• You should answer ALL questions.
Use no more than 15 sides of A4. Assignments longer than this are unlikely to be sufficiently concise.
• Hand write and scan your answers.
• Justify your answers and show the basis of your calculations.
• Assignments must be submitted through the online submission point on Blackboard.
• Your assignment will be checked for plagiarism through TurnItIn.
You should attach the coversheet (available on Blackboard) to your assignment be- fore submission.
• The DEADLINE for submission is 13:00 21 August.
Late Penalties
Assignments handed in after the deadline, without a pre-arranged extension will be subject to the following penalty:
A fixed absolute penalty of 10 marks is applied for each working day work is submitted after the agreed submission deadline. A mark of zero is applied to work submitted five or more working days after the agreed deadline if this threshold is not already reached.
Assignments will be marked out of 100 marks with reference to the stated marking criteria which are located at the end of this document. Section A is worth 30 marks. Section B is worth 35 marks. Section C is worth 35 marks.
Answer all of the following 6 questions. Each question is worth 5 marks.
1. Given the random sample {Xi}N , with each Xi ∼ Bernoulli (1/4), derive the expec-
tation and variance of Xi. Now derive the expectation and variance of the sample
mean X¯ = (1/N ) ΣN Xi. Appealing to the Central Limit Theorem, explain how X¯
is approximately/asymptotically distributed in large samples. [5 marks]
2. Let wi denote the wage, and let fi denote a female dummy variable, i.e., fi = 1 if individual i is female and zero otherwise. In a linear regression of wi on fi including an intercept parameter, assuming that fi is exogenous, what is the interpretation of the coefficient on the female dummy variable fi? If fi = 1 for all i = 1, 2, . . . , N , i.e., all individuals in the sample are female, what are the implications for the ordinary least squares (OLS) estimators for the slope and intercept parameters? [5 marks]
3. Given the random sample {yi, x1i}N
, each observation satisfies
yi = β0 + β1x1i + ui,
where yi is the dependent variable, x1i is the independent variable, ui is the error term, and β0 and β1 are parameters. Suppose that you inadvertently omit the inde- pendent variable in your specification and regress the dependent variable on just a constant, i.e., each observation instead satisfies yi = δ0 + vi, where yi is the depen- dent variable, vi is the error term, and δ0 is a parameter. If you estimate δ0 using OLS, when is the OLS estimator for δ0 an unbiased estimator for β0? [5 marks]
4. Under which conditions is measurement error problematic in terms of bias and in- consistency of the OLS estimator? Suggest an alternative estimation method that addresses the issue of measurement error. [5 marks]
5. You want to analyse the effect of a tax credit, i.e., the amount of money that taxpayers can subtract from taxes owed, on labour supply. You know that the tax amount that can be subtracted depends on the number of children that one has, and that in 2000 there have been large increases in this amount for those with children. If the following STATA output aims to use a differences-in-differences method for your analysis, ex- plain the function of each of the five STATA command lines below. Discuss any errors in the code and propose solutions for said errors. (Assume that the common trends assumption holds; the variable year identifies the year, kids measures the number of children that one has, and work is a dummy variable equal to 1 if in the labour force and 0 otherwise.) [5 marks]
6. Explain when one can use a sharp regression discontinuity design and give a con- crete example (different from the ones covered in class). [5 marks]
Given the random sample {yi, (x1i, x2i, . . . , xKi)}N
, each observation satisfies
yi = β1x1i + β2x2i + • • • + βKxKi + ui,
where yi is the dependent variable, (x1i, x2i, . . . , xKi) are the independent variables, ui is the error term, and (β1, β2, . . . , βK) are parameters.
1. Derive (but do not solve) the first order conditions for the ordinary least squares (OLS) estimator βˆ = (βˆ1, βˆ2, . . . , βˆK ). [4 marks]
Suppose that K = 2, i.e., that each observation satisfies yi = β1x1i + β2x2i + ui; we will refer to this multiple linear regression model for the rest of this section.
2. Consider estimating the linear models given by yi = δx2i + vi and x1i = ηx2i + wi, where vi and wi are error terms, and where δ and η are parameters. Derive the OLS estimators for δ and η. [6 marks]
3. Using the OLS estimators for δ and η from Part 2, consider the predictions yˆi and xˆ1i, and also the residuals vˆi and wˆi. Now suppose that vˆi = θwˆi + εi, where εi is an error term, and θ is a parameter. What does the OLS estimator for θ capture? [3 marks]
When K = 2, the OLS estimators for β1 and β2 are denoted by βˆ1 and βˆ2, with βˆ1 given by
ˆ ΣN
x2 ΣN
x1iyi − ΣN
x1ix2i ΣN
x2iyi
ΣN 2 ΣN 2
ΣN 2
and similarly for βˆ2. Suppose that E(ui|x1i, x2i) = 0 and Var(ui|x1i, x2i) = σ2.
