BUSINESS FORECASTING - PDF Free Download

BUSINESS FORECASTING FORECASTING WITH REGRESSION MODELS TREND ANALYSIS Prof. Dr. Burç Ülengin ITU MANAGEMENT ENGINEERING FACULTY FALL 2015

OVERVIEW The bivarite regression model Data inspection Regression forecast process Forecasting with simple linear trend Causal regression model Statistical evaluation of regression model Examples...

The Bivariate Regression Model The bivariate regression model is also known a simple regression model It is a statistical tool that estimates the relationship between a dependent variable(y) and a single independent variable(x). The dependent variable is a variable which we want to forecast

The Bivariate Regression Model General form Y f (X) Dependent variable Independent variable Specific form: Linear Regression Model Y 0 1X Random disturbance

The Bivariate Regression Model Y X 0 1 The regression model is indeed a line equation 1 = slope coefficient that tell us the rate of change in Y per unit change in X If 1 = 5, it means that one unit increase in X causes 5 unit increase in Y is random disturbance, which causes for given X, Y can take different values Objective is to estimate 0 and 1 such a way that the fitted values should be as close as possible

The Bivariate Regression Model Geometrical Representation Y Good fit Poor fit The red line is more close the data points than the blue one X

Best Fit Estimates Pop ulation regression model Y ^ Y b 0 0 b 1 1 X Samp le regression model X population sample error term ^ e Y - Y Ordinary Least Square Estimate - OLS min e 2 (Y - ^ Y) 2 (Y b 0 b 1 X) 2

Best Fit Estimates-OLS OrdinaryLeast SquareEstimate- OLS min e 2 (Y - ^ Y) 2 (Y b 0 b 1 X) 2 b 1 (XY n X Y) ( X n X 2 ) b 0 Y b 1 X

Misleading Best Fits X Y X Y X Y X Y e 2 =100 e 2 =100 e 2 =100 e 2 =100

THE CLASSICAL ASSUMPTIONS 1. The regression model is linear in the coefficients, correctly specified, & has an additive error term. 2. E() = 0. 3. All explanatory variables are uncorrelated with the error term. 4. Errors corresponding to different observations are uncorrelated with each other. 5. The error term has a constant variance. 6. No explanatory variable is an exact linear function of any other explanatory variable(s). 7. The error term is normally distributed such that: 2 ~ iid N 0, i

Regression Forecasting Process Data consideration: plot the graph of each variable over time and scatter plot. Look at Trend Seasonal fluctuation Outliers To forecast Y we need the forecasted value of X YT 1 b0 b1xt 1 Reserve a holdout period for evaluation and test the estimated equation in the holdout period

An Example: Retail Car Sales The main explanatory variables: Income Price of a car Interest rates- credit usage General price level Population Car park-number of cars sold up to time-replacement purchases Expectation about future For simple-bivariate regression, income is chosen as an explanatory variable

Bi-variate Regression Model Population regression model RCS t 0 1 DPI Our expectation is 1 >0 But, we have no all available data at hand, the data set only covers the 1990s. We have to estimate model over the sample period Sample regression model is RCS b b DPI e t 0 1 t t t t

Retail Car Sales and Disposable Personal Income Figures Disposable income $ 20500 20000 19500 19000 18500 18000 2200 2000 1800 1600 1400 Quarterly car sales 000 cars 17500 90 91 92 93 94 95 96 97 98 DPI RCS

OLS Estimate Dependent Variable: RCS Method: Least Squares Sample: 1990:1 1998:4 Included observations: 36 RCS b 0 b1dpi Variable Coefficient Std. Error t-statistic Prob. C 541010.9 746347.9 0.724878 0.4735 DPI 62.39428 40.00793 1.559548 0.1281 R-squared 0.066759 Mean dependent var 1704222. Adjusted R-squared 0.039311 S.D. dependent var 164399.9 S.E. of regression 161136.1 Akaike info criterion 26.87184 Sum squared resid 8.83E+11 Schwarz criterion 26.95981 Log likelihood -481.6931 F-statistic 2.432189 Durbin-Watson stat 1.596908 Prob(F-statistic) 0.128128

Basic Statistical Evaluation 1 is the slope coefficient that tell us the rate of change in Y per unit change in X When the DPI increases one $, the number of cars sold increases 62. Hypothesis test related with 1 H 0 : 1 =0 H 1 : 1 0 t test is used to test the validity of H 0 t = 1 /se( 1 ) If t statistic > t table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If t statistic < t table Do not reject H 0 or Pr > Do not reject H 0 t= 1,56 < t table or Pr = 0.1281 > 0.05 Do not Reject H 0 DPI has no effect on RCS

Basic Statistical Evaluation R 2 is the coefficient of determination that tells us the fraction of the variation in Y explained by X 0<R 2 <1, R 2 = 0 indicates no explanatory power of X-the equation. R 2 = 1 indicates perfect explanation of Y by X-the equation. R 2 = 0.066 indicates very weak explanation power Hypothesis test related with R 2 H 0 : R 2 =0 H 1 : R 2 0 F test check the hypothesis If F statistic > F table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If F statistic < F table Do not reject H 0 or Pr > Do not reject H 0 F-statistic=2.43 < F table or Pr = 0.1281 > 0.05 Do not reject H 0 Estimated equation has no power to explain RCS figures

Graphical Evaluation of Fit and Error Terms 2200000 2000000 1800000 400000 1600000 200000 1400000 0-200000 -400000 Residuls show clear seasonal pattern 90 91 92 93 94 95 96 97 98 Residual Actual Fitted

Model Improvement When we look the graph of the series, the RCS exhibits clear seasonal fluctuations, but PDI does not. Remove seasonality using seasonal adjustment method. Then, use seasonally adjusted RCS as a dependent variable.

