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For example, this mentioned here: link. I also saw this in my data.

I wonder - does anyone know a good reference where this is explained and justified more rigorously with some math and for some more-or-less wide class of processes?

Here are some plots:

Series with positive autocorrelation before differencing

Series with positive autocorrelation before differencing

Autocorrelation before differencing

Autocorrelation before differencing

Series after differencing

Series after differencing

Autocorrelation after differencing

Autocorrelation after differencing

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2 Answers 2

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Take the simple white noise process $Z_t$, $EZ_t=0$, $cov(Z_t,Z_{t-h})=0$, for all $h\neq 0$. Now take its difference $Y_t=Z_{t}-Z_{t-1}$and calculate the first lag autocovariance:

$$cov(Y_t,Y_{t-1})=cov(Z_t-Z_{t-1},Z_{t-1}-Z_{t-2})=-cov(Z_{t-1},Z_{t-1})=-var(Z_t)$$

Hence $corr(Y_t,Y_{t-1})=-1/2.$ (Since $var(Y_t)=2var(Z_t)$).

Now for any (causal) stationary process $X_t$ there exists such a white noise process $Z_t$ and coefficients $\psi_j$ such that $X_t=\sum_{j=0}^{\infty}\psi_jZ_{t-j}$. This is courtesy of the Wold decomposition. Thus

$$cov(X_t,X_{t+h})=\sum_{j=0}^\infty\psi_j\psi_{j+h}$$

For the differenced version $Y_t=X_t-X_{t-1}$ we have

$$Y_{t}=Z_{t}+(\psi_1-1)Z_{t-1}+\sum_{j=2}(\psi_{j}-\psi_{j-1})Z_{t-j}$$

and

$$cov(Y_t,Y_{t-1})=\psi_1-1+\sum_{j=2}^{\infty}(\psi_{j}-\psi_{j-1})(\psi_{j-1}-\psi_{j-2})$$

Now more often than not the coefficients $\psi_j$ are decreasing and less than one. So we have that $\psi_1-1<0$ and is larger than remaining sum. This would be one (very obvious) explanation why the first covariance is negative. More can be said with more careful analysis of the terms of the sum, but I think I managed to convey the general idea.

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  • $\begingroup$ Thanks a lot! This is very insightful and exactly what I needed. You may have a typo - $j+1$ should be $j-1$ and $j+2$ should be $j-2$ - I corrected it. $\endgroup$
    – Kochede
    Nov 26, 2013 at 4:02
  • $\begingroup$ By the way, in the same link above, author mentions that if you differenced series and got negative autocorrelation you may correct for it by adding extra MA-terms in your model for $Y_t$. While my first intuition when I see autocorrelation at a single lag is to add 1 AR-term (and this worked in my case). Do you have an idea what the author means? $\endgroup$
    – Kochede
    Nov 26, 2013 at 4:05
  • $\begingroup$ How did you say that $\psi_j$ is less than one? e.g.: ARMA(1, 1) with $\phi = 0.9, \theta = 0.5$. $\psi_j = 1.4(0.9)^{j-1}$ for $j \geq 1$. So first few terms are not less than one? $\endgroup$
    – shani
    Nov 2, 2018 at 2:39
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unwarranted differencing is like unwaranted drugs they can have nasty side effects . The spike in the second differences suggests an ma coefficient which will effectively countermand the unwarranted differencing. The aic/bic stuff just doesn't always work as it often suggests over-differencing and over-populated ARMA structure . In my experience it seldom works to identify a parsimonious model except in trivial cases due to non-gaussian complications.

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