This document discusses sales forecasting for an airline using time series modeling. It describes preparing the data by checking for volatility, non-stationarity, and seasonality. Several time series models are identified and compared using information criteria. The best model is found to be ARIMA(0,1,3) based on lowest MAPE error. Forecasts are generated for the next 12 months and graphically represented along with the actual historical sales values.

1. Sales Forecasting for Airline
Submitted By:
Anurag Shandilya
Ankur Khandelwal
Pullahbhatla Auproop
Srikanth Mallya
Shekhar Sinha
Soham Mukhopadhyay

2. Agenda
Introduction
Objective
Data Preparation
Modeling identification and estimates
Forecasting
Graphical Forecast

3. Introduction
 Time series relates to the values taken by the variable over
time and tabulated or plotted as chronologically ordered
numbers or data points to yield valid statistical inferences.
Components Description
Trend
Smooth long term movements for long period of time
Data moves steadily in one particular direction with little
fluctuation
Seasonal Variations Periodic movements with period of cycle <=1 year
Cyclic Movement Periodic movements with cycle >1year
Irregular Movement Random erratic movements

4. Objective
To project the airline travel for the next 12 months.
About the dataset:
The dataset contains two variables: DATE and AIR.
 DATE: contains sorted SAS date values recorded from Jan
1949 to Dec 1960.
 AIR: contains the sales value in that month

5. Data Preparation
1
• Check for Volatility
2
• Check for Non-Stationarity
3
• Check for Seasonality

6. Check for Volatility
 The plot of the data with time on horizontal axis and time
series on vertical axis provides and indication for volatility.
 A fan shaped or an inverted fan shaped plot shows high
volatility.
 For fan shaped plot , ‘log’ or square root transformation is
used to reduce volatility, while for inverted fan shaped plot,
‘exponential’ or ‘square’ transformation is used.

7. After log transformation, with reduced volatility(constant
variance)

8. Check for Non-Stationarity
• A non stationary data is completely memory less with no
fixed patterns. Such a data can’t be used for forecasting
• Augmented Dickey Fuller Test (ADF) used to check the non-
stationarity of data.
• Non-Stationarity can be removed by differencing.
• Data was found to be non-stationary & hence, differencing
of log transformed data was done to make data stationary.

9. Check for seasonality:
 The Auto Correlation function (ACF) gives the correlation
between y[t]-y[t-s] where ‘s’ is the period of lag.
 If ACF gives high values at fixed interval that interval can be
considered as the period of seasonality. A differencing of
same order will depersonalize the data.
 From the output of ACF it can be observed that the period
of seasonality is 12 years.

10. Model Identification and estimation
 Depending upon the number of future time points to be forecasted
, we set aside few of the most recent time points as the validation
sample.
 The rest of the data which is development sample, is used to
generate forecasts for the different models.
 MINIC(Minimum Information Criteria) generate the minimum
BIC(Bayesian Information Criteria) model after exploring all the
possible combinations of Auto Regressive and Moving Average lags
from 0 to 5.

11. Model Identification and estimation
MA MA0 MA1 MA2 MA3 MA4 MA5
AR0 -6.24946 -6.32141 -6.30667 -6.32826 -6.29985 -6.27302
AR1 -6.33466 -6.29602 -6.28026 -6.29538 -6.26474 -6.23915
AR2 -6.32028 -6.28278 -6.25595 -6.25609 -6.22534 -6.2063
AR3 -6.3503 -6.3141 -6.27621 -6.24523 -6.24194 -6.22247
AR4 -6.33057 -6.29054 -6.25187 -6.25848 -6.21998 -6.1958
AR5 -6.30796 -6.26784 -6.22782 -6.23527 -6.19898 -6.165
Minimum Information Criteria

12. Model Identification and estimation
 By observation we can see that minimum of the matrix is
the value -6.3503 corresponding to AR3 and MA 0
location(i.e. p=0 & =3).
 We consider all the models in the neighborhood of this
model and for each of them generate AIC(Akaike
Information Criteria) and SBC (Schwartz Bayesian Criteria)
and calculate and average of them.
 We select top 6-7 models based on relatively lower value of
the average and for each of them generate forecasts.

13. Forecasting
The forecasts generated (for the year 1960) for each of the 6
combination selected from AIC & SBC separately compared
with the actual values of the same time point stored in the
dataset.
‘MAPE’ (Mean Absolute Percentage Error) is calculated for 6
forecasted values for the year 1960.
Lowest MAPE value comes out to be for p=0 and q=3, hence
final forecasting will be done using this model.

14. Final Forecasted Values
Time Period Forecasted Value
Jan-61 428.4602827
Feb-61 406.2629999
Mar-61 478.50675
Apr-61 467.1975448
May-61 496.0186286
Jun-61 557.9997783
Jul-61 648.5088651
Aug-61 662.2020145
Sep-61 549.0387684
Oct-61 483.1252776
Nov-61 430.1473514
Dec-61 481.7337089

15. Graphical Representation
0
100
200
300
400
500
600
700
800
Jan-49
Jun-49
Nov-49
Apr-50
Sep-50
Feb-51
Jul-51
Dec-51
May-52
Oct-52
Mar-53
Aug-53
Jan-54
Jun-54
Nov-54
Apr-55
Sep-55
Feb-56
Jul-56
Dec-56
May-57
Oct-57
Mar-58
Aug-58
Jan-59
Jun-59
Nov-59
Apr-60
Sep-60
Feb-61
Jul-61
Dec-61
Actual Sales
Values
Forecasted Sales
Values
SalesValues

16. Appendix
AIC, SBC, MAP excel sheet is attached in mail
“SAS code for forecasting”