Seasonal Adjustment Sample: 1990:1 1998:4 Included observations: 36 Ratio to Moving Average Original Series: RCS Adjusted Series: RCSSA Scaling Factors: 1 0.941503 2 1.119916 3 1.016419 4 0.933083

Seasonally Adjusted RCS and 2200000 RCS 2000000 1800000 1600000 1400000 90 91 92 93 94 95 96 97 98 R C S R C SSA

OLS Estimate Dependent Variable: RCSSA Method: Least Squares Sample: 1990:1 1998:4 Included observations: 36 RCSSA b 0 b1dpi Variable Coefficient Std. Error t-statistic Prob. C 481394.3 464812.8 1.035674 0.3077 DPI 65.36559 24.91626 2.623411 0.0129 R-squared 0.168344 Mean dependent var 1700000. Adjusted R-squared 0.143883 S.D. dependent var 108458.4 S.E. of regression 100352.8 Akaike info criterion 25.92472 Sum squared resid 3.42E+11 Schwarz criterion 26.01270 Log likelihood -464.6450 F-statistic 6.882286 Durbin-Watson stat 0.693102 Prob(F-statistic) 0.012939

Basic Statistical Evaluation 1 is the slope coefficient that tell us the rate of change in Y per unit change in X When the DPI increases one $, the number of cars sold increases 65. Hypothesis test related with 1 H 0 : 1 =0 H 1 : 1 0 t test is used to test the validity of H 0 t = 1 /se( 1 ) If t statistic > t table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If t statistic < t table Do not reject H 0 or Pr > Do not reject H 0 t= 2,62 < t table or Pr = 0.012 < 0.05 Reject H 0 DPI has statistically significant effect on RCS

Basic Statistical Evaluation R 2 is the coefficient of determination that tells us the fraction of the variation in Y explained by X 0<R 2 <1, R 2 = 0 indicates no explanatory power of X-the equation. R 2 = 1 indicates perfect explanation of Y by X-the equation. R 2 = 0.1683 indicates very weak explanation power Hypothesis test related with R 2 H 0 : R 2 =0 H 1 : R 2 0 F test check the hypothesis If F statistic > F table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If F statistic < F table Do not reject H 0 or Pr > Do not reject H 0 F-statistic = 6.88 < F table or Pr = 0.012 < 0.05 Reject H 0 Estimated equation has some power to explain RCS figures

Graphical Evaluation of Fit and Error Terms 1900000 1800000 1700000 300000 200000 100000 1600000 1500000 0-100000 -200000 No seasonality but it still does not look random disturbance 90 91 92 93 94 95 96 97 98 Residual Actual Fitted Omitted Variable? Business Cycle?

Trend Models

Simple Regression Model Special Case: Trend Model Y t =b 0 +b 1 t+ t Independent variable Time, t = 1, 2, 3,..., T-1, T There is no need to forecast the independent variable Using simple transformations, variety of nonlinear trend equations can be estimated, therefore the estimated model can mimic the pattern of the data

Suitable Data Pattern NO SEASONALITY ADDITIVE SEASONALITY MULTIPLICTIVE SEASONALITY NO TREND ADDITIVE TREND MULTIPLICATIVE TREND

Chapter 3 Exercise 13 College Tuition Consumers' Price Index by Quarter 280 240 200 Holdout period 160 120 86 87 88 89 90 91 92 93 94 95 FEE

OLS Estimates Dependent Variable: FEE Method: Least Squares Sample: 1986:1 1994:4 Included observations: 36 fee b 0 b1t Variable Coefficient Std. Error t-statistic Prob. C 115.7312 1.982166 58.38624 0.0000 @TREND 3.837580 0.097399 39.40080 0.0000 R-squared 0.978568 Mean dependent var 182.8889 Adjusted R-squared 0.977938 S.D. dependent var 40.87177 S.E. of regression 6.070829 Akaike info criterion 6.498820 e 2 Sum squared resid 1253.069 Schwarz criterion 6.586793 Log likelihood -114.9788 F-statistic 1552.423 Durbin-Watson stat 0.284362 Prob(F-statistic) 0.000000

Basic Statistical Evaluation 1 is the slope coefficient that tell us the rate of change in Y per unit change in X Each year tuition increases 3.83 points. Hypothesis test related with 1 H 0 : 1 =0 H 1 : 1 0 t test is used to test the validity of H 0 t = 1 /se( 1 ) If t statistic > t table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If t statistic < t table Do not reject H 0 or Pr > Do not reject H 0 t= 39,4 > t table or Pr = 0.0000 < 0.05 Reject H 0

Basic Statistical Evaluation R 2 is the coefficient of determination that tells us the fraction of the variation in Y explained by X 0<R 2 <1, R 2 = 0 indicates no explanatory power of X-the equation. R 2 = 1 indicates perfect explanation of Y by X-the equation. R 2 = 0.9785 indicates very weak explanation power Hypothesis test related with R 2 H 0 : R 2 =0 H 1 : R 2 0 F test check the hypothesis If F statistic > F table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If F statistic < F table Do not reject H 0 or Pr > Do not reject H 0 F-statistic= 1552 < F table or Pr = 0.0000 < 0.05 Reject H 0 Estimated equation has explanatory power

Graphical Evaluation of Fit 300 250 200 Holdout period 150 100 86 87 88 89 90 91 92 93 94 95 FEE FEEF ACTUAL FORECAST 1995 Q1 260.00 253.88 1995 Q2 259.00 257.72 1995 Q3 266.00 261.55 1995 Q4 274.00 265.39

Graphical Evaluation of Fit and Error Terms 300 250 200 15 10 5 0-5 -10 Residuals exhibit clear pattern, they are not random Also the seasonal fluctuations can not be modelled -15 86 87 88 89 90 91 92 93 94 150 100 R es idual Ac tual Fitted Regression model is misspecified

Model Improvement Data may exhibit exponential trend In this case, take the logarithm of the dependent variable ln( Y ) t t ln( Y t ) b 0 0 t Y Ae 1 t b t e Calculate the trend by OLS After OLS estimation forecast the holdout period Take exponential of the logarithmic forecasted values in order to reach original units 1 1 t t

