金声玉亮2.0模型评测报告

问题

There is a single choice question about econometrics. Answer the question by replying A, B, C or D.
Question: If two variables, $x_t$ and $y_t$ are said to be cointegrated, which of the following statements are true?

i) $x_t$ and $y_t$ must both be stationary

ii) Only one linear combination of $x_t$ and $y_t$ will be stationary

iii) The cointegrating equation for $x_t$ and $y_t$ describes the short-run relationship

between the two series

iv) The residuals of a regression of $y_t$ on $x_t$ must be stationary
A. (ii) and (iv) only
B. (i) and (iii) only
C. (i), (ii), and (iii) only
D. (i), (ii), (iii), and (iv)
Answer:

答案

模型	模型答案	正确答案
金声玉亮2.0	A. (ii) and (iv) only Explanation: - (i) is incorrect because cointegration does not require both variables to be stationary. In fact, cointegration typically involves non-stationary variables (e.g., variables that are integrated of order 1, I(1)) that, when combined in a specific linear way, produce a stationary series. - (ii) is correct because cointegration implies that there exists a specific linear combination of the non-stationary variables that results in a stationary series. This linear combination is unique up to a scalar multiple. - (iii) is incorrect because the cointegrating equation describes the long-run equilibrium relationship between the variables, not the short-run relationship. The short-run dynamics are typically captured by the error correction model (ECM) derived from the cointegrating relationship. - (iv) is correct because if $x_t$ and $y_t$ are cointegrated, then the residuals of a regression of $y_t$ on $x_t$ (which represent the deviations from the long-run equilibrium) must be stationary. This is a key property of cointegration, as it implies that the deviations from the equilibrium relationship are temporary and will revert to zero over time.	A
deepseekr1	A. (ii) and (iv) only Explanation: - Statement i is false because cointegration requires variables to be integrated of the same order (typically non-stationary in levels, e.g., I(1)), not stationary themselves. - Statement ii is true for two variables, as cointegration implies a unique stationary linear combination (one cointegrating vector). - Statement iii is false since the cointegrating equation captures the long-run equilibrium, not the short-run relationship (modeled via error correction). - Statement iv is true because stationarity of residuals from a regression of \(y_t\) on \(x_t\) confirms cointegration (Engle-Granger method). Answer: A	A
gpto1	A	A
GPT-4o	A. (ii) and (iv) only	A
llama3.1-70B-Instruct	A	A
llama3.1-405B-Instruct	A. (ii) and (iv) only Explanation: i) Cointegration does not require both variables to be stationary. In fact, cointegration is often used to analyze the relationship between non-stationary variables. ii) Cointegration implies that there exists a linear combination of the variables that is stationary. This linear combination is unique, so only one linear combination of $x_t$ and $y_t$ will be stationary. iii) The cointegrating equation describes the long-run relationship between the two series, not the short-run relationship. iv) The residuals of a regression of $y_t$ on $x_t$ must be stationary if the variables are cointegrated. This is because the residuals represent the deviations from the long-run relationship, and these deviations should be stationary. Therefore, only statements (ii) and (iv) are true, making option A the correct answer.	A
DeepSeek-Coder-V2	A	A
Qwen2-72B-Instruct	A. (ii) and (iv) only	A
baichuan4	A. (ii) and (iv) only	A
doubao	A Cointegration does not require both variables to be stationary (i is false). Only one linear combination of the non-stationary variables is stationary (ii is true). The cointegrating equation describes the long-run relationship not the short-run (iii is false). And the residuals of a regression of one on the other must be stationary (iv is true). So only (ii) and (iv) are correct.	A
minimax	A. (ii) and (iv) only Cointegration implies that two or more non-stationary time series variables are related in the long run, which means that a linear combination of them is stationary. This is captured by statement (ii). Statement (iv) is also true because the residuals from the regression of one variable on the other(s) should be stationary if the variables are cointegrated. This is essentially what the cointegrating equation represents. Statements (i) and (iii) are not necessarily true for cointegrated series.	A
kimi	A. (ii) and (iv) only Cointegrated variables are non-stationary on their own, but some linear combination of them is stationary. This is the essence of cointegration. The statement that the residuals of a regression of $y_t$ on $x_t$ must be stationary is also true, as the regression is used to identify the stationary linear combination.	A
wenxinyiyan	A. (ii) and (iv) only Explanation: Cointegration, often referred to as cointegration of order d or CI(d), is a concept closely related to stationarity in time series analysis. When two (or more) non-stationary time series variables are combined in a specific, linear way such that the resulting series is stationary, these variables are said to be cointegrated. i) The statement "$x_t$ and $y_t$ must both be stationary" is incorrect. In fact, if they were both stationary, there would be no need to discuss cointegration. Cointegration arises when individual series are non-stationary, but a linear combination of them is. ii) The statement "Only one linear combination of $x_t$ and $y_t$ will be stationary" is correct. This is the essence of cointegration. While the individual series may be non-stationary, there exists a unique (or at least one) linear combination that is stationary. iii) The statement "The cointegrating equation for $x_t$ and $y_t$ describes the short-run relationship between the two series" is incorrect. The cointegrating equation actually describes a long-run equilibrium relationship between the two series. In the short run, the series may deviate from this equilibrium, but in the long run, they are expected to revert to it. iv) The statement "The residuals of a regression of $y_t$ on $x_t$ must be stationary" is correct, but with an important caveat. This is true if $x_t$ and $y_t$ are cointegrated and the regression is properly specified to capture this cointegration. In such a case, the residuals (which represent deviations from the long-run equilibrium) should be stationary. If they are not, it may suggest that the regression model is misspecified or that the series are not actually cointegrated. Therefore, the correct answers are (ii) and (iv) only, which corresponds to option A.	A
zhipu	A. (ii) and (iv) only Explanation: - (i) is false because for two variables to be cointegrated, they must individually be non-stationary (i.e., integrated of order 1, I(1)), but their linear combination must be stationary (I(0)). - (ii) is true because if $x_t$ and $y_t$ are cointegrated, there exists a unique linear combination of them that is stationary. - (iii) is false because the cointegrating equation describes the long-run equilibrium relationship between the two series, not the short-run relationship. - (iv) is true because for $x_t$ and $y_t$ to be cointegrated, the residuals from regressing $y_t$ on $x_t$ must be stationary. Therefore, the correct answer is A. (ii) and (iv) only.	A
gpto1	A	A

