A36779

3SMFE 06 23062 Level H

LH Statistical Methods in Finance and Economics

3SMFE4 06 27099 Level M

LM Statistical Methods in Finance and Economics

May/June Examinations 2023-24

1. (a)  Briefly answer the following questions.                                                                                        [8]

(i)  What is the consequence of specifying a model with a variable in logarithmic form, if in the true population model, the variable is actually in its level form?

(ii)  Is a regression with a low R2  useless? Explain. What does a low R2  imply about the specified regression model?

(iii)  Suppose you have N observations on a variable with constant mean μ but a het- eroskedastic variance.  What is the heteroskedasticity-consistent (White corrected) estimate of the variance? How does it compare to the usual estimate of this variance that ignores heteroskedasticity?

(b)  True or False? Explain your decision for the following statements.                                            [6]

(i)  By including one more independent variable in the regression, you will eliminate the possibility of omitted variable bias from excluding that variable.

(ii)  The central limit theorem states that sums of random variables, properly standard- ized, converge in distribution to the standard normal distribution.

(iii)  The OLS estimators are no longer BLUE (best linear unbiased estimators) under the situation of the heteroskedasticity.

(c)  Suppose you estimate the consumption function of Yi = α1 + α2Xi + ei and the savings

function of Zi = β1+ β2Xi + ui, where Y denotes consumption, Z denotes savings, X

denotes income, α’s and β’s are parameters and e and u are the random error terms.

Furthermore, X = Y +Z, that is, income is equal to consumption plus savings, and vari-ables are all in numerical terms.                   [11]

(i)  What is the relationship, if any, between the OLS estimators of α2 and β2?  Show your calculations.

(ii)  Will the residual (error) sum of squares be the same for the two models of Yi = α1+ α2Xi+ei and Zi = β1+β2Xi+ui? Explain your answer.

(iii)  Can you compare the R2  terms of the two models? Explain your answer.

2. (a)  Consider a time series of observations, It = (y1,x1), (y2,x2), ··· , (yT ,xT ).   Given that T = 29,yT = 10,yT — 1  = 12,xT+2  = xT+1  = xT = 5, and xT — 1  = 6,  provide the one- period and two-period forecasts for yT+1  and yT+2, respectively for the following models. In each case, vt are assumed to be independent random errors distributed as vt ~N(0, 4).         [20]

(i)  The random walk with drift yt = 5+yt — 1 +vt .

(ii)  The ARDL model yt = 6+0.6yt — 1+0.3xt +0. 1xt — 1vt .

(iii)  The error correction model Δyt = —0.4(yt — 1 —15—xt — 1)+0.3Δxt +vt .

(b)  Consider a time  series process of the form. yt = 2yt — 1 — yt —2 + α + vt , where vt ~ N(0, σ2 ), and α is a constant. Show that this process is integrated with order two.            [5]

3. (a)  From the data for the period of the first quarter in 1971 to the fourth quarter in 1988 for Canada, the following results were obtained:

where M 1 is the most liquid part of money supply including paper and checking account

and etc., GDP refers to the gross domestic product measured in billions of Canadian dollars, ln is the natural logarithmic term andˆ(u)t represents the residuals from model (1).     [13]

(i)  Interpret the fitted regression results for model (1).

(ii)  Based on the results from models  (1) and (2), do you suspect that model (1) is spurious? Explain your answer.

(iii)  From the above results, comment on cointegration. You need to explain your answer clearly.

(iv)  Now use the same notation and consider the following model:

Comment on this model. Is model (1) spurious? Explain your answer.

(b)  What is the difference between the tests of unit roots and the tests of cointegration?              [3]

(c)  The first-difference transformation to eliminate autocorrelation assumes that the parame- ter of autocorrelation ρ is —1.  Is the statement true or false?  Briefly justify your answer.          [2]

(d)  Write down the error correction model in ARDL(1,1) format for two time series processes,

yt and xt , where yt is the dependent variable at time t. Use β to denote long-run equililib-rium parameter. Show that the model would not work if another lag of the error correction term,for example, yt —2 — βxt —2, is added to the model.           [7]

4. The following model explores the factors influencing fundraising amounts on the web platforms Indiegogo and Kickstarter.   It  includes  148  projects from 2010 to 2015, all related to social enterprise.

ln(percentage)i = β1+β2ln(goal)i+β3producti+β4photo i+β5photosqi+β6malei+β7USUKi+ei where βs are parameters, e is the error term and i stands for the i-th project. percentage rep-resents the funded amount as a ratio of the target funding value, in percentage terms, goal is the target funding value that a project aims to achieve in US dollars, product indicates whether the project involves producing a physical product (Yes=1, No=0), photo is the number of photos posted by the project initiator on the platform’s website, photosq is the square of photo, male in-dicates whether the project initiator is male (Yes=1, No=0), USUK indicates whether the project initiator is from the US or the UK (Yes=1, No=0).                                                                                 [25]

Estimation results

Signif .  codes:    0   ‘ *** ’  0 . 001   ‘ ** ’  0 . 01   ‘ * ’  0 . 05   ‘ . ’  0 .1

Results from Analysis of Variances (ANOVA)

Variance-Covariance Table

(a)  Detail the steps and calculate the values of SSR(due to regression), R2 , adjusted R2  and

the standard error of the model.                                                                                                  [3]

(b)  Detail the  steps and calculate the missing values of A, B and C in the tables on the

Estimation results and the ANOVA.                                                                                          [1.5]

(c)  Report the fitted model. You need to include the standard errors, significance level, sam-

ple size, R2  and adjusted R2 .                                                                                                   [1.5]

(d)  Detail the steps and find the marginal effects of photo on ln percentage with 8 and 20

photos posted on the website, respectively. What do the results suggest?                               [2]

(e)  Interpret the fitted model in terms of goal variables.                                                                  [1]

(f)  Is this a good model in general? Perform an appropriate test on the overall significance of

the model using α = 5%.                                                                                                         [3]

(g)  A regression of the squared residuals on the same set of explanatory variables gives

R2 = 0.043, perform an appropriate test for heteroskedasticity.                                               [3]

(h)  Assuming you still suspect heteroscedasticity exists, and the variability of the funded per-

centage are larger than those projects which produced a physical product, propose an

appropriate method to test this type of heteroscedasticity. Detail the steps.                             [3]

(i)  Find the 95% interval estimate for a nonlinear function that λ = -βphoto/(2×βphotosq)

for the lnpercentage model.                                                                                                    [7]