Stationarity and nonstationarity

ARIMA Models in R

David Stoffer

Professor of Statistics at the University of Pittsburgh

Stationarity

A time series is stationary when it is "stable", meaning:

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Given data, $ \ x_1,...,x_n \ $ we can estimate by averaging

For example, if the mean is constant, we can estimate it by the sample average $\bar x$

Pairs can be used to estimate correlation on different lags:

$(x_1, x_2), (x_2, x_3), (x_3, x_4), ...$ for lag 1

$(x_1, x_3), (x_2, x_4), (x_3, x_5), ...$ for lag 2

Reasonable to assume stationary, but perhaps some slight trend.

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To estimate autocorrelation, compute the correlation coefficient between the time series and itself at various lags.

Here you see how to get the correlation at lag 1 and lag 6.

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Not stationary, but differenced data are stationary.

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Stationarity around a trend, differencing still works!

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First log, then difference

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ARIMA Models in R