Suitable Data Pattern NO SEASONALITY ADDITIVE SEASONALITY MULTIPLICTIVE SEASONALITY NO TREND ADDITIVE TREND MULTIPLICATIVE TREND

Original and Logarithmic Transformed Data 5.8 5.6 5.4 5.2 5.0 4.8 86 87 88 89 90 91 92 93 94 95 300 250 200 150 100 LOG(FEE) FEE 4.844187 127.000 4.844187 127.000 4.867534 130.000 4.912655 136.000 4.912655 136.000 4.919981 137.000 4.941642 140.000 4.976734 145.000 4.983607 146.000 LFEE FEE

OLS Estimate of the Logrithmin Trend Model Dependent Variable: LFEE Method: Least Squares Sample: 1986:1 1994:4 Included observations: 36 ln( fee) b 0 b1t Variable Coefficient Std. Error t-statistic Prob. C 4.816708 0.005806 829.5635 0.0000 @TREND 0.021034 0.000285 73.72277 0.0000 R-squared 0.993783 Mean dependent var 5.184797 Adjusted R-squared 0.993600 S.D. dependent var 0.222295 S.E. of regression 0.017783 Akaike info criterion -5.167178 Sum squared resid 0.010752 Schwarz criterion -5.079205 Log likelihood 95.00921 F-statistic 5435.047 Durbin-Watson stat 0.893477 Prob(F-statistic) 0.000000

Forecast Calculations obs FEE LFEEF FEELF=exp(LFEEF) 1993:1 228.0000 5.405651 222.6610 1993:2 228.0000 5.426684 227.3940 1993:3 235.0000 5.447718 232.2276 1993:4 243.0000 5.468751 237.1639 1994:1 244.0000 5.489785 242.2052 1994:2 245.0000 5.510819 247.3536 1994:3 251.0000 5.531852 252.6114 1994:4 259.0000 5.552886 257.9810 1995:1 260.0000 5.573920 263.4648 1995:2 259.0000 5.594953 269.0651 1995:3 266.0000 5.615987 274.7845 1995:4 274.0000 5.637021 280.6254

Graphical Evaluation of Fit and Error Terms 5.6 5.4 5.2 0.04 0.02 0.00 Residuals exhibit clear pattern, they are not random 5.0 4.8-0.02-0.04 86 87 88 89 90 91 92 93 94 R es idual Ac tual Fitted Regression model is misspecified Also the seasonal fluctuations can not be modelled

Model Improvement In order to deal with seasonal variations remove seasonal pattern from the data Fit regression model to seasonally adjusted data Generate forecasts Add seasonal movements to the forecasted values

Suitable Data Pattern NO SEASONALITY ADDITIVE SEASONALITY MULTIPLICTIVE SEASONALITY NO TREND ADDITIVE TREND MULTIPLICATIVE TREND

Multiplicative Seasonal Adjustment Included observations: 40 Ratio to Moving Average Original Series: FEE Adjusted Series: FEESA Scaling Factors: 1 1.002372 2 0.985197 3 0.996746 4 1.015929

Original and Seasonally Adjusted Data 280 240 200 160 120 86 87 88 89 90 91 92 93 94 95 FEE FEESA

OLS Estimate of the Seasonally Adjusted Trend Model Dependent Variable: FEESA Method: Least Squares Sample: 1986:1 1995:4 Included observations: 40 Variable Coefficient Std. Error t-statistic Prob. C 115.0387 1.727632 66.58749 0.0000 @TREND 3.897488 0.076240 51.12152 0.0000 R-squared 0.985668 Mean dependent var 191.0397 Adjusted R-squared 0.985291 S.D. dependent var 45.89346 S.E. of regression 5.566018 Akaike info criterion 6.319943 Sum squared resid 1177.261 Schwarz criterion 6.404387 Log likelihood -124.3989 F-statistic 2613.410 Durbin-Watson stat 0.055041 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 300 250 200 15 10 5 0-5 Residuals exhibit clear pattern, they are not random There is no seasonal fluctuations -10 86 87 88 89 90 91 92 93 94 95 150 100 R es idual Ac tual Fitted Regression model is misspecified

Model Improvement Take the logarithm in order to remove existing nonlinearity Use additive seasonal adjustment to logarithmic data Apply OLS to seasonally adjusted logrithmic data Forecast holdout period Add seasonal movements to reach seasonal forecasts Take an exponential in order to reach original seasonal forecasts

Suitable Data Pattern NO SEASONALITY ADDITIVE SEASONALITY MULTIPLICTIVE SEASONALITY NO TREND ADDITIVE TREND MULTIPLICATIVE TREND

Logarithmic Transformation and Additive Seasonal Adjustment Sample: 1986:1 1995:4 Included observations: 40 Difference from Moving Average Original Series: LFEE Adjusted Series: LFEESA Scaling Factors: 1 0.002216 2-0.014944 3-0.003099 4 0.015828 =log(fee)

Original and Logarithmic Additive Seasonally Adjustment Series 5.8 300 5.6 250 5.4 200 5.2 150 5.0 4.8 100 86 87 88 89 90 91 92 93 94 95 FEE LFEESA

OLS Estimate of the Logarithmic Additive Seasonally Adjustment Data Dependent Variable: LFEESA Method: Least Squares Sample: 1986:1 1995:4 Included observations: 40 Variable Coefficient Std. Error t-statistic Prob. C 4.822122 0.004761 1012.779 0.0000 @TREND 0.020618 0.000210 98.12760 0.0000 R-squared 0.996069 Mean dependent var 5.224171 Adjusted R-squared 0.995966 S.D. dependent var 0.241508 S.E. of regression 0.015340 Akaike info criterion -5.468039 Sum squared resid 0.008942 Schwarz criterion -5.383595 Log likelihood 111.3608 F-statistic 9629.026 Durbin-Watson stat 0.149558 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 5.8 5.6 5.4 5.2 0.04 0.02 0.00 Residuals exhibit clear pattern, they are not random 5.0 4.8-0.02 There is no seasonal fluctuations -0.04 86 87 88 89 90 91 92 93 94 95 R es idual Ac tual Fitted Regression model is misspecified

