THE UNIVERSITY OF MANCHESTER ECON20222 Quantitative Methods 2021/22 Semester 2 Exam 

The exam is set as if it was a 1.5 hour sit-down exam. Release Date: 30 May 2022, 2:00pm Submission Deadline: 1 June 2022, 2:00pm INSTRUCTIONS SPECIFIC TO THIS EXAM: �� Answer all questions. Marks are as indicated. You can earn a maximum of 100 marks. �� You have to submit typed responses. Your exam will not be accepted if it is not typed. You can include images of algebra or graphs if you wish. �� Where relevant, questions include word limits, inside [ ] at the end of each question part. These are limits, not targets. Excellent answers can be shorter than the word limit. If you go beyond the word limit the additional text will be ignored. Where a question includes a word limit you HAVE to include a word count for your answer (excluding formulae). You could use https://wordcounter.net to obtain word counts. �� The questions in Section A relate to the work presented in Fetzer, T. (2021) Subsi- dising the Spread of Covid-19: Evidence from the UK’s Eat-Out-To-Help-Out Scheme. Economic Journal, ueab074, https://doi.org/10.1093/ej/ueab074. This paper has been made available to you at the end of the Semester. �� When answering questions in Section B marks will be deducted for ignoring the instruc- tions given. You should consider the order in which you answer the questions. �� Tables and Figures are in the Appendix at the end of the Exam Paper. ©The University of Manchester ECON20222 Section A The questions in this section are based on the work presented in Fetzer, T. (2021) Subsidising the Spread of Covid-19: Evidence from the UK’s Eat-Out-To-Help-Out Scheme. Economic Journal, ueab074, https://doi.org/10.1093/ej/ueab074. You were asked to read this paper before this examination. Some relevant tables are reproduced in the Appendix. In your answers you can also refer to other material in the above article. 1. Briefly describe the circumstances which allow the author to consider a difference-in- difference research design to answer the question of whether or not the Eat-Out-to-Help- Out (EOHO) scheme caused higher covid infection rates. [No technical language, max 150 words.] [10 MARKS] 2. The author says that he is concerned about reverse causality. Briefly explain what the concern is. [No technical language, max 50 words.] 3. Describe briefly the key features of the data. [Max 50 words] [5 MARKS] [4 MARKS] 4. Equation (1) of the paper is the author’s Base Model. A well-established framework for estimating treatment effects is Easy Wooldridge (13.16). Is the author’s (1) fundamentally the same or does it differ in any substantial way? If the latter, briefly explain how. Explain how the treatment dummy varies over time and geographic location. [No technical language, max 50 words.] [6 MARKS] 5. The regression results presented in Table 1 suggest that the effect of the EOHO scheme was a 10% increase in Covid cases. Do you agree? [5 MARKS] 6. How does the author go about providing evidence to justify the parallel trends assumption? [max 150 words] [5 MARKS] 7. How does the author use rainfall data to strengthen the causal interpretation of the results? [ max 200 words.] 8. Would you judge the EOHO scheme a success? [max 150 words.] [10 MARKS] [5 MARKS] 1 of 8 ECON20222 Section B 9. In the simple regression model y=α+βx+u whereCov(u,x)=0, a very simple way to derive the OLS estimator in the population is to run the Covariance Operator through the model: Cov(y, x) = αCov(1, x) + βCov(x, x) + Cov(u, x). The estimator requires Cov(u, x) = Cov(1, x) = 0. Explain why can we write Cov(1, x) = 0. [Hint: In general, the definition of the covariance Cov(W,Z) is E{[W − E(W)][(Z − E(Z )}.] [3 MARKS] 10. Consider the following regression models relating log real hourly wage (log w) to age (a), whether the individual has a 1st class degree (D1 = 1), or an upper second (D2 = 1), or otherwise (D3 = 1), and whether the individual has a PG degree (P = 1) or not. log w = α + βa + (100γ)(a/10)2 + δ1D1 + δ2D2 + δ3P + u, q = 1, 2. u is an error term that satisfies the usual exogeneity assumptions. The index q denotes the type of UG degree undertaken; either q = 1: Science, Technology, Engineering and Maths (STEM), or q = 2: Law, Economics, and Management (LEM). The model is estimated for q = 1, 2 separately. This model, amongst others, was estimated by Walker, Ian and Yu Zhu, “Differences by degree: Evidence of the net financial rates of return to undergraduate study for England and Wales,” Economics of Education Review, 2011, 30 (6), 1177 to 1186. They use LFS data 2005-2009. The following was taken from their Table 5a. To answer this question you do not need to look at their paper. 