1) Consider the following regression model Yi = &beta

Homework answers / question archive / 1) Consider the following regression model Yi = β0 + β1 log(Xi) + ui, i = 1,

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.

(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

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,

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.

(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 (α1, α2, β1, β2). Let (uˆ, vˆ) be the residuals of the OLS estima- tions. Prove that uˆ = vˆ.

(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(ρ − 1)t + ρyt−1 + 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 = ρyt−1 + γ1zt + γ2zt−1 + γ3zt−2 + γ4zt−3 + εt with |ρ| < 1,

where εt is an MA(1) process, i.e.,

εt = ut + θut−1.

ut is a white noise process generated randomly from a normal distribution with mean zero and constant variance that is independent of yt−1, yt−2,, ..., zt, zt−1, ... . 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 = ρyt−1 + 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)

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)

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.

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,

where c is a positive constant and X = n−1 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.

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

i=1

your answers.

(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

(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.

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.