1.  Consider the following linear model;

yi  = β0 + β1 xi + β2 zi + β3wi + ui                                                        (1)

(a)  Explain,  in the context of the  model  provided, what is  meant  by the term  het-

eroscedasticity of the error terms. [5 MARKS]

(b)  If the model above does indeed suffer from heteroscedasticity of the error terms,

then the OLS estimator of the model parameters is no longer BLUE. Do you agree? Explain your answer. [5 MARKS]

(c) You are told that the form of the heteroscedasticity affecting the model is known and that,

Var(ui ) = σ 2wi xi(2) .

Show that, by using ordinary least squares, it is possible to estimate the parameters of an amended model which does not suffer from heteroscedasticity?  What is the name of the resulting estimator? [15 MARKS]

2. A data set contains the following variables, each of which has been observed for 753 women.

(a)  Consider the following model for labour force participation

inlfi  = β0 + β1nwifeinci + β2 agei + ui

to be estimated using the dataset just described.  Given the binary nature of the dependent variable, explain why OLS is not the Best Linear Unbiased Estimator. What other issues, if any, need addressing? [10 MARKS]

(b) Suggest an alternative estimation method and discuss how it will address the issues you note in part (a). [15 MARKS]

3.  In the below you have been provided with plots of the time series of Tomato Prices (Per KG) used in the Consumer  Prices  Index and the  Producer’s  Price  Index for  Furniture exported to the EU.

The below regression output is obtained when estimating the following model: furnituret  = β0 + β1Tomatoest + et

Call:

lm(formula  =  Furniture  ~Tomatoes)

Residuals:

Min           1Q   Median           3Q         Max

-9 .2704  -2 .0108  -0 .6317    1 .5452  18 .6197

Coefficients:

Estimate  Std .  Error   t  value       Pr(>|t|)

(Intercept)           -24 .42805        1 .50435    -16 .24     <2e-16  ***

Tomatoes                     1 .32501       0 .01681     78 .82     <2e-16  ***

---

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

Residual  standard  error:  4 .064  on  312  degrees  of  freedom Multiple  R-squared:    0 .9522,Adjusted  R-squared:    0 .952    F-statistic:    6213  on  1  and  312  DF,   p-value:  <  2 .2e-16

(a)  Discuss whether or not the two time series are likely to be stationary. [10 MARKS]

(b)  Explain whether or not the results of the regression presented in the question should be relied upon. [8 MARKS]

(c)  Is it possible that there is a relationship between tomato prices and furniture prices in light of the evidence presented? [7 MARKS]

4.  Consider the following model of wages.

wagei  = β0 + β1 educi + β2 experi + β3marriedi + β4 kidsi + ei                     (3)

Where wagei  is the wage earned by the ith  individual, educi  is their years of completed education, experi  is the number of years of work experience an individual has accumu- lated, marriedi  is a dummy variable taking a value of 1 if the individual is married and kidsi  is the number of children of school age the individual has caring responsibilities for.

(a)  Explain how a structural break in the model across genders might affect the model

presented in (3).  What might be the differences in two of the parameters of the model presented in (3) that you expect to see? [8 MARKS]

(b)  Explain how you might test whether or not there is a structural break in the model

above, across gender.

To implement your test you may assume that there is a dummy variable labelled “female” in the dataset. [14 MARKS]

(c)  How might you extend your tests above if you thought there might be structural breaks across other groups defined using dummy variables? [3 MARKS]