forecastmethods_techniques

7/25/2019 ForecastMETHODS_TECHNIQUES

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All Rights Reserved, Indian


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Those who have knowledge dont predict.

Those who predict dont have knowledge.


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I think there is a world market for may be 5 computer

- Thomas Watson, Chairman of IBM 19

Computers in future weigh no more than 1.5 tons

- Popular Mechanics, 1949

640K ought to be enough for everybody

- Bill Gates, 1981???


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Forecasting

Forecasting is a process of estimation of an unknown evensuch as demand for a product.

Forecasting is commonly used to refer time series data.

Time series is a sequence of data points measured at sucintervals.


5/116 All Rights Reserved, Indian

Corporate

Strategy

Product and Market

Planning

Business

Forecasting

Product and Market Strate

Aggregate Production

Planning

Master ProductionPlanning

Materials

Requirement

Planning

Aggregate

Forecasting

ItemForecasting

Demand

forecasting at SKU

Level

Resource Planning -

Medium to Long Range

Planning

Production Capacity - ShoRange Planning

Capacity Requirement

Planning



Forecasting methods

Qualitative Techniques.

Expert opinion (or Astrologers)

Quantitative Techniques.

Time series techniques such as exponential smoothing

Casual Models.

Uses information about relationship between system elements

(e.g regression).



Why Time Series Analysis ?

Time series analysis helps to identify and explain:

Any systemic variation in the series of data which is due to

Cyclical pattern that repeat.

Trends in the data. Growth rates in the trends.



Time Series Components

Trend Cyclical

Seasonal Irregular



Trend Component

Persistent upward or downward pattern

Due to consumer behaviour, population, economy, techno


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Cyclical Component

Repeating up and down movements.

Due to interaction of factors influencing economy such

Usually 2-10 years duration.


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Seasonal Component

Regular pattern of up and down movements.

Due to weather, customs, festivals etc.

Occurs within one year.


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Seasonal Vs Cyclical

When a cyclical pattern in the data has a period of lesyear, it is referred as seasonal variation.

When the cyclical pattern has a period of more than orefer to it as cyclical variation.


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Irregular Component

Erratic fluctuations

Due to random variation or unforeseen events

White Noise


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Time(a) Trend

Time(d) Trend with seasonal pattern

Time(c) Seasonal pattern

Time(b) Cycle

Demand

Demand

Dem

and

Demand

Randommovement

More

than

one

year gap

Less

than

one

year gap


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Time Series DataAdditive and Multiplicati

Random

t

Cyclical

t

ySeasonalit

t

Trend

t RCST tY

ModelgForecastinAdditive1.

Random

t

Cyclical

t

ySeasonalit

t

Trend

t RCST tY

ModelgForecastintiveMultiplica2.


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Time Series Data Decomposition

ySeasonalit

t

Trend

t ST tY

ModelgForecastintiveMultiplica

t

tt

T

YS

Yt / Tt is called

deseasonalized data


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Time Series Techniques

Moving Average.

Exponential Smoothing.

Auto-regression Models (AR Models).

ARIMA (Auto-regressive Integrated Moving Averag


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Moving Average Method


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Moving Average (Rolling Average)

Simple moving average. Used mainly to capture trend and smooth short term

Most recent data are given equal weights.

Weighted moving average Uses unequal weights for data


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Data

Demand for continental breakfast at the Die Another Da

Daily data between 1 October 201423 January 2015 (


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Simple moving average

iperiodtimetoingcorrespondDataY

1tperiodforForecastF

Yn

1F

i

1t

t

1nti

i1t

The forecast for period t+1 (Ft+1

) is given by the average of

recent data.


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Simple moving average

iperiodtimetoingcorrespondDataY

1tperiodforForecastF

Yn

1F

i

1t

t

1nti

i1t

The forecast for period t+1 (Ft+1

) is given by the average of

recent data.


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Demand for Continental Breakfast at DAD Hosp

Moving Average

0

10

20

30

40

50

60

Actual Demand Forecast

t

17ti

i1t Y7

1F


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Measures of aggregate error

Mean absolute error MAE

Mean absolute percentage

error MAPE

Mean squared error MSE

Root mean squared error

RMSE

n

ttE

nMAE

1

1

n

t t

t

Y

E

nMAPE

1

1

n

ttE

nMSE

1

21

n

ttE

nRMSE

1

21

Et = Ft - Yt


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DAD ForecastingMAPE and RMS


error MAPE

0.1068 or 10.68%


RMSE

5.8199


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Exponential Smoothing

Form of Weighted Moving Average Weights decline exponentially.

Largest weight is given to the present observation, le

immediately preceding observation and so on.

