Bill, an undergraduate student in Accounting and Finance, is examining the relationship between innovation and firm size in the chemical industry. The following simple regression was estimated
using the data on 28 companies:
Where variable rd denotes annual expenditures on research and development (R&D), and sales denotes annual sales, both are in millions of dollars. is the dependent variable y, and is the independent variable x. The disturbance terms, , are assumed to be normally and independently distributed with a mean of zero and constant variance s^{2}. He has completed a preliminary analysis of the sample data and produced the following sample information:
Lower case letters indicate that the variables are measured as deviations from their respective sample means i.e = .
Use the above sample information to answer all following questions. Show allcalculations explicitly.
 Calculate estimates, b_{0} and b_{1}, for the unknown equation coefficients, and .
[10 marks]
 Write down the regression equation and interpret the estimated coefficients.
[5 marks]
 Calculate the estimated standard errors of the estimated coefficients.
[5 marks]
 Perform a test of the null hypothesis H_{0}: = 0 against the alternative hypothesis.
[5 marks]
 Compute the value of R^{2} for the estimated regression. Briefly interpret the meaning of R^{2}.
[5 marks]
[Total 30 marks]

 Define the four assumptions that need to hold under the GaussMarkov Theorem(include the relevant mathematical expressions in your answer).
[10 marks]
 What is the main contribution of the GaussMarkov Theorem to modelling?
[5 marks]
[Total 15 marks]

 Explain the problem of heteroskedasticity.
[5marks]
 What are the consequences of heteroskedasticity for OLS?
[5marks]
 How the presence of heteroskedasticity can be tested?
[5 marks]
[Total 15 marks]
Q4.
6.1The linear regression model below is estimated to analyse the demand for petrol in the UK (c_{t}),
disposable income (y_{t}) and the price index of petrol (p_{t}) are included as explanatory variables. The
data contain 43 observations from 1968 to 2010.
. reg c y p Source  SS df MS Number of obs = 43 ————+——————————— F(2, 40) = 175.66 Model  .879902606 2 .439951303 Prob> F = 0.0000 Residual  .100181351 40 .002504534 Rsquared = 0.8978 ————+——————————— Adj Rsquared = 0.8927 Total  .980083958 42 .023335332 Root MSE = .05005 —————————————————————————— c  Coef. Std. Err. t P>t [95% Conf. Interval] ————+————————————————————— y  .731198 .0765292 9.55 0.000 .5765268 .8858692 p  .1007774 .034183 2.95 0.005 .1698637 .031691 _cons  18.91766 .6265087 30.20 0.000 20.18388 17.65144 dwstat // DurbinWatson test DurbinWatson dstatistic( 3, 43) = .1266305 
 The residuals of the regression are plotted in the figure below. The pattern of residuals implies that the assumptions for simple linear regression might not be satisfied. Discuss the potential problem.
[10marks]
b) Suppose the errors in the above case follow the AR(1) model. For instance, we assume that
Perform a DurbinWatson test (at both 5% and 1% significant levels) and interpret the results.
[10 marks]
Type of serviceAcademic paper writing
Type of assignmentCoursework
SubjectStatistics
Pages / words11 / 3000
Number of sources5
Academic levelUndergraduate
Paper formatHarvard
Line spacingDouble
Language styleUK English