ASHISH RANJAN

ANIRBAN GHOSH

AYAN DAS

MADHUMITA GHOSH

SOMDEEP SEN

Data Description

The dataset contains ten years sales of an airlines industry ; from 1949 to 1960

Monthly breakup of each year sales has been provided in thousands as ‘AIR’ variable

Objective

To predict the sales of the year 1961 through Time Series Analysis using SAS

Time Series relates to data varying over a period of time

Time Series generally includes four components

Components Description

Trend•Smooth long term movements for long period of time

•Data moves steadily in one particular direction with little fluctuation

Seasonality •Periodic movements with period of cycle <=1 year

Cyclicality •Periodic movements with cycle >1 year

Irregularity •Random erratic movements

Volatility Check

Non Stationarity Check

Check for Seasonality

Creation of Development & Validation Sample

Selection of P & Q

Generating final Forecast

Plot created by gplot option with time & sales provides an indication

A Japanese fan shaped or an inverted fan shaped plot are indicators of high volatility

For fan shaped plot we use log or square root

For inverted fan shape we use square or exponential

During Analysis:

The initial graph was fan shaped & hence log/square root was used for transformation

Among the two log provided a better result & hence it was chosen

A non stationary data is completely memory less with no fixed patterns

Such a data can’t be used for forecasting

Non-stationary is checked by using Augmented Dickey Fuller Test (ADF)

Here the null hypothesis(H0) is that the data is non stationary

If the P-value<α we reject H0 to claim that the data is stationary

If the P-value>α we can’t reject H0 to claim that the data is non-stationary

Such data can be converted to Stationarity by differencing

During Analysis:

Initial check using ADF showed non-Stationarity

Therefore differencing was used to convert the data in to a stationary one

Note: differencing was used for the log of the variables

Autocorrelation function gives the correlation between Y(t) & Y(t-s); S is the period of lag

If ACF gives high values at fixed interval, then it can be considered as period of seasonality

A differencing of same order would de- seasonalize the data

During Analysis:

It was found that ACF gave high values at fixed intervals of 12 (so, S=12)

Hence differencing was done at an interval of 12

Note: differencing was used for the log of the variables

Depending upon the no. of future time point to forecast some time points are set aside

These data are the validation sample; the rest of are called the development sample

The development sample is used to generate forecast for different models

During Analysis:

The development & validation sample created was named as D & S respectively

•AIC: Akaike information criterion

•BIC: Bayesian information criterion

•SBC: Schwarz criterion

The ‘minic’ function under proc ARIMA gives the minimum BIC model

All possible combinations of P&Q from 0-5 are explored

For each AIC & SBC are generated & the corresponding averages are calculated

Out of that 5-6 models based on the relative lower value of the average are selected

For each of them separate forecasts are generated

During Analysis:

The (3,0) combination gives the minimum value of BIC

Therefore all the possible 15 combinations except (0,0) are considered

Among those 6 combinations providing relative lower avg. of AIC & SBC were used for forecasting

Please the link to view avg. of AIC & SBC: http://bit.ly/1oZ1R4F

Forecasts are generated from each combinations of AIC & SBC

These are separately compared with actual values of same time using MAPE

The combination having minimum MAPE is selected

During Analysis:

MAPE was found to be minimum for (0,3)

Final forecast was done for that combination

Note: Here it needs to mentioned that before making the final forecasting both

the actual and predicted values were converted to original form using

exponential

Please the link to view min MAPE details: http://bit.ly/1oZ1R4F

Date Sales (In thousands)

Jan-61 444.82

Feb-61 420.47

Mar-61 453.76

Apr-61 499.38

May-61 511.44

Jun-61 579.87

Jul-61 674.35

Aug-61 657.19

Sep-61 551.07

Oct-61 500.22

Nov-61 423.30

Dec-61 469.02

0.00

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800.00

Jan

-49

May

-49

Sep

-49

Jan

-50

May

-50

Sep

-50

Jan

-51

May

-51

Sep

-51

Jan

-52

May

-52

Sep

-52

Jan

-53

May

-53

Sep

-53

Jan

-54

May

-54

Sep

-54

Jan

-55

May

-55

Sep

-55

Jan

-56

May

-56

Sep

-56

Jan

-57

May

-57

Sep

-57

Jan

-58

May

-58

Sep

-58

Jan

-59

May

-59

Sep

-59

Jan

-60

May

-60

Sep

-60

Jan

-61

May

-61

Sep

-61

SALE

S in

Th

ou

san

ds

Date

Actual/Forecasted

Actual Forecasted