Autoregressive Model Some cases the growth model may be more suitable to the data Y Y t t a 0 0 a 1 1 Y Y t1 t1 e t t Population model Sample model If data exhibits the nonlinearity, the autoregressive model can be adjusted to model exponential pattern Ln(Y ) t Ln(Y ) t a 0 0 a 1 1 Ln(Y Ln(Y t1 t 1 ) ) e t t Population model Sample model

OLS Estimate of Autoregressive Model Dependent Variable: FEE Method: Least Squares Sample(adjusted): 1986:2 1995:4 Included observations: 39 after adjusting endpoints fee t b 0 b 1 fee t1 Variable Coefficient Std. Error t-statistic Prob. C 0.739490 2.305654 0.320729 0.7502 FEE(-1) 1.016035 0.011884 85.49718 0.0000 R-squared 0.994964 Mean dependent var 192.7179 Adjusted R-squared 0.994828 S.D. dependent var 45.45787 S.E. of regression 3.269285 Akaike info criterion 5.256940 Sum squared resid 395.4643 Schwarz criterion 5.342251 Log likelihood -100.5103 F-statistic 7309.767 Durbin-Watson stat 1.888939 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 300 250 10 5 0 200 150 100-5 -10 Clear seasonal pattern 87 88 89 90 91 92 93 94 95 R es idual Ac tual Fitted Model is misspecified

Model Improvement To remove seasonal fluctuations Seasonally adjust the data Apply OLS to Autoregressive Trend Model Forecast seasonally adjusted data Add seasonal movement to forecasted values

OLS Estimate of Seasonally Adjusted Autoregressive Model Dependent Variable: FEESA Method: Least Squares Sample(adjusted): 1986:2 1995:4 Included observations: 39 after adjusting endpoints Variable Coefficient Std. Error t-statistic Prob. C 1.125315 0.811481 1.386743 0.1738 FEESA(-1) 1.013445 0.004181 242.4027 0.0000 R-squared 0.999371 Mean dependent var 192.6894 Adjusted R-squared 0.999354 S.D. dependent var 45.27587 S.E. of regression 1.151024 Akaike info criterion 3.169101 Sum squared resid 49.01968 Schwarz criterion 3.254412 Log likelihood -59.79748 F-statistic 58759.08 Durbin-Watson stat 1.335932 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 300 250 4 200 2 150 100 0-2 No seasonal pattern in the residuals 87 88 89 90 91 92 93 94 95 R es idual Ac tual Fitted Model specification seems more corret than the previous estimates

Seasonal Autoregressive Model If data exhibits sesonal fluctutions, the growth model should be remodeled Y Y t t a 0 0 a 1 1 Y Y ts ts e t t Populationmodel Sample model If data exhibits the nonlinearity and sesonality together, the seasonal autoregressive model can be adjusted to model exponential pattern Ln(Y ) t Ln(Y ) t a 0 0 a 1 1 Ln(Y Ln(Y ts ts ) ) e t t Populationmodel Sample model

New Product Forecasting Growth Curve Fitting For new products, the main problem is typically lack of historical data. Trend or Seasonal pattern can not be determined. Forecasters can use a number of models that generally fall in the category called Diffusion Models. These models are alternatively called S-curves, growth models, saturation models, or substitution curves. These models imitate life cycle of poducts. Life cycles follows a common pattern: A period of slow growth just after introduction of new product A period of rapid growth Slowing growth in a mature phase Decline

New Product Forecasting Growth Curve Fitting Growth models has its own lower and upper limit. A significant benefit of using diffusion models is to identfy and predict the timing of the four phases of the life cycle. The usual reason for the transition from very slow initial growth to rapid growth is often the result of solutions to technical difficulties and the market s acceptance of the new product / technology. There are uper limits and a maturity phase occurs in which growth slows and finally ceases. 35 30 25 20 15 10 5 0 SALES 1 2 3 4 5 6 7 8 9 10 TIME

GOMPERTZ CURVE Gompertz function is given as Y Le t ae bt where L = Upper limit of Y e = Natural number = 2.718262... a and b = coefficients describing the curve The Gompertz curve will range in value from zero to L as t varies from zero to infinity. Gompertz curve is a way to summarize the growth with a few parameters.

GOMPERTZ CURVE An Example HDTV: LCD and Plazma TV sales figures YEAR HDTV 2000 1200 2001 1500 2002 1770 2003 3350 2004 5500 2005 9700 2006 15000 16000 14000 12000 10000 8000 6000 4000 2000 0 00 01 02 03 04 05 06 07 08 09 10 HDTV

GOMPERTZ CURVE An Example Dependent Variable: HDTV Method: Least Squares Sample (adjusted): 2000 2006 Included observations: 7 after adjustments Convergence achieved after 61 iterations HDTV=C(1)*EXP(-C(2)*EXP(-C(3)*@TREND)) Coefficient Std. Error t-statistic Prob. C(1) 332940 850837 0.391 0.716 C(2) 6.718 2.023 3.321 0.029 C(3) 0.128 0.087 1.477 0.214 R-squared 0.992 Mean dependent var 5431.429 Adjusted R-squ 0.988 S.D. dependent var 5178.199 S.E. of regress 559.922 Akaike info criterion 15.791 Sum squared 1254049 Schwarz criterion 15.76782 Log likelihood -52.26849 Durbin-Watson stat 0.704723 350000 300000 250000 200000 150000 100000 50000 0 00 05 10 15 20 25 30 35 40 45 50 HDTVF

LOGISTICS CURVE Logistic function is given as Y t 1 L ae bt where L = Upper limit of Y e = Natural number = 2.718262... a and b = coefficients describing the curve The Logistic curve will range in value from zero to L as t varies from zero to infinity. The Logistic curve is symetric about its point of inflection. The Gompertz curve is not necessarily symmetric.

