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Specimen Exam: Sketched solutions

Section A
1.
(a) A model of uncovered interest rate parity was estimated over the period 1st
quarter 1992 to 4th quarter 2017 (104 observations) with standard errors in
parentheses:
ln+1 = 0.031 + 0.923ln + 0.873(
$ − ) + +1 (1)
(0.021) (0.046) (0.514)

Total sum of squares = 0.988 Residual sum of squares = 0.159

where = spot exchange rate of the U.K. pound relative to the US dollar

$ = US three-month interest rate
= UK three-month interest rate

(i) Construct a 95 per cent confidence interval for the constant. [5%]

0.031 - 1.96*0.021 < α < 0.031 + 1.96*0.021
-0.01016 < α < 0.07216

(ii) Test the null hypothesis that the coefficient on ln equals 1. What do you
conclude? [5%]

t102 = (0.923-1)/0.046 = -1.67391

(iii) Calculate the 2. [10%]

=1-(0.159/0.988) = 0.839069

(iv) Calculate the standard error of the equation. [10%]

̂= 0.159/102= 0.001559
̂ = 0.0395

(b) The equation now has the coefficient on ln impose equal to 1 and the constant
set to zero, giving the equation:

∆ln+1 = (
$ − ) + +1

(i) The residual sum of squares has increase to 0.187. With the F-critical value
being 3.09, undertake a test for the joint hypothesis that the coefficient on ln
equals 1 and the constant is zero. [10%]

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=
(. − . )
⁄
.
⁄
= .

(ii) Calculate the standard error of the equation. [10%]

̂= 0.187/103= 0.00182
̂ = 0.0426

(iii) Comment on problems using 2 as a measure of the goodness of fit when the
dependent variable is reparameterised. [10%]


= −




Initially = ∑( − ̅)
but become = ∑(∆ − ∆̅̅ ̅̅ )


(c) The residuals in equation (1) are believed to be:

= −1 + where ~(0,
2 )

(i) How would you test to see if the residuals had such a relationship? [10%]

LM-test for autocorrelation.

(ii) What are the implications for the estimated coefficients if the residuals followed
such a pattern? [10%]

Coefficients are unbiased but inefficient

(d) It is believed a risk premium term should be included, making the equation:
∆ln+1 = (
$ − ) + + (2)

where is measured with error as ̃ = + with ~(0,
2).
Carefully explain what are the implication for the estimated coefficient .[20%]


Consider a simple regression model
Yi = β0 + β1RISKi + ui
If RISKi is measured with errors
̃ = RISKi + ei

Assume that ei is uncorrelated with RISKi so E(RISKi ei) = 0.

The regression equation estimated is
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= + ̃ + −

= + ̃ +

The problem of endogeneity occurs as

(̃) = (( + )( − ))

(̃) = − () ≠



2. (a) What is meant by panel data? What advantages are there with panel estimation?
[15%]

This is a dataset where the same cross section units are observed over time, giving the
equation

'
it i it it
y X u    i = 1, 2, …, N
t = 1, 2, …, T

Advantages of panel data

1. Increased sample size so number of associated benefits
2. More variability in the data
3. Incorporates dynamics in the analysis
4. Potential instrumental variables
5. Enables techniques to be employed to overcome missing observations
6. Greater heterogeneity in the data – potentially makes the equation more robust

(b) Cabral and Mollick (“Mexican real wages and the U.S. economy”, Economic
Modelling 2017) estimated the following relationship over the period 1996 to 2014
across Mexican states using fixed effects:

ln (/) = + 1 ln
∗ + 2 ln + 0 ln + 1 ln + 2∆ ln +

where
(/)= Real wage in state i at time t

∗ = US real GDP at time t
= Average years of schooling in state i at time t
= Real exchange rate at time t (where an increase is a depreciation)
= Foreign direct investment in state i at time t
= Population of state i at time t

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i) If was measured in months of schooling, i.e. multiplied by 12,
explain what happens to the constant. [10%]

  *1 2 0 1 2ln / ln ln ln ln lnio t it t it it ititW P Y EDU RER FDI POP             

To have
2
ln12
it
EDU  as a regressor, the above equation needs to modified as

  *1 2 2 2
0 1 2
ln / ln ln ln12 ln12
ln ln ln
io t itit
t it it it
W P Y EDU
RER FDI POP
    
   
   
   



Hence

  *2 1 2 0 1 2ln / ln12 ln ln12 ln ln lnio t it t it it ititW P Y EDU RER FDI POP                
With constant being
2
ln12
io
 

ii) What is meant by ∆ ln ? [10%]


1
ln ln ln
it it it
POP POP POP  = Population growth in state i at time t

(c) The authors state “To minimize endogeneity problems, we initially lag all regressors
by one year in a fixed effects model approach.”
i) What is meant by fixed effects? [10%]

Usually means the least squares dummy variable model, allowing for the
individuality of each cross sectional element, the intercept varies but the slopes
remain the same.
This is known as the fixed effect model as the intercept varies over cross
sections but not over time. The slope coefficients are constant over cross
sections.

ii) What are endogeneity problems? [10%]

