Gauss-Markov Theorem
Under the following assumptions, the ordinary least squares (OLS) estimators are the best linear unbiased estimators conditional on X for time series data.
- The stochastic process {(Xt1, Xt2, … , Xtk, Yt): t = 1, 2, … , n} follows the linear model y = beta0 + beta1Xt1 + … + betakXtk + ut where {u: t = 1, 2, … , n} is the sequence of errors of disturbances.
- In the sample (and therefore in the underlying time series process), no independent variable is constant nor a perfect linear combination of the others.
- For each t, the expected value of the error ut, given the explanatory variables for all time periods, is zero. Mathematically, E(ut|X) = 0, t = 1, 2, … , n
- Conditional on x, the variance of ut is the same for all t: Var(ut|x) = Var(ut) = omicron-squared, t = 1, 2, … , n.
- Conditional on X, the errors in two different time periods are uncorrelated: Corr(ut, us| X) = 0 for all t is not equal to s.
