Question

In the context where sample mean of the dependent variable \( Y \) lies closer to the observed \( Y_1 \) than its least square predictor \( \hat{Y}_1 \), ............

• A. the goodness of fit \( (R^2) \) is negative
• B. the goodness of fit \( (R^2) \) is equal to 1
• C. the goodness of fit \( (R^2) \) is equal to 0
• D. the goodness of fit \( (R^2) \) must lie between 0 and 1

Hint:

The goodness of fit \( R^2 \) measures how well the model explains the variation in the dependent variable, and it can be negative if the model fits worse than the simple mean.

Answer 1

The Correct Option is A

Step 1: Understand the concept of goodness of fit.
The goodness of fit, often represented by \( R^2 \), is a measure of how well the regression model fits the observed data. It is calculated as the square of the correlation between the observed and predicted values of the dependent variable.
Step 2: Analyze the options.
- Option (A) is correct. If the sample mean of the dependent variable \( Y \) is closer to the observed value than the least square predictor \( \hat{Y}_1 \), it indicates that the model is performing worse than just using the sample mean. In such cases, the goodness of fit \( R^2 \) can be negative, indicating that the model is worse than a horizontal line representing the mean.
- Option (B) is incorrect because the goodness of fit \( R^2 \) equals 1 only when the regression line perfectly fits the data, which is not always the case.
- Option (C) is incorrect because \( R^2 = 0 \) indicates that the model explains none of the variation in the dependent variable, not that it lies closer to the observed value.
- Option (D) is incorrect because although \( R^2 \) is usually between 0 and 1, it can be negative when the model fits worse than the simple average model.
Final Answer: \[ \boxed{\text{the goodness of fit } (R^2) \text{ is negative}} \]