LOGISTICS or GOMPERTZ CURVES? The answer lies in whether, in a particular situation, it is easier to achieve the maximum value the closer you get to it, or whether it becomes more difficult to attain the maximum value the closer you get to it. Are there factors assisting the attainment of the maximum value once you get close to it, or Are there factors preventing the attainment of the maximum value once it is nearly attained? If there is an offsetting factor such that growth is more difficult to maintain as the maximum is approached, then the Gompertz curve will be the best choice. If there are no such offsetting factors hindering than attainment of the maximum value, the logistics curve will be the best choice.

LOGISTICS CURVE An Example HDTV: LCD and Plazma TV sales figures Dependent Variable: HDTV Method: Least Squares Sample (adjusted): 2000 2006 Included observations: 7 after adjustments Convergence achieved after 1 iteration HDTV=C(1)/(1+C(2)*EXP(-C(3)*@TREND)) 160000 120000 YEAR HDTV 2000 1200 2001 1500 2002 1770 2003 3350 2004 5500 2005 9700 2006 15000 Coefficient Std. Error t-statistic Prob. C(1) 149930.000 350258.500 0.428 0.691 C(2) 199.182 432.110 0.461 0.669 C(3) 0.517 0.073 7.048 0.002 R-squared 0.997 Mean dependent var ####### Adjusted R-squ 0.995 S.D. dependent var ####### S.E. of regress 370.451 Akaike info criterion 14.965 Sum squared 548936 Schwarz criterion 14.942 Log likelihood -49.377 Durbin-Watson stat 1.632 80000 40000 0 00 05 10 15 20 25 30 35 40 45 50 HDTV HDTVF

LOGISTICS versus GOMPERTZ CURVES 350000 300000 250000 200000 150000 100000 50000 0 00 05 10 15 20 25 30 35 40 45 50 H D T V F _ G H D T V F _ L

BUSINESS FORECASTING FORECASTING WITH MULTIPLE REGRESSION MODELS

CONTENT DEFINITION INDEPENDENT VARIABLE SELECTION,FORECASTING WITH MULTIPLE REGRESSION MODEL STATISTICAL EVALUATION OF THE MODEL SERIAL CORRELATION SEASONALITY TREATMENT GENERAL AUTOREGRESSIVE MODEL ADVICES EXAMPLES...

MULTIPLE REGRESSION MODEL DEPENDENT VARIABLE, Y, IS A FUNCTION OF MORE THAN ONE INDEPENDENT VARIABLE, X 1, X 2,..X k GENERAL FORM Y f (X,X LINEAR FORM - POPULATIONREGRESSION Y LINEAR FORM - SAM PLE REGRESSION Y b 0 0 1 b 1 1 2 X X,...X 1 1 b k 2 2 ) X X 2 2...... b k k X X k k e

SELECTING INDEPENDENT VARIABLES FIRST, DETERMINE DEPENDENT VARIABLE SEARCH LITERATURE, USE COMMONSENSE AND LIST THE MAIN POTENTIAL EXPLANATORY VARIABLES IF TWO VARIABLE SHARE THE SAME INFORMATION SUCH AS GDP AND GNP SELECT THE MOST RELEVANT ONE IF A VARITION OF A VARIABLE IS VERY LITTLE, FIND OUT MORE VARIABLE ONE SET THE EXPECTED SIGNS OF THE PARAMETERS TO BE ESTIMATED

AN EXAMPLE: SELECTING INDEPENDENT VARIABLES LIQUID PETROLIUM GAS-LPG- MARKET SIZE FORECAST POTENTIAL EXPLANATORY VARIABLES POPULATION PRICE URBANIZATION RATIO GNP or GDP EXPECTATIONS LPG t 0 1 1 POP 0 t 2 2 GDP 0 t 3 3 UR 0 t 4 4 PRICE 0 t t

PARAMETER ESTIMATES-OLS ESTIMATION Pop ulation Regression Model Y 0 1 X1 2 X2... k X k Samp le Regression Model ^ Y b 0 b X1 b 1 2 X2... b k X k Error term e ^ Y - Y Ordinary Least Square min e 2 (Y ^ (Y - Y) b 0 2 Estimate- OLS b X1 b 1 2 X2... b IT IS VERY COMPLEX TO CALCULATE b s, MATRIX ALGEBRA IS USED TO ESTIMATE b s. k X k ) 2

FORECASTING WITH MULTIPLE REGRESSION MODEL Ln(SALES t ) = 23 + 1.24*Ln(GDP t ) - 0.90*Ln(PRICE t ) IF GDP INCREASES 1%, SALES INCRESES 1.24% IF PRICE INCREASES 1% SALES DECRAESES 0.9% PERIOD GDP PRICE SALES 100 1245 100 230 101 1300 103? Ln(SALES t ) = 23 + 1.24*Ln(1300) - 0.90*Ln(103) Ln(SALES t ) = 3.63 e 3.63 = 235

EXAMPLE : LPG FORECAST 800000 600000 400000 200000 0 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 TU PSATAY

LOGARITHMIC TRANSFORMATION 13.5 13.0 12.5 12.0 11.5 11.0 10.5 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 LSATA

LSATA LSATA SCATTER DIAGRAM 14 14 13 13 12 12 11 11 UNEXPECTED RELATION 10 13.6 13.8 14.0 14.2 14.4 14.6 10 0 2 4 6 8 10 12 LGNP LP

LSATA=f(LGNP) Dependent Variable: LSATA Method: Least Squares Sample: 1968 1997 Included observations: 30 LSATA LGNP 1 0 Variable Coefficient Std. Error t-statistic Prob. C -44.91150 3.097045-14.50140 0.0000 LGNP 4.081938 0.220265 18.53195 0.0000 R-squared 0.924616 Mean dependent var 12.47858 Adjusted R-squared 0.921924 S.D. dependent var 0.736099 S.E. of regression 0.205681 Akaike info criterion -0.260637 Sum squared resid 1.184535 Schwarz criterion -0.167224 Log likelihood 5.909555 F-statistic 343.4333 Durbin-Watson stat 0.485414 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 14 13 12 0.4 0.2 0.0 11 10-0.2-0.4 NOT RANDOM -0.6 68 70 72 74 76 78 80 82 84 86 88 90 92 94 96 R es idual Ac tual Fitted

