Stephen Gutierrez, a risk analyst, is modeling the daily change in a fixed-income portfolio's 99% 1-day Value at Risk (VaR), measured in basis points. His initial regression model, `Model 1`, includes `ΔInterestRate`, `ΔCreditSpread`, and `VIX`. He then adds `GoldPriceChange` to form `Model 2`, believing it captures broader market sentiment. The outputs are below, along with the correlation between `VIX` and `GoldPriceChange`.
**Additional Information:** The correlation between `VIX` and `GoldPriceChange` is 0.75. The fixed-income portfolio's VaR is primarily sensitive to interest rate and credit spread movements, with `VIX` considered a general market risk proxy. `GoldPriceChange` is not typically considered a direct driver of fixed-income VaR.
Which of the following is the most appropriate assessment of `Model 2` and the inclusion of `GoldPriceChange`?
A.The inclusion of `GoldPriceChange` is problematic due to multicollinearity, as evidenced by the increased p-value of `VIX` and its weak theoretical link to fixed-income VaR, suggesting `Model 1` is preferred.
B.`Model 2` is superior because `GoldPriceChange` is statistically significant and the R-squared increased, indicating an improved overall fit for the VaR model.
C.The marginal increase in adjusted R-squared suggests `GoldPriceChange` provides negligible additional explanatory power, but its statistical significance warrants its retention for robustness.
D.The observed change in `VIX`'s significance indicates `VIX` was never truly a robust predictor, and `GoldPriceChange` effectively replaces it as a better sentiment proxy.
Rationale:
The most appropriate assessment acknowledges that while `GoldPriceChange` shows individual statistical significance (p-value of 0.046), its inclusion is problematic. The high correlation (0.75) between `VIX` and `GoldPriceChange` indicates multicollinearity. A key symptom of this is the change in `VIX`'s coefficient: its p-value increased significantly from 0.045 (statistically significant) to 0.183 (not statistically significant), and its standard error increased from 0.005 to 0.006. This suggests that `VIX`'s individual contribution becomes obscured by the presence of `GoldPriceChange`. Furthermore, the problem explicitly states that `GoldPriceChange` is not typically considered a direct driver of fixed-income VaR, making its economic relevance weak despite statistical significance. `Model 1` is therefore preferred for its parsimony, stronger theoretical grounding, and more stable coefficient estimates for `VIX`.
The option that `Model 2` is superior because `GoldPriceChange` is statistically significant and the R-squared increased is incorrect because it focuses solely on the individual statistical significance of the new variable and the marginal increase in R-squared. It fails to consider the negative implications of multicollinearity on other theoretically relevant variables like `VIX`, the negligible increase in adjusted R-squared, and the weak economic justification for `GoldPriceChange` in a fixed-income VaR model.
The option about the marginal increase in adjusted R-squared suggesting negligible additional explanatory power but warrants retention for robustness is incorrect. While it correctly identifies the negligible improvement in adjusted R-squared, it incorrectly concludes that its statistical significance still warrants retention. The retention of a variable should consider not just individual significance, but also its theoretical relevance, its impact on other coefficients (due to multicollinearity), and the overall parsimony of the model.
The option stating that the observed change in `VIX`'s significance indicates `VIX` was never truly a robust predictor is incorrect. The change in `VIX`'s significance is a classic symptom of multicollinearity rather than an indictment of `VIX`'s standalone predictive power. In `Model 1`, `VIX` was statistically significant. The high correlation between `VIX` and `GoldPriceChange` causes their effects to be conflated, leading to inflated standard errors and reduced t-statistics for both variables when they are included together, making them individually appear less significant than they might be in isolation or in combination with less correlated predictors.
