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ECON0019: QUANTITATIVE ECONOMICS AND ECONOMETRICS

创建日期：2024-07-08 15:12:13

所属分类：经济Economics

标签： ECON0019

QUANTITATIVE ECONOMICS AND ECONOMETRICS

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ECON0019: QUANTITATIVE ECONOMICS AND ECONOMETRICS

Answer BOTH questions.

1. In “Does Trade Cause Growth?” (American Economic Review, 1999), Jeffrey Frankel and David Romer study the effect of trade on income. Their simple specification is

logYi = α + βTi+ γWi+ εi ,

where Yi is per capita income of country i, Ti is international trade, Wi is within-country trade, and εi reflects other determinants of income. Since εi is likely to be correlated with the trade variables, Frankel and Romer decide to use instrumental variables to estimate the coefficients β and γ. As instruments, they use a measure of country’s geographic position (its proximity to other countries) Pi and the country size Si.

(a) Explain in detail (step by step) how one can construct the instrumental variable estimates βˆ and ˆγ when one has data on Yi, Ti, Wi, Pi, and Si for a random sample of countries.

(b) Provide formal conditions for these estimates to be consistent. Which of them can be tested?

(c) Explain the economic intuition why the conditions from Question 1(b) may be satisfied in this context. Also give at least one economic justification why at least one of them may be violated.

(d) Suppose that, unfortunately, data on within-country trade Wi are not available. In order to be able to estimate β, the researchers add another assumption (on top of those you proposed in item (b)): that Wi follows the model

Wi = η + λSi+ νi ,

and Pi is uncorrelated with νi. Explain in detail (step by step) how one can estimate β from the data on Yi, Ti, Pi, and Si only and why that estimator will be consistent.

(e) Suppose now it is known that γ = 0. Further, the true effects of international trade on income are heterogeneous across countries, denoted βi, such that the true model for per capita income is

logYi = α + βiTi+ εi.

Suppose the researchers estimate the regression of logYi on Ti (and a constant), using Pi as the single instrument. Under what condition would they asymptotically recover the average causal effect E[βi]? Provide an economic justification for why this condition may be violated in this context. Explain how to interpret the estimand of this IV procedure in that case and the direction in which it may differ from the average causal effect.

2. The transmission of human capital across generations has drawn attention for many decades in Economics. Mikael Linhdahl, Marten Palme, Sofia Sadgren-Massih and Anna Sjogren (“A Test of the Becker-Tomes Model of Human Capital Transmission Using Microdata on Four Generations”, Journal of Human Capital, 2014) use Swedish data to examine this question across several generations employing years of schooling as a measure for human capital.

(a) A simple version for the model entertained in their article, focussing on particular family, is:

St = β0 + β1St − 1 + Et

where St is years of schooling for generation t and Et is “ability” for that generation. Assume that |β1 | < 1 so that stationarity and weak dependence hold. How would you test whether Et is serially correlated in this particular context?

(b) Their model also postulates that ability is transmitted across generations according to:

Et = α0 + α1Et − 1 + Vt.

Assume that Vt is iid across generations, with mean zero and variance given by σ2 > 0. If α1 ≠ 0 is contemporaneous exogeneity satisfied? Justify your answer. If one has data on T generations, suggest a consistent estimator for α1 × β1 and a test for the null hypothesis that α1 × β1 = 0. Explain your answer.

(c) Assume now that one has data on a cross-section with only two generations (“parent” and “child”). Suppose that the data on years of schooling for the child only provides the number of years when those are less than 12 years and, otherwise, one can only know that an individual had 12 or more years of schooling. The data on years of schooling for the parent is nonetheless complete, i.e., one observes the number of schooling years without the restriction above. Let HEC be equal to 1 if the child has 12 or more years of education, and zero otherwise. Denote by SP the years of schooling for the parent. Consider the following model:

HEC = 1(γ0 + γ1 ln SP + U ≥ 12).

Assume that U ∼ Ⅵ(0, 1) and notice that the regressor is the logarithm of SP. Write down the log-likelihood function for the model above and a random sample of N families. How would you estimate

E[∂P(HEC = 1|SP)/∂SP ]?

(d) Under the data scenario above (i.e., years of schooling for the child is censored at 12 years), let SC be the child’s years of schooling. A friend is interested in the following regression:

SC = β0 + β1SP + E

using this cross-sectional data. She suggests using only the uncensored observations (i.e., those for which SC < 12) and estimate the regression above with OLS. Will the estimator consistently estimate β0 and β1? Explain your answer.

(e) Still on the regression from question 2(d), another friend instead suggests using a selection model to estimate β0 and β1. Explain how you would construct such an estimator in this particular setting.

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