LSATA=f(LP) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): 1969 1997 Included observations: 29 after adjusting endpoints LSATA LP 1 0 Variable Coefficient Std. Error t-statistic Prob. C 11.70726 0.081886 142.9694 0.0000 LP 0.190128 0.015096 12.59492 0.0000 R-squared 0.854551 Mean dependent var 12.53724 Adjusted R-squared 0.849164 S.D. dependent var 0.674006 S.E. of regression 0.261768 Akaike info criterion 0.223756 Sum squared resid 1.850107 Schwarz criterion 0.318052 Log likelihood -1.244459 F-statistic 158.6319 Durbin-Watson stat 0.187322 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 14 13 12 0.5 11 0.0-0.5-1.0 NOT RANDOM 70 72 74 76 78 80 82 84 86 88 90 92 94 96 R es idual Ac tual Fitted

LSATA=f(LGNP,LP) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): 1969 1997 Included observations: 29 after adjusting endpoints LSATA 0 1LGNP 2LP Variable Coefficient Std. Error t-statistic Prob. C -30.808410 7.715902-3.992846 0.0005 LGNP 3.066655 0.556533 5.510284 0.0000 LP 0.045318 0.028281 1.602436 0.1211 R-squared 0.932905 Mean dependent var 12.53724 Adjusted R-squared 0.927744 S.D. dependent var 0.674006 S.E. of regression 0.181176 Akaike info criterion -0.480999 Sum squared resid 0.853443 Schwarz criterion -0.339555 Log likelihood 9.974488 F-statistic 180.7558 Durbin-Watson stat 0.364799 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 14 13 0.4 12 0.2 11 0.0-0.2 NOT RANDOM -0.4 70 72 74 76 78 80 82 84 86 88 90 92 94 96 R es idual Ac tual Fitted

WHAT IS MISSING? GNP AND PRICE ARE THE MOST IMPORTANT VARIABLES BUT THE COEFFICIENT OF THE PRICE IS NOT SIGNIFICANT AND HAS UNEXPECTED SIGN RESIDUAL DISTRIBUTION IS NOT RANDOM WHAT IS MISSING? WRONG FUNCTION-NONLINEAR MODEL? LACK OF DYNAMIC MODELLING? MISSING IMPORTANT VARIABLE? POPULATION?

LSATA=f(LGNP,LP,LPOP) Dependent Variable: LSATA Method: Least Squares 0 1LGNP 2LP Sample(adjusted): 1969 1997 Included observations: 29 after adjusting endpoints LSATA 3LPOP Variable Coefficient Std. Error t-statistic Prob. C -50.913420 3.992134-12.75343 0.0000 LGNP 0.755445 0.337894 2.235746 0.0345 LP -0.131508 0.021528-6.108568 0.0000 LPOP 4.955945 0.486887 10.17885 0.0000 R-squared 0.986958 Mean dependent var 2.53724 Adjusted R-squared 0.985393 S.D. dependent var 0.674006 S.E. of regression 0.081461 Akaike info criterion -2.049934 Sum squared resid 0.165899 Schwarz criterion -1.861342 Log likelihood 33.72405 F-statistic 630.6084 Durbin-Watson stat 0.398661 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 13.5 13.0 12.5 0.2 0.1 12.0 11.5 11.0 0.0-0.1-0.2 NOT RANDOM 70 72 74 76 78 80 82 84 86 88 90 92 94 96 R es idual Ac tual Fitted

WHAT IS MISSING? GNP, POPULATION AND PRICE ARE THE MOST IMPORTANT VARIABLES. THEY ARE SIGNIFICANT THEY HAVE EXPECTED SIGN RESIDUAL DISTRIBUTION IS NOT RANDOM WHAT IS MISSING? WRONG FUNCTION-NONLINEAR MODEL? LACK OF DYNAMIC MODELLING? YES. MISSING IMPORTANT VARIABLE? YES, URBANIZATION

LSATA=f(LGNP,LP,LPOP,LSATA t-1 ) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): 1969 1997 Included observations: 29 after adjusting endpoints LSATA t 0 1 LGNP Variable Coefficient Std. Error t-statistic Prob. C -16.185910 3.832897-4.222893 0.0003 LGNP 0.523657 0.150971 3.468585 0.0020 LP -0.033964 0.013483-2.518934 0.0188 LPOP 1.279753 0.419566 3.050182 0.0055 LSATA(-1) 0.619986 0.060756 10.20446 0.0000 R-squared 0.997557 Mean dependent var 2.53724 Adjusted R-squared 0.997150 S.D. dependent var 0.674006 S.E. of regression 0.035983 Akaike info criterion -3.655968 Sum squared resid 0.031074 Schwarz criterion -3.420227 Log likelihood 58.01154 F-statistic 2450.048 Durbin-Watson stat 2.118752 Prob(F-statistic) 0.000000 t 2 LP t 3 LPOP t 4 LSATA t1

Graphical Evaluation of Fit and Error Terms 14.0 13.5 13.0 12.5 12.0 0.05 11.5 11.0 0.00-0.05-0.10 RANDOM 70 72 74 76 78 80 82 84 86 88 90 92 94 96 R es idual Ac tual Fitted

Basic Statistical Evaluation 1 is the slope coefficient that tell us the rate of change in Y per unit change in X When the GNP increases 1%, the volume of LPG sales increases 0.52%. Hypothesis test related with 1 H 0 : 1 =0 H 1 : 1 0 t test is used to test the validity of H 0 t = 1 /se( 1 ) If t statistic > t table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If t statistic < t table Do not reject H 0 or Pr > Do not reject H 0 t= 3,46 < t table or Pr = 0.002 < 0.05 Reject H 0 GNP has effect on RCS

