Stationarity Assumption

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1. Stationarity Assumption

Let’s go back to hare data and how we talked about autocorrelation of lag1.

  layout(matrix(1:2, 1, 2))       # make two plots side by side (1row 2col)
  plot(hare, type="o")
  plot( hare[2:31], hare[1:30], ylab="This Year", xlab="Lat Year" )

  cor(  hare[2:31], hare[1:30] )   #- lag 1

## [1] 0.7025777

We assumed cor(\(X_2,X_1\)) is same as cor(\(X_{31},X_{30}\)). That’s a big assumption! We have asumed other things without thinking about them.

Weak Stationarity

Series of r.v. \(\{X_1,\ldots,X_n\}\) is called weakly stationary if

• Mean \(E(X_t)\) does not depend on \(t\).
• Variance \(V(X_t)\) does not depend on \(t\).
• cor\((X_t,X_{t+h})\) does not depend on \(t\).
  

Strong Stationarity

Series of r.v. \(\{ X_1, \ldots, X_n \}\) is called strongly stationary if

• Joint pdf of \(\{X_1,\ldots, X_n\}\) are identical to joint pdf of \(\{X_{t+1},\ldots, X_{t+n}\}\) for all \(t\). This is pretty strong assumption.
  

When we say “stationary”, we mean weak stationarity.

2. Stationary?

    plot(hare,type="o")

    data(color)
    plot(color,type="o")

    data(larain)
    plot(larain,type="o")

    data(oilfilters)
    plot(oilfilters,type="o",ylab="Sales")

Stationary?

3. Warning: How it looks depends on your scope

How ‘stationary’ it looks depend on scale of the plot.

    plot(hare,type="o")

    plot(hare,type="o", xlim=c(1917,1922))    # zoom in

4. Example of Non-sationariy process

• Series with Trend
• Series with non-constant variance
• Random Walk
  
  

Summary

• For ACF and ACVF to be plotted and analyzed, the series must be (weakly) stationary.
• Weak Stationaritymeans
  • \(E(X_t)\) is constant over time.
  • \(V(X_t)\) is constant over time.
  • ACF: \(\rho(h) = cor(X_t,X_{t-|h|})\) does not depend on time.