ECON20110 Econometrics Semester 1 2018/19 Paper
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ECON20110 Econometrics
Semester 1 2018/19 Paper
1. A researcher is interested in the relationship between alcohol consumption (average units of alcohol consumed per week) and how much an individual earns (annual salary in ↔1,000’s) and their level of education (reported in years). The researcher proposes the following linear model:
log(consumptioni ) = β0 + β1 log(incomei ) + β2 educationi + ui (1)
where ui is an unobserved error term.
(a) How would the researcher interpret the parameter coefficients β1 and β2 ?
[5 MARKS]
(b) In terms of the variables included in model (1), state the Gauss-Markov Theorem
along with all the required assumptions.
[10 MARKS]
(c) Do you think that OLS will uncover the causal relationship of income and education on alcohol consumption? Explain. [Hint: Discuss what kind of factors are contained in the error term u. Are these factors likely to be correlated with income and education?]
[10 MARKS]
2. Consider the following model:
y = Xβ + u (2)
┌ β0 ┐
where y is a (n× 1) vector containing observations on the dependent variable, β = ' β1 ' ,
' β2 '
X is a (n × 3) matrix. The first column of X is a column of ones whilst the second and third columns contain observations on two explanatory variables (x1 and x2 respectively). u is (n × 1) vector of error terms. The following are obtained:
┌ |
192.0 |
259.6 |
1153.1 ┐ |
┌ |
1234.718 ┐ |
X ′ X = ' |
259.6 |
402.8 |
1371 . 1 ' ; |
X ′ y = ' |
1682.376 ' |
' |
1153.1 |
1371.1 |
8107.6 ' |
' |
7345.581 ' |
┌ 0.2792 -0.1055 -0.0219 ┐
(X′ X)一 1 = ' -0 . 1055 0 .0457 0 .0073 ' ; u = 0 .08069 ,
' -0.0219 0.0073 0.0020 '
where u is standard error of the regression.
(a) (i) How many observations are there, i.e. what is the value of n?
[2 MARKS]
(ii) Calculate the OLS estimates of β, and the standard error of βˆ1 .
[10 MARKS]
(b) For each of the following hypothesis tests, clearly state the null and alternative
hypotheses, the test statistic and its distribution, the decision rule, and your conclu- sions. Use a 5% significance level in both cases.
(i) Test the null hypothesis that explanatory variable x1 has no effect on the de- pendent variable y.
[5 MARKS]
(ii) The R2 from the OLS regression of model (2) is 0.7763, test the overall signif-
icance of the regression.
[8 MARKS]
3. A national fast-food restaurant is struggling and has asked you to advise on how to turn around their fortunes and increase their monthly sales. Of particular interest to them is how price sensitive their customers are and whether their advertising spending is value for money. They have provided you with data from 75 of their stores across the country with the following information (unit of measurement in brackets):
❼ sales - monthly sales revenue in a given restaurant (↔1,000)
❼ price - average price of all menu items sold (↔1)
❼ advert - monthly spend on advertising (↔1,000)
pricesq and advertsq are the squared values of price and advert respectively. Using the provided data you estimate 4 separate models. The R output from the OLS estimation of each of these models follows:
Model A
Call:
lm(formula = sales ~ price + advert, data = data)
Coefficients:
(Intercept) price advert
Estimate Std. Error t value Pr(>|t|)
118.9136 6.3516 18.722 < 2e-16 -7.9079 1.0960 -7.215 4.42e-10
1.8626 0.6832 2.726 0.00804
Residual standard error: 4.886 on 72 degrees of freedom
Multiple R-squared: 0.4483,Adjusted R-squared: 0.4329
F-statistic: 29.25 on 2 and 72 DF, p-value: 5.041e-10
Model B
Call:
lm(formula = sales ~ advert, data = data)
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 74.1797 1.7990 41.234 <2e-16
advert 1.7326 0.8903 1.946 0.0555
Residual standard error: 6.37 on 73 degrees of freedom Multiple R-squared: 0.04932,Adjusted R-squared: 0.0363 F-statistic: 3.787 on 1 and 73 DF, p-value: 0.0555
Model C
Call:
lm(formula = sales ~ price + pricesq + advert, data = data)
Coefficients:
(Intercept) price pricesq advert
Estimate Std. Error t value
206.0446 -38.6449 2.6952 1.7478 |
85.8045 30.2060 2.6469 0.6923 |
2.401 -1.279 1.018 2.525 |
Pr(>|t|)
0.0190
0.2049
0.3120
0.0138
Residual standard error: 4.885 on 71 degrees of freedom
Multiple R-squared: 0.4562,Adjusted R-squared: 0.4332
F-statistic: 19.85 on 3 and 71 DF, p-value: 1.891e-09
Model D
Call:
lm(formula = sales ~ price + advert + advertsq, data = data)
Coefficients:
(Intercept) price advert advertsq
Estimate Std. Error t value
109.7190 -7.6400 12.1512 -2.7680 |
6.7990 1.0459 3.5562 0.9406 |
16.137 -7.304 3.417 -2.943 |
Pr(>|t|)
< 2e-16
3.24e-10
0.00105
0.00439
Residual standard error: 4.645 on 71 degrees of freedom
Multiple R-squared: 0.5082,Adjusted R-squared: 0.4875
F-statistic: 24.46 on 3 and 71 DF, p-value: 5.6e-11
(a) Consider Model A, interpret each of the three estimated coefficients. Comment on
whether the magnitudes and signs of the estimates are reasonable.
[12 MARKS]
(b) Comment on the statistical significance (individually) of the price and pricesq vari-
ables in Model C. Are these two variables jointly significant? For the joint test, clearly state the null and alternative hypotheses, the test statistic and distribution, your decision rule, and your conclusion. [Hint: Model B is the restricted model.]
[8 MARKS]
(c) Using model D, what is the optimal average monthly advertising spend?
[5 MARKS]
4. Consider the following model for crime levels in city i in time period t based on the amount of police officers employed by the city in the previous period t - 1.
Crimeit = β0 + β1 Policei,t一1 + uit (3)
(a) (i) Explain what it means for the variable Police to be endogenous. Comment on
the consequences of using OLS to estimate β1 in model (3) if we assume the variable Police is endogenous.
[5 MARKS]
(ii) You also have infomation on the number of firefighters employed by city i in period t - 1. Explain how this additional information could be used in Instru- mental Variable (IV) estimation. Comment on the conditions required for an instrumental variable to be suitable and detail how these conditions may be tested.
[10 MARKS]
(b) A number of city councils are keen to reduce crime levels in their city and embark
on a policy of putting more police officers on the street. Explain how you could use the difference-in-difference estimator to estimate the effectiveness of such a policy. Carefully explain what information you would require to perform the analysis, how you would estimate the policy effect, and any required assumptions.
[10 MARKS]
2022-05-25