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Answer: t-statistic
## Explanation When testing a hypothesis about a single regression coefficient (in this case, testing whether β = 1), the appropriate test statistic is the **t-statistic**. ### Why t-statistic? - The t-statistic is used for testing hypotheses about individual regression coefficients - Formula: t = (estimated coefficient - hypothesized value) / standard error - In this case: t = (0.86 - 1) / 0.80 = -0.14 / 0.80 = -0.175 ### Why not the other options? - **Chi-square test statistic**: Used for testing variance or goodness-of-fit, not individual coefficients - **F test statistic**: Used for testing multiple coefficients simultaneously or overall model significance - **Sum of squared residuals**: A measure of model fit, not a test statistic for coefficient hypotheses ### Statistical Context In linear regression, when testing H₀: β = β₀ against H₁: β ≠ β₀, the test statistic follows a t-distribution with n-k-1 degrees of freedom (where n is sample size and k is number of predictors).
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An analyst is testing a hypothesis that the beta, β, of stock CDM is 1. The analyst runs an ordinary least squares regression of the monthly returns of CDM, R_CDM, on the monthly returns of the S&P 500 index, R_m, and obtains the following relation: R_CDM = 0.86 R_m − 0.32
The analyst also observes that the standard error of the coefficient of R_m is 0.80. In order to test the hypothesis H₀: β = 1 against H₁: β ≠ 1, what is the correct statistic to calculate?
A
t-statistic
B
Chi-square test statistic
C
F test statistic
D
Sum of squared residuals
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