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#### 1) Consider the following regression model   Yi  = β0 + β1 log(Xi) + ui,          i = 1,

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1) Consider the following regression model

Yi  = β0 + β1 log(Xi) + ui,          i = 1, ..., n.

The error term is i.i.d. with zero mean and variance equal to σ2 and the error and regressors are independent.  We assume Xi  > 0 for all i.

 }i=1
(a)  (4  marks)  Suppose  you  are  given  a  random  sample  of  observations  {(Yi, Xi)  n     .

Derive the OLS estimator of β1  and provide an interpretation of this parameter.

 i
(b)  (4 marks) Unfortunately you do not observe Xi, instead you observe X  which equals

 X∗
i   = Xi exp(εi),

where εi  has zero mean and is independent of ui  and Xi  (and therefore log (Xi)). You

n

are  given  a  random  sample  of  observations  {(Yi, X)}

. Formulate  the  estimable

i      i=1

regression model and show that the OLS estimator of β1  is inconsistent, resulting in an underestimate in absolute size of the coefficient on average.

2.  Consider the regression model

y = β0 + β1x1 + β2x2 + u,  with E(u) = 0.

 }i=1
You may assume that the regressors are non-stochastic.  Suppose the error has a normal distribution and exhibits heteroskedasticity, i.e., V ar(ui) varies with x1i  and/or x2i, for all i. You are provided with a random sample {(yi, x1i, x2i)  n     .

(a)  (4  marks)  Discuss  how,  with  the  help  of  heteroskedasticity  robust  standard  errors,

 0
you  would  test  the  null  hypothesis  H(a)

:  β1

=  1  against  the  one-sided  alternative

 H(a)
A     : β1  > 1 when you use OLS to estimate β1.  In your answer explain the need for robust standard errors.

 1i
(b)  (4 marks) Suppose you are told that V ar (ui) = c2x2

where c is an unknown constant

and x1i  = 0 for all i. Discuss how this knowledge about the form of heteroskedasticity allows you to (i) obtain a more efficient estimator of β1  and (ii) hence conduct a test of the null hypothesis H0  : β1  = 0 that is more powerful.

3.  A researcher tries two regression models to describe the relationship between a variable

X > 0 and a variable Y  > 0:

log Y  = α1 + α2 log X + u,                                             (3.1)

log(XY ) = β1 + β2 log X + v.                                             (3.2)

(a)  (2 marks) Explain whether model (3.2) is a restricted version of (3.1) or not.

(b)  (2 marks) Using a random sample of size n, the researcher obtains (αˆ1, αˆ2, βˆ1, βˆ2) as the OLS estimators for 1, α2, β1, β2). Let (uˆ,) be the residuals of the OLS estima- tions. Prove that = .

(c)  (4 marks) Assume that the regression equation (3.1) describes the true relationship between X and Y and there exists a variable W uncorrelated with Y. What are the properties of the OLS estimator for γ2 if we use the following regression specification.

log Y  = γ1 + γ2 log X + γ3W + w?                                    (3.3) Explain your answer intuitively.

 }t=1
4.      (a)  (4 marks) What are the conditions for a series to be covariance stationary?  Consider the following time series model for {xt   T     :

 −                   t             t−1         t−2
xt  = xt   1 + λ2ε  + λε      + ε     ,

where εt  is i.i.d with mean zero and variance σ2, for t = 1, ..., T , and x0  = 0.  Discuss whether there are any values of λ so that xt  is covariance stationary.

 }t=1
(b)  (4 marks) Consider the following time series model for {yt   T     :

yt  = θ0 + θ1 − 1)t + ρyt1 + εt,

where  εt  is  i.i.d  with  mean  zero  and  variance  σ2,  t  =  1, ..., T ,  and  y0  =  0.  How  can we  test  whether  ρ  =  1?   Clearly  specify  the  null  and  alternative  hypothesis,  the  test statistic, and the rejection rule.

5.  Consider the following regression model

yt  = ρyt1 + γ1zt + γ2zt1 + γ3zt2 + γ4zt3 + εt      with |ρ| < 1,

where εt  is an MA(1) process, i.e.,

εt  = ut + θut1.

ut  is a white noise process generated randomly from a normal distribution with mean zero and constant variance that is independent of yt1, yt2,, ..., zt, zt1, ... .  You have a random sample and can use T  observations, which we label t = 1, 2, ..., T  for convenience, for the estimation.  Assume there is no perfect multicollinearity.

