代做IB2B20 Financial Econometrics Summer 2024代写Web开发

IB2B20

Financial Econometrics

Summer 2024

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:

RSS = 4862, and when the  regression  only includes a constant and increases to 5026. Number of observations = 191.

where ER t  is the excess return on a stock, ERt(M) 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. RSS is the residual sum  of squares (or SSR, the sum of squared residuals).

a)   Provide an interpretation of the estimated coefficient on the SMBt variable. (1 mark)

b)   What is an appropriate null hypothesis for the estimated coefficient on the term ? Test this hypothesis against a two-sided alternative, 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一k一1 at the 1%, 5% and 10% significance level are equal to 2.602, 1.973, and 1.653, respectively). (2 marks)

c)   Do the variables SMBt and HMLt 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一k一1 are 2.331, 3.044 and 4.721 at the 10%, 5% and 1% significance level respectively). (4 marks)

d)   Explain carefully the concept of a p-value and why it might be useful when testing the significance of an estimated coefficient. If the p-value on the estimated coefficient on the  HMLt variable was 0.015, how would we interpret this? (3 marks)

Question 2

a)   For either the two variable OR the multiple variable linear regression model, show that the OLS (ordinary least squares) estimators () are unbiased, under the following multiple linear regression (MLR) assumptions: MLR.1 - linear in parameters, MLR.2 – random sampling, MLR.3 - no perfect collinearity and MLR.4 - the errors have a conditional mean of zero, E(U Ix) = 0. What does an estimator being unbiased imply? (5 marks)

b)    For the multiple variable linear regression model, show that using the four assumptions in part (a) and the fifth MLR assumption, MLR.5: homoscedasticity or constant variance where (5 marks)

Question 3

Consider again the estimation results from equation (1) (from Question 1).

a)   In model (1), we may have omitted a variable measuring risk associated with momentum.  Why might it be problematic to omit this variable? What assumption of the multiple linear regression model would be eventually be violated? (3 marks)

b)  What is the likely sign of any bias, arising from omitting momentum, of the OLS estimated coefficient for HMLt? Is the true parameter closer to zero than the estimated coefficient, or further from zero? Justify any assumptions you make and provide mathematical formulas where appropriate. (4 marks)

c)   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一k一1 is equal to 1.653). (3 marks)

Question 4

We want to estimate the following linear regression model by OLS:

y = β0 + β1x1 + β2x2 + u              (2)

We are concerned about the potential problem of heteroscedasticity in the regression error.

a)   How would you test for the presence of heteroscedasticity using White’s test? State carefully the null hypothesis of the test and write down any regression equations you would need to estimate. (4 marks)

b)   What is the easiest way to deal with the presence of heteroskedasticity?  Describe conceptually how this solution works and provide mathematical expressions where appropriate. (3 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. (3 marks)

Section B

Answer ANY TWO questions

Question 5

a) Consider the following MA (2) process:

Write down the set of equations for the one-step, two-step, and three-step ahead forecasts of yt. What happens to the forecast beyond three-steps ahead? Provide a detailed description and use mathematical formulas where appropriate. (6 marks)

b) If you had a time-series of one-step ahead forecasts (produced recursively, forecasting one-step ahead at each point in time say), and the observed outcomes of the variable forecast, how would you evaluate the performance of the forecasts? How would you compare its performance to another model or benchmark? Provide a detailed description of each stage, any evaluation criteria and/or test statistics used, and where appropriate use mathematical formula. (6 marks)

c) Describe in detail how, in a time-series regression such as that described in equation (1) (from Section A Question 1), the strict exogeneity required in time-series analysis might be violated. Why might regressions with lagged dependent variables, such as an AR (1), violate strict exogeneity? Use mathematical formulas where appropriate. (8 marks)

d) What is the distinction between a deterministic and a stochastic trend? Give an example of a process that has a stochastic trend, showing why it is non-stationary and where the stochastic part arises (Hint: what is the mean and variance of the process?). (10 marks)

Question 6

a)   Is the MA (1) process given below weakly dependent?

Show why this is or is not the case, with specific reference to the conditions required of the definition of weak dependence. (8 marks)

b)   We wish to test for the presence of a unit  root in the natural logarithm of a time- series, yt. Outline the testing procedure, stage by stage, of how you would test for a unit root. What regressions would you run, what transformations would you use, what are the null and alternative hypotheses and test statistics to be used? What is different regarding the critical values and what needs to be considered when obtaining them? Why might we include a time-trend in such an analysis? Provide a detailed description and use mathematical formulas where appropriate. (12 marks)

c)   Describe, using an empirical example or simulation experiment, what is meant by a spurious regression. How might cointegration among two or more variables resolve the spurious regression problem and how would this cointegration relationship be incorporated into an Error correction model (ECM)? Provide a detailed example and outline the basic concepts using mathematical formulation where appropriate. (10 marks)

Question 7

We are interested in the effect of a given independent X variable (say the unemployment rate) on a dependent variable Y (say the crime rate). The data set comprises of X and Y defined over a cross sectional units i = 1,2, … . , N (for example cities) and over time t = 1,2, … . , T.

a)   Define a pooled regression model you might run, specifying the assumptions and potential advantages/disadvantages of   using pooled regression analysis. Provide mathematical formulation where appropriate. (10 Marks)

b)   What is the basic idea underlying a fixed effects estimation approach and how might this approach address potential problems of  using a pooled regression? Show the stages deriving the specification of the fixed effects regression to be run and outline the  assumptions  underlying the fixed effects estimation.   Provide  mathematical formulation where appropriate. (12 Marks)

c)   Explain the concept of first differences. How does first differences estimation differ from pooled regression and fixed effects? What criteria might you use to decide which of the two approaches, fixed effects and first differencing, to use for estimation? Provide mathematical formulation where appropriate. (8 marks)