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# P1.T2.215. Properties of linear regression (Stock & Watson)

#### David Harper CFA FRM

##### David Harper CFA FRM
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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