econmatrics

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DEPARTMENT OF ECONOMICS UNIVERSITY OF VICTORIA

ECONOMICS 365: Econometrics I Section - A01

Winter Session I, 2018: Assignment 2 Dr. Nilanjana Roy Due Date: Monday, November 5th, 2018 Total Marks: 45 marks Instructions: Please show all your work clearly. Your assignment must be deposited in the box marked “Econ365” opposite the Economics department main office by 4 pm on the due day. No late assignments will be accepted. Please make sure that everything is properly stapled, and that you have put your name, student number, and course number to make sure that your grade is recorded properly. Question 1: (6 marks) Consider the following statements. State whether each statement is TRUE or FALSE. Provide a brief but clear explanation. Each part is worth 3 marks.

(i) In the classical K-regressor, n-observation multiple linear regression model y = Xβ+u, with u ~ N(0, σu2𝐈𝐈𝐧𝐧), X being nonstochastic and full column rank, 𝛃𝛃�, the least squares estimator of 𝛃𝛃, is the best among all linear estimators of 𝛃𝛃.

(ii) In the Multiple Linear Regression model y=Xβ + u where u ~ N(0, 𝜎𝜎𝑢𝑢2𝑰𝑰𝒏𝒏) and the regressor matrix X contains a lagged dependent variable, the least squares estimator of β is still unbiased. (Use a simple example of an appropriate regression model to explain this clearly)

Question 2: (14 marks) Consider the linear regression model: 𝑦𝑦𝑖𝑖𝑖𝑖 = 𝛼𝛼𝑖𝑖 + 𝑋𝑋𝑖𝑖𝑖𝑖𝛽𝛽 + 𝑢𝑢𝑖𝑖𝑖𝑖, 𝑖𝑖 = 1, . . , 𝑁𝑁 and 𝑡𝑡 = 1, … , 𝑇𝑇 (1) where 𝑦𝑦𝑖𝑖𝑖𝑖 is the observation on the dependent variable, y, for the i

th unit at the tth time point (e.g. y of country 1 in time period 1 is given by 𝑦𝑦11), 𝑋𝑋𝑖𝑖𝑖𝑖 is the observation on the single explanatory variable, X, for the ith unit at the tth time point and similarly for the error term 𝑢𝑢𝑖𝑖𝑖𝑖. 𝛼𝛼𝑖𝑖 is the intercept term for the i

th unit (so it is not constant) and captures unobserved factors that are specific to the unit. For simplicity, consider 𝑁𝑁 = 3 and 𝑇𝑇 = 2 in answering the questions below.

a) Write down the model in equation (1) for all units and all times, arranging them by unit first and then time. So the first equation will be for unit 1 and time period 1, the second equation will be for unit 1 and time period 2 and then for unit 2 and time period 1 and so on. (3 marks)

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b) Then show that for each unit i, we can write a model of the form 𝑦𝑦𝑖𝑖 = 𝛼𝛼𝑖𝑖𝜄𝜄 + 𝑋𝑋𝑖𝑖𝛽𝛽 + 𝑢𝑢𝑖𝑖

by appropriately defining all the vectors (i.e. write down their elements) and noting their dimensions. Indicate clearly the scalar(s). (4 marks)

c) Write down models of the form in (b) above for 𝑖𝑖 = 1, 2 𝑎𝑎𝑎𝑎𝑎𝑎 3. Note that this is a more compact way of writing the system of equations in (a) by using vectors. (1 mark)

d) Using part (c), we can write down the full system from part (a) in vector-matrix form as 𝑦𝑦 = 𝐷𝐷𝛼𝛼 + 𝑋𝑋𝛽𝛽 + 𝑢𝑢 (2)

where 𝐷𝐷 =

   





   





ι

ι ι









00

00 00

What is the dimension of the D matrix? Define the vectors in equation (2) above using the vectors you provided for part (c) and provide their dimensions. Identify clearly the scalar(s). (5 marks)

e) Equation (2) can then be written as 𝑦𝑦 = 𝑍𝑍𝑍𝑍 + 𝑢𝑢 where 𝑍𝑍 is a partitioned matrix given by [𝐷𝐷 𝑋𝑋]. Define 𝑍𝑍. (1 mark) Note: We can now apply least squares method to equation (2) (making the usual assumptions about the error term 𝑢𝑢𝑖𝑖𝑖𝑖) to get an estimator of 𝑍𝑍. This should give you an idea of how 𝛽𝛽 was estimated in equations (1) and (2) in Baird et al. (2011) since those equations are similar to equation (1) in this question but with a different N and different Ts for the different units making up the N units.

Question 3: (25 marks)

(a) This question is based on Baird, Sarah, Jed Friedman, and Norbert Shady, (2011), “Aggregate Income Shocks and Infant Mortality in the Developing World”, The Review of Economics and Statistics, 93(3): 847-856. The article can be found at the link https://www.mitpressjournals.org/doi/10.1162/REST_a_00084

In order to answer the questions below you will need to focus on sections I, II, IIIA and IV. (Note that the working paper version of this article is cited in the reference section of the article and covers more details of their analysis)

i. What is the main research question that this paper is addressing? (2 marks)

ii. Why is the health measure that the authors have chosen better for the empirical work of interest compared to other health measures such as life expectancy? (2 marks)

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iii. Based on the article explain clearly, using your own words, why it is hard to know a priori the sign of 𝛽𝛽 in equations (1) or (2)? (5 marks)

iv. 1st row 2nd column of Table 2 in the paper gives the estimate of the effect of log(per capita GDP) on infant mortality per 1000 children born based on a particular specification of equation (1). Explain clearly how this number (-23.06) implies that “a 1% decrease in per capita GDP is associated with a 0.24 increase in infant mortality per 1,000 children born”. (4 marks)

v. What conclusion do the authors reach about the research question? (2 marks)

(b) The file 365hw2_data.xlsx gives the data on infant mortality per 1,000 children born (IMR) and on GDP per capita in current U.S. dollars adjusted for differences across countries in purchasing power parity (PPP) for 2015 (GDP) for 217 countries. The data have been obtained from the World Development Indicators http://databank.worldbank.org/data/reports.aspx?source=world-development-indicators

i. Consider the simple linear regression model 𝑖𝑖𝑖𝑖𝑖𝑖𝑖𝑖 = 𝛼𝛼 + 𝛽𝛽 log(𝐺𝐺𝐷𝐷𝐺𝐺𝑖𝑖) + 𝑢𝑢𝑖𝑖 ∀𝑖𝑖

Estimate the model using the data in the given file and Least Squares method and report the sample regression function. (4 marks)

ii. Interpret the estimate of 𝛽𝛽 obtained from estimating the model in (i) above. (1 mark)

iii. What are two main differences between the model in this question and the model in equation (1) in the paper in Q3(a)? (3 marks)

iv. Would you say that the result from this very simple model is similar to that from equation (1) with a linear time trend in the paper in Q3(a)? Explain very briefly. (2 marks)