Basic Statistical Evaluation R 2 is the coefficient of determination that tells us the fraction of the variation in Y explained by X 0<R 2 <1, R 2 = 0 indicates no explanatory power of X-the equation. R 2 = 1 indicates perfect explanation of Y by X-the equation. R 2 = 0.9975 indicates very strong explanation power Hypothesis test related with R 2 H 0 : R 2 =0 H 1 : R 2 0 F test check the hypothesis If F statistic > F table Reject H 0 or Pr < (exp. =0.05) Reject H 0 If F statistic < F table Do not reject H 0 or Pr > Do not reject H 0 F-statistic=2450 < F table or Pr = 0.0000 < 0.05 Reject H 0 Estimated equation has power to explain RCS figures

SHORT AND LONG TERM IMPACTS If we specify a dynamic model, we can estimate short and a long term impact of independent variables simultaneously on the dependent variable Short term effect of x y y y (1 y t 0 2 2 0 y )y 0 (1 1 2 1 x ) x 0 t t 0 1 2 x 1 2 y t y x 1 (1 t t1 2 ) x in equilibriu m conditions y t y t-1 y Long term effect of x

AN EXAMPLE: SHORT AND LONG TERM IMPACTS Short Term Impact Long Term Impact LGNP 0.523657 1.3778 LP -0.033964-0.0892 LPOP 1.279753 3.3657 If GNP INCREASES 1% AT TIME t, THE LPG SALES INCREASES 0.52% AT TIME t IN THE LONG RUN, WITHIN 3-5 YEARS, THE LPG SALES INCREASES 1.38%

SESONALITY AND MULTIPLE REGRESSION MODEL SEASONAL DUMMY VARIABLES CAN BE USED TO MODEL SEASONAL PATTERNS DUMMY VARIABLE IS A BINARY VARIABLE THAT ONLY TAKES THE VALUES 0 AND 1. DUMMY VARIABLES RE THE INDICATOR VARIABLES, IF THE DUMMY VARIABLE TAKES 1 IN A GIVEN TIME, IT MEANS THAT SOMETHING HAPPENS IN THAT PERIOD.

SEASONAL DUMMY VARIABLES THE SOMETHING CAN BE SPECIFIC SEASON THE DUMMY VARIABLE INDICATES THE SPECIFIC SEASON D1 IS A DUMMY VARIABLE WHICH INDICATES THE FIRST QUARTERS» 1990Q1 1» 1990Q2 0» 1990Q3 0» 1990Q4 0» 1991Q1 1» 1991Q2 0» 1991Q3 0» 1991Q4 0» 1992Q1 1» 1992Q2 0» 1992Q3 0» 1992Q4 0

FULL SEASONAL DUMMY VARIABLE REPRESANTATION DATE D1 D2 D3 1990 Q1 1 0 0 1990 Q2 0 1 0 1990 Q3 0 0 1 1990 Q4 0 0 0 1990 Q1 1 0 0 1991 Q2 0 1 0 1991 Q3 0 0 1 1991 Q4 0 0 0 1992 Q1 1 0 0 1992 Q2 0 1 0 1992 Q3 0 0 1 1992 Q4 0 0 0 BASE PERIOD

COLLEGE TUITION CONSUMERS' PRICE INDEX BY QUARTER 280 240 200 160 120 86 87 88 89 90 91 92 93 94 95 FEE

COLLEGE TUITION CONSUMERS' PRICE INDEX BY QUARTER QUARTERLY DATA THEREFORE 3 DUMMY VARIABLES WILL BE SUFFICIENT TO CAPTURE THE SEASONAL PATTERN DATE D1 D2 D3 1990 Q1 1 0 0 1990 Q2 0 1 0 1990 Q3 0 0 1 1990 Q4 0 0 0

COLLEGE TUITION PRICE INDEX TREND ESTIMATION Dependent Variable: LOG(FEE) Method: Least Squares Sample(adjusted): 1986:3 1995:4 Included observations: 38 after adjusting endpoints LFEE t 0 1t 2D1 t 3D1 t1 4D1 t2 Variable Coefficient Std. Error t-statistic Prob. C 4.832335 0.006948 695.4771 0.0000 @TREND 0.020780 0.000232 89.57105 0.0000 D1-0.011259 0.007202-1.563344 0.1275 D1(-1) SEASONAL -0.029526 PATTERN 0.007198 MODELLED -4.101948 0.0003 D1(-2) -0.017082 0.007010-2.436806 0.0204 R-squared 0.995921 Mean dependent var 5.244170 Adjusted R-squared 0.995427 S.D. dependent var 0.231661 S.E. of regression 0.015666 Akaike info criterion -5.352558 Sum squared resid 0.008099 Schwarz criterion -5.137087 Log likelihood 106.6986 F-statistic 2014.429 Durbin-Watson stat 0.161634 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 5.8 5.6 5.4 0.04 0.02 5.2 5.0 4.8 0.00-0.02-0.04 NOT RANDOM 87 88 89 90 91 92 93 94 95 R es idual Ac tual Fitted

COLLEGE TUITION PRICE INDEX AUTOREGRESSIVE TREND ESTIMATION Dependent Variable: LOG(FEE) Method: Least Squares Sample(adjusted): 1986:3 1995:4 Included observations: 38 after adjusting endpoints LFEE LFEE D1 D1 t 0 1 t1 2 t 3 t1 4D1 t2 Variable Coefficient Std. Error t-statistic Prob. C 0.050887 0.022969 2.215524 0.0337 LOG(FEE(-1)) 0.997510 0.004375 227.9958 0.0000 D1-0.031634 0.002833-11.16704 0.0000 D1(-1) -0.035335 0.002833-12.47301 0.0000 D1(-2) -0.006775 0.002761-2.454199 0.0196 R-squared 0.999368 Mean dependent var 5.244170 Adjusted R-squared 0.999292 S.D. dependent var 0.231661 S.E. of regression 0.006165 Akaike info criterion -7.217678 Sum squared resid 0.001254 Schwarz criterion -7.002206 Log likelihood 142.1359 F-statistic 13051.60 Durbin-Watson stat 1.605178 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 5.8 5.6 5.4 0.02 0.01 0.00 5.2 5.0 4.8-0.01-0.02 RANDOM 87 88 89 90 91 92 93 94 95 R es idual Ac tual Fitted

