# Time-series stationarity

If I difference a time series and take out trend and seasonality ... does it mean we are left with only irregularity on which we plot the acf and pacf to arrive at the MA and AR order?

Do 1st difference, 2nd difference always detrend the series, or do we need to detrend separately?

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To take out seasonality you need to first check if you time series is stationary. If it is I suppose your logic holds. – dfhgfh Jan 15 '13 at 14:47

Even if the original process is a stationary ARMA process plus a trend, differencing will still not have the effect of detrending the series. For example, let $x_t$ be such a series: $x_t = \mu(t) + y_t$ where $y_t$ is a stationary ARMA process. Let $z_t$ denote the differenced process: $z_t = x_t - x_{t-1} = \mu(t) - \mu(t-1) + y_t - y_{t-1}$. Then $E[z_t] = \mu(t) - \mu(t-1) + E[y_{t}] - E[y_{t-1}] = \mu(t) - \mu(t-1) \neq 0.$ Differencing multiple times will likewise not yield a stationary series.
Differencing is most useful for analyzing series with unit roots, e.g., AR processes with coefficients equal to (or close to) unity. Consider the simplest such case in which $y_t$ follows a random walk, i.e., in which $y_t = y_{t-1} + e_t$. This process is non-stationary. However, the differenced series is stationary. Ref: en.wikipedia.org/wiki/Unit_root – Arthur Small Jan 16 '13 at 5:05