Requires smoothing constant ()

Ranges from 0 to 1


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Next forecast = (present actual value) + (1-) prese


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Simple Exponential Smoothing Equati

Smoothing Equations

11

1 *)1(*

YF

FYF ttt

Ft+1 is the forecasted value at time t+1


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Simple Exponential Smoothing Equations

Smoothing Equations

....)1()1( 32

21 tttt YYYF


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Exponential Smoothing Forecast

0

10

20

30

40

50

60

18-09-2014 08-10-2014 28-10-2014 17-11-2014 07-12-2014 27-12-2014 16-01-2015 05-02-20

Demand Exponential Smoothing Forecast

ttt FYF *)8098.01(*8098.01


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DAD ForecastingMAPE and RMSE



error MAPE

0.0906 or 9.06%


RMSE

5.3806

= 0.8098


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Choice of

Forsmooth data, try a high value of a, forecast responsiv

current data.

Fornoisy data try low a forecast more stableless respon


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Optimal

11

1

2

)1(

/

ttt

n

t

tt

FYF

nFYMin


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Double exponential Smoothing Holts m

Simple exponential smoothing may produce consistently biasthe presence of a trend.

Holt's method (double exponential smoothing) is appropriate has a trend but no seasonality.

Systematic component of demand = Level + Trend


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Holts Method

Holts Equations

Forecast Equation

11

11

)1()()(

)()1()(

tttt

tttt

TLLTii

TLYLi

ttt TLF

1

Equation for

intercept or

level

Equation for

Slope (Trend)

i i l l f d


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Initial values of Lt and Tt

L1is in general set to Y1.

T1can be set to any one of the following values (or use regression to ge

)1/()(

3/)()()()(

11

3423121

121

nYYT

YYYYYYTYYT

n


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0

10

20

30

40

50

60

0 20 40 60 80 100 120

Demand Forecast

Double exponential smoothing

= 0 8098; = 0 05

DAD Forecasting MAPE and RMSE


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DAD ForecastingMAPE and RMSE

Double Exponential Smoothing


error MAPE

0.0930 or 9.3%


RMSE

5.5052

= 0.8098; = 0.05


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Forecasting Accuracy


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Forecasting Accuracy

The forecast error is the difference between the forecast value and the a

the corresponding period.

tperiodfor timeForecastF

tperiodat timevalueActualY

tperiodaterrorForecastE

FYE

t

t

t

ttt


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Forecasting Power of a Model

Theils coefficient (U Statistic)


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Theils coefficient (U Statistic)

n

ttt

n

t tt

YY

FYU

1

21

1

2

11

)(

)(

The value U is the relative forecasting power of the method against nave tec

the technique is better than nave forecast If U > 1, the technique is no bettforecast.


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Theils coefficient for DAD Hos

Method U

Moving Average with 7 periods 1.221

Exponential Smoothing 1.704

Double exponential Smoothing 1.0310

Th il C ffi i t


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Theils Coefficient

1-n

1t

2

1

21-n

1t

11t

1

2

1

2

1

2 F

U2,1

t

tt

t

t

n

t

t

n

t

t

n

t

tt

Y

YY

Y

Y

FY

FYU

U1 is bounded between 0 and 1, with values closure to zero indicating

IfU 2 = 1, there is no difference between nave forecast and the foreca

IfU 2 < 1, the technique is better than nave forecast

IfU 2 > 1, the technique is no better than the nave forecast.


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S l Eff t


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Seasonal Effect

Seasonal effect is defined as the repetitive and predictable pbehaviour in a time-series around the trend line.

In seasonal effect the time period must be less than one year,

weeks, months or quarters.

Seasonal effect


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Seasonal effect

Identification of seasonal effect provides better understandinseries data.

Seasonal effect can be eliminated from the time-series. This prdeseasonalization or seasonal adjusting.


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Seasonal Adjusting

tttt

tt

tt

TSTY

modeladditiveinadjustmentSeasonal

100TT

STeffectSeasonal

modetivemultiplicainadjustmentSeasonal

tt

t

SS

Y

Seasonal Index


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Seasonal Index

Method of simple averages

Ratio-to-moving average method



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Average the unadjusted data by period ( for example daily or month

Calculate the average of daily (or monthly) averages.

Seasonal index for day i (or month i) is the ratio of daily avera

month i) to the average of daily (or monthly) averages times 100.

