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I have a time series consisting of 490 days and for each day I have the residual of a forecasting model. I wanted to check if the residuals somehow correlate and calculated the ACF at lag 1 which is acf(1)=0.42 Here is the ACF plot: ACF

Now I wanted to check the linear relationship between lag=0 (denoted as x1 in the scatterplot) and lag=1 (x2 in the scatterplot) so I plotted a scatterplot, but I don't see any linear relationship. Why do I have a correlation of 0.42 at lag 1 but I don't see the linearity between lag 0 and 1?

Am I missing a point here?

Update: I uploaded a CSV file with the time series here

Scatterplot

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    $\begingroup$ The green line has pretty much the 0.42 slope you see in the acf. That seems too match actually quite well. Maybe you expected this to be more clearly on a line, rather than a "blob" of seemingly independent data points? Probably a robust fit (rlm()) would reduce the slope of green line and be closer to your visual assessment. $\endgroup$ – Georg M. Goerg Mar 15 '18 at 11:44
  • $\begingroup$ Can you comment about what the axis exactly mean? $\endgroup$ – user1420303 Mar 15 '18 at 12:12
  • $\begingroup$ @GeorgM.Goerg: I tried rlm() and plotted the residuals vs fitted plot i.imgur.com/3hIGRtl.png - how can i interprete the red line? as a comparison, this is the residuals vs fitted plot of lm(): i.imgur.com/F7m3dyQ.png $\endgroup$ – UDE_Student Mar 15 '18 at 12:41
  • $\begingroup$ @user1420303: Do you mean the scatterplot? if yes, then x1 consists of timeseries with lag 0 and x2 is the "shifted" timeseries of lag=1. (as mentioned my initial timeseries has 490 values. Hence for the comparison, i create two "new" timeseries. The first one (x1) consists of values between 1:489 and the second one (x2) consists of values 2:490) $\endgroup$ – UDE_Student Mar 15 '18 at 12:44
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    $\begingroup$ There are other effects that are masking your "view" . Daily,weekly.holiday.monthly to name a few and possibly level shifts or local time trends AND definitely anomalies. The only time your approach would confirm the .42 is if there were no other (significant) effects. If you post your data in a csv file I will try and help further. see stats.stackexchange.com/search?q=user%3A3382+daily+sales+data for more info on modelling daily data. $\endgroup$ – IrishStat Mar 15 '18 at 14:51
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upon receipt of the 497 daily values I obtained a plot enter image description here which supported my "guess" that there were other/latent factors present ( i.e. one clear level shift and one masked by anomalies ) . I used AUTOBOX my tool of choice and it identified a model enter image description here enter image description here excerpting some pulse indicators. The final equation does indeed include an AR(1) and some possible seasonal/monthly dummies along with a bunch of anomalies.

The residual plot enter image description heresuggest sufficiency ( supported by the acf of the residuals) enter image description here

Hope this helps you and others regarding interpreting unconditional statistics either visual or written.

The forecast for the next 21 days is shown here enter image description here

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