Time Series – Help Interpreting ACF and PACF Plots in Time Series Analysis

autocorrelationtime series

My ACF- and PACF plots are illustrated below:

Normal view

Zoomed

The first one is in original scale and the second picture is zoomed. What process would you classify this? AR, MA or ARMA? =)

Thank you for any help =)

Best Answer

Neither displays any geometric decay nor any obvious orders from cut-off points, I would say ARMA. I would fit both an AR and MA and then inspect the acf/pacf of the residuals, maybe they would provide some insight. :-)

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Interpreting ACF and PACF Plots in Time Series Analysis

There is no apparent structure in the plots that you show.

The lag order of those negative partial autocorrelations that lie outside the bands are not multiple of each other (they are lags, 22, 56, 62, 78, 94) i.e., they do not arise after a regular number of lags as for example 12, 24, 36, 48, so I wouldn't infer any pattern based on that from the plot.

As a complement you may apply a runs test, which is a test for independence that may be useful to capture runs of positive or negative values, which would suggest some pattern in the data.

As regards the significance of some of the autorrelations I see that they arise at large orders. You should think if those autocorrelations make sense or may be expected in the context of your data. Is it sensible to expect that the value observed 56 observations ago will affect the current observation? If we had quarterly data, it would be worth inspecting significant correlation at lags 8 and 12 because they are multiples of the periodicity of the data and may reflect some seasonal pattern that we could explain in the context of the data. But I wouldn't concern that much if significant lags arose at lags 9, 11 or much higher lags for which I didn't have an explanation that will justify it as a regular pattern.

Time Series Analysis – How to Analyze ACF and PACF Plots

Looking at your ACF and PACF is useful in the full context of your analysis as well. Your Ljung-Box Q-statistic; p-value; confidence interval, ACF and PACF should be viewed together. For instance the Q test here:

acf, ci, Q, pvalue = tsa.acf(res1.resid, 
  nlags=4, confint=95,  qstat=True, 
  unbiased=True)

Here - our Q test for autocorrelation is an overall gut check of our graphical interpretation.

Draft notes on Time Series analysis in Statsmodels: http://conference.scipy.org/proceedings/scipy2011/pdfs/statsmodels.pdf

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Interpreting ACF and PACF Plots in Time Series Analysis

Time Series Analysis – How to Analyze ACF and PACF Plots

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