acf(c(0,1,2,3,4,5),plot=FALSE)
Autocorrelations of series ‘c(0, 1, 2, 3, 4, 5)’, by lag
0 1 2 3 4 5
1.000 0.500 0.057 -0.271 -0.429 -0.357
Why does the ACF output becomes negative as lag increases? My understanding is that no matter what the lag is, the series is in general increasing. Therefore the auto-correlation should be positive. For example, at lag 2, we are calculating the correlation function of the two series [0,1,2,3] and [2,3,4,5], where the positive correlation still holds. Where do I get this wrong?
Update
Here is my intuitive understanding of the acf of an monotonically increasing sequence:
ACF of a sequence is a function $\gamma(k)$ of the lag, k. By definition, this function indeed measures the correlation between $y_t$ and $y_{t-k}$. The misunderstanding comes from the understanding of correlation. A monotonically increasing sequence is not stationary, so the mean is not stable. In another word, the sequence does not exhibit mean reverting behavior. This distorts my usual understanding of correlation (when we think about the mean level at 0). Since the mean increases over time, those observations come earlier are more likely to be lower than the sample mean, thus inducing a negative sample acf when lag is larger.
