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?
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$\begingroup$ Auto correlation has a precise definition. As far as I know, "trend" does not. So, the answer will depend on how you define trend. $\endgroup$– Peter FlomCommented Oct 4, 2013 at 12:45
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2 Answers
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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.
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$\begingroup$ Thanx, I have a question about your answer. What exactly means "If there is a "trend" in the data then the acf will suggest non-stationarity."? That there is a correlations in acf? $\endgroup$– emanueleCommented Oct 4, 2013 at 13:06
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1$\begingroup$ If there is a trend then for any of the possible cases that I mentioned there will be a correlation between sucessive values which is also called auto-correlation $\endgroup$ Commented Oct 4, 2013 at 14:51
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$\begingroup$ So, in few words, we can states that no auto-correlation, then no trends. Isn't it? $\endgroup$– emanueleCommented Oct 4, 2013 at 15:06
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1$\begingroup$ Not necessarily so ! ... Because if there is a "trend" AND there are unusual values (pulses) or other Gaussian Violations ( e.g. time varying variance/time varying parameters et.al. ) the variance is inflated. Since the acf is the ratio of the covariance to the (possibly inflated) variance there is a downwards bias to the acf causing one to incorrectly accept the hypothesis of a non-significant(null) acf. This is called the "Alice in Wonderland effect" where everything looks rosy BUT it is not because the glasses your are looking through are not opaque. $\endgroup$ Commented Oct 4, 2013 at 15:47
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$\begingroup$ oops.. I meant to say the glasses you are looking through are opaque and consequently your view may be (is) distorted. $\endgroup$ Commented Oct 4, 2013 at 17:32
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Autocorrelation function (ACF) is an theoretical object related to the population moments. What happens when these moments do not exist as finite?
Sample autocorrelation function (SACF) is a descriptive statistic and is a function of sample moments, mainly sample mean. What is a breakpoint value for the sample mean? Is it small or large? If you know these then you would know what are dangers related to inference from these sample values.
Objects calculated from the sample always exist, though it will be probable that these estimates do diverge when underlying population moments are not really finite or process goes through somekind of change.
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