问题


There is a single choice question about econometrics. Answer the question by replying A, B, C or D.
Question: If a Johansen "trace" test for a null hypothesis of 2 cointegrating vectors is applied to a system containing 4 variables is conducted, which eigenvalues would be used in the test?
A. All of them
B. The largest 2
C. The smallest 2
D. The second largest
Answer:

答案

模型 模型答案 正确答案
金声玉亮2.0

C. The smallest 2

In a Johansen test for cointegration, the trace test statistic is calculated as the sum of the eigenvalues of the matrix Π (or the matrix obtained from the reduced rank regression), but with a specific adjustment depending on the null hypothesis being tested. When testing the null hypothesis of r cointegrating vectors against the alternative of more than r cointegrating vectors, the trace statistic involves summing the (n-r) smallest eigenvalues, where n is the number of variables in the system.

In this case, since the null hypothesis is that there are 2 cointegrating vectors (r=2) and the system contains 4 variables (n=4), the trace test would involve the 2 smallest eigenvalues (n-r = 4-2 = 2). Therefore, the correct answer is C. The smallest 2.		 C
deepseekr1

The Johansen trace test evaluates the null hypothesis of \( r \) cointegrating vectors against the alternative of more than \( r \). The test statistic is calculated as:
\[
\text{Trace Statistic} = -T \sum_{i=r+1}^{k} \ln(1 - \lambda_i),
\]
where \( \lambda_i \) are the eigenvalues (ordered from largest to smallest) and \( k \) is the number of variables.

For the question:
- \( H_0 \): 2 cointegrating vectors (\( r = 2 \))
- System with \( k = 4 \) variables

The trace statistic sums eigenvalues **from \( r+1 = 3 \) to \( k = 4 \)**. These correspond to the **two smallest eigenvalues** (since eigenvalues are ordered as \( \lambda_1 > \lambda_2 > \lambda_3 > \lambda_4 \)).

**Answer:** C. The smallest 2		 C
gpto1 C C
GPT-4o C. The smallest 2 C
llama3.1-70B-Instruct

The correct answer is C		 C
llama3.1-405B-Instruct

C		 C
DeepSeek-Coder-V2 C C
Qwen2-72B-Instruct C. The smallest 2 C
baichuan4 C C
doubao A C
minimax A. All of them C
kimi C. The smallest 2 C
wenxinyiyan C

Explanation:
In the Johansen test for cointegration, the null hypothesis typically specifies the number of cointegrating vectors (also known as the cointegration rank) that exist among the variables in the system. The test involves comparing the eigenvalues of a certain matrix to critical values from a pre-specified distribution.

When conducting a Johansen "trace" test for a null hypothesis of 2 cointegrating vectors in a system containing 4 variables, the test focuses on the smallest eigenvalues because these correspond to the weakest relationships (or the least significant cointegrating vectors) in the system. Specifically, to test the null hypothesis of 2 cointegrating vectors, we would look at the sum of the smallest 2 eigenvalues (hence the name "trace" test, which involves summing eigenvalues).

Therefore, the correct answer is C: The smallest 2 eigenvalues would be used in the test.		 C
zhipu The correct answer is D. The second largest.

In a Johansen "trace" test, the test statistic is constructed using the eigenvalues of the estimated long-run variance matrix. When testing for a null hypothesis of \(r\) cointegrating vectors, you look at the sum of the \(r+1\) largest eigenvalues to construct the trace statistic.

For the null hypothesis of 2 cointegrating vectors in a system with 4 variables, you need to test against the alternative of 3 cointegrating vectors. Therefore, the relevant eigenvalue to consider is the second largest one.

So, the correct choice is:
D. The second largest		 C
gpto1 C C