IB2B20_A
Financial Econometrics
Summer 2023
Section A
Answer ALL questions in this section
Question 1
Consider the following results, of a linear regression estimated by ordinary least squares (OLS), investigating the determinants of the excess returns of an individual company’s stock:
where ERt is the excess return on a stock, is the excess return on an aggregate stock market index, SMBt is the small minus big and HMLt the high minus low risk factors of Fama and French. The first coefficient is the estimate of the intercept and the coefficients next to the variable names are the estimated respective slope parameters. The numbers in brackets beneath the estimated coefficients are their standard errors.
a) Provide an interpretation of the estimated coefficient on the variable. (2 marks)
b) What is the key assumption for least squares when considering bias of OLS estimators? Explain carefully and use the assumption to show unbiasedness of the coefficients. (5 marks)
c) Describe, providing mathematical expressions, the difference between the adjusted and unadjusted R-squared. Does a low R-squared invalidate a regression? Explain carefully. (3 marks)
Question 2
Consider again the estimation results in equation (1) (from Question 1).
a) If you were to test the null hypothesis that the estimated coefficient on the term is equal to 1, against the one-sided alternative that it is less than 1, would you reject or fail to reject the null at any of the three 1%, 5% and 10% significance levels? Explain clearly how you reached the conclusion. (Hint: the critical values of the t-student distribution tn−k−1 at the 1%, 5% and 10% significance level are equal to 2.327, 1.645, and 1.281, respectively) (3 marks)
b) Do the independent variables in equation (1) help or not help explain ERt? Construct a joint test of this hypothesis and provide mathematical formula where appropriate. (Hint: the critical values of the F-student distribution Fq,n−k−1 with degrees of freedom q=3 and n-k-1 = 120 are 2.13, 2.68 and 3.78 at the 10%, 5% and 1% significance level respectively) (5 marks)
c) State briefly the general principle of the joint hypothesis test using an F-statistic.
Question 3
Consider again the estimation results from equation (1) (from Question 1).
a) Say we have omitted from the regression a risk factor measuring momentum. What are the relevant facts we need to consider when determining if this omission causes the estimated coefficient on the SMBt variable to be biased? What might be direction of the bias of the OLS estimated coefficient for SMBt? (6 marks)
b) Construct the 90% confidence interval for the coefficient on the term HMLt (three decimals are enough). State in one sentence how you interpret this confidence interval. Provide mathematical formula where appropriate. (Hint: the 90th critical value c of the t-student distribution tn−k−1 is equal to 1.64. (4 marks)
Question 4
We want to estimate the following linear regression model by OLS:
We are concerned about the potential problem of heteroscedasticity in the regression error.
a) What happens to OLS estimators and their variances if we introduce heteroscedasticity but retain all other assumptions of the classical linear regression model? (2 marks)
b) Suppose we test for the presence of heteroscedasticity in the model’s residuals, where the p-value from the White test is 0.03. How would you interpret this result? State carefully the null and alternative hypothesis of the test. How would you address the issue of heteroscedasticity if it were present? Provide mathematical expressions where appropriate. Write down any regression equations you would need to estimate. (4 marks)
c) Describe and define the concept of a weakly dependent time-series. Give an example and explain why this property is important for time-series regression analysis. Explain carefully and provide mathematical expressions where appropriate. (4 marks)
Section B
Answer ANY TWO questions
Question 5
a) Consider the following AR (1) process:
Write down the set of equations for the one-step, two-step, and three-step ahead forecasts of yt. Provide a detailed description and use mathematical formulas where appropriate. (7 marks)
b) Outline how you would evaluate the relative forecast performance of two models (for example an AR(1) as in part a) and an AR(2)) using a “pseudo out-of-sample” forecasting exercise. Provide a detailed description of each stage and use mathematical formula where appropriate. (10 marks)
c) IGNORE THIS QUESTION – NOT COVERED IN 2024 COURSE. Consider the following GARCH (1,1) process:
Write down a set of equations in and and their lagged values, which could be employed to produce one-step, two-step and three-step ahead forecasts for the conditional variance of yt. How would you evaluate the performance of the forecasts of the conditional variance? Provide a detailed description and use mathematical formula where appropriate. (13 marks)
Question 6
a) Is the following process stationary?
State why this is or is not the case, with specific reference to the definition of weak stationarity regarding the mean, variance, and covariance. Use mathematical derivations where appropriate. (8 marks)
b) We wish to test for the presence of a unit root in the natural logarithm of a time-series, yt. To this end we estimate, using linear regression, the following equation (using 244 observations):
Where we report t-statistics in the brackets, ∆yt = yt − yt−1 and t is a linear deterministic time trend.
Outline the testing procedure, stage by stage, of how you would test for a unit root. Does yt contain a unit root? Why would we include a time-trend in such a regression? Provide a detailed description and use mathematical formulas where appropriate. (Hint: the 10% and 5% Dickey-Fuller critical values, where a time trend is included, are -3.13 and -3.43 respectively) (12 marks)
c) Describe, using an empirical example or simulation experiment, what is meant by a spurious regression. How might the use and application of cointegration among two or more variables resolve the spurious regression problem? Provide a detailed example and outline the basic concept using mathematical formulation where appropriate. (10 marks)
Question 7
a) Assume two cross-sectional data sets are available, one at a point in time before an event and one after. The event is the building of a factory, and the data is for house prices and a dummy variable defining whether a house is located close to or not close to the newly built factory. Each cross-sectional data set is drawn independently of one another. Describe how you would apply the pooled cross-sectional difference-in-difference approach to examine the effect on house prices of being located close to a newly built factory. Outline the regression(s) estimated, null and alternative hypotheses to be used and provide an interpretation. Use mathematical formulas where appropriate. What would be the implications if the two data sets were not independently drawn? (10 marks)
b) Describe the fixed effects estimator for panel data. What criteria might you use to decide whether using a fixed effects estimator is appropriate for estimation? Provide mathematical formulation where appropriate. (12 marks)
c) IGNORE THIS QUESTION – NOT COVERED IN 2024 COURSE. We estimate a panel model with just one independent variable, using both fixed effects (FE) and random effects (RE) methods, so that we have two estimates of the same coefficient, = 0.533 and = 0.379 with associated standard errors of 0.159 and 0.130 respectively. Construct a test to choose which of the two estimators you would use. Explain in detail the implementation, provide an interpretation, and use mathematical formulation where appropriate. (Hint: the critical values of the distribution are equal to 2.70, 3.84 and 6.63 at the 10%, 5% and 1% significance level, respectively.) (8 marks)