Statistical unit root tests are a common way to determine objectively whether (additional) differencing is necessary. These are statistical hypothesis tests of stationarity that are designed to determine whether differencing is required.
The augmented Dickey-Fuller (ADF) test evaluates the null hypothesis that a time series sample has unit root against the alternative of stationarity. It regresses the differenced time series on a time trend, the first lag, and all lagged differences, and computes a test statistic from the value of the coefficient on the lagged time series value. statsmodels makes it easy to implement (see companion notebook).
Formally, the ADF test for a time series, yt, runs the linear regression:
Where α is a constant, β is a coefficient on a time trend, and p refers to the number of lags used in the model. The α=β =0 constraint implies a random walk, whereas only β=0 implies a random walk with drift. The lag order is usually decided using the AIC and BIC information criteria introduced in Chapter 7, Linear Models.
The ADF test statistics uses the sample coefficient, γ, that, under the null hypothesis of unit-root non-stationarity equals zero, and is negative otherwise. It intends to demonstrate that, for an integrated series, the lagged series value should not provide useful information in predicting the first difference above and beyond lagged differences.