ECON20110/30370 Econometrics Semester 2 2020/21

1.  In the context of a linear model, presented below in matrix form

Y = Xβ + u,

where X is a n × q matrix, β is a q × 1 column vector, and both Y and u are n × 1 column vectors.

(a)  Explain how estimation of the parameter vector by Maximum Likelihood would differ

from estimation by Ordinary Least Squares. [10 MARKS]

(b)  If the errors in the model are not Normally distributed are you able to make an

argument that the Ordinary Least Squares estimator is asymptotically normally dis- tributed? Be sure to make clear any arguments you rely on. [9 MARKS]

(c) Show how, in the context of the above model, Weighted Least Squares can be used to address the issue of heteroscedasticity. [6 MARKS]

2. Your friend has asked you to help them understand any potential problems that might arise in a model they have been asked to analyse.  The model your friend has estimated is based on the Capital Asset Pricing Model and is:

SRt  = β0 + β1 (RMt - RFt ) + β2 ∆SIt + ut

where the daily returns for the stock are denoted by SRt , RMt   is the market rate of return, based on an index of stocks and RFt  is the risk free rate of return.  ∆SIt  is the daily change in the level of a sentiment index, SIt , which is designed to summarise how a group of analysts feel about the stock in question.  Typically SIt  goes up when the company’s stock is considered to be underpriced and goes down when it is considered overpriced.

Your friend was searching online and found mention of various potential problems relating to regressions including heteroscedasticity and autocorrelation in regressions. Your friend has sent you an email asking for your help in identifying whether this is something they should be concerned about, as they are unfamiliar with the two terms.

(a)  Explain the difference between heteroskedasticity and autocorrelation in the context of the model above. [8 MARKS]

(b) Your friend claims that neither heteroscedasticity nor autocorrelation will mean the parameter estimates, when estimated by Ordinary Least Squares, are inconsistent. Your friend then claims he cannot see why they should worry about these problems, even if they exist. Discuss these claims. [6 MARKS]

(c)  Explain to your friend how they can test for the presence of autocorrelation and heteroscedasticity in their model. [11 MARKS]

3. The University has provided you with a dataset relating to 1,000 former students.  Of a long list of variables in the data, one variable is IPi , which has a value of 1 if the ith student undertook an industrial placement and 0 otherwise. In modelling the impact of a range of student characteristics on the probability of an individual student under-taking an industrial placement:

(a)  Explain why an Ordinary Least Squares model is inappropriate. [8 MARKS]

(b)  Propose an alternative modelling approach. [14 MARKS]

(c)  Explain,  briefly, the difference  between the  meaning of the  parameters from the model you propose and those from a linear model estimated using Ordinary Least Squares. [3 MARKS]

4. You are working for a marketing consultancy, who have been asked by Netflix to attempt to find ways to increase their total number of subscribers.

You have just returned from a presentation where your team leader presented a regression they had run on some data they had found online.

Your team leader had estimated the linear model:

netflixt  = β0 + β1 Bookst + et                                                           (1)

where netflixt  is the total number of Netflix subscribers in period t and Bookst  is an index of the volume of sales of books, periodicals and magazines in period t, compiled by the Office for National Statistics.

Your team leader presented the graphs and regression output,  below, as part of their presentation.

Call:

lm(formula = data$Netflix  ~ data$Books)

Residuals:

Min         1Q Median         3Q       Max

-39702   -9956   -1357     4335   53309

Coefficients:

Estimate Std . Error t value Pr(>|t|)

(Intercept) -82287 .9       18522 .0   -4 .443 9 .95e-05  ***

data$Books       1858 .9           209 .9     8 .855 4 .07e-10  ***

---

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

Residual  standard  error: 24300  on 32 degrees  of freedom

Multiple R-squared:   0 .7102,Adjusted R-squared:   0 .7011

F-statistic: 78 .41  on  1  and 32 DF,   p-value: 4 .069e-10

At the end of the meeting your team leader said that they were going to recommend that Netflix follow the strategy of promoting reading, as this will cause their subscription numbers to grow. The rest of your team, who have not studied econometrics, agree with your team leader’s suggestion.

You now need to draft a memo for your team which identifies and explains why this regression might be spurious and what you might be able to do to make the results more reliable for use in the context given. [25 MARKS]