BUSINESS FORECASTING

Transcription

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

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

3 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

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

5 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

6 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

7 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

8 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

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

10 THE CLASSICAL ASSUMPTIONS 1. The regression model is linear in the coefficients, correctly specified, & has an additive error term. 2. E() = 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

11 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

12 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

13 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

14 Retail Car Sales and Disposable Personal Income Figures Disposable income $ Quarterly car sales 000 cars DPI RCS

15 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 DPI R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 8.83E+11 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

16 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 = > 0.05 Do not Reject H 0 DPI has no effect on RCS

17 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 = 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 = > 0.05 Do not reject H 0 Estimated equation has no power to explain RCS figures

18 Graphical Evaluation of Fit and Error Terms Residuls show clear seasonal pattern Residual Actual Fitted

19 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.

20 Seasonal Adjustment Sample: 1990:1 1998:4 Included observations: 36 Ratio to Moving Average Original Series: RCS Adjusted Series: RCSSA Scaling Factors:

21 Seasonally Adjusted RCS and RCS R C S R C SSA

22 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 DPI R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid 3.42E+11 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

23 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 = < 0.05 Reject H 0 DPI has statistically significant effect on RCS

24 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 = 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 = < 0.05 Reject H 0 Estimated equation has some power to explain RCS figures

25 Graphical Evaluation of Fit and Error Terms No seasonality but it still does not look random disturbance Residual Actual Fitted Omitted Variable? Business Cycle?

26 Trend Models

27 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

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

29 Chapter 3 Exercise 13 College Tuition Consumers' Price Index by Quarter Holdout period FEE

30 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 R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion e 2 Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

31 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 = < 0.05 Reject H 0

32 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 = 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 = < 0.05 Reject H 0 Estimated equation has explanatory power

33 Graphical Evaluation of Fit Holdout period FEE FEEF ACTUAL FORECAST 1995 Q Q Q Q

34 Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random Also the seasonal fluctuations can not be modelled R es idual Ac tual Fitted Regression model is misspecified

35 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

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

37 Original and Logarithmic Transformed Data LOG(FEE) FEE LFEE FEE

38 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 R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

39 Forecast Calculations obs FEE LFEEF FEELF=exp(LFEEF) 1993: : : : : : : : : : : :

40 Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random R es idual Ac tual Fitted Regression model is misspecified Also the seasonal fluctuations can not be modelled

41 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

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

43 Multiplicative Seasonal Adjustment Included observations: 40 Ratio to Moving Average Original Series: FEE Adjusted Series: FEESA Scaling Factors:

44 Original and Seasonally Adjusted Data FEE FEESA

45 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 R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

46 Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random There is no seasonal fluctuations R es idual Ac tual Fitted Regression model is misspecified

47 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

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

49 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: =log(fee)

50 Original and Logarithmic Additive Seasonally Adjustment Series FEE LFEESA

51 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 R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

52 Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random There is no seasonal fluctuations R es idual Ac tual Fitted Regression model is misspecified

53 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

54 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 FEE(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

55 Graphical Evaluation of Fit and Error Terms Clear seasonal pattern R es idual Ac tual Fitted Model is misspecified

56 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

57 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 FEESA(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

58 Graphical Evaluation of Fit and Error Terms No seasonal pattern in the residuals R es idual Ac tual Fitted Model specification seems more corret than the previous estimates

59 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

60 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

61 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 SALES TIME

62 GOMPERTZ CURVE Gompertz function is given as Y Le t ae bt where L = Upper limit of Y e = Natural number = 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.

63 GOMPERTZ CURVE An Example HDTV: LCD and Plazma TV sales figures YEAR HDTV HDTV

64 GOMPERTZ CURVE An Example Dependent Variable: HDTV Method: Least Squares Sample (adjusted): 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) C(2) C(3) R-squared Mean dependent var Adjusted R-squ S.D. dependent var S.E. of regress Akaike info criterion Sum squared Schwarz criterion Log likelihood Durbin-Watson stat HDTVF

65 LOGISTICS CURVE Logistic function is given as Y t 1 L ae bt where L = Upper limit of Y e = Natural number = 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.

66 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.

67 LOGISTICS CURVE An Example HDTV: LCD and Plazma TV sales figures Dependent Variable: HDTV Method: Least Squares Sample (adjusted): Included observations: 7 after adjustments Convergence achieved after 1 iteration HDTV=C(1)/(1+C(2)*EXP(-C(3)*@TREND)) YEAR HDTV Coefficient Std. Error t-statistic Prob. C(1) C(2) C(3) R-squared Mean dependent var ####### Adjusted R-squ S.D. dependent var ####### S.E. of regress Akaike info criterion Sum squared Schwarz criterion Log likelihood Durbin-Watson stat HDTV HDTVF

68 LOGISTICS versus GOMPERTZ CURVES H D T V F _ G H D T V F _ L

69 BUSINESS FORECASTING FORECASTING WITH MULTIPLE REGRESSION MODELS

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

71 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 b X X,...X 1 1 b k 2 2 ) X X b k k X X k k e

