*AIMs: Define and interpret a sample regression function, regression coefficients, parameters, slope and the intercept. Describe the key properties [assumption] of a linear regression.*

**Questions:**

215.1. We regressed the monthly returns of Apple (AAPL) against the S&P 500 ($SPX) for the last thirty-six months ending January 31st; Apple's monthly return is the dependent variable (Y, regressand), the index's monthly return is the independent variable (X, regressor) and the number of paired observations, n = 36. In regard to the dependent variable, Apple's average monthly return over the period was +4.84% with a standard deviation of 6.686%. In regard to the independent variable, the average monthly return of the index was +1.69% with a standard deviation of 4.687%. The covariance between the two series, Covariance(X,Y), was 0.00216. What is the equation for the sample regression line? (note: I did use actual data, trying to keep it real folks!

*)*

**Bonus for finding the correlation and R^2**a. AAPL = 0.01 + 0.33*SPX

b. AAPL = 0.02 + 0.67*SPX

c. AAPL = 0.03 + 0.98*SPX

d. AAPL = 0.04 + 1.29*SPX

215.2. A dataset consists of the price of gasoline (Price), the

*regressor*, and the weekly household demand for gas in terms of gallons (Quantiity), the

*regressand*. An ordinary least squares (OLS) regression line produces the following demand function:

**Quantity = 11 - 1.5*Price**.

One of the datapoints in the scatterplot is a household that "demands" 8.0 gallons when the price is $3.00 per gallon; i.e., Quantity(i) = 8.0 gallons, Price(i) = $3.00. What is the residual of this observation, u(i)?

a. -1.5

b. zero

c. +1.5

d. Impossible, the observation must lie on the line

215.3. Each of the following is a key property [assumption], according to Stock & Watson, of a linear regression EXCEPT for:

a. The conditional distribution of the error term, u(i), given X(i), has a mean of zero

b. The variance of the conditional distribution of the error term given X(i), variance[u(i) | X(i) = x], converges to ZERO as sample (n) and X(i) increase

c. Each observation [X(i), Y(i)] for i = 1, ....n, is independent and identically distributed (i.i.d.)

d. Large outliers are unlikely; i.e., X and Y have nonzero finite kurtosis

**Answers:**