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I want to see if I am on the right track analysing my ACF and PACF plots:

enter image description here

Background: (Reff: Philip Hans Franses, 1998)

  1. As both ACF and PACF show significant values, I assume that an ARMA-model will serve my needs

  2. The ACF can be used to estimate the MA-part, i.e q-value, the PACF can be used to estimate the AR-part, i.e. p-value

  3. To estimate a model-order I look at a.) whether the ACF values die out sufficiently, b.) whether the ACF signals overdifferencing and c.) whether the ACF and PACF show any significant and easily interpretable peaks at certain lags

  4. ACF and PACF might suggest not only one model but many from which I need to choose after considering other diagnostic tools

Having that in mind, I would go ahead and say that the most obvious model seems to be ARMA (4,2) as ACF values die out at lag 4 and PACF shows spikes at 1 and 2.

Another way to analyze would be an ARMA(2,1) as I see two significant spikes in my PACF and one significant spike in my ACF (after which the values die out starting from a much lower point (0.4)).

Looking at my in-sample-forecast results (using a simple Mean Absolute Percentage Error) ARMA (2,1) delivers much better results then ARMA(4,2). So I use ARMA(2,1)!

Can you confirm my method and findings of analyzing ACF and PACF plots?

Help appreciated!

EDIT:

Descriptive Statistics:

count  252.000000
mean    29.576151
std      7.817171
min     -0.920000
25%     26.877500
50%     30.910000
75%     34.915000
max     47.430000

Skewness of endog_var: [-1.35798399]

Kurtsosis of endog_var: [ 5.4917757]

Augmented Dickey-Fuller Test for endog_var: (-3.76140904255411, 0.0033277703768345287, {'5%': -2.8696473721448728, '1%': -3.4487489051519011, '10%': -2.5710891239349585}

Time-Series:

enter image description here

Residuals (ARMA (2,1):

enter image description here

ACF/PACF of Residuals:

enter image description here

EDIT II:

Data:

14.37561
23.95561
25.41561
13.88561
23.31561
33.12561
35.30561
35.78561
37.21561
35.23561
37.34561
38.28561
39.03561
36.34561
39.08561
39.34561
38.80561
40.10561
34.13561
35.42561
27.29561
34.13561
39.89561
47.77561
40.57561
36.15561
33.66561
30.97561
24.90561
23.41561
0.31561
8.45561
37.36561
33.40561
13.97561
11.62561
35.07561
36.15561
37.09561
36.95561
37.85561
32.31561
35.41561
36.35561
37.34561
35.90561
37.40561
36.44561
37.37561
36.16561
35.24561
38.47561
39.18561
39.61561
29.55561
35.50561
38.05561
40.32561
44.39561
37.65561
46.27561
29.41561
40.41561
33.44561
37.04561
35.34561
25.24561
30.23561
15.40561
26.79561
35.38561
40.22561
43.14561
36.96561
41.93561
11.30561
6.87561
32.92561
34.54561
38.27561
36.40561
25.44561
37.26561
26.39561
31.13561
35.90561
38.41561
33.66561
33.16561
31.96561
30.34561
37.77561
32.25561
33.21561
38.37561
36.63561
40.78561
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31.49561
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37.34561
30.73561
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18.20561
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18.49561
25.41561
27.92561
29.42561
25.91561
27.56561
28.69561
29.89561
31.47561
29.34561
25.35561
21.98561
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33.87561
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10.77561
18.69561
30.19561
33.89561
29.81561
27.55561
22.37561
20.32561
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31.89561
32.10561
27.67561
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26.96561
21.27561
34.68561
34.13561
35.80561
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33.42561
9.28561
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30.29561
29.56561
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33.40561
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26.35561
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29.99561
28.87561
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14.07561
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24.92561
26.40561
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29.08561
30.18561
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16.15561
21.96561
32.29561
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26.53561
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29.76561
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33.05561
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36.88561
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27.00561
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35.45561
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26.35561
26.44561
28.72561
30.85561
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20.32561
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34.01561
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20.23561
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9.87561
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21.27561
26.22561
31.47561
31.11561
32.97561
32.34561
29.36561
32.40561
31.16561
32.05561
31.78561
32.34561
33.87561
31.80561
29.90561
30.09561
32.36561
28.15561
26.30561
15.32561
31.03561
33.47561
33.44561
33.71561
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35.38561
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41.37561
33.40561
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20.30561
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29.01561
32.36561
28.33561
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11.87561
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0.42561
29.29561
23.39561
19.36561
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  • $\begingroup$ Data look a bit left-skew, perhaps nonstationary. It looks to me like there's some potential issues with the residuals, perhaps even conditional heteroskedasticity. $\endgroup$ – Glen_b Jan 22 '15 at 16:13
  • $\begingroup$ In my opinion the skewness suggests anomalous values (pulses) which can only be confirmed by analysis of the original data. $\endgroup$ – IrishStat Jan 22 '15 at 21:05
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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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The sole reliance on the ACF and PACF using tools suggested in the mid 60's is sometimes but seldomly correct except for simulated data. Model Identification tools like AIC/BIC almost never correctly identify a useful model but rather show what happens when you don't read the small print regarding the assumptions. I would suggest that you start as simply as possible BUT not too simply and estimate a tentative model ; AR(1) as suggested by Glen_b . The residuals/analysis from this tentative model can be used to compute yet another ACF and PACF suggesting potential model augmentation or model simplification. Note that interpretation ala your references REQUIRE that the current series/residuals are free of any deterministic structure i.e. Pulses, Level Shifts, Local Time Trends and Seasonal Pulses and furthermore that the series has constant error variance and that the parameters of the tentative model are invariant over time. If you wish you can post your data and I will attempt to help you form a useful model.

