N1611 - Financial Econometrics
Self-study Questions #1
1. Consider the following regression:
Rt;XY = a + βRt;M + ut
where Rt;XY is the excess return on stock XY and Rt;M is the excess return on the market.
(a) Given certain assumptions the least squares estimators a and β are said to be BLUE. Explain what BLUE means.
(b) State the assumptions that must hold so that the least squares estimators a and β are BLUE.
2. A researcher has modelled the log returns on the FTSE All Shares Index, denoted by rt;FTSE , as a function of an interest rate, it , and GDP growth rate, gt :
rt;FTSE = β1 + β2it + β3gt + ut
She estimates the model using quarterly data from 1978Q1 to 2000Q4 (92 observations) and obtains the following:
Variable Coe¢ cient Std. Error t-Statistic p-value
|
Constant |
0.012 |
0.022 0.571 |
0.569 |
|
it |
0.098 |
0.198 0.492 |
0.624 |
|
gt |
1.190 |
0.685 1.739 |
0.086 |
|
R-squared |
0.033 |
Mean dependent var |
0.029 |
|
Adjusted R-squared |
0.011 |
S.D. dependent var |
0.061 |
|
S.E. of regression |
0.060 |
Akaike info criterion |
-2.747 |
|
Sum squared resid |
0.324 |
Schwarz criterion |
-2.664 |
|
Log likelihood |
129.351 |
F-statistic |
1.512 |
|
Durbin-Watson stat |
1.931 |
Prob(F-statistic) |
0.226 |
(a) Use the above output to test whether GDP growth rate (gt ) is signiÖcantin this regression.
(b) Test the hypothesis that the interest rate (it ) and the GDP growth rate (gt ) jointly have no e§ecton log returns (rt;FTSE ). Explicitly state your null and alternative hypotheses.
(c) Set out the procedure that could be used to test
H0 : β2 = 0; β3 = 1
H1 : β2 ≠ 0 and/or β3 ≠ 1
3. Consider the extended CAPM model:
Rp;t = β1 + β2 RM;t + β3 RSMB;t + β4 RHML;t + β5 RMOM;t + ut
where the variables are defined as:
Rp is the excess return on a portfolio of stocks,
RM is the market risk premium,
RSMB is the return of small company stocks minus that of big company stocks,
RHML is the return of the third most expensive stocks sorted by the market price/book value ratio minus the cheapest third, and
RMOM is a momentum factor (the di§erence in current returns between a portfolio composed of the top 30% of stocks ranked by returns over the preceding year and a portfolio composed of stocks from the bottom 30%).
Suppose we have a quarterly data for all variables above between January 1980 and December 2012.
(a) Suppose we wish to test within the extended CAPM model above that there is a di§erent level of (excess) returns in quarter 4 of each year. Set out the model you would estimate and the test you would carry out.
(b) How can we test within this model whether in at least one quarter of the year there is a di§erent level of (excess) returns?
4. Consider the following exponential regression model:
Yt = AXt(β)eut
(a) Show how we can transform the above model, so that it can be estimated using the ordinary least squares (OLS) method.
(b) Brieáy discuss the interpretation of the estimated slope coe¢ cient in the trans- formed model.