Financial Risk Manager Part 1

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Consider the following 2 regression models:

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

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

A researcher determines that the two models have identical R-squared values. This most likely implies that:

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NNikitesh

Last updated: February 2, 2026 at 10:22

Model 2 must have a lower value of adjusted R-squared

Model 2 must have a higher value of adjusted R-squared

Model 1 and 2 will also have identical values of adjusted R-squared

Variable $x_3$ is statistically significant

Explanation:

When two regression models have identical R-squared values, it means the additional variable $x_3$ has zero explanatory power for $y$ and its coefficient estimate $\beta_3$ must be exactly zero. The adjusted R-squared formula is:

$\bar{R}^2 = 1 - \left( \frac{n - 1}{n - k - 1} \right)(1 - R^2)$

where $k$ represents the number of regressors. Since Model 2 has more regressors (k increases from 2 to 3) while R-squared remains the same, the denominator $(n - k - 1)$ decreases, making the fraction $\frac{n - 1}{n - k - 1}$ larger. This causes the adjusted R-squared for Model 2 to be lower than for Model 1. Therefore, Model 2 must have a lower value of adjusted R-squared.

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