10.2 How much to difference
Differencing too little leaves a non-stationary series, recognisable by a sample ac.f. that dies away only very slowly. Differencing too much is also a mistake, with its own signature.
Theorem 10.2.1 (Over-differencing).
- (i)
- If \(X_t = a + bt + Y_t\), where \(Y_t\) is stationary, then \(\nabla X_t = b + \nabla Y_t\) is stationary.
- (ii)
- If \(Y_t\) is already white noise, then \(\nabla Y_t = Y_t - Y_{t-1}\) is a non-invertible MA(1) process with \(\beta = -1\), and \(\rho (1) = -\frac 12\).
Proof. (i) \(\nabla (a + bt) = b\) (Proposition 2.4.4), and \(\nabla Y_t = Y_t - Y_{t-1}\) is a fixed linear combination of a stationary process, so its mean is constant and \(\Cov (\nabla Y_t, \nabla Y_{t+k}) = 2\gamma _Y(k) - \gamma _Y(k + 1) - \gamma _Y(k - 1)\) depends only on \(k\).
(ii) \(\nabla Y_t = Y_t - Y_{t-1}\) is MA(1) with \(\beta = -1\), so by Section 6, \(\rho (1) = -1/(1 + 1) = -\frac 12\), the boundary value of Theorem 6.2.1, and \(\theta (B) = 1 - B\) has its root at \(B = 1\), on the unit circle: not invertible. □
So if a differenced series has a sample \(r_1\) close to \(-0.5\), it has probably been differenced once too often. The practical rule is to difference until the sample ac.f. dies away reasonably quickly, and no further. (Formal tests for a unit root, such as the Dickey–Fuller test, are available in software.)
Example 10.2.2. Show that the ARIMA(0,1,1) model \(X_t = X_{t-1} + Z_t - 0.4Z_{t-1}\) can be written as \[ X_t = \sum _{j=1}^\infty 0.6(0.4)^{j-1}X_{t-j} + Z_t , \] so that the predictable part of \(X_t\) is an exponentially weighted average of the past.
Solution. The model is \((1 - B)X_t = (1 - 0.4B)Z_t\), invertible since \(|0.4| < 1\). So \begin {align*} \pi (B) &= \frac {1 - B}{1 - 0.4B} = (1 - B)(1 + 0.4B + 0.16B^2 + \dots ) \\ &= 1 - 0.6B - 0.24B^2 - 0.096B^3 - \dots , \end {align*}
and \(\pi _j = 0.6(0.4)^{j-1}\): the coefficient of \(B^j\) is \(0.4^j - 0.4^{j-1} = -0.6(0.4)^{j-1}\). These weights are positive, decrease geometrically, and add to \(0.6/(1 - 0.4) = 1\), so \(\sum \pi _jX_{t-j}\) is a weighted average of past values, with the most recent weighted most. This is exactly the forecast of simple exponential smoothing with smoothing constant \(0.6\) (Section 14).
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