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L3 Stationarity in Time Series

Notation

  • Mean function: \(\mu_{t} = E[Y_{t}]\) (also includes the time point, \(t\))
    > At a particular time point \(t\) what is the expected value of the variable?
  • Variance Function
    • \(\sigma_{t}^{2} =\gamma_{0}\)
    • Assumption: Variance should be finite \(\lt \infty\)
  • Auto-covariance Function
    • \(\gamma_{t,s} = Cov(Y_{t},Y_{s})\): Covariance between value at two time points \(t\) and \(s\).
    • Auto → comes from the same series (vs two different variables \(X,Y\) in rest of statistics… here both the variables come from the same time series)
  • Auto-correlation Function ^0a3745
    • \(\rho_{t,s}=C\cup(Y_{t},Y_{s}) = \dfrac{\gamma_{t,s}}{\sigma_{t}\sigma _{s}}\)

Stationarity

  • Joint PDF of a Time series

    • \(F_{X_{1}}(x_{1})\) = marginal CDF
    • \(f_{X_{1}}(x_{1})\) = marginal PDF
    • Joint CDF/PDF for multiple variables \(X_{1},X_{2},\dots,X_{n}\)
    • \(f_{Y_{t},Y_{s}}(y_{t},y_{s}) \neq f_{Y_{t}(y_{t})}\times f_{Y_{s}}(y_{s})\) because the variables are NOT independent!

  • Problems in Handling Joint PDF

    • Putting a single distribution on the entire structure is difficult (since distributions (or moments) may change for all \(t\))
    • A simplification to solve this problem → STATIONARITY
  • Stationarity
    • Most common assumption in TS analysis
    • Probability laws do not change with time (?)
    • Process is in Statistical equilibrium
  • Types of Stationarity
    • Strict (or Strong)
      • Observed sample: \(\{ Y_{t_{1}}, Y_{t_{2}},\dots, Y_{t_{n}} \}\)
      • Stochastic process: \(Y(w,t)\)
      • We defined the joint CDF \(F_{Y_{t_{1}},Y_{t_{2}},\dots, Y_{t_{n}}}\)
      • First order stationary in distribution
        • \(F_{Y_{t_{1}}}(y_{1}) = F_{Y_{t_{1}+k}}(y_{1})\) for any \(t_{1}\) and \(k\)
      • Second order stationary
        • \(F_{Y_{t_{1}},Y_{t_{2}}}(y_{1},y_{2}) = F_{Y_{t_{1}+k},Y_{t_{2}+k}}(y_{1},y_{2})\)
        • for any \(t_{1},t_{2}\) and \(k\)
      • \(n\)-th order stationary
        • for any \(t_{1},\dots, t_{n}\)
        • should hold true if you shift the time origin by \(k\)
      • For any of the above, it should remain consistent if the time origin is shifted.
      • STRONG → JOINT DISTRIBUTION stays the same even if the time origin (time stamp) is shifted by \(k\).