we can use data to estimate it
-recall that the error term comes from the population equation:
-recall that residuals come from the estimated equation:
-errors aren’t observable and residuals are calculated from data

13. 2.5 Estimating the Error Variance
These two formulas combine to give us:
-Furthermore, an unbiased estimator of σ2 is:
-But we don’t have data on ui, we only have data on uihat

14. 2.5 Estimating the Error Variance
A true estimator of σ2 using uihat is:
-Unfortunately this estimator is biased as it doesn’t account for two OLS restrictions:

15. 2.5 Estimating the Error Variance
-Since we have two different OLS restrictions, if we have n-2 of the residuals, we can then calculate the remaining 2 residuals that satisfy the restrictions
-While the errors have n degrees of freedom, residuals have n-2 degrees of freedom
-an unbiased estimator of σ2 takes this into account:

16. Theorem 2.3 (Unbiased Estimation of σ2)
Using assumptions SLR.1 through SLR.5,

17. Theorem 2.3 Proof
If we average (2.59) and remember that OLS residuals average to zero we get:
Subtracting this from (2.59) we get:

18. Theorem 2.3 Proof
Squaring, we get:
Summing, we get:

19. Theorem 2.3 Proof
Expectations and statistics give us:
Given (2.61), we now prove the theorem since

20. 2.5 Standard Error
-Given (2.57) and (2.58) we now have unbiased estimators of the variance of B1hat and B0hat
-furthermore, we can estimate σ as
-which is called the STANDARD ERROR OF THE REGRESSION (SER) or the STANDARD ERROR OF THE ESTIMATE (Shazam)
-although σhat is not an unbiased estimator, it is consistent and appropriate to our needs

21. 2.5 Standard Error
-since σhat estimates u’s standard deviation, it in turn estimates y’s standard deviation when x is factored out
-the natural estimator of sd(B1hat) is:
-which is called the STANDARD ERROR of B1hat (se(B1hat))
-note that this is a random variable as σhat varies across samples
-replacing σ with σhat creates se(B0hat)

22. 2.6 Regression through the Origin
-sometimes it makes sense to impose the restriction that when x=0, we also expect y=0
Examples:
-Calorie intake and weight gain
-Credit card debt and monthly payments
-Amount of House watched and how cool you are
-Number of classes attended and number of notes taken

23. 2.6 Regression through the Origin
-this restriction sets our typical B0hat equal to zero and creates the following line:
-where the tilde’s distinguish this problem from the typical OLS estimation
-since this line passes through (0,0), it is called a REGRESSION THROUGH THE ORIGIN
-since the line is forced to go through a point, B1tilde is a BIASED estimator of B1

24. 2.6 Regression through the Origin
-our estimate of B1tilde comes from minimizing the sum of our new squared residuals:
-through the First Order Condition, we get:
-which solves to
-note that all x cannot equal zero and if xbar equals zero, B1tide equals B1hat