72 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

73 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 POP 0 t 2 2 GDP 0 t 3 3 UR 0 t 4 4 PRICE 0 t t

74 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

75 FORECASTING WITH MULTIPLE REGRESSION MODEL Ln(SALES t ) = *Ln(GDP t ) *Ln(PRICE t ) IF GDP INCREASES 1%, SALES INCRESES 1.24% IF PRICE INCREASES 1% SALES DECRAESES 0.9% PERIOD GDP PRICE SALES ? Ln(SALES t ) = *Ln(1300) *Ln(103) Ln(SALES t ) = 3.63 e 3.63 = 235

76 EXAMPLE : LPG FORECAST TU PSATAY

77 LOGARITHMIC TRANSFORMATION LSATA

78 LSATA LSATA SCATTER DIAGRAM UNEXPECTED RELATION LGNP LP

79 LSATA=f(LGNP) Dependent Variable: LSATA Method: Least Squares Sample: Included observations: 30 LSATA LGNP 1 0 Variable Coefficient Std. Error t-statistic Prob. C LGNP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

80 Graphical Evaluation of Fit and Error Terms NOT RANDOM R es idual Ac tual Fitted

81 LSATA=f(LP) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): Included observations: 29 after adjusting endpoints LSATA LP 1 0 Variable Coefficient Std. Error t-statistic Prob. C LP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

82 Graphical Evaluation of Fit and Error Terms NOT RANDOM R es idual Ac tual Fitted

83 LSATA=f(LGNP,LP) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): Included observations: 29 after adjusting endpoints LSATA 0 1LGNP 2LP Variable Coefficient Std. Error t-statistic Prob. C LGNP LP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

84 Graphical Evaluation of Fit and Error Terms NOT RANDOM R es idual Ac tual Fitted

85 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?

86 LSATA=f(LGNP,LP,LPOP) Dependent Variable: LSATA Method: Least Squares 0 1LGNP 2LP Sample(adjusted): Included observations: 29 after adjusting endpoints LSATA 3LPOP Variable Coefficient Std. Error t-statistic Prob. C LGNP LP LPOP R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

87 Graphical Evaluation of Fit and Error Terms NOT RANDOM R es idual Ac tual Fitted

88 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

89 LSATA=f(LGNP,LP,LPOP,LSATA t-1 ) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): Included observations: 29 after adjusting endpoints LSATA t 0 1 LGNP Variable Coefficient Std. Error t-statistic Prob. C LGNP LP LPOP LSATA(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) t 2 LP t 3 LPOP t 4 LSATA t1

90 Graphical Evaluation of Fit and Error Terms RANDOM R es idual Ac tual Fitted

91 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 = < 0.05 Reject H 0 GNP has effect on RCS

92 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 = 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 = < 0.05 Reject H 0 Estimated equation has power to explain RCS figures

93 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 y )y 0 ( x ) x 0 t t 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

94 AN EXAMPLE: SHORT AND LONG TERM IMPACTS Short Term Impact Long Term Impact LGNP LP LPOP 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%

95 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.

96 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

97 FULL SEASONAL DUMMY VARIABLE REPRESANTATION DATE D1 D2 D Q Q Q Q Q Q Q Q Q Q Q Q BASE PERIOD

98 COLLEGE TUITION CONSUMERS' PRICE INDEX BY QUARTER FEE

99 COLLEGE TUITION CONSUMERS' PRICE INDEX BY QUARTER QUARTERLY DATA THEREFORE 3 DUMMY VARIABLES WILL BE SUFFICIENT TO CAPTURE THE SEASONAL PATTERN DATE D1 D2 D Q Q Q Q

100 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 D D1(-1) SEASONAL PATTERN MODELLED D1(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

101 Graphical Evaluation of Fit and Error Terms NOT RANDOM R es idual Ac tual Fitted

102 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 LOG(FEE(-1)) D D1(-1) D1(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

103 Graphical Evaluation of Fit and Error Terms RANDOM R es idual Ac tual Fitted

104 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 LFEE(-1) LFEE(-2) DYNAMIC PART OF THE MODEL LFEE(-3) LFEE(-4) D D1(-1) SEASONAL PART OF THE MODEL D1(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

105 GAP SALES FORECAST LSALES SALES

106 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 LSALES(-1) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) SEASONALITY IS NOT MODELLED

107 Graphical Evaluation of Fit and Error Terms NOT RANDOM R es idual Ac tual Fitted

108 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 LSALES(-1) D D1(-1) D1(-2) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

109 Graphical Evaluation of Fit and Error Terms RANDOM R es idual Ac tual Fitted

110 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 y y t1 ts

111 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 LSALES(-4) R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic) t 0 1 LSALES t4

112 Graphical Evaluation of Fit and Error Terms R es idual Ac tual Fitted