Saturday, May 16, 2015

Intuition of Stationary Condition of a Time Series

ARIMA Forecasting Part-I

The  forecasting methods can be broadly divided into two categories: quantitative and qualitative methods. The quantitative forecasting literature is dominated by two sub- categories of methods: non-causal time series models and the causal econometric approaches. The difference between them is whether the forecasting model identifies any causal relationship between the dependent variable and its influencing factors. De Gooijer & Hyndman (2006) reviewed the past 25 years of research into time series forecasting and found one third of all papers concerned used time series forecasting. They have classified the papers according to the models as: exponential smoothing, ARIMA, Seasonality, State space and structural models and the Kalman filter, Nonlinear models, Long memory models, ARCH/GARCH models and Count data forecasting. Forecasting in the past four decades is dominated by the integrated autoregressive moving-average models (ARIMAs) proposed by Box and Jenkins (1970). Depending on the frequency of the time series, either simple ARIMA or seasonal ARIMA (i.e. SARIMA) models are mostly used.

Prior to understanding the ARIMA, following are few things to understand:

1. A basic understanding and intuition of stationary is required.
2. Understanding  of an Autoregressive process (AR) process.
3. Understanding of Moving Average (MA) process.
4. Understanding of mixed model of ARMA.

Basic intuition on Stationary property of a time series.

Here is a video in which, I tried to explain the stationary property of a time series in the least technical way just to shed the light or develop a basic intuition.


De Gooijer, J. G., & Hyndman, R. J. (2006). 25 Years of Time Series Forecasting. International Journal of Forecasting, 22(3), 443–473. doi:10.1016/j.ijforecast.2006.01.001

Box, G. E. P. and G. M. Jenkins (1970) Time series analysis: forecasting and control, Holden-   Day,          San Francisco.


  1. Fantastic articles is post by you in this blog. You give a nice thing. Thank you for such a nice article. Every word of this blog helps me to give detail to me.
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  2. Dear Shishir,

    Firstly, sorry if I am bothering you.
    My problem is I use Stata software for building Seasonal Arima model (sorry again because I don't use R software) and I don't know how to write the full model in equation from the Stata results.
    I show you the results below. Hope you can help me. Thank you in advance, Shishir!

    arima sxh_m, arima(1,0,0) sarima(1,0,1,12)
    ARIMA regression

    Sample: 2000m1 - 2014m12 Number of obs = 180
    Wald chi2(3) = 738.45
    Log likelihood = -959.4371 Prob > chi2 = 0.0000

    | OPG
    sxh_m | Coef. Std. Err. z P>|z| [95% Conf. Interval]
    sxh_m |
    _cons | 101.9032 51.05939 2.00 0.046 1.828592 201.9777
    ARMA |
    ar |
    L1. | .7795168 .0356272 21.88 0.000 .7096887 .8493449
    ARMA12 |
    ar |
    L1. | .9334173 .1179918 7.91 0.000 .7021577 1.164677
    ma |
    L1. | -.802917 .1693845 -4.74 0.000 -1.134905 -.4709294
    /sigma | 49.30807 1.341556 36.75 0.000 46.67866 51.93747
    Note: The test of the variance against zero is one sided, and the two-sided confidence interval is truncated at zero.

  3. Dear Nguyễn Đỗ Quốc Thống,
    Thank you for the query. Can you see my blog on ARIMA and SARIMA. I have the equations written there...

  4. I understood what stationery is, but I dont understand why we need a stationery series for time series forecasting. Can you explain? Thanks.

  5. Stationary makes data unit root free. Well, for the forecasting stationary may/maynot require it depends upon your requirements. I am for sure for proper forecasting you should be able to predict out of sample data.