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Home » SAS Statistics » SAS: Time Series Forecasting - ARIMA


SAS: TIME SERIES FORECASTING - ARIMA

Rajat 1 Comment SAS Statistics, Time Series

In this tutorial, we will cover how to perform ARIMA with SAS, along with an
explanation of how it works.

Hope you have gone through the Part-1 of this series

Table of Contents
 1. Data Preparation Steps For ARIMA Modeling
    1. Step 1 : Check the time series
    2. Step 2 : Check the volatility of the series
    3. Step 3 : Treatment of Volatile Series
    4. Step 4 : Check For Non-Stationarity
    5. Step 5 : Make Non-Stationary Data Stationary
    6. Step 6 : Check Seasonality
    7. Step 7 : Split Data into Training and Validation

--------------------------------------------------------------------------------


DATA PREPARATION STEPS FOR ARIMA MODELING

 1. Check if there is variance that changes with time - Volatility. For ARIMA,
    the volatility should not be very high.
 2. If the volatility is very high, we need to make it non-volatile.
 3. Check for Stationary - a series should be stationary before performing
    ARIMA.
 4. If data is non-stationary, we need to make it stationary.
 5. Check for Seasonality in the data

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> Library - SASHELP
> Data set - AIR


STEP 1 : CHECK THE TIME SERIES

As a matter of practice, we first plot the time series and have a cursory look
upon it. It can be done directly in SAS using following code :

> proc sgplot data = sashelp.AIR;
> series x = date Y = AIR;
> run;
> quit;

It would give you the following plot in the result window :

SAS : Time Series Modeling



It is clear from the chart above that the series of AIR is having an increasing
trend and consistent pattern over time. The peaks are at a constant time
interval which is indicative of presence of seasonality in the series.

> This is a non-stationary series for sure and hence we need to make it
> stationary first.

Practically, ARIMA works well in case of such types of series with a clear trend
and seasonality. We first separate and capture the trend and seasonality
component off the time-series and we are left with a series i.e. stationary.
This stationary series is forecasted using ARIMA and then final forecasting
incorporates the pre-captured trend and seasonality.


We would understand it in details further in Step-3.



STEP 2 : CHECK THE VOLATILITY OF THE SERIES

Volatility is the degree of variation of a time-series over time. For ARIMA, the
volatility should not be very high. For checking the volatility of time-series,
we do a scatter plot using the following SAS code :

> Proc gplot data=SAShelp.AIR;
> plot Date * AIR;
> Run;Quit;

It would give you the following plot in the result window :

Check the volatility of Series

The highlighted area is showing the diverging pattern (Fan shaped) of the
scatter plot and hence depicting that the data is volatile. Ideally, the
highlighted pattern should be parallel for ARIMA modeling.


STEP 3 : TREATMENT OF VOLATILE SERIES

We need to make the series non-volatile and move ahead. We would transform the
AIR series and remove volatility. Generally a hit and trail method for
transformation is used, but we would suggest to not to waste your time.

Box-Cox Transformation can be used to help you out and recommend the suitable
transformation.

> Proc Transreg Data = sashelp.AIR;
> Model BOXCOX (AIR) = Identity(Date);
> Run;

You get following plot along with Lamba value, which is "0" in this case.



Now based on this Lambda value, you can decide the transformation. Take help
from the table provided below.


Box cox Transformation

In our case, it is suggesting a log transformation, so we do the same. In a new
data (Masterdata) we create a new variable (Log_AIR).

Data Masterdata;
Set SAShelp.AIR;
Log_AIR = log(AIR);
Run;


We can check the volatility again of the transformed series, just to be sure,
using scatter plot as elaborated above.


STEP 4 : CHECK FOR NON-STATIONARITY

Now on the transformed series, we check whether the series is stationary or
non-stationary. For performing ARIMA , a series should be stationary, however if
the series is non-stationary, we make it stationary (For more explanation on
stationarity, read Part 1 of this series).
Rather than identifying the series's stationarity visually as we have done in
step 1, we now use Augmented Dickey-Fuller Unit Ratio Test for the same.
Unit Root - Homogeneous Non-Stationarity Data
Dickey-Fuller test

> The Dickey-Fuller test is used to test the null hypothesis that the time
> series exhibits a lag d unit root against the alternative of stationarity.

Null Hypothesis : Non-Stationary
Alternative Hypothesis : Stationary

There are three types by which you can calculate test statistics of
dickey-fuller test.


 1. Zero Mean - No Intercept. Series is a random walk without drift.
 2. Single Mean - Includes Intercept. Series is a random walk with drift.
 3. Trend - Includes Intercept and Trend. Series is a random walk with linear
    trend.

> All the above test statistics are computed from the OLS regression model.


Drawback of ADF Test
Uncertainty about what test version to use, i.e. about including the intercept
and time trend terms. Inappropriate exclusion or inclusion of these terms
substantially affects test reliability.


> Using of prior knowledge (for instance, as result of visual inspection of a
> given time series) about whether the intercept and time trend should be
> included is the mostly recommended way to overcome the difficulty mentioned.

