Statistical testing

Technical
Author

Krishnakant A

Purpose

  • To include all kinds of statistical test that I have come across in reading literature.
  • Would also be including tests used in time series analysis as well

List

T-statistic

  • Used when sample size is small and/or standard deviation is unknown

  • Z-table is used to find out the p-value and/or z-score at a given significance/confidence level

  • Its usually given by this formula

\[ t_{\hat\beta} = \frac{\hat\beta - \beta_0}{s.e(\hat\beta)}\] where \(s.e(\hat\beta)\) is the standard error of the estimator \(\hat\beta\) for \(\beta\) and \(\beta_0\) is a non-random , know constant , which may or may not match actual unknown parameter value \(\beta\)

  • t-ratio statistic of the sample skewness is \[ t = \frac {\hat S( r)}{\sqrt{6/T}} \] asymptotically a standard normal random variable

  • t-ratio statistic of the sample excess kurtosis is \[ t = \frac{\hat{K(r) - 3}}{\sqrt{24/T}} \] which is asymptotically a standard normal random variable

  • Jarque-Bera test(JB) combines the two prior tests \[ JB = \frac {\hat S^2( r)}{{6/T}} + \frac{[\hat{K(r) - 3}]^2}{{24/T}} \] which is asymptotically distributed as chi-squared random variable with 2 degrees of freedom

  • all the above tests in this case are goodness of fit tests