Simple Methods and Procedures Used in Forecasting

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Simple Methods and Procedures Used in Forecasting The project prepared by : Sven Gingelmaier Michael Richter Under direction of the Maria Jadamus-Hacura

What Is Forecasting? Prediction of future events and conditions are called forecasts, and the act of making such prediction is called forecasting. (WordNet Dictionary ) Sales will be $200 million!

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

Time Series Analysis 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

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

Linear Trend Analysis Linear Trend 950000 900000 850000 y = -10405t + 860988 R 2 = 0,8497 800000 750000 700000 650000 600000 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 empircal data Linear (empircal data)

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 )

SUMMARY OUTPUT Regression Statistics 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

Linear Trend Analysis Linear trend equation: ) Y = 860988,43 10405,27* t Y ) - Estimated or predicted value of born baby s Interpretation of slope coefficient : Here b 1 = 10405,27 tells us that the average value of born baby s decreases by 10405 on average in each year.

Measures of fit -The Coefficient of Determination R2 -Standard Error of Estimate Su - Coefficient of random variation V

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 R 2 =0,8496. It means that 84 % of the total variation of the number of born baby s is explained by the trend model.

Standard Error of Estimate S u = 24085,46 It is the standard deviation around the trend line of the predicted values of Y.

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.

Predicted Value We estimate the value of born baby s in the year 2008 by extrapolation trend function for t = 19 : ) Y = 860988, 43 10405, 27*19 = 663288,34 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 )

Exponential Smoothing 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.

Brown's Linear (double) Exponential Smoothing 950.000 900.000 850.000 800.000 750.000 700.000 forecast 650.000 600.000 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

Brown's quadratic (triple) smoothing model 950000 900000 850000 800000 750000 700000 650000 600000 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

Holt's method double exponential smoothing 950000 900000 forecast 850000 800000 750000 700000 650000 600000 1 3 5 7 9 11 13 15 17 19 21 23 actual smoothed data

Nonlinear smoothing model 950.000 900.000 forecast 850.000 800.000 750.000 700.000 650.000 600.000 1 3 5 7 9 11 13 15 17 19 21 23 actual smoothed data

Summary of Results MAE Forecasted value for 2008 ex post error absolute value of ex post error Brown's Linear (double) Exponential Smoothing 19932 676589-1861 1861 Brown's quadratic ( triple) smoothing model 29244 698999-24271 24271 Holt's method double exponential smoothing 17831 672391 2337 2337 Nonlinear smoothing model 16726 677927-3199 3199 Real value of born baby s in the year 2008 674728

Summary of Results ( graphically ) 705000 700000 695000 690000 forecasted value real value 685000 680000 675000 670000 665000 660000 655000 Brown's Linear (double) Exponential Smoothing Brown's quadratic (ie, triple) smoothing model Holt's method double exponential smoothing Nonlinear smoothing model

General Comparison (graphically) 710000 700000 Forecasted value for 2008 real value 690000 680000 670000 660000 650000 35000 30000 25000 20000 15000 MAE absolute value of ex post error 640000 Brown's Linear (double) Exponential Smoothing Brown's quadratic (ie, triple) smoothing model Holt's method double exponential smoothing Nonlinear smoothing model Trend model 10000 5000 0 Brown's Linear (double) Exponential Smoothing Brown's quadratic (ie, triple) smoothing model Holt's method double exponential smoothing Nonlinear smoothing model Trend model

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