Financial Risk Manager Part 1

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Consider the following 2 regression models built to estimate a common phenomenon:

Model 1: $y_t = \beta_1 + \beta_2 x_{2t} + \beta_3 x_{3t} + u_t$

Model 2: $y_t = \beta_1 + \beta_2 x_{2t} + \beta_3 x_{3t} + \beta_4 x_{4t} + u_t$

$R^2$ and the adjusted $R^2$ for the two models are given below:

Model	$R^2$	Adjusted $R^2$
Model 1	0.75	0.73
Model 2	0.81	0.72

Which one of the following is most likely correct?

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NNikitesh

Last updated: February 2, 2026 at 10:22

The coefficient estimate $\beta_4$ is zero

The researcher must have made a mistake because adjusted $R^2$ for model 2 must be greater than the adjusted $R^2$ for model 1

Variable $x_4$ is statistically significant

The coefficient estimate $\beta_4$ is non-zero but not significant

Explanation:

The fact that R-squared is higher for model 2 (0.81 vs 0.75) but the adjusted R-squared is lower for model 2 (0.72 vs 0.73) suggests that the extra variable $x_4$ is not statistically significant.

Key points:

R-squared always increases when adding more variables, even if they are irrelevant (0.81 > 0.75).
Adjusted R-squared penalizes for adding insignificant variables. The decrease from 0.73 to 0.72 indicates $x_4$ doesn't contribute enough explanatory power to justify its inclusion.
Option A is incorrect because if $\beta_4$ was exactly zero, R-squared would not increase.
Option B is incorrect because adjusted R-squared can decrease when adding insignificant variables.
Option C is incorrect because if $x_4$ were statistically significant, adjusted R-squared would typically increase.
Option D is correct because the pattern suggests $\beta_4$ is non-zero (R-squared increased) but not statistically significant (adjusted R-squared decreased).

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