ECON108 Assignment 1
A. Ullah Due Date: April 19, 2018
1. Consider a simple econometric model (mean or intercept model) for one variable as yi = µ+ui
,
where µ =E(yi), is an unknown parameter, yi
is i-th sample observation on an economic variable
y, and ui are errors, i = 1, · · · , n.
(a) What is the meaning of parameter µ in this model? Write assumptions on ui
.
(b) Derive Method of Moments (MM) and Ordinary Least Squares (OLS) estimators of µ.
Are they identical? Also, write an estimator of variance of yi
, σ
2
.
(c) Explain the meaning that the OLS estimator, denoted by ˆµ is the best linear unbiased
estimator (BLUE) of µ.
2. (a) Consider a linear regression model for two variables as yi = β0 + xiβ1 + ui = µ(xi) + ui
,
i = 1, · · · , n, where µ(xi) = E(yi
|xi) is assumed to be linear, x is a scalar variable, and
ui are errors. Derive the formulas for Ordinary Least Squares (OLS) and Method of
Moments (MM) estimators βˆ
1 and βˆ
1 of β0 and β1, respectively. Are they identical?
Also, write an estimator ˆσ
2 of the variance of error σ
2
.
(b) State assumptions on errors ui
in the model in (a), and write the OLS estimator of ui as
uˆi
. Show that (i) Puˆi = 0, (ii) Puˆixi = 0, (iii) Puˆi(yi − y¯) = 0, where ˆyi = βˆ
0 + xiβˆ
1,
(iv) sample average of y = sample average of ˆy.
(c) Show that βˆ
1 is an unbiased estimator of β1, and ˆσ
2
is unbiased estimator of σ
2
. Further,
show the variance of the estimator βˆ
1 and how you would calculate it in practice. Also,
write (no derivation) of the variance of βˆ
0. Explain the meaning that βˆ
1 is BLUE (GaussMarkov
Theorem).
(d) Define and interpret measure of goodness of fit and write it as R2 = SSE/SST = 1 −
SSR/SST, where SSE is explained sum of squares, SST is total sum of squares, and SSR
is residual sum of squares. Also, write the meaning of degrees of freedom and adjusted
R2
.
(e) Explain the meaning of βˆ
1 in the following predicted regressions (i) ˆy = βˆ
0 + xβˆ
1, (ii)
yˆ = βˆ
0 + log xβˆ
1, (iii) logdy = βˆ
0 + xβˆ
1, (iv) logdy = βˆ
0 + log xβˆ
1.
(f) Write a linear regression model in k variables x1,
· · · , xk with an intercept, and its assumptions.
Write (no derivation) first order conditions for obtaining OLS estimators of
your regression coefficients, and give an interpretation of estimator (partial effect or ceteris
paribus) of regression coefficients. Define goodness of fit.
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(g) What is the meaning of ”collinearity” and ”perfect collinearity” and their effects on OLS
estimators.
(h) Explain omitted variable bias issue in OLS estimators. What is the effect on variances of
estimators in such a case. (see Ch. 3 (3-4b)).
(i) What is the meaning of homoskedasticity assumption versus heteroskedasticity.
(j) Describe sampling probability distributions (small and large samples (asymptotic) of a
standardized OLS estimator of a regression coefficient(ch.4 (4-1) & Ch.5 (5-2)), meaning
of p-value, consistency of a regression coefficient estimator (Ch.5, 5-1), and F-test statistic
for testing significance of all k variables (Ch.4 (4-5)).
(k) Ch.2: Q.3 by hand.
3. Ch.3: Q. C2, Ch. 4: Q. C11 (i and ii)
4. Consider the following population of eight individuals with daily wages y in dollars and age x
as
y x
70 20
90 30
80 20
100 30
100 40
200 40
80 50
90 50
(a) Using this data calculate unconditional expectations (averages) E(y) and E(x), unconditional
variances V(x) and V(y), and Cov(y,x) & correlation. Also, for x = 20, 30, 40, 50,
calculate conditional expectation(average) E (y|x) = µ(x) and conditional variance V (y|x)
= σ
2
(x). Now calculate a column of eight values of u = y − µ(x) corresponding to x =
20, 30, 40, 50. Using this calculate unconditional E(u) & V(u) and conditional E(u|x) &
V (u|x). Is V(y|x) = V (u|x)?
(b) Are your calculated unconditional expectations (average) E(y) & E(u), and unconditional
variances V (y) & V (u) in (a) above identical with the average of conditional
E (y|x) = µ(x), E(u|x), V(y|x), and V(u|x) taken over x values. That is, show that your
unconditional results are equal to the averages of the corresponding conditional results
over x values.
(c) Plot the data y and conditional expectation (average) of y against x = 20, 30, 40, 50
on the same graph, and separately conditional variance of y against x values. Based on
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these plots write your econometric regression model between y and x. Is it a linear or
nonlinear model for this given data? Is the conditional variance model heteroskedastic or
homoskedastic?
Notes: Question 1 is a special case of two variables case in Q.2 with β1 = 0. In Q.2, for
most parts see Ch.2 , done in Econ 107 or equivalent course. Q.4 relates to basic statistics
of unconditional and conditional averages and variances used in Econ 107 .
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