4. Prove that βˆ1 is unbiased. [7 marks]
5. Suppose that x2i = λx1i for some fixed constant λ > 0. What is the implication for the existence of βˆ1 and βˆ2? [4 marks]
6. Suppose that x1i and x2i are dummy variables, i.e., they can only take on values
of 0 and 1, and also that x2i = 1 − x1i. Show that βˆ1 = ΣN x1iyi/ ΣN 2 and
βˆ2 = ΣN x2iyi/ ΣN x2 . Why does this occur? [4 marks]
7. When x1i and x2i are dummy variables with x2i = 1 x1i, re-parametrize the model so that yi = γ0 + γ1x1i + ui, where γ0 = β2 and γ1 = β1 β2, and derive the standard error of γˆ1, the OLS estimator for γ1. [7 marks]
You are interested in estimating the effect of smoking during pregnancy on child birth weight. You have individual data on a sample of 1 million mothers from various regions within a country for a given year and you want to estimate the following model:
ln(bwght) = β0 + β1packs + β2age + u, (1)
where ln(bwght) is the natural logarithm of the child birth weight, packs is the weekly average number of packs of cigarettes smoked by the mother during pregnancy, age is the mother’s age at the time of birth, u is an error term, and β0, β1, and β2 are parameters. We will refer to this equation as model (1) for the rest of the section.
1. Assume that packs is exogenous. An ordinary least squares (OLS) regression of model (1) gives an estimate of β1 equal to 0.299 with a standard error of 0.450. How would you interpret this parameter estimate? [5 marks]
2. Discuss whether it is reasonable to assume that packs is exogenous. Show whether the OLS estimator of β1 is expected to be biased upwards or downwards. [6 marks]
Assume that the exogeneity assumption for packs is not holding. You therefore consider instrumenting packs with excise taxes on cigarettes (denoted by cigtax), which vary across regions within a country based on local laws.
3. Present the two conditions that are required for cigtax to be a relevant and exogenous instrument for packs and discuss whether they are likely to hold. [7 marks]
4. Explain how you can test whether cigtax is a strong instrument. [5 marks]
5. Show how you can implement the two stage least squares estimation of model (1) using cigtax as an instrument for packs. [7 marks]
6. Discuss how your answer to part [5.] changes if the excise tax is set at country level instead. [5 marks]
1st (70+)
• For essay-based subjects:
– Excellent knowledge and understanding of the subject and understanding of theoretical methodological issues.
– A coherent argument that is logically structured and supported by evidence.
– Demonstrates a capacity for intellectual initiative/independent thought and an ability to engage with the material critically.
– Use of appropriate material from a range of sources extending beyond the read- ing list.
– High quality organisation and style of presentation (including referencing); mini- mal grammatical or spelling errors; written in a fluent and engaging style.
– A very high level of skill in problem solving, which demonstrates powers of criti- cal analysis (NB: where problem solving is an important key learning outcome).
• For mathematical subjects:
– Perfect, or near-perfect answers to a high proportion of the parts of the questions attempted, and a firm grasp of the central issues covered.
– Answers are presented fluently and logically.
– Explanations, where required, show evidence of an excellent comprehension of the material.
– Interpretations, where required, often display a strong critical appreciation of the material.
– Excellent use of common standard mathematical notation and conventions. 2:1 (60–69)
• For essay-based subjects:
– Very good knowledge and understanding of the subject and displays awareness of underlying theoretical and methodological issues.
– A generally critical, analytical argument that is reasonably well structured and well-supported.
– Some critical capacity to see the implications of the question, though not able to ‘see beyond the question’ enough to develop an independent approach.
– Some critical knowledge of relevant literature; use of works beyond the pre- scribed reading list; demonstrating some ability to be selective in the range of material used and to synthesise rather than describe.
– Well presented: no significant grammatical or spelling errors; written clearly and concisely; fairly consistent referencing and bibliographic formatting.
– A very good ability to apply principles effectively in the solution of factual prob- lems and to deal with problems in an orderly manner, with realism and discrimi- nation (NB: where problem solving is an important key learning outcome).
– A very good knowledge of much of the important material, possibly excellent in places, but with a limited account of some significant topics, or with some omissions/misunderstandings.
– There is a good fluency and logical structure to most of the answers.
– Explanations, where required, show evidence of good comprehension of the material though there may be some limited understanding of some areas.
