world:quiz3
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| where $(\epsilon_i)$ are iid, centered, with variance $\sigma^2$. | where $(\epsilon_i)$ are iid, centered, with variance $\sigma^2$. | ||
| Which of the following statements are always true? | Which of the following statements are always true? | ||
| - | * 1. Assume that $R^2=1$ in a simple linear regression, it means that all the $(x_i,y_i)$ are on the same line. | + | * 1. Assume that $R^2=1$ in a simple linear regression, it means that all the $(x_i,y_i)$ are on the same line. **TRUE** |
| - | * 2. Even if the adjusted coefficient of determination $R_a^2=1$, it may happen sometimes that $Y \neq \hat Y$. | + | * 2. Even if the adjusted coefficient of determination $R_a^2=1$, it may happen sometimes that $Y \neq \hat Y$. **FALSE** |
| - | * 3. The adjusted coefficient of determination may be negative. | + | * 3. The adjusted coefficient of determination may be negative. **TRUE** |
| ===== Exercise 2: residuals and hat matrix ===== | ===== Exercise 2: residuals and hat matrix ===== | ||
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| - | * 4. We always have $\sum_{i=1}^n \hat \epsilon_i=0$, even when none of the vectors $X_i$ is equal to the vector $\mathsf{1}$. | + | * 4. We always have $\sum_{i=1}^n \hat \epsilon_i=0$, even when none of the vectors $X_i$ is equal to the vector $\mathsf{1}$. **FALSE** |
| - | * 5. $\hat Y$ is always orthogonal to $\hat \epsilon$, even when none of the vectors $X_i$ is equal to the vector $\mathsf{1}$. | + | * 5. $\hat Y$ is always orthogonal to $\hat \epsilon$, even when none of the vectors $X_i$ is equal to the vector $\mathsf{1}$. **TRUE** |
| - | * 6. Define $\tilde \beta \in \argmin_{\beta} \sum_{i=1}^n |Y_i-x'_i \beta|$. Then the SSE associated with $\hat \beta$ is not always smaller that the SSE associated to $\tilde \beta$. | + | * 6. Define $\tilde \beta \in \argmin_{\beta} \sum_{i=1}^n |Y_i-x'_i \beta|$. Then the SSE associated with $\hat \beta$ is not always smaller that the SSE associated to $\tilde \beta$. **FALSE** |
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world/quiz3.1759356734.txt.gz · Last modified: 2025/10/02 00:12 by rdouc
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