Single Linear Regression - Estimation

Question 1

Interpretation of Beta1

Question 2

OLS Etimators

Answer 2

Beta1=(Sample covariance)/(Sample variance of X)
Beta0= Y(sample)-Beta1X(sample)

Question 3

R-squared definition

Answer 3

The fraction of the sample variance of Yi that is explained or predicted by Xi
= ESS/TSS or 1-SSR/TSS
Bounded between 0 and 1 where if 1 all the variation in Y is explained all in X

Question 4

TSS, ESS + SSR relationship

Answer 4

ESS = explained sum of squared
TSS = total sum of squares
SSR = sum of squared residuals

TSS = ESS + SSR

Question 5

Least Squares Assumption #1

Independence

Answer 5

Assumption that the conditional expectation of ui (error) on X is zero meaning the correlation is zero. This then means that E(Y|X)=Beta0+XiBeta1 (since the error term conditional is 0)

Question 6

Least Squares Assumtpion #2

IID

Answer 6

The dataset are independent and identically distributed - ie. all terms are independent and have the same probability distribution

Question 7

Least Squares Assumption #3

No outliers

Answer 7

Large outliers are unlikely - in econometrics we look at the data first and remove legitimate outliers

Question 8

What do the three least squares assumptions tell us about the OLS estimators (Beta0 and Beta1)

Answer 8

1. Unbiased - the expected value of beta is equal to the true value of beta - E(Beta0-sample)=Beta0
2. Consistent - as the sample size n gets large, the Beta values will get closer to their true values with high probability
3. Under CLT when n is large the marginal distributions of Beta0 and Beta1 are normal