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L4 Weak vs strong stationarity

  • Weak (Covariance) stationarity
  • Stationarity in a wide sense

If its first and second moments are unaffected by a change of time origin.

  • We are not putting any assumption in a distribution sense.
\[ Cov(Y_{t},Y_{t+k}) = \gamma_{t,t+k} \simeq Y_{k} \]

Should only depend on the lag, \(k\). Should be completely free from \(t\)

  • \(E(Y_{t}) = \mu, \forall t\)
  • \(Var(Y_{t}) = \sigma^{2} \lt \infty, \forall t\)
  • \(Cov(Y_{t},Y_{t+k}) = \gamma_{k}, \forall t\)
  • \(Corr(Y_{t},Y_{t+k}) = \rho_{k}, \forall t\)

When one says, "I am analyzing a stationary time series" they are actually referring to a weak stationary series.

Example 2

\[ Y_{t} = e_{t} \]

\(e_{t} \sim i.i.d(0,\sigma^{2})\) , which is the error.

  1. \(E(Y_{t})= E(e_{t}) = 0\)
  2. \(Var(Y_{t})= Var(e_{t}) =\sigma^{2}\)
  3. \(Cov(e_{t},e_{t+k}) = 0\), \(k \neq 0\) since IID

So, \(Y_{t} = e_{t}\) is indeed stationary.

Example 3

\[ Y_{t} = e_{t} + 0.5 e_{t-1} \]

\(e_{t} \sim i.i.d(0,\sigma^{2})\) , which is the error. Is the process stationary?

  1. \(E(Y_t) = E(e_{t}) + 0.5 E(E_{t-1}) = 0\)
  2. \(Var(Y_t) = Var(e_{t}) + 0.25 Var(E_{t-1}) + 2 Cov(e_{t, 0.5e_{t-1}}) = 0\) which gives \(\sigma^{2} + 0.25 \sigma^{2} + 0 = 1.25 \sigma^{2}\) which is free from \(t\).
  3. \(Cov(e_{t} + 0.5e_{t-1}, e_{t+k} + 0.5e_{t+k-1})\)… is?

Example 4

\[ Y_{t} = e_{1} + e_{2} + e_{3} + \dots + e_{t} \]
  1. \(E(Y_{t}) = 0\)
  2. \(Var(Y_{t}) = t\sigma^{2}\) which is time dependent… thus, \(Y_{t}\) is not stationary.

Example 5

\[ Y_{t} = a + bt + e_{t} \]
  1. \(E(Y_{t}) = E(a+bt+ e_{t}) = a + bt\) which is time dependent…, thus, \(Y_{t}\) is not stationary.

Strong Vs Weak Stationary

  • Strict stationarity → Joint Distribution should be time-invariant.
  • Weak stationarity → Only two moments should be time-invariant.

Strong \(\not\implies\) weak

Strong stationary \(\not\implies\) weak stationary.

  • There is a IID Cauchy process, which is strictly stationary but its first two moments don't exist, thus it is not weak stationary.
  • Nonlinear function of a strict stationary variable is still strictly stationary, but not weak (e.g. if \(Y_{t}\) is strictly stationary then \(Y_{t}^{2}\) is strict but not weak stationary as the variance of \(Y_{t}^{2}\) may not exist.)

  • Weak Stationary \(\implies\) strong stationarity

    • Since for a multivariate normal distribution, the first two moments characterize the entire distribution. Knowing that the first two moments are time invariant will imply that the entire joint distribution is also time invariant.
    • Thus, if the process \(\{ X_{t} \}\) is a Gaussian TS, then weak stationary \(\implies\) string stationary.

  • Non-stationarity

    • If a process lacks stationary or statistical equilibrium \(\implies\) Non-stationary
    • Three points amount to this
      • Trend
      • Seasonality (cyclicality)
      • Heteroscedasticity (changing variance)