COLLEGE TUITION PRICE INDEX GENERALIZED AUTOREGRESSIVE TREND ESTIMATION Dependent Variable: LFEE Method: Least Squares Sample(adjusted): 1987:1 1995:4 Included observations: 36 after s adjusting endpoints LFEEt 0 ilfeet i 2D1t 3D1 t1 4D1t 2 i1 Variable Coefficient Std. Error t-statistic Prob. C 0.048752 0.024114 2.021760 0.0529 LFEE(-1) 1.126366 0.182970 6.156010 0.0000 LFEE(-2) 0.292152 0.256488 1.139051 0.2643 DYNAMIC PART OF THE MODEL LFEE(-3) -0.344963 0.253185-1.362491 0.1839 LFEE(-4) -0.076855 0.181751-0.422857 0.6756 D1-0.043879 0.005597-7.840118 0.0000 D1(-1) -0.048562 SEASONAL 0.010241 PART OF -4.742040 THE MODEL 0.0001 D1(-2) -0.005369 0.009855-0.544814 0.5902 R-squared 0.999502 Mean dependent var 5.263841 Adjusted R-squared 0.999377 S.D. dependent var 0.221681 S.E. of regression 0.005532 Akaike info criterion -7.363447 Sum squared resid 0.000857 Schwarz criterion -7.011554 Log likelihood 140.5420 F-statistic 8025.362 Durbin-Watson stat 1.892211 Prob(F-statistic) 0.000000

GAP SALES FORECAST 4000000 3000000 16 2000000 15 14 13 1000000 0 12 11 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 LSALES SALES

SIMPLE AUTOREGRESSIVE REGRESSION MODEL Dependent Variable: LSALES Method: Least Squares Sample(adjusted): 1985:2 1999:4 Included observations: 59 after adjusting endpoints LSALES t 0 LSALES t1 Variable Coefficient Std. Error t-statistic Prob. C 0.613160 0.484163 1.266433 0.2105 LSALES(-1) 0.958714 0.036128 26.53623 0.0000 R-squared 0.925115 Mean dependent var 13.43549 Adjusted R-squared 0.923802 S.D. dependent var 0.848687 S.E. of regression 0.234272 Akaike info criterion -0.031358 Sum squared resid 3.128350 Schwarz criterion 0.039067 Log likelihood 2.925062 F-statistic 704.1714 Durbin-Watson stat 2.159164 Prob(F-statistic) 0.000000 SEASONALITY IS NOT MODELLED

Graphical Evaluation of Fit and Error Terms 16 15 14 13 0.4 0.2 12 11 0.0-0.2-0.4-0.6 NOT RANDOM 86 87 88 89 90 91 92 93 94 95 96 97 98 99 R es idual Ac tual Fitted

AUTOREGRESSIVE REGRESSION MODEL WITH SEASONAL DUMMIES Dependent Variable: LSALES Method: Least Squares Sample(adjusted): 1985:3 1999:4 Included observations: 58 after adjusting endpoints LFEE t 0 LSALESt 1 2D1 t 3D1 t1 4D1 t2 Variable Coefficient Std. Error t-statistic Prob. C 0.299734 0.111564 2.686656 0.0096 LSALES(-1) 0.994473 0.008213 121.0873 0.0000 D1-0.547251 0.018685-29.28766 0.0000 D1(-1) -0.175405 0.018732-9.364126 0.0000 D1(-2) 0.033281 0.018458 1.803073 0.0771 R-squared 0.996547 Mean dependent var 13.46547 Adjusted R-squared 0.996287 S.D. dependent var 0.823972 S.E. of regression 0.050210 Akaike info criterion -3.062940 Sum squared resid 0.133616 Schwarz criterion -2.885316 Log likelihood 93.82526 F-statistic 3824.335 Durbin-Watson stat 1.828642 Prob(F-statistic) 0.000000

Graphical Evaluation of Fit and Error Terms 16 15 14 0.2 0.1 13 12 11 0.0-0.1-0.2 RANDOM 86 87 88 89 90 91 92 93 94 95 96 97 98 99 R es idual Ac tual Fitted

ALTERNATIVE SEASONAL MODELLING FOR NONSEASONAL DATA, THE AUTOREGRESSIVE MODEL CAN BE WRITTEN AS y IF THE LENGTH OF THE SEASONALITY IS S, THE SESONAL AUTOREGRESSIVE MODEL CAN BE WRITTEN AS y t t 0 0 1 1 y y t1 ts

SEASONAL LAGGED AUTOREGRESSIVE REGRESSION MODEL Dependent Variable: LSALES Method: Least Squares Sample(adjusted): 1986:1 1999:4 Included observations: 56 after adjusting endpoints LSALES Variable Coefficient Std. Error t-statistic Prob. C 0.329980 0.169485 1.946953 0.0567 LSALES(-4) 0.990877 0.012720 77.89949 0.0000 R-squared 0.991180 Mean dependent var 3.50893 Adjusted R-squared 0.991016 S.D. dependent var 0.804465 S.E. of regression 0.076248 Akaike info criterion 2.274583 Sum squared resid 0.313945 Schwarz criterion -2.202249 Log likelihood 65.68834 F-statistic 6068.330 Durbin-Watson stat 0.434696 Prob(F-statistic) 0.000000 t 0 1 LSALES t4

Graphical Evaluation of Fit and Error Terms 16 15 14 13 0.2 0.1 12 11 0.0-0.1-0.2 86 87 88 89 90 91 92 93 94 95 96 97 98 99 R es idual Ac tual Fitted