# ARIMA forecast with seasonality and trend, strange result

as I am stepping into forecasting with ARIMA models, I am trying to understand how I can improve a forecast based on ARIMA fit with seasonality and drift.

My data is the following time series ( over 3 years, with clear trend upwards and visible seasonality, which seems to be not supported by autocorrelation at lags 12, 24, 36??).

    > bal2sum3years.ts
Jan     Feb     Mar     Apr     May     Jun     Jul     Aug
2010 2540346 2139440 2218652 2176167 2287778 1861061 2000102 2560729
2011 3119573 2704986 2594432 2362869 2509506 2434504 2680088 2689888
2012 3619060 3204588 2800260 2973428 2737696 2744716 3043868 2867416
Sep     Oct     Nov     Dec
2010 2232261 2394644 2468479 2816287
2011 2480940 2699780 2760268 3206372
2012 2951516 3119176 3032960 3738256


The model that was suggested by auto.arima(bal2sum3years.ts) gave me the following model:

    Series: bal2sum3years.ts
ARIMA(0,0,0)(0,1,0)[12] with drift

Coefficients:
drift
31725.567
s.e.   2651.693

sigma^2 estimated as 2.43e+10:  log likelihood=-321.02
AIC=646.04   AICc=646.61   BIC=648.39


However, the acf(bal2sum3years.ts,max.lag=35) does not show acf coefficients higher than 0.3. The seasonality of the data is, however, pretty obvious - spike at the beginning of every year. This is what the series looks like on the graph:

The forecast using fit=Arima(bal2sum3years.ts,seasonal=list(order=c(0,1,0),period=12),include.drift=TRUE) , called by function forecast(fit), results in the next 12months's means being equal to the last 12 months of the data plus constant. This can be seen by calling plot(forecast(fit)),

I have also checked the residuals, which are not autocorrelated but have positive mean ( non zero).

The fit does not model the original time series precisely, in my opinion ( blue the original time series, red is the fitted(fit):

The guestion is, is the model incorrect? Am I missing something? How can I improve the model? It seems that the model literally takes the last 12 months and adds a constant to achieve the next 12 months.

I am a relative beginner in time series forecasting models and statistics.

• "The fit does not model the original time series precisely, in my opinion" -- why would you expect it to?? – Glen_b Apr 18 '13 at 7:19
• @Glen_b, this opinion was based on the differences that I see when I look at the plot. If I am trying to forecast, for instance, monthly sales for accounting purposes, the error could be significant... – zima Apr 18 '13 at 15:27
• "the differences I see when I look at the plot" is another way of saying "does not model the time series precisely". This is not in dispute. Your expression of a desire for a better forecast is the same desire every forecaster has. In many cases it can be very important. Nevertheless, this desire doesn't put more information into the data. Every ARIMA model - indeed, any time series model of relevance to this task - has a nonzero error term. There will always be mismatch between data and fit. Is there something that makes you think your model has missed something that can be modeled? – Glen_b Apr 18 '13 at 21:37
• I have just thought about something.. Maybe ARIMA model is indeed not able to reflect the data due to not taking into account the nature of the data - user activity on the website. I think there might be other events affecting the numbers, not just seasonality - such as special events, promotions.. Maybe other prediction methods (not ARIMA), but more complex ones involving Machine Learning techniques, are able to better predict the values. I will look into that. – zima Apr 25 '13 at 8:56
• Quite plausible. If so, you should be able to identify such failure in the residuals. Note that both ARIMA models and structural time series models can incorporate things like special events and promotions via regression terms; time series regression models are fairly common. – Glen_b Apr 25 '13 at 9:09

From the appearance of your data, after seasonal differencing, there may well be no substantive remaining seasonality. That peak at the start of each year, and the subsequent pattern through the rest of the year is quite well picked up by an $I_{[12]}$ model; the model has incorporated "obvious seasonality".