We run Proc ARIMA with Stationarity = (ADF) option to do so :

> PROC ARIMA DATA= Masterdata ;
> IDENTIFY VAR = log_Air STATIONARITY= (ADF) ;
> RUN;
> QUIT;

There are many outputs of the above code, a part of which is used for checking
stationarity:

ARIMA : Check Stationary

Important Note :

> Check Tau Statistics (Pr < Tau) in ADF Unit Root Tests table. It should be
> less than 0.05 to say data is stationary at 5% level of significance.


STEP 5 : MAKE NON-STATIONARY DATA STATIONARY

Post establishing the non-stationarity of the series, we need to make the series
stationary. Differencing process is used for making the series stationary.

> Differencing : Transformation of the series to a new time series where the
> values are the differences between consecutive values


Differencing Procedure may be applied consecutively more than once, giving rise
to the "first differences", "second differences", etc.

Differencing Orders :

1st order : ∇xt = xt - xt-1

2nd order : ∇2xt = (∇xt - ∇xt-1) = xt - 2xt-1 + xt-2

> It is unlikely that more than two differencing orders would ever be required.

Note : If there is a physical explanation for a trend or seasonal cycle : use
regression to make series stationary.


For that we use the output of the Step-3 code itself. While we have run the code
above, we have got "Autocorrelation Check for White Noise" along with "
Augmented Dickey-Fuller Unit Root Tests". Looking at "Autocorrelation Check for
White Noise", we decide the order(s) of differencing required.

Stationary : Order of Differencing

A heat map has been made using Excel for demonstration, SAS output is black and
white only.
The first row of the above autocorrelation matrix shows correlation of
time-series with 1st to 6th lags, second row show the same for 7th to 12th
lags...and so on ... The same is visible in ACF chart provided in Step-3
visuals.

> We can see that in above matrix the highest auto-correlation exists with 1st
> lag, it starts decreasing but again increases to attain a local peak at 12th
> lag.


STEP 6 : CHECK SEASONALITY

Highest Correlation with 1st Lag indicates towards the presence of trend and
that with 12th lag indicates an annual seasonality. Hence we need to do
differencing at first and Twelfths orders. We perform differencing and check the
stationarity again.

> PROC ARIMA DATA= masterdata ;
> IDENTIFY VAR = Log_Air (1,12) STATIONARITY= (ADF) ;
> RUN;quit;

We have used 1 and 12 in bracket to define the 1st and 12th order of
differencing.

Check whether data is stationary


> Check Tau Statistics (Pr < Tau) in ADF Unit Root Tests table again and see if
> the value <0.05 to say data is stationary at 5% level of significance.


How this differencing actually worked :
1. First order (1) Differencing removes the trend, but Seasonality still exists.
2. Second Order (12) Differencing removes the seasonality.


How to do it with MS Excel:

> First subtract first lag from each observation and plot it. Then in this new
> series subtract 12th lag from each observation.


STEP 7 : SPLIT DATA INTO TRAINING AND VALIDATION

Now we can break the data into Training and Validation samples.We cannot use
random sampling like we do in regression models to split the data. Instead, we
can use recent data for validation and remaining data be used to train the
model. We would develop ARIMA model and forecast on Testing part and would check
the results on Validation part.

> Data Training Validation;
> Set Masterdata;
> If date >= '01Jan1960'd then output Validation;
> Else output Training;
> Run;

Next Step - Follow Part 3 of this series to learn how to train ARIMA model on a
training dataset using SAS.

> This article was originally written by Rajat Agarwal, later Deepanshu gave
> final touch to the post. Rajat is an analytics professional with more than 8
> years of work experience in diverse business domains. He has gained expert
> knowledge in Excel and SAS. He loves to create innovative and imaginative
> dashboards with Excel. He is founder and lead author cum editor at Ask
> Analytics.


Related Posts
 * How to Calculate Correlation in SAS (with Examples)
 * SAS : Calculate AUC of Validation Data
 * SAS: Time Series Forecasting - ARIMA [Part 3]
 * How to Build a Random Forest Model in SAS
 * How to Build a Decision Tree in SAS

SAS Tutorials : Top 100 SAS Tutorials
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1 Response to "SAS: Time Series Forecasting - ARIMA"
 1. AnonymousJuly 24, 2017 at 6:22 AM
    
    despite doing everything - using MINIC, my autocorrelation is still
    significant, what should I do
    
    Autocorrelation Check of Residuals
    
    To Chi- Pr >
    Lag Square DF ChiSq --------------------Autocorrelations--------------------
    
    6 19.46 4 0.0006 -0.045 -0.094 0.282 -0.178 0.079 0.279
    12 41.37 10 <.0001 -0.260 0.024 0.311 -0.199 0.005 -0.121
    18 64.20 16 <.0001 -0.275 0.212 -0.075 -0.193 0.207 -0.072
    24 92.59 22 <.0001 -0.125 0.151 -0.180 -0.168 0.272 -0.250
    
    ReplyDelete
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