Example: DAD Exa

mple - Demand for Continental Bre


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Example: DAD Example - Demand for Continental Bre

5 October1 November 2014 Data

DAY

Week

(5-11 October)

Week 2

(12-18 October)

Week 3

(19-25 OCT)

Sunday 41.00 40.00 40.00

Monday 30.00 44.00 43.00

Tuesday 40.00 49.00 41.00

Wednesday 40.00 50.00 46.00

Thursday 40.00 45.00 41.00

Friday 40.00 40.00 40.00

Saturday 40.00 42.00 32.00

Seasonality Index


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Seasonality Index

DAY

Week1

(5-11 October)

Week 2

(12-18 October)

Week 3

(19-25 OCT)

Week 4

(26 Oct-1 Nov)

Daily

Average

Sunday 41.00 40.00 40.00 41.00 40.50

Monday 30.00 44.00 43.00 41.00 39.50

Tuesday 40.00 49.00 41.00 40.00 42.50

Wednesday 40.00 50.00 46.00 43.00 44.75

Thursday 40.00 45.00 41.00 46.00 43.00

Friday 40.00 40.00 40.00 45.00 41.25

Saturday 40.00 42.00 32.00 45.00 39.75

41.61

Average of daily

averages

Deseasonalized Data


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Deseasonalized Data

Date Demand Seasonal Index De-seasonalized Data

10/5/2014 41 97.34% 42.12

10/6/2014 30 94.94% 31.60

10/7/2014 40 102.15% 39.16

10/8/2014 40 107.55% 37.19

10/9/2014 40 103.34% 38.70

10/10/2014 40 99.14% 40.3510/11/2014 40 95.54% 41.87

Trend


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Trend

Trend is calculated using regression on deseasonalized d

Deseasonalized data is obtained by dividing the actual

seasonality index.


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SUMMARY OUTPUT

Regression Statistics

Multiple R 0.268288

R Square 0.071978Adjusted R Square 0.036285

Standard Error 3.715395

Observations 28

ANOVA

df SS MS F Significa

Regression 1 27.83729 27.83729 2.016587 0.1674

Residual 26 358.9082 13.80416

Total 27 386.7455

Coefficients Standard Error t Stat P-value Lower 9

Intercept 39.81731 1.442768 27.59786 8.7E-21 36.851

Day 0.123437 0.086923 1.420066 0.167469 -0.055

Forecast


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Ft = Tt x St

F29 = T29 x S29

F29 = (39.81 + 0.1234 x 29) x 0.9733 = 42.2372

Y29 = 46


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A t l ti F ti (ACF)


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All Rights Reserved Indian

Auto-correlation Function (ACF)

A k-period plot of autocorrelations is called autofunction (ACF) or a correlogram.

Auto-Correlation


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Auto-correlation of lag k is auto-correlation between

To test whether the autocorrelation at lag k is significa

from 0, the following hypothesis test is used:

H0: k = 0

HA: k 0

For any k, reject H0 if | k| > 1.96/n. Where n is thobservations.

ACF for Demand for Continental

B kf t


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Breakfast

Partial Auto-correlation


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Partial auto correlation of lag k is an auto correlation betweewith linear dependence between the intermedia values (Yt-k+1, Yremoved.

Partial Auto correlation Function


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Partial Auto-correlation Function

A k-period plot of partial autocorrelations is caautocorrelation function (PACF).

Partial Auto-Correlation


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Partial auto-correlation of lag k is auto-correlation between Yt aremoval of linear dependence of Yt+1 to Yt+k-1.

To test whether the partial autocorrelation at lag k is significantlythe following hypothesis test is used:

H0:pk = 0

HA:pk 0

For any k, reject H0 if |pk| > 1.96/n. Where n is the number of

PACF Demand for Continental Breakfast at


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PACF Demand for Continental Breakfast at

DAD hospital

Stationarity


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A time series is stationary, if:

Mean () is constant over time.

Variance () is constant over time.

The covariance between two time periods (Yt) and (only on the lag k not on the time t.

We assume that the time series is stationary before

applying forecasting models

ACF Plot of non-stationary and stationary


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process

Non-stationary Stationary

White Noise


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White noise is a data uncorrelated across time that fodistribution with mean 0 and constant standard deviat

In forecasting we assume that the residuals are white N


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Residual White Noise


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Auto-Regression


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Auto-regression is a regression of Y on itself observed

time points.

That is, we use Yt as the response variable and Yt-1, Yt-2explanatory variables.

AR(1) Parameter Estimation


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n

t

t

n

t

tt

n

t

tt

n

t

t

ttt

Y

YY

YY

YY

2

21

2

1

2

21

2

2

1

OLS

Estimate

AR(1) Process


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Yt = Yt-1 + t

Yt = (Yt-2 + t-1) + t

Yt = t X0 +

t-1 1 + t-2 2++

1 t-1 + t

1

0

0

t

i

itit

t XY

Auto-regressive process (AR(p))


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Assume {Yt} is purely random with mean zero and consdeviation (White Noise).

Then the autoregressive process of orderp or AR(p) proc

tptpttt YYYY ...22110

AR(p) process models each future observationas a function p previous observations.


