For a sample of 5 observations the following model was estimated

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Question 1 (15 marks)
For a sample of 5 observations the following model was estimated:
2
01 2 y x xe ?? ?? ?? ? (1)
The Eviews output for this estimation is given in Table 1.
Table 1: Regression Results for Equation (1)
Dependent Variable: Y
Sample: 1 5
Variable Coefficient Std. Error t-Statistic Prob.
C 0.981556 0.070343 13.95395 0.0051
X 1.374640 0.326153 4.214707 0.0519
X^2 -1.354467 0.309784 -4.372292 0.0485
S.E. of regression 0.072819R-squared 0.905311
Sum squared resid 0.010605Adjusted R-squared 0.810622
However, it is suspected in equation (1) the restriction given by ?2 1 ? ?? holds. That is, that
the model should be specified as:
2
01 1 y x xe ?? ?? ?? ? (2)
(i) Using calculus, derive the following equations for the OLS estimators of 0 1 ? ? and in
equation (2): 0 1 b y bz ? - and ? ? i i
1 2 2
i
n
=
n
zy zy b
z z
? ?
? ? where ? ? 2 z xx ? ? .Show your working.
(6 marks)
(ii) Table 2 provides information for the model in Equation (2). Use this and the equations
you derived in (i) above to calculate the OLS estimates 0 1 b b and in Equation (2) for the
sample of 5 observations. (3 marks)
Table 2
obs y X 2 x ? ? 2 x ? x ? ? 2 x ? x ? ? 2 x ? x y ? ? 2 x ? x y ? ?
2 2 x ? x ? ?2 2 x ? x yˆ
1 1 0 0 0 0 0 0 0 0 0.992336
2 1.2 0.3 0.09 0.21 0.39 0.252 0.468 0.0441 0.1521 1.276283
3 1.4 0.5 0.25 0.25 0.75 0.35 1.05 0.0625 0.5625 1.330369
4 1.2 0.8 0.64 0.16 1.44 0.192 1.728 0.0256 2.0736 1.208677
5 1 1 1 0 2 0 2 0 4 0.992336
sum 5.8 2.6 1.98 0.62 4.58 0.794 5.246 0.1322 6.7882
mean 1.16 0.52 0.396 0.124 0.916 0.1588 1.0492 0.02644 1.35764
(iii) Now test the restriction 02 1 H :? ? ?? . On the basis of the results of this test which model
do you prefer – the model given by equation (1) or the model given by equation (2)?
(6 marks)
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Question 2 (15 marks)
Thomas Piketty, Emmanuel Suez and Stefanie Stantcheva (NBER Working Paper No.
17616; 2011) examine evidence from 18 OECD countries that there is a strong negative
correlation between top tax rates and top 1% income shares since 1960. The file TOP1.WF1
includes data on income shares of the top 1% (Top1Share) from 18 OECD countries
combined with top income tax rate data (TopMTR) for two different periods. The first is 1960
and the second period is 2005. The variables for the 1960 period are called TopMTR1960 and
Top1Share1960 and the variables for the period 2005 are called TopMTR2005 and
Top1Share2005.
(i) Use a scatter plot to plot the two variables TopMTR1960 and Top1Share1960.
What does the plot show? (3 marks)
(ii) Use a scatter plot to plot the two variables TopMTR2005 and Top1Share2005.
What does the plot show? Use the same scales for both axes used in the plot
for (i). (3 marks)
(iii) Generate the variables (1 1960) ?TopMTR and (1 2005) ?TopMTR and
describe what these variables measure. Estimate an elasticity for Top1Share
with respect to (1 ) ?TopMTR for each of the two periods. In both cases is the
estimated elasticity consistent with what you expected from looking at the
plots? (6 marks)
(iv) Use a scatter plot to plot the change in top income shares from the 1960s to the
2000s against the change in the top marginal tax rate. Describe what this plot
shows and discuss whether or not is it consistent with the estimation results
you obtained in (iii) above. (3 marks)