The endogeneity problem is that ( ) 0
it it
E X  where itX denotes one of the regressors

iii) Why would lagging regressors minimise the endogeneity problem? [10%]

More likely for
1
( ) 0
it it
E X   as X is pre-determined so should not be affected by it

iv) Explain why it does not make sense to lag
∗. [10%]

US real GDP at time t is unlikely to be affected by real wage in Mexican state i at
time t
(d) The results are as follows:
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Model 1 standard error Model 2 standard error
Constant -10.84 3.714 5.04 1.191
0.544 0.188 1.293 0.109

∗ 0.637 0.148
-0.05 0.025 -0.155 0.018
0.005 0.002 0.005 0.002
-0.271 0.094 -0.121 0.096

n 574 574
R2 0.896 0.875

i) Comment on the estimated coefficients of Model 1. [10%]


Variable

∗ Expected to be positive –
could be due to increase in
Mexican trade with US
Correctly signed
and statistically
significant
Increase years of schooling
increases productivity
leading to an increase in
wage
Correctly signed
and statistically
significant
Increase in real exchange
rate reduces
competitiveness of
Mexican goods so wages
go down.
Correctly signed
and statistically
significant
Increase in FDI into state i
results in increased
demand for labour so
increase in real wage
Correctly signed
and statistically
significant
∆ Growth in population
reduces wage as increase
supply of potential workers
Correctly signed
and statistically
significant


ii) Compare the results from Model 1 and Model 2. [10%]

Dropping
∗ results in changes in magnitude of coefficient, all increasing in
effect, and drop in R2

iii) Do you conclude that US real GDP determines the real wage in Mexico?
[10%]

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Tests, ie t or F, would show that US real GDP impacts on the real wage in Mexico .
Could be the only trending variable so see what happens when a time trend is
included.

3.
(a) What are the assumptions that are required for the ordinary least squares
estimator to be the best linear unbiased estimator? [10%]

Yi = 0 + 1X1i + 2X2i + ... + kXki + ui

The assumptions concerning ui are as follows:-
1. E(ui) = 0
2. E(ui2) = 2
3. E(uiuj) = 0 for i j
4. E(Xjui) = 0  i and j
5. Xis are stationary

(b) The following wage equation was obtained using a cross-section of 2269
individuals aged between 20 and 65, with standard errors are given in
parentheses

iiiiiii uOldMiddleMaleTenureSW 
)003.0(
057.0
)007.0(
032.0
)053.0(
166.0
)055.0(
100.0
)009.0(
047.0
)070.0(
839.9ln
Total sum of squares = 3714 Residual sum of squares= 1763

Wi Annual wage of individual i
Si Years of education after the age of 15
Tenurei Number of years with the current employer
Male 1 if individual i is a male and 0 otherwise
Young 1 if individual i is aged between 20 to 34 and 0 otherwise
Middle 1 if individual i is aged between 35 to 49 and 0 otherwise
Old 1 if individual i is aged between 50 to 65 and 0 otherwise

i) What is the interpretation of the constant? [5%]

Constant represents wage for an individual who has 0 years of education
after age of 15, 0 years with the current employer, is a female and is
young.

ii) Undertake a test of the null hypothesis that the coefficient on Middle equals
zero. What do you conclude at the 5% significance level? [10%]

t =0.032/0.007 = 4.57

iii) Rather than Middle and Old including in the wage equation, if the dummies for
Young and Middle had been included as regressors what would happen to the
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value of the constant? What would be the coefficients on Young and on Middle?
[15%]

Constant would increase.
9.839 + 0.057 = 9.896
Coefficients would be:
Young -0.059
Middle -0.025

i) Construct a 95% confidence interval of the coefficient on Si. [10%]

0.047 - 1.96*0.009 < β < 0.047 + 1.96*0.009

0.02936 < β < 0.06464

ii) Calculate R2 and undertake a test for the overall significance of the regression.
[10%]

R2 = 0.2503
− =
.
⁄
.
⁄
= .

(c) A wage equation with Si, Tenurei, Youngi, and Middlei as explanatory variables was
estimated separately for 1241 females and for 1028 males, with the residual sum of
squares equaling 940 and 603 respectively. When estimating the model with the
combined data the residual sum of squares equaled 1664. Test the null hypothesis
that the coefficients are the same for males and for females. [10%]

− =
− ( + )
⁄
( + )
⁄
= .

(d) It is claimed that the number of years, Tenurei, is a function of the wage received
i) Carefully explain what are the implications for estimating the wage
equation by ordinary least squares. [10%]

The coefficients will suffer from simultaneous equation bias.

ii) Briefly discuss how to overcome any econometric problems that arise.[5%]

Use IV estimation. Want an instrument that is unrelated to Wi but correlated
with Tenurei.

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e) If data are obtained on the same 2269 individuals in 5 years’ time and 10 years’
time, carefully explain how the wage equation should be modified to incorporate
the new information. What are the advantages of using panel datasets? [15%]

Discussion of use of fixed effects dummies.

Large differences between the estimated coefficients from the Fixed Effects
Model and the Random Effects Model. If the data are from a fixed sample
then the Fixed Effects Model is more appropriate.

If the cross-sectional units are random selections from a larger
distribution, then Random Effects Model is more suitable.