ECMT 5001 Principles of Econometrics 2020 S1
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Principles of Econometrics ECMT 5001
Take-home Final Exam
2020 S1
1. As a researcher, you have n independent and identically distributed observations on yi , x1i and x2i , where i = {1, 2, 3, · · · , n}. The population from which the sample has been drawn is large. The population model can be described as follows.
● For each observation, the dependent variable, yi is a linear function of two regressors and an error term.
yi = x1i 一x2i＋ ui
● The error term has a standard normal distribution: ui ～N(0, 1）.
● Three unobserved variables, vji for j = {1, 2, 3} all have standard normal distributions so vij ～N(0, 1）.
● The regressors, x1i and x2i are related to the three unobserved variables as follows:
x1i = v1i＋ v3i
x2i = v2i 一 v3i
(a) Explaining your reasoning carefully, as n gets larger:
i. what will the sample variance of each of the observed variables tend to? [2 marks]
ii. what will the covariance of the two regressors tend to? [1 marks]
iii. what will the OLS coefficient estimates tend to in regression model (1)? [1 marks] yi = B0 ＋B1x1i＋ B2x2i＋ ui (1)
iv. what will the R2 of regression model (1) tend to? [1 marks]
v. what will the OLS coefficient estimates in regression model (2) tend to? [1 marks] x1i = y0 ＋y1x2i＋ e1i (2)
vi. what will the OLS coefficient estimates in regression model (3) tend to? [1 marks] x2i = 80 ＋81x1i＋ e2i (3)
vii. given that 
1i and 
2i are the residuals from OLS estimation of regression models (2) and
(3) respectively, what will the OLS coefficient estimates in regression model (4) tend to?
[1 marks]
yi = α0 ＋α1 
1i＋ α2 
2i＋ vi
viii. what will the R2 equal in regression model (4) tend to?
(4) [1 marks]
ix. if you impose the linear restriction B1 = B2 = B on regression model (1) by suitably transforming the regressors, what would the OLS estimate of B tend to? [1 marks]
2. An economist wishes to predict annual family spending Y using income X1 , size X2 , and annual savings X3 · The following intermediate calculations have been obtained for a sample of n = 100 families. All monetary figures are in thousands of dollars.
= 10 X¯1 = 12 X¯2 = 5 X¯3 = 1
n n n n
E Yi2 = 11, 400 E X1,iYi = 13, 000 E X2,iYi = 6, 000 E X3,iYi = 500,
i= 1 i= 1 i= 1 i= 1
and the fitted values of regressions A and B:
A : 
i = 1 ·7 ＋0 ·4X1,i＋ 0 ·7X2,i (excluding X3）
B : 
i = 4 · 3 ＋0 · 3X1,i＋ 0 ·6X2,i 一 0 ·9X3,i (including X3）·
(a) Calculate the residual sum of squares and R2 from regression A. What percentage of the variation in Y is explained by regression A? Justify your answer and show all derivations. [4 marks]
(b) Calculate the residual sum of squares and R2 from regression B. What percentage of the variation in Y is explained by regression B? Justify your answer and show all derivations. [4 marks]
(c) Has the inclusion of X3 in the regression analysis reduced the residual sum of squares? What proportional change in previously unexplained variation in Y is achieved by adding X3 to the analysis? Explain your answer. [2 marks]
(d) You wish to learn whether or not to include annual savings in your regression analysis via statistical testing with a significance level of 1%, using answers to the earlier parts of this question. (i) Formulate the null and alternative hypotheses (ii) Describe the test statistic, (iii) Compute the critical region, and (iv) Describe the decision rule and the outcome. Justify your answers. [4 marks]
3. A manufacturer of booklets packages them in boxes of 100. It is known that, on the average, a booklet weighs 30 grams, with a standard deviation of 1.5 grams. The manufacturer is interested in calculating
Prob「100 booklets weigh more than 3.3 kgl ,
a number that would help detect whether too many booklets are being put in a box. Explain how you would approximate the value of this probability reliably. Carefully describe your steps and indicate the use of any relevant theorems or assumptions. [5 marks]
2022-05-31