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Answer: The high correlations between each pair of index returns indicate that multicollinearity exists between the variables in this regression.
## Explanation **D is correct.** This is an example of multicollinearity, which arises when one of the regressors is very highly correlated with the other regressors. In this case, all three regressors are highly correlated with each other, so multicollinearity exists between all three. Since the variables are not perfectly correlated with each other, this is a case of imperfect, rather than perfect, multicollinearity. **Why other options are incorrect:** - **A is incorrect:** The p-value of a regression coefficient indicates whether the coefficient is statistically significant. The coefficient value (0.3533) alone doesn't determine statistical significance - the high p-value (0.8382) for Russell 3000 shows it's not statistically significant. - **B is incorrect:** Adjusted R², like R², measures the percentage of the variation in the data that can be explained by the model as a whole. It gives no indication of the statistical significance of the individual regression coefficients. - **C is incorrect:** A p-value less than the test size indicates that a regression coefficient is statistically significant. A high p-value (0.9452) actually indicates the coefficient is NOT statistically significant. **Key evidence of multicollinearity:** - Russell 1000 and Russell 3000 correlation: 0.998 (extremely high) - Russell 1000 and Russell 2000 correlation: 0.813 (high) - Russell 3000 and Russell 2000 correlation: 0.845 (high) - All individual p-values are high (>0.05), yet overall R² is high (0.905) - This classic pattern indicates multicollinearity where individual variables appear insignificant but the model overall has good explanatory power.
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| Regression statistics |
|---|
| Multiple R |
| R-squared |
| Adjusted R-squared |
| Standard error |
| Observations |
| Regression output | Coefficients | Standard error | t-stat | P-value |
|---|---|---|---|---|
| Intercept | 0.0023 | 0.0006 | 3.5305 | 0.0005 |
| Russell 1000 | 0.1093 | 1.5895 | 0.0688 | 0.9452 |
| Russell 2000 | 0.1055 | 0.1384 | 0.7621 | 0.4470 |
| Russell 3000 | 0.3533 | 1.7274 | 0.2045 | 0.8382 |
| Correlation matrix | Portfolio returns | Russell 1000 | Russell 2000 | Russell 3000 |
|---|---|---|---|---|
| Portfolio returns | 1.000 | |||
| Russell 1000 | 0.937 | 1.000 | ||
| Russell 2000 | 0.856 | 0.813 | 1.000 | |
| Russell 3000 | 0.945 | 0.998 | 0.845 | 1.000 |
Based on the regression results, which statement is correct?
A
The estimated coefficient of 0.3533 indicates that the returns of the Russell 3000 Index are more statistically significant in determining the portfolio returns than the other two indexes.
B
The high adjusted R² indicates that the estimated coefficients on the Russell 1000, Russell 2000, and Russell 3000 Indexes are statistically significant.
C
The high p-value of 0.9452 indicates that the regression coefficient of the returns of the Russell 1000 Index is more statistically significant than the other two indexes.
D
The high correlations between each pair of index returns indicate that multicollinearity exists between the variables in this regression.