Columbia University MA in Economics
GR 5411 Econometrics I Seyhan Erden
Midterm Exam
(please specify the page number of each question when you are submitting your problem set to Gradescope)
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1. (50p) Growth.dta1 contains data on average growth rates over 1960-1995 for 65 countries, along with variables that are potentially related to growth.
Variable
Definition
Country_name
Name of country
growth
Average annual percentage growth of real Gross Domestic Product (GDP)* from 1960 to 1995.
rgdp60
The value of GDP* per capita in 1960, converted to 1960 US dollars
tradehare
The average share of trade in the economy from 1960 to 1995, measured as the sum of exports plus imports, divided by GDP; that is, the average value of (X + M)/GDP from 1960 to 1995, where X = exports and M = imports (both X and M are positive).
yearsshcool
Average number of years of schooling of adult residents in that country in 1960
rev_coups
Average annual number of revolutions, insurrections (successful or not) and coup d’etats in that country from 1960 to 1995
assasinations
Average annual number of political assassinations in that country from 1960 to 1995 (per million population)
(a) (5p) Assume the population regression is in the form of 𝑌 = 𝑋𝛽 + 𝜀, where 𝑌 is growth and 𝑋 contains a column of ones and all of the rest of the variables listed above. Run this regression, report your results (if you are writing on a Word document, to make the writing look like a Stata output, please use the Word font: Bold, Courier New Size 9)
(b) (6p) Using the matrix commands in Stata find (𝑋′𝑋)+,
(c) (6p) Using the matrix commands in Stata, create following matrices/vectors; 𝑃 =
+, .
𝑖(𝑖′𝑖) 𝑖′ and 𝑀 = 𝐼 − 𝑃 . Just show these commands, you do not need to list these
..
matrices in your solutions.
(d) (6p) Calculate 𝑦4𝑀.𝑦.
(e) (6p) Calculate 𝛽5𝑋4𝑀.𝑋𝛽5.
(f) (4p) Using (d) and (e) calculate 𝑅8
(g) (6p) Suppose you want to test if all coefficients (except constant) are jointly significant using the matrix form of hypothesis testing: 𝐻.: 𝑅4𝛽 = 𝑞. Write what is 𝑅 ? (hint: once you create R in part (h), you can copy/paste here, if you are writing your answers on a Word document, use the font Bold,Courier New, Size 9)
(h) (6p) Create 𝑅 and 𝑞 with Stata.
(i) (5p) Write the F test in matrix form and find it with Stata. Compare it to part (a) output to
make sure you did this correctly.
1 These data were provided by Professor Ross Levine of Brown University and were used in his paper with, Thorsten Beck and Norman Loayza “Finance and the Sources of Growth” Journal of Financial Economics, 2000, Vol. 58, pp. 261- 300.
2
2.
(30p) Let 𝑋< ∼ 𝑁(0, 𝜎8) for some 𝜎8 > 0 unknown. Suppose we observe an iid random sample, 𝑋,, . . . , 𝑋C. Estimate 𝜎8 by C
𝜎D 8 = 1 G 𝑋 8 𝑛<