# Issues on analyzing a time series, ACF and PACF quite the same

I'm attending a econometric course and I'm new to time series analysis. They gave me a time series to analyze and I'm trying to apply all the things I learnt (and understood) to an actual time series.

The time series is about the Harmonised Index of Consumer Prices in Austria from Janaury 2000 to September 2017, with base=100 2015 get it from Eurostat.

I provide the dataset so you can reply it:

x<-ts(c(74.28,74.85,74.92,74.78,74.71,75.00,74.78,74.85,75.14,75.35,75.64,75.93,75.96,76.21,76.33,76.71,76.86,76.96,76.85,76.66,76.95,77.10,77.07,77.27,77.40,77.53,77.63,77.98,78.12,78.11,77.98,78.19,78.19,78.36,78.36,78.57,78.71,78.85,79.06,78.98,78.88,78.89,78.79,79.01,79.22,79.20,79.40,79.62,79.67,80.02,80.34,80.24,80.53,80.68,80.44,80.75,80.68,81.06,81.22,81.62,81.60,81.87,82.26,82.08,82.09,82.30,82.13,82.26,82.73,82.77,82.63,82.87,82.84,83.12,83.34,83.82,83.83,83.84,83.71,83.97,83.85,83.80,83.91,84.22,84.28,84.54,84.94,85.30,85.45,85.45,85.39,85.42,85.63,86.20,86.61,87.14,86.89,87.16,87.95,88.18,88.66,88.90,88.64,88.48,88.82,88.79,88.56,88.42,87.90,88.33,88.49,88.65,88.71,88.62,88.26,88.62,88.80,88.90,89.08,89.35,88.98,89.16,90.09,90.28,90.23,90.22,89.76,90.00,90.35,90.68,90.68,91.30,91.24,91.97,93.09,93.64,93.56,93.53,93.19,93.34,93.90,94.11,94.17,94.40,93.88,94.38,95.55,95.82,95.63,95.62,95.13,95.47,96.55,96.85,96.92,97.15,96.54,96.87,97.80,97.82,97.94,97.77,97.15,97.37,98.31,98.30,98.33,99.08,98.01,98.30,99.22,99.35,99.44,99.43,98.77,98.79,99.69,99.70,99.84,99.88,98.48,98.75,100.15,100.28,100.45,100.40,99.89,99.73,100.25,100.35,100.32,100.95,99.85,99.76,100.82,100.93,101.04,101.04,100.49,100.33,101.33,101.72,101.79,102.55,101.91,102.13,102.94,103.27,103.21,103.06,102.47,102.46,103.93),frequency = 12)


Usually I use Gretl, but in this case I'm gonna use R cause I found easier to explain the code.

I follow the Box and Jenkins procedure. The first things I do is to visualize the time series. In this case, it's:

Firstly I wonder if there any kind of outliers (for example Addivite, Level shift, Innovation Outliers of Temporary change), that's not the case, I suppose. Obsviously it's not mean stationary and I have some doubts on the variance of the series. So I'm gonna consider a log-transformation and a difference of order 1.

d_log_x<-diff(log(x))


Now I have the time series with no trend. Now I should see the error part and the seasonality. The latter shows that seasonaliry increases during time:

At this point I remember that if the case that seasonality is evolutive, we prefer to consider a multiplicative model for time series. Anyway I'm gonna check now the periodgram of this series and that's what I get:

This should show me (if I'm not wrong) that we have a seasonaly every 6 months and 3 months.

Now I think of using a season differences, but which? I behave as follows: I see the standard deviations of the series with seasonal differences and I choose the lowest:

dif<-as.vector(c())
for(i in 1:48){dif<-rbind(dif, sd(diff(d_log_x,lag=i)))}
[1] 12 36 24 48  6 18


So seasonal difference with order 12 gaves me the lowest sd.

sddlog<-diff(d_log_x, lag=12)
plot(sddlog)


Now I can use acf and pacf to detect the best model, here they are:

Now my question are:

• I don't know how to interpret it. I mean I would say that's only a seasonal arma ( but didn't I erase it?) and acf and pacf are so similar. I see both of them (seasonally speaking) with significant value at 12 lag and 24th lag (acf is not significant but almost it is) and then everything disappears. I would say it's a ARIMA(0,0)(0,1), but the PACF should not die so brutally, it should go to zero slowly.
• Maybe should I use another kind of filter to eliminate the seasonality?
• Is it proper, maybe, to use a dummy variable?

i took your 213 monthly values into AUTOBOX , a piece of software that I have helped to develop. It suggested that there was a significant change in parameters at or around period 126. The final model is here containing both memory and dummy indicators. A level shift is in general an intercept change. The stats are here and here . The Actual/cleansed graph is informative while the ACTUAL/Fit/Forecast is a nice summary with a plot of the model residuals here supporting sufficiency. The plot of the forecasts is here suggesting increasing uncertainty going forward .

The ACF,of the residuals is here

In summary your approaches were interesting and it appears that between uour course and this you have experienced quite a bit.

BY the way the CHOW test for constancy of parameters is here

If you look closely at the plot of the original series you can visually detect what AUTOBOX discovered is that there appears to be either a structural change in parameters or a structural change in error variance (more visual variability a t a higher level. This linkage between the variance anD the level of the series CAN often be mis-interpreted as justification for a Box-Cox log transformation
When (and why) should you take the log of a distribution (of numbers)? .

• Could you make it clearer how this post answers the questions? – whuber Nov 15 '17 at 20:50
• It points out that a log transform is not needed and there is a need for the "dummies" that he speculated about. So he was certainly in the right track about a number of things and wrong in another..Why don't we see what the OP says or does with my response. – IrishStat Nov 15 '17 at 20:55
• I just looked through your post again and I still can't see where those points are made! – whuber Nov 15 '17 at 20:57
• Thank you Irish State, but what you wrote is very new to me. I think CPA has to do with Change Point Analysis, I've never heard of it. You said your program (AUTOBOX) found an outlier, in particular a level shift. I've studied it but our approach was different. So if you say that my time series has a level shift outlier I think I'm gonna build a regression model against a dummy variable where this latter has 1s after 126. – Mario Migliaccio Nov 15 '17 at 23:05
• But I was interested in the way I approached to the solutions, about differencing either seasonal and non-seasonal and then looking at the acf and pacf and spectral analysis. – Mario Migliaccio Nov 15 '17 at 23:13