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Model Identification in AR(p)Process

Pure AR Model Identification


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Model ACF PACF

AR(1) Exponential Decay: Positive side if 1

> 0 and alternating in sign starting onnegative side if 1 < 0.

Spike at lag 1, then cuts off t

Spike positive if 1 > 0 and nside if 1 < 0.

AR(p) Exponential decay: pattern depends onsigns of 1, 2, etc

Spikes at lags 1 to p, then cutzero.

tptpttt YYYY ...22110

Partial Auto-Correlation


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Partial auto-correlation of lag k is auto-correlation between Yt anremoval of linear dependence of Yt+1 to Yt+k-1.

To test whether the autocorrelation at lag k is significantly differfollowing hypothesis test is used:

H0: k = 0

HA: k 0

For any k, reject H0 if | k| > 1.96/n. Where n is the number of

PACF FunctionDAD ContinentaBreakfast


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First t

PAC a

differe

from

zero.

ACF Function - DAD Continental

Breakfast


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Actual Vs Fit- Continental Breakfas

Data


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Data

AR(2) Model


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MAPE = 8.8%



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(Yt 41.252) = 0.663 (Yt-1 41.252) 0.204 (Yt-2 41.252)

Yt = (Xt - )


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Moving Average Process


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A moving average process is a time series regression modelthe value at time t, Yt, is a linear function of past errors.

First order moving average, MA(1), is given by:

Yt = 0 + 1t-1 + t

Moving Average Process MA(q)


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{Yt} is a moving average process of order q (written MA(q)) if

constants 0, 1, . . . q

We have.,

tqtqtttY ...2210

MA(q) models each future observation as a function ofq previous errors

Pure MA Model Identification


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Model ACF PACF

MA(1) Spike at lag 1 then cuts of to zero. Spikepositive if 1 > 0 and negative side if

1 < 0.

Exponential decay. On neg1> 0 on positive side i

MA(q) Spikes at lags 1 to q, then cuts off to zero. Exponential decay or sine w

pattern depends on signs o

ACF Function - DAD ContinentalBreakfast


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SPSS Output


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Actual Vs Fitted Value


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Residual Plot


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ARMA(

p,q) Model Identification


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ARMA(p,q) models are not easy to identify. We usually start wAR and MA process. The following thump rule may be used.

The final ARMA model may be selected based on parameters sRMSE, MAPE, AIC and BIC.

Process ACF PACF

ARMA(p,q) Tails off after q lags Tails off to zero after p

lags


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ACF PACF

ARMA(2,1) Model Output


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AR(2) Model


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MAPE = 8.8%



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Forecasting Model Evaluation

Akaikes information criteria


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Akaike s information criteria

AIC = -2LL + 2m

Where m is the number of variables estimated in the model

Bayesian Information Criteria

BIC = -2LL + m ln(n)

Where m is the number of variables estimated in the model and n is the numbeobservations

Final Model Selection


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Model AR(2) MA(1) ARMA(2,1)

BIC 3.295 3.301 3.343

AR(2) has the

least BIC


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ARIMA

ARIMA has the following three components:


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ARIMA has the following three components:

Auto-regressive component: Function of past values of the time

Integration Component: Differencing the time series to make it a st

Moving Average Component: Function of past error values.

Integration (d)


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Used when the process is non-stationary.

Instead of observed values, differences between observed values are

When d=0, the observations are modelled directly. If d = 1,between consecutive observations are modelled. If d = 2, the dif

differences are modelled.

ARIMA (p, d, q)


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The q and p values are identified using auto-correlation

and Partial auto-correlation function (PACF) respectivelidentifies the level of differencing.

Usually p+q


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Differencing is a process of making a non-stationary

stationary process.

In differencing, we create a new process Xt, where Xt = Yt

ARIMA(p,1,q) Process


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tqtqtt

ptpttt

XXXX

...

...

22110

22110

Where Xt = YtYt-1

Ljung-Box Test

Ljung-Box is a test on the autocorrelations of residuals. The test sta


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m

k

k

m kn

rnnQ

1

2

)2(

n = number of observations in the time series.

k = particular time lag checkedm = the number of time lags to be tested

rk = sample autocorrelation function of the kth residual term.

Q statistic is approximate chi-square distribution with mpq degrees of freedom are correctly specified.

H0: The model does not exhibit lack of fit

HA: The model exhibits lack of fit

AR(2)

Ljung-Box Test


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P-value is 0.740, thus the model doesnt show

lack of fit

Box-Jenkins Methodology

Identification: Identify the ARIMA model using ACF & PAC


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y gwould give the values of p, q and d.

Estimation: Estimate the model parameters (using maximum

Diagnostics: Check the residual for any issue such as not pr

Noise.

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