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EC 303: Empirical Economic Analysis

创建日期：2024-04-20 15:30:19

所属分类：经济Economics

标签： EC 303

Empirical Economic Analysis

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EC 303: Empirical Economic Analysis

Due Thursday November 11, 11:59 pm.
1 Theoretical Problems
Problem 1 (10 points). Suppose X has a Uniform distribution with support [a, b]. Explain how you
estimate θ = (a, b) by method of moments (MoM).
Problem 2 (10 points). Suppose Xi ∼ Gamma(α, β). To see what gamma distribution is please refer
to page 195 of your textbook. What roles do these 2 parameters play? How do they change the shape of
distribution? Explain how you estimate θ = (α, β) by method of moments (MoM). (Hint: Use pdf of gamma
distribution to find the first and second moments and then construct sample analogs)
Problem 3 (20 points, 5 points each). Suppose we have the following estimator σˆ2 = c
∑n
i=1(Xi−X)2,
where c ∈ R is a constant that may depend on the sample size n.
a. Express the expected value of σˆ2 as a function of c, σ2, and sample size.
– Hint: Use the fact that
∑
(Xi −X)2 =
∑
X2i − (
∑
Xi)
2
n , and that E[Y
2] = V (Y ) + E[Y ]2.
b. What does the value of c need to be for this estimator to be unbiased?
c. If instead we use c′ = 1n , what would the bias of the estimator be? Does this biased estimator tend to
over- or under-estimate the true value σ2? How does the bias change as n increases?
d. Taking as given that V[σˆ2] = c2σ4[2(n− 1)], what value of c minimizes the MSE? What does this tell
you about your estimator?
Problem 4 (15 points, 5 points each). Work on the problems from the textbook chapter 7: 10, 16, 24.
All questions will be graded.
Problem 5 (20 points, 10 points each). Explain how you estimate λ and ν2:
a. Suppose X has a Poisson distribution with parameter λ. Explain how you estimate λ by using first
moment and then by using second moment. Are they the same?
b. Let {Y1, Y2, ..., Yn} denote an i.i.d. sample from a population with a F (ν1, ν2) distribution where ν1
and ν2 are the numerator and denominator degrees of freedom, respectively. If
E(Yi) = ν2/(ν2 − 2).
Derive an estimator of ν2 using the method of moments.
2 Empirical Problems
Problem 2: Comparing performances of estimators. In this problem, you are going to compare per-
formances of three estimators with stata simulations.
1
a. (5 points) Suppose true data generating process is normal with N(6, 3). Consider the following three estimators:
sample mean θˆ1 = X¯, sample median θˆ2 = X˜, and θˆ3 =
n+1
2n X
(n) where X(n) is the largest order
statistic i.e. maximum in the sample. Pick sample size (n) and number of samples or simulation
size (m). For the values you picked, how do these three estimators compare in terms of variances?
Which estimator is more variable? Which estimator is less variable? Rank them from the best to the
worst. Interpret your results. How do results change when you increase/decrease n and m? Interpret.
(Hint: In order to answer this question, you need to generate random samples of size n from the given
distribution and repeat m times.)
b. (5 points) Repeat the same process above when true data generating process is Uniform with U [0, 2]
c. (5 points) Is there any estimator that outperforms others in terms of variability (lower variance) in both cases?
Is there a single estimator that stands out in both cases?
d. (5 points) Now, repeat the same procedure in parts a and b with a fourth estimator θˆ4. Use 10 percent trimmed-
mean (ignore top and bottom 5 percent of the distribution) as an estimator. Compare performance
of 4 estimators with normal and uniform data generating processes. Rank them from the best to the
worst.
e. (5 points) Is θˆ4 more robust estimator? Why? How does it perform compared to best estimators in normal and
uniform cases?

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1300A ENGG 1300 OLET1139 STAT 3503 STAT 206 FEM11149 ELEN 4720 COMP20003 SOLA1070 CS574 C11CS LINB29H3 MAST90083 ECON7200 STAT 610 LINB29 DSM-5 COMP3161 EEEN40010 MATH 4431 K353 STA 3386 IERG4300 COMS W4167 ENG5298 AMME 2302 CS3910 COMPSCI 320 STATS 369 GR5242 COSC 328 CSCI435 MGTS1301 INFS2200 COMP3900 CIS 375 Module AE03 COMP3121 CS260 HAM503 MGMT5050 INT3075 COSC 285 GEOL0029 GEGE2X01 STAT1201 MATH1061 MMF1941H STAT 5060 CS 520 COMP 4320 MSIN0104 000T FINS3648 ECMT2160 ECON4003 Phys 129L CSE 142 CS 3130 MGEB01 EECS 340 Finance (20443/20444) ECMM149 COMP5310 EEEN60180 CSC 111 AT2 CSCI 4430 STAB57 MA/CSC 427 MATH322 D100 CSCI544 CITS3004 CI6227 CSE 565 APCO 1P01 852L1 ELEC S411F COMP3207 STATS330 ECON7530 FIT5149 CSI4104 CMPT 371 COMP201 SEHS4696 FI4003 MATH6182 CG1 WS MATH401 MGT 205 QTM 110 COMP 0137 MACE12201 FIT5210 ECE 544 BISM7206 HUDM 6055 ECON3203 STATS 210 MSCI 570 MATH6173 COMPSCI4039 Accounting COM3503 AOE2024 ECON213 PSYC10004 Control Data 100/200A PHIL2617 PHYS3116 MIE250 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