Week 10 Quiz
1. The following is correct about the maximum likelihood estimator
(a) The MLE estimator does not work with non-linearity in parameters
(b) The MLE estimator does not work with non-linearity in variables
(c) The MLE estimator makes an assumption about the probability density of the data
(d) The MLE estimator does not require an assumption about the probability density of
the data
Hint: MLE approach is employed for non-linearity in parameters and variables which OLS
cannot handle. This requires assumption about the probability density of the data. Then the
join probability distribution is maximized.
2. The following is incorrect about the likelihood function
(a) The log-transformation does not affect the location coordinates of MLE parameters
estimates: Monotonicity.
(b) Log transformation eases the computational requirements of finding the joint-
probability maximizing point through differentiation
(c) Likelihood function is not differentiable and hence requires log transformation
(d) For a linear model, both OLS and MLE yield the same parameter estimates for the
model coefficients
Hint: Log transformation has monotone equivariance property, hence it does not affect the
location of estimates. It eases the computational burden of maximizing the joint probability
through differentiation. For a linear model, both OLS and MLE yield the same estimates.
3. If a coin is tossed 100 times, and 62 Heads are observed. What is the MLE estimate of
probability ‘p’ for observing head in %.
(a) 50%-55%
(b) 55%-60%
(c) 45%-50%
(d) 65%-70%
(e) None of the above
𝑛 62
Hint: 𝑝̂ = 𝑁 = 100 = 62%
4. Consider the following loglikelihood function. F(β)= -2β^2+4β+5
Find the MLE estimate of parameter β
(a) 0-2
(b) 2-4
(c) 4-6
(d) 6-8
(e) None of the above
𝑑 𝑑2
Hint: (−2β2 + 4β + 5 ) = −4β + 4 = 0, β = 1 also, (−2β2 + 4β + 5 ) =
𝑑𝑥 𝑑𝑥 2
−4 (𝑚𝑎𝑥𝑖𝑚𝑎)
5. Consider the following loglikelihood function. F(β)= 3β^4-96β+7
Find the MLE estimate of parameter β
(a) 0-2
(b) 2-4
(c) 4-6
(d) 6-8
(e) None of the above
𝑑 𝑑2
Hint: (3β4 − 96β + 7) = 12β3 − 96 = 0, 𝛃 = 𝟐 also, (3β^4 − 96β + 7 ) = 36𝛽 2 =
𝑑𝑥 𝑑𝑥 2
144 (𝑚𝑖𝑛𝑖𝑚𝑎). The Likelihood function maxima can not be obtained.
6. Which of the following is not associated with quantile regression
(a) Increasing (decreasing) outliers in higher (lower) tails does not affect median
(b) Quantile regression minimizes weighted errors
(c) Quantile regression models conditional quantiles whereas OLS models conditional
mean
(d) If data is heteroscedastic or exhibits nonlinearity, quantile regression is a more suitable
choice
(e) None of the all
Hint: Increasing the higher quantile extreme values and decreasing the lower quantile
extreme values does not affect median. Quantile regression estimates are obtained by
minimizing weighted absolute errors. Quantiles (e.g., median) are robust to heteroscedasticity
and non-linearity in the data.
7. Which of the following is not associated with quantile regression
(a) If one minimizes the sum of squares of the differences from a centre of the
distribution, the resulting centre is mean
(b) If one minimizes the sum of absolute differences from a centre of the distribution, the
resulting centre is median
(c) If one minimizes the sum of weighted absolute differences from a centre of the
distribution, weights being 25% with positive deviations and 75% with negative
deviations, the resulting centre is third quartile, that is 75-percentile
(d) None of the all
Hint: Minimizing the sum of squares of differences from the centre, provides mean location
as the centre. Similarly minimizing absolute differences from the centre results in the median
location as the centre. If one minimizes the sum of weighted absolute differences from the
centre, where positive deviations are assigned 25% weight and negative deviations are
assigned 75% of the weight, then the resulting centre is 1st quartile that is 25-percentile.
Question 8-10: A researcher runs a cross-sectional regression of returns on a number of
stocks from NSE, on size (classified as Small, Medium, and Large). Here, medium is the
reference category. For example, a small stock will have a value of 1 with Small
classification and 0 otherwise for the remaining two (Medium and Large classifications).
The following results are obtained from a quantile regression with the following
quantiles ϴ =0.25, 0.5, and 0.75
Return=β0+ β1(Small)+ β2(Large)+error
Question 8: If one picks only small stocks, what is Q1(25%) of the univariate distribution of
returns (identify the interval in which correct value lies).
(a) 0%-0.7%
(b) 0.7%-1.4%
(c) 1.4%-2.1%
(d) 2.1%-2.8%
(e) 2.8%-3.5%
Hint: 25-percentile return for small stock β0+β1=2.0%+0.50%= 2.50%
Question 9: If one picks only large stocks, what is the median of the univariate distribution of
returns (identify the interval in which correct value lies).
(a) 0%-0.7%
(b) 0.7%-1.4%
(c) 1.4%-2.1%
(d) 2.1%-2.8%
(e) 2.8%-3.5%
Hint: 50-percentile return for small stock β0+β2=4.0%+(-1.0%)= 3.0%
Question 10: If one computes the differences in small and large stocks (small-minus- large),
what is Q3 (75-percentile) from the univariate distribution of these differences (identify the
interval in which correct value lies).
(a) 0%-0.7%
(b) 0.7%-1.4%
(c) 1.4%-2.1%
(d) 2.1%-2.8%
(e) 2.8%-3.5%
Hint: 75-percentile return for small-large stock β1-β2=1.5%-(-1.5%)= 3.0%
