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计算代写-A20520

时间：2021-01-14

A20520 Page 5 Turn Over

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%]

A20520 Page 5 Turn Over

=

(. − . )

⁄

.

⁄

= .

(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

A20520 Page 5 Turn Over

= + ̃ + −

= + ̃ +

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

A20520 Page 5 Turn Over

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:

A20520 Page 5 Turn Over

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%]

A20520 Page 5 Turn Over

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

A20520 Page 5 Turn Over

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.

A20520 Page 5 Turn Over

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.

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%]

A20520 Page 5 Turn Over

=

(. − . )

⁄

.

⁄

= .

(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

A20520 Page 5 Turn Over

= + ̃ + −

= + ̃ +

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

A20520 Page 5 Turn Over

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:

A20520 Page 5 Turn Over

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%]

A20520 Page 5 Turn Over

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

A20520 Page 5 Turn Over

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.

A20520 Page 5 Turn Over

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.