What is the relation between the autocorrelation and the trend? Can a trend exist in a time series of independent variables? And in time series with a non-zero autocorrelation, does a trend always exist?
Solved – Autocorrelation and trends
autocorrelationmathematical-statisticstime seriestrend
Related Solutions
@whuber gave a nice answer. I would just add, that you can simulate this very easily in R:
op <- par(mfrow = c(2,2), mar = .5 + c(0,0,0,0))
N <- 500
# Simulate a Gaussian noise process
y1 <- rnorm(N)
# Turn it into integrated noise (a random walk)
y2 <- cumsum(y1)
plot(ts(y1), xlab="", ylab="", main="", axes=F); box()
plot(ts(y2), xlab="", ylab="", main="", axes=F); box()
acf(y1, xlab="", ylab="", main="", axes=F); box()
acf(y2, xlab="", ylab="", main="", axes=F); box()
par(op)
Which ends up looking somewhat like this:
So you can easily see that the ACF function trails off slowly to zero in the case of a non-stationary series. The rate of decline is some measure of the trend, as @whuber mentioned, although this isn't the best tool to use for that kind of analysis.
I think the author is probably talking about the residuals of the model. I argue this because of his statement about adding more fourier coefficients; if, as I believe, he is fitting a fourier model, then adding more coefficients will reduce the autocorrelation of the residuals at the expense of a higher CV.
If you have trouble visualizing this, think of the following example: suppose you have the following 100 points data set, which comes from a two-coefficient fourier model with addeded white gaussian noise:
The following graph shows two fits: one done with 2 fourier coefficients, and one done with 200 fourier coefficients:
As you can see, the 200 fourier coefficients fits the DATAPOINTS better, while the 2 coefficient fit (the 'real' model) fits the MODEL better. This implies that the autocorrelation of the residuals of the model with 200 coefficients will almost surely be closer to zero at all lags than the residuals of the 2 coefficient model, because the model with 200 coefficients fits exactly almost all datapoints (i.e., the residuals will be almost all zeros). However, what would you think will happen if you leave, say, 10 datapoints out of the sample and fit the same models? The 2-coefficient model will predict better the datapoints you leaved out of the sample! Thus, it will produce a lower CV error as opossed to the 200-coefficient model; this is called overfitting. The reason behind this 'magic' is because what CV actually tries to measure is prediction error, i.e., how well your model predicts datapoints not in your dataset.
- In this context, autocorrelation on the residuals is 'bad', because it means you are not modeling the correlation between datapoints well enough. The main reason why people don't difference the series is because they actually want to model the underlying process as it is. One differences the time series usually to get rid of periodicities or trends, but if that periodicity or trend is actually what you are trying to model, then differencing them might seem like a last resort option (or an option in order to model the residuals with a more complex stochastic process).
- This really depends on the area you are working on. It could be a problem with the deterministic model also. However, depending on the form of the autocorrelation, it can be easily seen when the autocorrelation arises due to, e.g., flicker noise, ARMA-like noise or if it is a residual underlying periodic source (in which case you would maybe want to increase the number of fourier coefficients).
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Best Answer
The autocorrelation(acf) function summarizes the correlation of different lags and is a descriptive statistic. If there is a "trend" in the data then the acf will suggest non-stationarity. However a non-stationary acf does not necessarily suggest a "trend". If the series is impacted by one or more level/step shifts the the acf will suggest non-stationarity (symptom) but the cause is simple a shift in the mean at one or more points in time. "Trends" in time series can sometimes be adequately handled by incorporating a deterministic structure. There are a few forms ( the software package needs to help here and/or the skilled analyst ) to determine which form is the most adequate. One form is to incorporate a predictor series(x1) such as 1,2,3,4,..n which would imply one and only one trend. Another might be to incororporate two input series (x1 and x2) where x1=1,2,3,4,,..n and x2=0,0,0,0,0,1,2,3,..n reflecting two trends where the second trend starts at period 6 in this example. Of course there could be multiple "trends" or breakpoints in trend.
An alternative way of incorporating a "trend" is to model the data as some variety of a differencing model of the form (1-B)Y(T) = constant + [theta(B)/phi(B)]*A(T) which suggests one and only 1 trend . Good analytics will suggest the "most correct" approach to an individual data set. If you have exogenous/causal/supporting/input series then the "trend" in Y could well be explained by one or more of these series. The acf is of little or no help in helping you decide which "trend model" is appropriate.