課程名稱︰統計學與計量經濟學原理暨實習上
課程性質︰必修
課程教師︰張勝凱
開課學院：社科院
開課系所︰經濟系
考試日期（年月日）︰2019/01/02
考試時限（分鐘）：180
試題 :
Problem 1 (points 5,5,5)
You are given the following information obtained from a random sample of
5 observations.
Assume the population has a normal distribution.
20,18,17,22,18
You want to determine whether or not the mean of the polpulation from
which this sample was taken is significant less than 21.
1. State the null and alternative hypotheses.
2. Determine the p-value.
3. Test whether or not the mean of the population is signicantly less than
21 at α= 0.1 .
Problem 2 (points 7,8)
The regression line:
y_i = β_0 + β_1 * x_i + u_i
Given a random sample {(x_i,y_i),i=1,...100}, and we have
Σx_i = 90, Σx_i ^2 = 400, Σy_i = 100, Σy_i ^2 = 1200, Σx_i*y_i = 600
1. Using OLS to estimate the regression line, compute the OLS estimates
β_0 <hat> and β_1 <hat>.
2. Compute SST, SSE, SSR and R^2.
Problem 3 (points 7,8,5)
For the joint pmf of random variables X and Y in the table below:
┌──┬──┬──┐
│ │ x=1│ x=2│
├──┼──┼──┤
│ y=0│ 0.3│ 0.4│
│ y=1│ 0.1│ 0.2│
└──┴──┴──┘
1. Find the conditional expectation function E(Y|X=1) and E(Y|X=2).
2. For the best linear prediction E*(Y|X) = α+βX, find α and β.
3. Are X and Y stochastically independent?
Problem 4 (points 5,5,5,5)
Consider a linear model of wage equation:
wage = β_0 + β_1 * edu + u
u = e ×edu
where e is a random variable with E(e)=0 and Var(e)= (σ_e)^2, and e is
independent of edu.
Suppose the sample you have for monthly earnings (wage) and years of
schooling (edu) is the random sample, and there is a variation in the
explanatory variable in your sample.
Let β_0 <hat> and β_1 <hat> be an OLS estimators of above equation.
1. Find E(u|edu) and Var(u|edu).
2. Are β_0 <hat> and β_1 <hat> unbiased estimators for β_0 and β_1 ?
Why or Why not.
3. Is the homoskedasticity assumption hold for the above linear model?
If yes, please provide the explanations; if not, write the transformed
equation that has a homoskedastic error term.
4. Are β_0 <hat> and β_1 <hat> best linear unbiased estimators (BLUE) for
β_0 and β_1 ? Why or Why not.
Problem 5 (points 7,8)
1. Consider the following model:
Y_i = β_0 + u_i
Suppose you have the size n i.i.d. sample of Y_i. Derive the OLS estimator
for β_0.
2. Consider the following model:
Y_i = β_1 * X_i + u_i
Suppose you have the size n i.i.d. sample of (X_i,Y_i). Derive the OLS
estimator for β_1.
Problem 6 (points 3,4,4,4)
Suppose that 8% of college graduates go to get ph.D.s (within 10 years of
graduation).
We want to prove that the rate is lower among ABC University graduates. We
sampled 400 ABC University graduates and found 24 of them have ph.D.s.
1. State the null and alternative hypotheses.
2. Using the critical value approach, test the hypotheses at the significance
α=0.025 .
3. Find the p-value of the test, test the hypotheses at the significance
α=0.1 .
4. Find the typeⅡ error using μ_a =0.04 and the significance α=0.025 .
=========================================================
Answers:
Problem 1
1. H_0: μ≧21, H_A: μ＜21
2. between 0.05 and 0.025
3. yes
Problem 2
1. β_1 <hat> = 1.5987, β_0 <hat> = -0.4388
2. SST= 1100, SSE= 815.3605, SSR= 284.6395, R^2= 0.7412 (小數點後可能差一點)
Problem 3
1. E(Y|X=1)= 0.25 , E(Y|X=2)= 1/3
2. β= 0.083, α= 0.1667
3. (check any point then you can show) not
Problem 4
1. E(u|edu)= 0 , Var(u|edu)= edu^2 * (σ_e)^2
2. (check A1~A4, 上課筆記) both are unbiased
3. not, correct: wage/edu = β_0/edu + β_1 + e
4. by Gauss-Markov Theorem, 符合A1~A5即為BLUE，但由3.不滿足A5(homoskedasticity)
∴not
Problem 5
1. y <bar>
2. Σ(x_i*y_i)/Σ(x_i)^2
Problem 6
1. H_0: μ≧0.08 , H_A: μ＜ 0.08
2. do not reject H_0
3. reject H_0
4. 0.0853