Series transformations

Transformations are often useful for stabilizing a series before estimating models. This is particularly important for ARIMA models, which require series to be stationary before models are estimated. A series is stationary if the global level (mean) and average deviation from the level (variance) are constant throughout the series.

While most interesting series are not stationary, ARIMA is effective as long as the series can be made stationary by applying transformations, such as the natural log, differencing, or seasonal differencing.

Variance stabilizing transformations. Series in which the variance changes over time can often be stabilized using a natural log or square root transformation. These are also called functional transformations.

• Natural log. The natural logarithm is applied to the series values.
• Square root. The square root function is applied to the series values.

Natural log and square root transformations cannot be used for series with negative values.

Level stabilizing transformations. A slow decline of the values in the ACF indicates that each series value is strongly correlated with the previous value. By analyzing the change in the series values, you obtain a stable level.

• Simple differencing. The differences between each value and the previous value in the series are computed, except the oldest value in the series. This means that the differenced series will have one less value than the original series.
• Seasonal differencing. Identical to simple differencing, except that the differences between each value and the previous seasonal value are computed.

When either simple or seasonal differencing is simultaneously in use with either the log or square root transformation, the variance stabilizing transformation is always applied first. When simple and seasonal differencing are both in use, the resulting series values are the same whether simple differencing or seasonal differencing is applied first.