模型

模型答案

正确答案

金声玉亮2.0

A. (ii) and (iv) only

Explanation:

- (i) is incorrect because cointegration does not require both variables to be stationary. In fact, cointegration typically involves non-stationary variables (e.g., variables that are integrated of order 1, I(1)) that, when combined in a specific linear way, produce a stationary series.

- (ii) is correct because cointegration implies that there exists a specific linear combination of the non-stationary variables that results in a stationary series. This linear combination is unique up to a scalar multiple.

- (iii) is incorrect because the cointegrating equation describes the long-run equilibrium relationship between the variables, not the short-run relationship. The short-run dynamics are typically captured by the error correction model (ECM) derived from the cointegrating relationship.

- (iv) is correct because if $x_t$ and $y_t$ are cointegrated, then the residuals of a regression of $y_t$ on $x_t$ (which represent the deviations from the long-run equilibrium) must be stationary. This is a key property of cointegration, as it implies that the deviations from the equilibrium relationship are temporary and will revert to zero over time.

deepseekr1

A. (ii) and (iv) only

**Explanation:**
- **Statement i** is false because cointegration requires variables to be integrated of the same order (typically non-stationary in levels, e.g., I(1)), not stationary themselves.
- **Statement ii** is true for two variables, as cointegration implies a unique stationary linear combination (one cointegrating vector).
- **Statement iii** is false since the cointegrating equation captures the **long-run** equilibrium, not the short-run relationship (modeled via error correction).
- **Statement iv** is true because stationarity of residuals from a regression of $y_t$ on $x_t$ confirms cointegration (Engle-Granger method).

