ECO220 – Quantitative Methods in Economics
Problem Set 2

Y =




0不默认

Pr（Y = 1）= Pr（默认）=θ0

1.写下大小为n的样本的似然函数。找到最大的

2.解释为什么最大似然估计是随机的（事前）。使用

3.解释估计量的无偏性和一致性的概念。显示是否

4.解释贝叶斯方法和最大似然方法之间的基本差异，以推断默认概率或未知数

5.为参数θ0提供一个扩散优先级的表达式。解释为什么你

2个
6.假设θ0〜Beta（α0，​​β0），即对于未知的默认概率θ0，我们选择Beta分布作为先验分布。证明Beta分布是似然函数L的共轭先验

y | θ
</ s> </ s> </ s> </ s> </ s> </ s> </ s> </ s> </ s> </ s>
。至少提供

7.找到未知参数θ0的贝叶斯估计量。写贝叶斯

8.证明后验分布π

θ| ÿ
</ s> </ s> </ s> </ s> </ s> </ s> </ s> </ s> </ s> </ s>

9.除了报告默认概率θ0的点估计之外，它还很有用

P（θL≤θ0≤θU）= 0.95

10.假设在n个随机选择的100个人中，默认有10个人

（“预计还款额”）。该测试在预测“默认”时的有效率为90％。

Regression analysis (10 points)
Given the data on the joint distribution n
(yt
, xt)
T
t=1 o
where the variable yt captures
household consumption and xt
is measured household disposable income, we are interested
in estimating the unknown parameters β0 and β1 of the consumption function:
yt = β0 + β1xt + ut
The parameter β0 refers to autonomous consumption and β1 is the marginal propensity
to consume. We assume that there are cross-sectional variations in income Var (xt) > 0.
Suppose that the error terms are normally independently distributed and independent of
the regressors such that:
ut ∼ N (0, 1)
1. Show that yt
| xt ∼ N (β0 + β1xt
, 1). Find the maximum likelihood estimators for
the unknown parameters β0 and β1. Discuss the necessary assumptions to implement
maximum likelihood. (1pt)
2. Relax the assumption that the error terms are normally distributed. Suppose that
the error terms are independently and identically distributed with zero mean and
unknown variance such that:
ut ∼ IID
0, σ2

1Predicting default when the individual does not default.

Additionally, suppose the errors are orthogonal to the regressor xt such that:
E [xtut
] = 0
Find the moments estimators of the unknown parameters β0 and β1. Discuss the
necessary assumptions to implement the method-of-moments. (1pt)
3. Find the estimates of the unknown parameters β0 and β1 by minimizing the 2-norm
of the error vector u. Discuss the necessary assumptions to implement the ordinary
least-squares method and provide a geometrical interpretation for the least-squares
estimates. (1pt)
4. Write the appropriate matrix version of the regression model and write the matrix expression for the ordinary least-squares estimates. Specify the appropriate
dimension of all the vectors and matrices in the model. (1.5pt)
5. Suppose we observe a sample of size n = 3 such that {(y1, 2),(y2, 2),(y3, 2)}, i.e.
there are no variations in household income in our sample. Demonstrate that the
parameters β0 and β1 are not identified from the data using the ordinary leastsquares method. (2pts)
6. Discuss the difference between the concept of identification by a given data set
and an estimation method and the concept of asymptotic identification by a given
estimation method. Provide a necessary and sufficient condition for the parameters
β0 and β1 to be asymptotically identified. (1pt)
7. Provide another example of a linear regression model (distinct from the example presented in class) where the vector of parameter δ = (δ1, δ2) is not identified asymptotically using ordinary least-squares but is identified by a given dataset using ordinary
least-squares. (1pt)
8. Explain why the maximum likelihood, method-of-moments and least-squares estimators of the unknown parameters (β0, β1) are stochastic. (0.5pt)
5
9. Explain using your own words, how you would proceed to estimate the unknown
parameters (β0, β1) in the Bayesian framework. (0.5pt)
10. Explain the concept of data-generating process and discuss its importance in econometrics. (0.5pt)