2 of 8 ECON20222 γ (x100) 1st class Upper 2nd Lower 2nd and below PG degree –0.110 0.075 0.090 0.066 STEM LEM –0.103 (0.254) 0.138 (0.011) –0.150 (0.013) 0.236 (0.062) 0.185 (0.034) 0.094 (0.031) ECON20222 Constant 0.435 β 0.107 (0.138) (0.006) (0.007) (0.025) (0.018) (0.018) (a) (6 points) Consider the following 3 statements: S1 Instead of specifying the age-squared variable as (a/10)2 you re-estimate the regression with a2. The estimate and standard error for the STEM regression would be –0.00110 (0.00007). S2 In neither model is a quadratic in age justified on the evidence given. S3 The model is misspecified (“wrong”) because there should also be a term δ4D3. State whether each statement is TRUE or FALSE. If it is false, write one sentence explaining why. (b) (5 points) It is clear that the relationship between E(log w) and age is a quadratic. Compute the age at which E(log w) is a maximum for men who obtain a first class degree and complete a PG degree for both subject groups. Does your calculation change for men who get a lower second or worse (and therefore do not qualify for a PG degree)? Explain why in one sentence. (c) (6 points) Compute E(logw|a=25,q=2)−E(logw|a=25,q=1) and E(log w|a = 65, q = 2) − E(log w|a = 65, q = 1). Express your answers as pecentages. Comment briefly. (d) (6 points) Someone suggests that E(log w|a, q = 2) − E(log w|a, q = 1) is a con- stant. How would you amend the model above so that this is imposed upon the model? Make sure you define any new variables/parameters. [Hint: another com- monly used greek symbol for a regression parameter is λ.] [23 MARKS] 3 of 8 x,z y = β0 + β1x + u x = π0 + π1z + v. (a) (6 points) The investigator wants to estimate β1 using IV, but is concerned that the instrument z might be “weak”. Which of the following statements are a consequence of having a “weak intrument”: S1 A large standard error on β1,IV compared with its OLS counterpart. S2 A very low R2 when regressing x on z and a constant using OLS. S3 An insignificant OLS estimate π��1. Simply write down “S1”,“S2” or “S3”, or any other combination, or “NONE”. Do not explain why. (b) (5 points) The IV estimator in the population is given by: β1 = Cov(z,y) Cov(z, x) When z is a binary variable, use this formula to derive an expression for the IV estimator in the sample in terms of y ̄1, y ̄0, x ̄1, x ̄0, where y ̄1 is the sample average of y for the z = 1 sub-sample, y ̄0 is the sample average of y for the z = 0 sub-sample etc. (c) (3 points) Consider the following statement: S4 When x is a binary variable, it is always the case that the IV estimator can be ̄ ̄ ̄ written as y ̄1 − y ̄0, where y ̄1 denotes the sample average for y when x = 1 etc. State whether S4 is TRUE or FALSE. If it is false, write one sentence explaining why. [14 MARKS] 12. Consider the following two time series: restaurant visits (vt) and the series of rainfall (rt) (in the UK) through the pandemic period, including the period of the Eat-Out-Help-Out Scheme. Both are observed every week. vt is the number of visits per week and rt is average weekly rainfall. a) Explain what it would mean for the rt series to granger-cause the series vt. Support your explanation with a simple model you could estimate. [Max 100 words] [5 MARKS] �� ECON20222 11. Consider the following model: 4 of 8 b) Without estimating the above model, make an argument why you think that rt does (or does not) granger-cause vt. [Max. 150 words] [5 MARKS] 5 of 8 ECON20222 ECON20222 6 of 8 1 Appendix ECON20222 7 of 8 END OF EXAMINATION ECON20222 8 of 8