EDIT AFTER DATA WAS REPORTED :

365 values were delivered and analyzed, yielding the following AR(1) model with identified Pulses and 2 Level Shifts .enter image description here . note that this had been a popular guess . The residuals from this model are plotted hereenter image description here . There is a suggestion of variance hetero-scedasticity but this is a symptom and one needs to find the correct cure which we will ultimately find. Proceeding the acf of the residuals shown here enter image description here exhibits a suggestion of model inadequacy. A closer look at the table of the acf of the residuals is here enter image description here suggesting structure at lags 7 and 14. Putting the the two clues together ( sample size of 365 and significant weekly i.e. lag 7 structure ) I decided to investigate whether or not this was indeed daily data. New users often omit very important information when they define their data on the mistaken premise that the computer should be smart enough to figure everything out. Note that the lag 7 and lag 14 clues were swamped in the OP'S ACF and PACF plots. The presence of deterministic structure in the residuals increase the error variance thus suppressing the acf. Once the outliers/pulses/level shifts were identified the acf revealed the presence of an autoregressive structure /daily indicators which then needed to be accounted for.

I then analyzed the data allowing the software to proceed with the clue that it was daily data. With only 365 values it is not possible to properly construct models containing seasonal/holiday predictors BUT that is possible with more than 1 year of data.

The model that was found is presented here enter image description here containing 5 daily dummies , two Level Shifts , a number of pulses and an arima model of the form (1,0,0)(1,0,0) . The plot of the residuals no longer evidences the non-constancy structure as a better model is in place.enter image description here . Thenter image description heree acf of the residuals is much cleaner . The Actual /Cleansed graph highlights the unusual pulse points. enter image description here . THe lesson here is that when one analyzed the data without the critical piece of information that it was a daily time series there were a ton of pulses reflecting an inadequate representation (or perhaps the advanced knowledge of the daily clue ) . The Actual/Fit and Forecast is presented hereenter image description here .

It would be interesting to see what others would do with the same data set. Note that all analyses were conducted in a hands-free mode using software that is commercially available.

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  • 1
    $\begingroup$ early morning mis read ... Don't normally see the lag(0) in my graphs $\endgroup$ – IrishStat Jan 22 '15 at 13:45
  • 1
    $\begingroup$ It tricked me at first as well. $\endgroup$ – Glen_b Jan 22 '15 at 13:48
  • $\begingroup$ Thanks for your answer. As someone without experience in the field of time-series forecasting it is hard to fully understand the procedure of choosing the right model as there is no officially right way to go. Unfortunately i am not allowed to post my raw data. I hope that the additional information is useful (see 'EDIT:') $\endgroup$ – Peter Knutsen Jan 22 '15 at 14:57
  • $\begingroup$ You can scale/mask your data before you present it. Looking at the plot it appears there might be some unusual values which if untreated downwards biases the acf and the pacf incorrectly suggesting sufficiency. There is a visual suggestion of a downwards trend followed by no trend but that is just a guess at this moment. $\endgroup$ – IrishStat Jan 22 '15 at 17:52
  • $\begingroup$ i just added some data which you might use.. $\endgroup$ – Peter Knutsen Jan 22 '15 at 22:58
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It looks to me like you're counting the spikes at lag 0.

Your PACF shows one reasonably large spike at lag 1, suggesting AR(1). This will of course induce a geometric-like decrease in the ACF (which, broadly speaking, you see). You seem to be trying to fit the same dependence twice - both as AR and MA.

I'd have just tried AR(1) on that to start with and seen if there was anything left worth worrying over.

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  • $\begingroup$ Peter; my answer had a typo in it (I had AR(1) correct in the last para, but typed MA(1) in the second paragraph), which is fixed now. $\endgroup$ – Glen_b Jan 22 '15 at 13:48
  • $\begingroup$ Thanks for your answer. Counting from lag 0 is of course a cardinal mistake! I tried AR(1) and the result was not as good as ARMA(2,1)! $\endgroup$ – Peter Knutsen Jan 22 '15 at 14:26
  • $\begingroup$ It may well be the case that it's not as good - nevertheless, the AR(1) would be the place to start. What did the PACF of residuals look like, for example? What does the original series look like? There's much that might be going on that can't be gleaned easily from an ACF and PACF of the data. $\endgroup$ – Glen_b Jan 22 '15 at 14:39
  • $\begingroup$ Thanks. I posted some additional information which might lead to new insights. $\endgroup$ – Peter Knutsen Jan 22 '15 at 15:02

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