See also: Confidence Intervals and Hypothesis Testing

Motivation. Assume we have our estimators for our sample size using OLS, . Now, assuming we have the true population data (impossible in real life) and take samples of size from the whole population, we get different tuple of estimators . If we plot these on a graph, we get an approximate bell curve. This is due to the Central Limit Theorem. Knowing this fact, we can deduce if there is a correlation between and .

Remark. is the minimum required for CLT. is a conservative requirement for CLT to apply. Remark. We will only look at since it is the more important parameter.

Hypothesis Testing

def. The Null hypothesis in regression is , i.e. there is no correlation.

def. Regression T-test. See Student’s t-test. A T-test is a test for rejecting the null hypothesis. let the T-statistic . Then

  • The cutoff value is determined by how powerful (=) you want the test to be. This is determined by the Student’s T-Distribution.
    • This is because
  • This is Student’s t-test but with only one random variable.
  • Normally, we set the cutoff , i.e. two standard deviations away. This is around an test.

Table of common T-Statistic values

Value of TSignificanceSidedness

A Large-sample critical value is simply using the idea that as the i.e. the statistic becomes a normal distribution. Thus the cutoff values are using the standard normal table.

Intuition. Untitled

Confidence Interval



standard error of regression: mean squared of residuals; also the estimator for the error

Standard error of…

  • Residuals → standard error of regression
  • Varaince of is also the variance of CLT limit with the same sample size

Remark. Substantive significance does not imply statistical significance, nor does vice versa. The two are irrelevant.