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When I execute both functions on the same vector, I get slightly different graphs. I think I know what is happening but I want to confirm. acf() shows the spike at lag 0 which is 1 naturally, this makes the rest of the graph look much smaller. Acf() in contrast does not show the spike at lag 0 and starts at lag 1. Am I right in my assumptions?

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The difference is actually explained in the forecast package manual:

The functions improve the acf, pacf and ccf functions. The main differences are that Acf does not plot a spike at lag 0 when type=="correlation" (which is redundant) and the horizontal axes show lags in time units rather than seasonal units.

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