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Inference about the slope: t-Test

  • t-test for a population slope
    • Is there a linear relationship between x and y?
  • Null and alternative hypotheses
    • H0:  β1 = 0 (no linear relationship)
    • H1:  β1 ≠ 0 (linear relationship does exist)
  • Test statistic
  • Where:
    • b1 = Sample regression slope coefficient
    • β1 = Hypothesized slope
    • sb1 = Estimator of the standard error of the slope

  • A risk analyst performs a simple linear regression on return data comprising three variables evolving in time and obtains, amongst others, the following statistics:


  • Based on these data at a 95% confidence level, the analyst should conclude that:
  1. The intercept and “X Variable 2” are statistically significant
  2. “X Variable 1” and “X Variable 3” are statistically significant
  3. “X Variable 1”, “X Variable 2” and “X Variable 3” are all statistically not significant
  4. More information is required, such as the corresponding p-values, before any meaningful deductions may be made

A is correct. (Relatively) small standard errors and high t-stats are one indication of indicate statistical significance
B is incorrect. – (Relatively) large standard errors and low t-stats are one indication of indicate statistical significance
C is incorrect. Negative t-stats are not an indication of statistical insignificance
D is incorrect. The p-values are redundant information if the t-stat is provided. That is, the p-values tell one nothing more than the t-stats do.


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