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, {z_i} (i=1,2,3,…) is a stochastic process with z_i 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(z_i) does not depend on i, and ii) Cov(z_i,z_i-j) exists, is finite, and depends only on j but not on i.
• White Noise. A covariance stationary process, {z_i}, is white noise if E(z_i)=0 and Cov(z_i,z_i-j)=0 for j≠0. We typically assume that the error term in most estimating equations is white noise.
  
• Ergodicity. A stationary process is said to be ergodic if:
  • lim_n->∞|E[f(z_i,…z_i+k)g(z_i+n,…z_i+n+l)]| =|E[f(z_i,…z_i+k)||E[g(z_i+n,…z_i+n+l)]|
  • This means that as the observations from time series become further and further apart, they become independent.
    

  • E(x_i|z_i-1,z_i-2,z_i-3,…z₁)=x_i-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 x_i be an element z_i. The scalar process {x_i} is a martingale with respect to {z_i} if:
• Random Walk. A random walk is a specific type of martingale made up of the sum of a white noise process. Let {g_i} be a white noise process. Then a random walk process {z_i} would equal the following:

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