– Interpretations, where required, show some evidence of a critical appreciation of the material.
– Some good use of common standard mathematical notation and conventions.
2:2 (50–59)
– Good comprehension of the subject, though there may be some errors and/or gaps, and some awareness of underlying theoretical/methodological issues with little understanding of how they relate to the question.
– Capacity for argument is limited with a tendency to assert/state opinion rather than argue on the basis of reason and evidence; structure may not be evident.
– Tendency to be descriptive rather than critical, but some attempt at analysis.
– Some attempt to go beyond or criticise the ‘essential reading’ for the unit; dis- playing limited capacity to discern between relevant and non-relevant material.
– Adequately presented: writing style conveys meaning but is sometimes awk- ward; some significant grammatical and spelling errors; inconsistent referencing but generally accurate bibliography.
– An efficient attempt at solving problems, but a tendency to overlook a number of points (NB: where problem solving is an important key learning outcome).
– A reasonably good knowledge of several important topics, possibly showing some good understanding in places, but with a limited account of some sig- nificant topics, or with some significant omissions/misunderstandings.
– There is fluency and logical structure to some of the the answers.
– Explanations, where required, show evidence of good comprehension of the material though with limited understanding in some areas.
– Interpretations, where required, are generally standard but may in parts show some evidence of a critical appreciation of the material.
– Limited use of common standard mathematical notation and conventions. 3rd (40–49)
– Limited knowledge and understanding with significant errors and omissions and generally ignorant or confused awareness of key theoretical/methodological is- sues.
– Largely misses the point of the question, asserts rather than argues a case; un- derdeveloped or chaotic structure; evidence mentioned but used inappropriately or incorrectly.
– Very little attempt at analysis or synthesis, tending towards excessive descrip- tion.
– Limited, uncritical and generally confused account of a narrow range of sources.
– Satisfactorily presented: but not always easy to follow; frequent grammatical and spelling errors; limited attempt at providing references (e.g. only referencing direct quotations) and containing bibliographic omissions.
– Attempts to Identify relevant areas for focusing problem solving but makes sig- nificant mistakes in solutions indicative of either a lack of discrimination or an understanding of a principle (NB: where problem solving is an important key learning outcome).
• For mathematical subjects:
– A reasonable spread of relevant knowledge but showing a good grasp of only a minority of the material. Some questions may be answered well, others will have major omissions or misunderstandings. Some questions may not be attempted at all.
– There may be some evidence of a logical structure to the answers in some areas.
– Explanations, where required, are short and display a limited understanding of the material. Some explanations are not given.
– Interpretations, where required, are poor and do not show critical appreciation of the material.
– Very limited use of common standard mathematical notation and conventions.
• For essay-based subjects:
– Shows very limited understanding and knowledge of the subject and/or misses the point of the question.
– Incoherent or illogical structure; evidence used inappropriately or incorrectly.
– Unsatisfactory analytical skills.
– Limited, uncritical and generally confused account of a very narrow range of sources.
– Unsatisfactory presentation e.g. not always easy to follow; frequent grammat- ical and spelling errors and limited or no attempt at providing references and containing bibliographic omissions.
– Limited attempt to Identify relevant areas for focusing problem solving but makes significant mistakes in solutions indicative of either a lack of discrimination or an understanding of a principle (NB: where problem solving is an important key learning outcome).
• For mathematical subjects:
– Considerable deficiencies, or very partial attempts at questions, across large parts of the topics set, but with some relevant material at places.
– There is little evidence of a logical structure to the answers.
– Explanations, where required, are poor or missing.
– Interpretations, where required, are weak or missing and show almost no critical appreciation of the material.
– Limited or no use of common standard mathematical notation and conventions.
• For essay-based subjects
– Shows little or no knowledge and understanding of the subject, no awareness of key theoretical/methodological issues and/or fails to address the question.
– Unsuccessful or no attempt to construct an argument and an incoherent or il- logical structure; evidence used inappropriately or incorrectly.
– Very poor analytical skills.
– Limited, uncritical and generally confused account of a very narrow range of sources.
– Very poor quality of presentation and limited or no attempt at providing refer- ences and containing bibliographic omissions.
– Overlooks most of the points in a problem (NB: where problem solving is an important key learning outcome).
• For mathematical subjects:
– Substantial deficiencies, or no attempt, across large parts of the topics set, but with a little relevant material at places.
– There is little or no logical structure to the answers.
– Explanations, where required, are poor or missing.
– Interpretations, where required, are missing or wrong and show no critical ap- preciation of the material.
– Very limited or no use of common standard mathematical notation and conven- tions.
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