Simple Methods and Procedures Used in Forecasting

(1)

Simple Methods and

Procedures Used in

Forecasting

The project prepared by : Sven Gingelmaier Michael Richter Under direction of the Maria Jadamus-Hacura

(2)

Prediction of future events and conditions are called forecasts, and the act of making such prediction is called forecasting.

(WordNet Dictionary )

What Is Forecasting?

Sales will be $200 million!

(3)

Forecasting Methods Used in

the Project :

Forecasting Methods Used in

the Project :

Linear trend model

Exponential smoothing models :

- Brown´s linear exponential smoothing - Browns quadratic smoothing model

- Holt´s method double exponential smoothing - Nonlinear smoothing model

(4)

Time series, denoted by { Y_t : t ∈ N} , is a sequence of observations on particular variables.

Decomposition of time series data (classical decomposition):

Trend

Seasonal Trend

Cyclical Movements Irregular Components

Time Series Analysis

(5)

The data that has been analyzed in the Project are :

- number of born Baby´s in Germany - analyzed period starts from 1990 to

2007

- the Data was taken from the Website of the German Census Office

(6)

Linear Trend Analysis

Linear Trend

y = -10405t + 860988 R² = 0,8497

600000 650000 700000 750000 800000 850000 900000 950000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

empircal data Linear (empircal data)

(7)

Linear Trend Analysis

We applied Ordinary Least Squares Method ( OLS ) to estimate coefficients and the measures of fit of the linear

trend model .

We utilized Excel regression option for calculation . ( Tools / Data Analysis / Regression )

(8)

Multiple R 0,9217700

R Square 0,8496599

Adjusted R Square 0,8402637

Standard Error 24085,46 V= 3,16%

Observations 18

ANOVA

df SS MS F Significance F

Regression 1 52456625447 52456625447 90,42538644 5,50673E-08

Residual 16 9281751953 580109497,1

Total 17 61738377400

Coefficients Standard Error t Stat P-value Lower 95%

Intercept 860988,4379 11844,32006 72,69209493 1,35626E-21 835879,6012 t -10405,26832 1094,228689 -9,509226385 5,50673E-08 -12724,9295

SUMMARY OUTPUT

Regression Statistics

(9)

Linear Trend Analysis

860988, 43 10405, 27 *

Y ) = − t

Linear trend equation:

Interpretation of slope coefficient :

Here b₁= 10405,27 tells us that the average value of born baby´s decreases by 10405 on average in each year .

Y)

- Estimated or predicted value of born baby´s

(10)

Measures of fit

-The Coefficient of Determination R2 -Standard Error of Estimate Su

- Coefficient of random variation V

(11)

Coefficient of

Determination, R

²

The coefficient of determination is the portion of the total variation in the

dependent variable that is explained by variation in the independent variable

In our example R²=0,8496.

It means that 84 % of the total variation of the number of born baby´s is explained by the trend model .

(12)

Standard Error of

Estimate

S_u = 24085,46

It is the standard deviation around

the trend line of the predicted

values of Y.

(13)

Coefficient of random

variation

V = 3,16%

The value of standard error is around 3% of the mean of the number of born baby´s .

(14)

Predicted Value

We estimate the value of born baby´s in the year 2008 by extrapolation trend function for t = 19 :

860988, 43 10405, 27 *19 663288, 34

Y) = − =

The real number of born baby´s in Germany in the year 2008 is 674728 .

The ex post error of estimation is equal to :

674728 – 663288,34 = 11439,7

This error is less than estimated from the regression model . ( S_u = 24085,5 )

(15)

Exponential Smoothing

Methods

Methods Methods

Methods

Exponential smoothing has become very popular as a forecasting method for a wide variety of time series data.

The predicted value in this method is a weighted average of past observations . Weights decay geometrically as we go backwards in time .

(16)

Brown's Linear (double)

Exponential Smoothing

600.000 650.000 700.000 750.000 800.000 850.000 900.000 950.000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 actual smoothed data

forecast

(17)

Brown's quadratic

(triple) smoothing model

600000 650000 700000 750000 800000 850000 900000 950000

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 data forecasts

(18)

Holt's method double

exponential smoothing

600000 650000 700000 750000 800000 850000 900000 950000

1 3 5 7 9 11 13 15 17 19 21 23

actual smoothed data

forecast

(19)

Nonlinear smoothing

model

600.000 650.000 700.000 750.000 800.000 850.000 900.000 950.000

1 3 5 7 9 11 13 15 17 19 21 23

actual smoothed data

forecast

(20)

Summary of Results

674728

Real value of born baby´s in the year 2008

3199 -3199

677927 16726

Nonlinear smoothing model

2337 2337

672391 17831

Holt's method double exponential smoothing

24271 -24271

698999 29244

Brown's quadratic ( triple) smoothing model

1861 -1861

676589 19932

Brown's Linear (double) Exponential Smoothing

absolute value of

ex post error ex post

error Forecasted

value for 2008 MAE

(21)

Summary of Results

( graphically )

655000 660000 665000 670000 675000 680000 685000 690000 695000 700000 705000

Brown's Linear (double) Exponential

Smoothing

Brown's quadratic (ie, triple) smoothing

model

Holt's method double exponential

smoothing

Nonlinear smoothing model forecasted value real value

(22)

General Comparison

(graphically)

640000 650000 660000 670000 680000 690000 700000 710000

Smoothing

Brown's quadratic (ie, triple) smoothing

model

smoothing

Nonlinear smoothing model

Trend model Forecasted value for 2008 real value

0 5000 10000 15000 20000 25000 30000 35000

Smoothing

Brown's quadratic (ie,

triple) smoothing

model

smoothing

Nonlinear smoothing

model

Trend model MAE

absolute value of ex post error

(23)

Simple Methods and Procedures Used in Forecasting