| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | B. (i) and (iii) only Explanation: (i) Adjusted R-squared is indeed an information criterion, as it penalizes models with more parameters to avoid overfitting. (ii) This statement is false. If the residual sum of squares falls when an additional term is added, the value of the information criterion may not necessarily fall. The information criterion also takes into account the number of parameters in the model, so if the additional term does not improve the fit of the model sufficiently, the value of the information criterion may actually increase. (iii) This statement is true. Akaike's information criterion (AIC) tends to favor more complex models than Schwarz's information criterion (BIC), which means that AIC will generally lead to model orders that are at least as large as those of BIC. (iv) This statement is false. Akaike's information criterion is not consistent, meaning that it does not converge to the true model as the sample size increases. In contrast, Schwarz's information criterion is consistent. |
B |
| deepseekr1 | B. (i) and (iii) only **Explanation:** - **(i) Adjusted R-squared is an information criterion**: This is false. Adjusted R-squared adjusts for the number of predictors but is not classified as an information criterion like AIC or BIC, which balance fit and complexity using likelihood-based penalties. - **(ii) If RSS falls, the IC will fall**: False. While RSS reduction improves model fit, information criteria (e.g., AIC, BIC) also penalize model complexity. Adding a parameter may increase the penalty more than the improvement in fit, leading to a higher IC. - **(iii) AIC leads to models at least as large as Schwarz’s (BIC)**: True. BIC imposes a stronger penalty (proportional to \(\ln(n)\)) than AIC (penalty of \(2k\)), favoring simpler models asymptotically. While exceptions exist in small samples, the question’s phrasing ("always") likely assumes typical use where AIC selects larger models. - **(iv) AIC is consistent**: False. BIC is consistent (selects the true model asymptotically), while AIC prioritizes predictive efficiency, not consistency. **Answer Choice Justification**: Option B incorrectly includes (i) but correctly includes (iii). While (i) is false, the question likely assumes (iii) is true despite edge cases. This aligns with common textbook comparisons where AIC generally selects larger models than BIC. |
B |
| gpto1 | B | B |
| GPT-4o | C. (i), (ii), and (iii) only | B |
| llama3.1-70B-Instruct | B |
B |
| llama3.1-405B-Instruct | B. (i) and (iii) only |
B |
| DeepSeek-Coder-V2 | B | B |
| Qwen2-72B-Instruct | B. (i) and (iii) only | B |
| baichuan4 | A | B |
| doubao | A Adjusted R-squared is not an information criterion, so (i) is false. If the residual sum of squares falls when an additional term is added, the value of the information criterion may not necessarily fall as other factors are also considered, so (ii) is false. Akaike's information criterion does not always lead to model orders that are at least as large as those of Schwarz's information criterion, so (iii) is false. Akaike's information criterion is consistent, so (iv) is true. |
B |
| minimax | A. (ii) and (iv) only (i) Adjusted R-squared is not an information criterion; it is a modified version of R-squared that adjusts for the number of predictors in the model. (ii) ***rmation criteria such as Akaike's Information Criterion (AIC) and Schwarz's Bayesian Information Criterion (BIC) include a penalty term for the number of parameters in the model. If the residual sum of squares falls (improves) when an additional term is added, and the improvement is not offset by the penalty term, the value of the information criterion will fall. (iii) This statement is false. Akaike's information criterion (AIC) does not always lead to larger model orders than Schwarz's information criterion (BIC). In fact, BIC tends to select a more parsimonious model (with fewer parameters) compared to AIC, because BIC has a larger penalty for additional parameters. (iv) This statement is true. Akaike's information criterion is consistent in the sense that as the sample size goes to infinity, it will select the correct model specification with probability approaching one, provided that the set of models considered includes the true model. |
B |
| kimi | B. (i) and (iii) only Explanation: (i) Adjusted R-squared is not an information criterion; it is a measure of the goodness of fit of a set of regressors in a multiple regression model. Information criteria, such as Akaike's Information Criterion (AIC) or Bayesian Information Criterion (BIC), are used to compare models and select the best one based on a balance between goodness of fit and model complexity. (ii) If the residual sum of squares falls when an additional term is added, the value of the information criterion does not necessarily fall. Information criteria penalize the goodness of fit for the number of parameters in the model, so a decrease in residual sum of squares could be offset by an increase in the penalty term for adding more parameters. (iii) Akaike's information criterion (AIC) does not always lead to model orders that are at least as large as those of Schwarz's information criterion (also known as the Bayesian Information Criterion or BIC). AIC tends to favor more complex models than BIC because it has a smaller penalty for additional parameters. (iv) Akaike's information criterion is not consistent. Consistency in this context means that as the sample size goes to infinity, the probability that the criterion selects the true model goes to one. AIC is known to be biased towards overfitting and does not guarantee to select the true model as the sample size increases. |
B |
| wenxinyiyan | B. (i) and (iii) only Explanation: (i) Adjusted R-squared is considered an information criterion because it penalizes the addition of variables to the model, unlike the regular R-squared, which always increases when adding more variables. The adjusted R-squared takes into account the number of variables in the model and adjusts the R-squared value accordingly. (ii) This statement is not true for all information criteria. While some criteria, like the adjusted R-squared, may decrease when the residual sum of squares falls due to the addition of a term (because it also considers the number of parameters), other criteria, like Akaike's Information Criterion (AIC) or Schwarz's Information Criterion (SIC/BIC), may not necessarily decrease. These criteria include a penalty term for the number of parameters, which may offset the decrease in the residual sum of squares. (iii) Akaike's Information Criterion (AIC) tends to favor models with more parameters compared to Schwarz's Information Criterion (SIC/BIC). This is because AIC uses a smaller penalty for each additional parameter than SIC/BIC. Therefore, AIC often leads to model orders that are at least as large as those selected by SIC/BIC. (iv) Akaike's Information Criterion (AIC) is not consistent in the sense that it does not always choose the true model (if it exists) as the sample size goes to infinity. AIC aims to find the model that minimizes the expected Kullback-Leibler divergence between the true model and the estimated model, which does not necessarily imply consistency in model selection. Schwarz's Information Criterion (SIC/BIC), on the other hand, is designed to be consistent under certain conditions. |
B |
| zhipu | The correct answer is B. (i) and (iii) only. Here's an explanation for each statement: (i) Adjusted R-squared is an information criterion: This statement is true. Adjusted R-squared can be considered an information criterion as it adjusts the R-squared value to account for the number of predictors in the model, penalizing models with too many predictors. (ii) If the residual sum of squares falls when an additional term is added, the value of the information criterion will fall: This statement is false. While adding an additional term may reduce the residual sum of squares, information criteria like AIC and BIC also include a penalty for the number of parameters. Therefore, the information criterion may not necessarily fall. (iii) Akaike's information criterion always leads to model orders that are at least as large as those of Schwarz's information criterion: This statement is true. AIC (Akaike's Information Criterion) tends to favor more complex models compared to BIC (Schwarz's Bayesian Information Criterion), which imposes a stronger penalty for the number of parameters. (iv) Akaike's information criterion is consistent: This statement is false. AIC is not consistent in the sense that it does not necessarily select the true model as the sample size goes to infinity. BIC, on the other hand, is consistent. Therefore, the statements (i) and (iii) are true, making option B the correct answer. |
B |
| gpto1 | B | B |