Q1. (1 mark):  Consider the straight line through the origin given by  for some parameter  . Given a set of data points   for  , formulate a minimisation problem for the least squares estimator estimator  for the parameter  (Note, this is the OLS regression without a constant term).

Q2. (1 mark):  Using calculus, find an expression for the least squares estimator in terms of the   for  .for the OLS regression in Q1

Q3. (1 mark): Prove that the least squares estimator you derived in the previous question gives rise to a minimum. (Hint: consider the second derivative).

Q4. (1 mark)  Read the given data file data.xls into EVIEWS, and run a regression of KMLIT against the rest of the variables assuming homoskedastic errors. Copy and paste the EVIEWS output into the space below, and report the estimated equation with the standard errors below the coefficients.

Q5. (1 mark) Comment on the sign of each estimated coefficient in turn, and state whether this is what you expect. Ignore significance at this stage. 

Q6 (1 mark) Interpret the estimated effect of the Number of Cylinders (CYL) on kilometers travelledmark) Formulate and carry out an appropriate hypothesis test, to test whether Engine size affects a car’s mileage (i.e. how far it can travel per litre). Use the t-statistic approach, at the =0.05 level.

Q8 (2 mark) Formulate and carry out an appropriate hypothesis test for testing whether the number of cylinders affects a car’s mileage. Use the p-values approach, at the =0.05 level.

Q9 (4 marks) Test the following hypotheses about the coefficients on CYL (B1) and ENGCM3 (B2). Clearly specify the rejection region if you are using critical values, and clearly state your conclusions. When using p values, calculate and compare your p-values to the test size then state your conclusion.  

(Hint, assume the Central Limit Theorem Holds)

(a) H0: ,  H1: ,  with =0.05 using the critical-value approach.

(b) H0: ,  H1: ,  with =0.05 using the critical-value approach.

(c) H0: ,  H1: ,  with =0.05 using the p-value approach.

(d) H0: ,  H1: ,  with =0.05 using the p-value approach

Q10 (2 marks) You want to test whether a unit increase in a car’s weight (mass) has a greater detrimental effect on fuel efficiency than a unit increase in the power of the car’s engine, rather than the same effect. Formulate a hypothesis test to do this. Use re-parametrisation to convert the model to allow you to test this hypothesis using a simple t-test

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