Basic Econometric Concepts: Time Series

Today I will review a few basic concepts of time series econometrics. A time series is a stochastic process where observations appear in different time periods. For instance, {zi} (i=1,2,3,…) is a stochastic process with zi representing the GDP each quarter. Below are a few important definitions which are important to econometric estimation using time series data.

  • Covariance Stationary Processes. A process is covariance stationary if i) E(zi) does not depend on i, and ii) Cov(zi,zi-j) exists, is finite, and depends only on j but not on i.
  • White Noise. A covariance stationary process, {zi}, is white noise if E(zi)=0 and Cov(zi,zi-j)=0 for j0. We typically assume that the error term in most estimating equations is white noise.
  • Ergodicity. A stationary process is said to be ergodic if:
    • limn->∞|E[f(zi,…zi+k)g(zi+n,…zi+n+l)]| =|E[f(zi,…zi+k)||E[g(zi+n,…zi+n+l)]|
    • This means that as the observations from time series become further and further apart, they become independent.

    • E(xi|zi-1,zi-2,zi-3,…z1)=xi-1.
    • This means that given all the information from the past, our best guess at the value of x in this time period is the value of x last time period. For instance, Hall’s Martingale Hypothesis states that given a variety of macroeconomic variables, my best guess of aggregate consumption this quarter is equal to aggregate consumption last quarter.

    • z1=g1
    • z2=g1+g2
    • zi=g1+g2+…+gi
  • Martingale. Let xi be an element zi. The scalar process {xi} is a martingale with respect to {zi} if:
  • Random Walk. A random walk is a specific type of martingale made up of the sum of a white noise process. Let {gi} be a white noise process. Then a random walk process {zi} would equal the following:

For further information, see: Hayashi, Fumio (2000) Econometrics. Princeton University Press. USA.