**Answer:** A

gpto1

GPT-4o

A. (ii) and (iv) only

llama3.1-70B-Instruct

llama3.1-405B-Instruct

A. (ii) and (iv) only

Explanation:

i) Cointegration does not require both variables to be stationary. In fact, cointegration is often used to analyze the relationship between non-stationary variables.

ii) Cointegration implies that there exists a linear combination of the variables that is stationary. This linear combination is unique, so only one linear combination of $x_t$ and $y_t$ will be stationary.

iii) The cointegrating equation describes the long-run relationship between the two series, not the short-run relationship.

iv) The residuals of a regression of $y_t$ on $x_t$ must be stationary if the variables are cointegrated. This is because the residuals represent the deviations from the long-run relationship, and these deviations should be stationary.

Therefore, only statements (ii) and (iv) are true, making option A the correct answer.

DeepSeek-Coder-V2

Qwen2-72B-Instruct

A. (ii) and (iv) only

baichuan4

A. (ii) and (iv) only

doubao

A

Cointegration does not require both variables to be stationary (i is false). Only one linear combination of the non-stationary variables is stationary (ii is true). The cointegrating equation describes the long-run relationship not the short-run (iii is false). And the residuals of a regression of one on the other must be stationary (iv is true). So only (ii) and (iv) are correct.

minimax

A. (ii) and (iv) only

Cointegration implies that two or more non-stationary time series variables are related in the long run, which means that a linear combination of them is stationary. This is captured by statement (ii). Statement (iv) is also true because the residuals from the regression of one variable on the other(s) should be stationary if the variables are cointegrated. This is essentially what the cointegrating equation represents. Statements (i) and (iii) are not necessarily true for cointegrated series.

kimi

A. (ii) and (iv) only

Cointegrated variables are non-stationary on their own, but some linear combination of them is stationary. This is the essence of cointegration. The statement that the residuals of a regression of $y_t$ on $x_t$ must be stationary is also true, as the regression is used to identify the stationary linear combination.

wenxinyiyan

A. (ii) and (iv) only

Explanation:

Cointegration, often referred to as cointegration of order d or CI(d), is a concept closely related to stationarity in time series analysis. When two (or more) non-stationary time series variables are combined in a specific, linear way such that the resulting series is stationary, these variables are said to be cointegrated.

i) The statement "$x_t$ and $y_t$ must both be stationary" is incorrect. In fact, if they were both stationary, there would be no need to discuss cointegration. Cointegration arises when individual series are non-stationary, but a linear combination of them is.

ii) The statement "Only one linear combination of $x_t$ and $y_t$ will be stationary" is correct. This is the essence of cointegration. While the individual series may be non-stationary, there exists a unique (or at least one) linear combination that is stationary.

iii) The statement "The cointegrating equation for $x_t$ and $y_t$ describes the short-run relationship between the two series" is incorrect. The cointegrating equation actually describes a long-run equilibrium relationship between the two series. In the short run, the series may deviate from this equilibrium, but in the long run, they are expected to revert to it.

iv) The statement "The residuals of a regression of $y_t$ on $x_t$ must be stationary" is correct, but with an important caveat. This is true if $x_t$ and $y_t$ are cointegrated and the regression is properly specified to capture this cointegration. In such a case, the residuals (which represent deviations from the long-run equilibrium) should be stationary. If they are not, it may suggest that the regression model is misspecified or that the series are not actually cointegrated.

Therefore, the correct answers are (ii) and (iv) only, which corresponds to option A.

zhipu

A. (ii) and (iv) only

Explanation:
- (i) is false because for two variables to be cointegrated, they must individually be non-stationary (i.e., integrated of order 1, I(1)), but their linear combination must be stationary (I(0)).
- (ii) is true because if $x_t$ and $y_t$ are cointegrated, there exists a unique linear combination of them that is stationary.
- (iii) is false because the cointegrating equation describes the long-run equilibrium relationship between the two series, not the short-run relationship.
- (iv) is true because for $x_t$ and $y_t$ to be cointegrated, the residuals from regressing $y_t$ on $x_t$ must be stationary.

Therefore, the correct answer is A. (ii) and (iv) only.

gpto1