Answer all the questions.
1. Consider the regression model Yt = β1 + β2 Xt + Ut
where all the classical assumptions hold. Derive the normal equations of the least squares estimates β1 ^ and β2 ^ of the model. Using these normal equations, answer the following.
(a) Obtain expressions for β1 ^ and β2 ^ .
(b) Show that these LS estimates pass through (X¯, Y¯).
(c) The mean value of residuals U^ is zero.
(d) Residuals are uncorrelated with Xt and predicted Yt (i. e. ∑Ut ^ Xt = 0 and ∑Yt ^ Ut ^ = 0).
2. What are the assumptions of the Classical Linear Regression Model (2-3 lines per assumption).
Based on 240 monthly rates of return Fogler and Ganapathy obtained the following characteristic line for IBM stock in relation to market portfolio index,
Rit ^ = 0.7264 + 1.0598Rmt ; R2 = 0.4710
se = (0.3001) (0.0728); df = 238
F1,238 = 211.9
where Rit is the return of i th security (IBM in this case) and Rmt is the return of the market portfolio.
(a) A security whose beta coefficient is said to be a volatile security. Formulate and carry out appropriate hypothesis test to determine whether the IBM security is volatile or not.
(b) Is the intercept term significantly different from zero? If so what is the economic reasoning behind this result?
3. Consider the following two models:
Model I: Yi = β1 + β2 Xi = ui
Model II: yi *= αi + α2 xi * + ui
where y* and x* are standardized variables (i. e. yi * = (Yi - Y¯)/SY and xi * (Xi - X¯)/SX with SX and SY are standard deviations).
Show that α2 ^ = β2 ^ (SX /SY )
4. You have the following data for estimating the relationship between X and Y.
Y -2 -1 0 1 2
X -2 0 1 0 1
First adopt the model: Yi = α + βXi + ui
(a) Compute the least squares estimates of α and β.
(b) What is an unbiased estimate of Var(ui )? Use this formula to calculate the Var(ui ).
(c) If you adopt the model Xi = γ + δYi + vi , how would your results in (a) and (b) change?