(a)  (4 marks) Briefly  discuss  the  finite-sample  and  large-sample  properties  of  the  OLS

estimator for ρ from the regression model where θ = 0. (b)  (4 marks)  Let γ1  = γ2  = γ3  = γ4  = 0, so that

yt  = ρyt1 + εt      with |ρ| < 1.

Discuss how you can obtain a consistent estimator for ρ that recognizes θ = 0.  Show that this estimator is consistent.  Clearly annotate your proof.

6.  Consider the following model to determine the effectiveness of vaccination on the contain- ment of measles, a dangerously contagious but vaccine-preventable disease, in a develop- ing country:

measlesi  = α0 + α1 log(vaccinated)i + α2childreni + β3avginci + β4cityi + ui,       (6.1)

 u
where  the  variance  of  the  error  term  ui  is  assumed  constant  and  equal  to  σ2 , and  i de- notes  a  particular  district;  measlesi  is  the  number  of  reported  measles  cases  per  10,000 residents, vaccinatedi  is the reported number of vaccinated residents per 10,000 residents, childreni  is the number of children per 10,000 residents, avginci  is the average level of in- come, and cityi  equals to 1 if the district is in a city, 0 otherwise. You have a random sample of 450 districts.

(a)  (2 marks) What is the interpretation of the coefficient α1?  What is the expected sign of α1?

Epidemiological studies suggest that areas with high rates of infection are more heavily tar- geted with vaccination programmes, hence these areas will have more people vaccinated. The following model describes this relationship:

log(vaccinated)i  = γ0 + γ1measlesi + γ3vaccentrei + γ4comi + γ5cityi + vi,         (6.2)

 v
where  the  variance  of  the  error  term  vi  is  assumed  constant  and  equal  to  σ2, vaccentrei is  the  number  of  vaccination  centres  available  per  10,000  residents  in  district  i,  comi  is  a variable  that  measures  the  communication  and  media  environment  regarding  disease containment  issues.   Assume  that  children,  avginc,  city,  vaccentre, and  com are  exoge- nous.

(b)  (7 marks) Identify and discuss two likely problems with estimating α1  by OLS. Clearly indicate the direction of bias, if any.

(c)  (7  marks)  In  light  of  your  answers  in  (b),  describe  an  appropriate  estimator  for  α1.

Briefly discuss and justify the necessary assumptions, if any.

(d)  (4 marks) Another researcher argues that there should be no statistical difference be- tween the estimates for α1  obtained from OLS and from the estimator in (c). Describe a test that might help the researcher statistically justify the argument.  Clearly indicate the null, the alternative, test statistic, test distribution, and rejection rule. Are there any limitations of the test?

7.  Let  X1, X2, ..., Xn   be  a  random  sample  of  size  n  from  the  exponential  distribution  with

parameter θ.  The probability density function is specified as

f(x) =

1     x

θ e θ ,  x ≥ 0 and θ > 0.

Note that E(Xi) = θ and V ar(Xi) = θ2.

In this question we will consider various estimators of θ.

(a)  Let us first consider a sequence of estimators given by

bθc  = cX,

n

where c is a positive constant and X = n1  P Xi.

i=1

For example, bθ1  = X and bθ0.5  = 0.5X¯.

i.  (6 marks) For any given  c, determine the bias, variance, and mean squared error

(MSE) of the estimator bθc.

Note that MSE(bθ) = E(bθ − θ)2  = Bias2(bθ) + V ar(bθ).

ii.  (3 marks) Show that c  =    n   minimizes the MSE of bθ  .

n + 1                                           c

Hint:  You will need to look at the derivative of MSE(bθ) with respect to c.

n

iii.  (5 marks) Is the estimator bθc    = (n + 1)1  P Xi  unbiased?  Consistent?  Prove

i=1

(b)  (6 marks) Derive the MLE estimator for θ, ˆθMLE. Briefly discuss what desirable prop- erties this estimator has.

8.  Let us consider a dataset containing information on government expenditure on education (EducE),  gross  domestic  product  (GDP ),  population  (P op),  and  capital  stock  (Capital) for  153  countries  in  2019.    Economic  theory  suggests  that  government  expenditure  on education per capita has a positive effect on per capita GDP .  That is,

perGDPi  = β1 + β2perEducEi + β3perCapitali + ui,

where perGDPi  =  GDPi , perEducEi  =  EducEi , and perCapitali  =  Capitali , and i denotes a

particular country.

P opi

P opi

P opi

(a)  (6  marks)  Explain  intuitively  why  we  should  be  worried  about  the  presence  of  het- eroskedasticity  in  this  model.   Briefly  indicate  three  distinct  methods  to  potentially deal with heteroskedasticity in this situation.

(b)  (5  marks)  Discuss  how  you  would  implement  the  White  test  for  the  presence  of heteroskedasticity  in  this  case.   Briefly  discuss  how  the  White  test  differs  from  the Goldfeld-Quandt test.

(c)  (4 marks) Discuss  two  reasons  why  we  may  expect  the  OLS  estimator  for  β2  to  be neither unbiased nor consistent.  Provide supportive arguments for each.

(d)  (5  marks)  The  dataset  also  has  a  dummy  variable  Democratici   which  equals  1  if country i is democratic,  0 otherwise.  Discuss how you  could test  whether the  effect of government expenditure on education per capita is higher for democratic countries. Clearly indicate the regression you would run and the test you would use.

Comment:  You may assume that the OLS estimator of the regression you would run is consistent, but there may be heteroskedasticity in your model.

9.  We are interested in explaining the binary variable vhappy, a dummy variable that denotes whether an individual considers him/herself "very happy" or not (1 = yes, 0 = no).

We  consider  the  following  socio-demographic  variables:  occattend  and  regattend  (which are  dummy  variables  indicating  whether  the  individual  occasionally  or  regularly  attends church,  where  the  excluded  dummy  indicates  that  the  individual  never  attends  church), income  (family  income  in  1,000US\$),  unemp10  (dummy  indicating  whether  the  individual has been unemployed in the last 10 years), and educ  (years of education completed).  Let z  be  a  linear  function  of  the  variables  that  determine  the  probability  of  being  very  happy, where:

z = β0 + β1occattend + β2regattend + β3 log(income) + β4unemp10 + β5educ.

The  β  parameters  have  been  estimated  based  on  a  Linear  Probability  Model  (LPM)  and Logit Model.  Two sets of results (labelled specification (A) and (B)) are obtained based on a random sample of 9,963 observations from US. The usual standard errors for the Logit model and the robust standard errors for the LPM are reported in parentheses.

LPM (A)     LPM (B)     Logit (A)     Logit (B)

constant                   .080

(.032)

.088

 −
(.032)

2.074

 −
(.178)

2.032

(.176)

 −
occattend                 .005

(.010)

regattend                 .093

(.014)

.023

 −
(.051)

.420

(.065)

log(income)             .061

(.012)

 −
unemp10                 .097

(.010)

educ                          .008

(.002)

.064

 −
(.012)

.103

(.010)

.007

(.002)

.362

 −
(.070)

.487

(.050)

.036

(.008)

.379

(.069)

 −
.514

(.049)

.034

(.008)

R2                                       .023            .019

log L                                                      −5890.6     −5998.9

(a)  (5 marks) Discuss clearly how the parameter estimates for β are obtained for the Lin- ear Probability Model and the Logit model. No technical details expected, but intuition rewarded.

(b)  For this part use specification (A).

i.  (5  marks)  Obtain  for  both  models  the  predicted  probability  of  happiness  for  an individual  who  regularly  attends  church,  has  a  family  income  of  £150,000,  has not  been  unemployed  in  the  last  10  years  and  has  13  years  of  education.  Hint: log(150) ' 5.

ii.  (5 marks) Obtain for both models what the effect of an additional year of education is for this individual and state whether the impact of a further additional year is the same.

(c)  You would like to test the joint significance of the church attendance variables occat- tend  and regattend.

i.  (2 marks) Explain why you cannot conduct this test using the LPM based on the reported results.

ii.  (3 marks) Explain how you would conduct this test using the Logit Model based on the reported results.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.

Dennis V. Lindley, William F. Scott, New Cambridge Statistical Tables, (1995) © Cambridge University Press, reproduced with permission.