FORECASTING WITH REGRESSION MODELS TREND ANALYSIS BUSINESS FORECASTING Prof. Dr. Burç Ülengin ITU MANAGEMENT ENGINEERING FACULTY FALL 2011

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 Dependent variable Independent variable Specific form: Linear Regression Model Random disturbance

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

Best Fit Estimates population sample

Best Fit Estimates-OLS

Misleading Best Fits                                     X Y X Y X Y X Y  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:

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  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  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

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

OLS Estimate Dependent Variable: RCS Method: Least Squares Sample: 1990:1 1998:4 Included observations: 36 VariableCoefficientStd. Errort-StatisticProb. C DPI R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid8.83E+11 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

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  Do not reject H 0 t= 1, 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 = 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  Do not reject H 0 F-statistic= Do not reject H 0 Estimated equation has no power to explain RCS figures

Graphical Evaluation of Fit and Error Terms Residuls show clear seasonal pattern

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:

Seasonally Adjusted RCS and RCS

OLS Estimate Dependent Variable: RCSSA Method: Least Squares Sample: 1990:1 1998:4 Included observations: 36 VariableCoefficientStd. Errort-StatisticProb. C DPI R-squared Mean dependent var Adjusted R-squared S.D. dependent var S.E. of regression Akaike info criterion Sum squared resid3.42E+11 Schwarz criterion Log likelihood F-statistic Durbin-Watson stat Prob(F-statistic)

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  Do not reject H 0 t= 2,62 < t table or Pr = < 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 = 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  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

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

Trend Models

Simple Regression Model Special Case: Trend Model 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 Holdout period

OLS Estimates Dependent Variable: FEE Method: Least Squares Sample: 1986:1 1994:4 Included observations: 36 VariableCoefficientStd. Errort-StatisticProb. 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) e2e2

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  Do not reject H 0 t= 39,4 > t table or Pr = < 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 = 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  Do not reject H 0 F-statistic= 1552 < F table or Pr = < 0.05 Reject H 0 Estimated equation has explanatory power

Graphical Evaluation of Fit Holdout period ACTUAL FORECAST 1995 Q Q Q Q

Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random Also the seasonal fluctuations can not be modelled Regression model is misspecified

Model Improvement  Data may exhibit exponential trend  In this case, take the logarithm of the dependent variable  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

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

Original and Logarithmic Transformed Data LOG(FEE) FEE

OLS Estimate of the Logrithmin Trend Model Dependent Variable: LFEE Method: Least Squares Sample: 1986:1 1994:4 Included observations: 36 VariableCoefficientStd. Errort-StatisticProb. 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)

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

Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random Also the seasonal fluctuations can not be modelled Regression model is misspecified

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:

Original and Seasonally Adjusted Data

OLS Estimate of the Seasonally Adjusted Trend Model Dependent Variable: FEESA Method: Least Squares Sample: 1986:1 1995:4 Included observations: 40 VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random There is no seasonal fluctuations 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=log(FEE) Adjusted Series: LFEESA Scaling Factors:

Original and Logarithmic Additive Seasonally Adjustment Series

OLS Estimate of the Logarithmic Additive Seasonally Adjustment Data Dependent Variable: LFEESA Method: Least Squares Sample: 1986:1 1995:4 Included observations: 40 VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms Residuals exhibit clear pattern, they are not random There is no seasonal fluctuations Regression model is misspecified

Autoregressive Model  Some cases the growth model may be more suitable to the data  If data exhibits the nonlinearity, the autoregressive model can be adjusted to model exponential pattern

OLS Estimate of Autoregressive Model Dependent Variable: FEE Method: Least Squares Sample(adjusted): 1986:2 1995:4 Included observations: 39 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms Clear seasonal pattern 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

Dependent Variable: FEESA Method: Least Squares Sample(adjusted): 1986:2 1995:4 Included observations: 39 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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) OLS Estimate of Seasonally Adjusted Autoregressive Model

Graphical Evaluation of Fit and Error Terms No seasonal pattern in the residuals Model specification seems more corret than the previous estimates

Seasonal Autoregressive Model  If data exhibits sesonal fluctutions, the growth model should be remodeled  If data exhibits the nonlinearity and sesonality together, the seasonal autoregressive model can be adjusted to model exponential pattern

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.

GOMPERTZ CURVE  Gompertz function is given as 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.

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

GOMPERTZ CURVE An Example

LOGISTICS CURVE  Logistic function is given as 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.

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 YEAR HDTV

LOGISTICS versus GOMPERTZ CURVES

FORECASTING WITH MULTIPLE REGRESSION MODELS BUSINESS FORECASTING

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

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

PARAMETER ESTIMATES-OLS ESTIMATION IT IS VERY COMPLEX TO CALCULATE b’s, MATRIX ALGEBRA IS USED TO ESTIMATE b’s.

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%  PERIODGDPPRICESALES ?  Ln(SALES t ) = *Ln(1300) *Ln(103)  Ln(SALES t ) = 3.63  e 3.63 = 235

EXAMPLE : LPG FORECAST

LOGARITHMIC TRANSFORMATION

SCATTER DIAGRAM UNEXPECTED RELATION

LSATA=f(LGNP) Dependent Variable: LSATA Method: Least Squares Sample: Included observations: 30 VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms NOT RANDOM

LSATA=f(LP) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): Included observations: 29 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms NOT RANDOM

LSATA=f(LGNP,LP) Dependent Variable: LSATA Method: Least Squares Sample(adjusted): Included observations: 29 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms NOT RANDOM

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?

Dependent Variable: LSATA Method: Least Squares Sample(adjusted): Included observations: 29 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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) LSATA=f(LGNP,LP,LPOP)

Graphical Evaluation of Fit and Error Terms NOT RANDOM

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

Dependent Variable: LSATA Method: Least Squares Sample(adjusted): Included observations: 29 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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) LSATA=f(LGNP,LP,LPOP,LSATA t-1 )

Graphical Evaluation of Fit and Error Terms RANDOM

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  Do not reject H 0  t= 3,46 < t table or Pr = < 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 = 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  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

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 Long term effect of x

AN EXAMPLE: SHORT AND LONG TERM IMPACTS Short Term ImpactLong 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%

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 »1990Q11 »1990Q20 »1990Q30 »1990Q40 »1991Q11 »1991Q20 »1991Q30 »1991Q40 »1992Q11 »1992Q20 »1992Q30 »1992Q40

BASE PERIOD DATE D1D2D3  1990 Q  1990 Q  1990 Q  1990 Q  1990 Q  1991 Q  1991 Q  1991 Q  1992 Q  1992 Q  1992 Q  1992 Q FULL SEASONAL DUMMY VARIABLE REPRESANTATION

COLLEGE TUITION CONSUMERS' PRICE INDEX BY QUARTER

 QUARTERLY DATA THEREFORE 3 DUMMY VARIABLES WILL BE SUFFICIENT TO CAPTURE THE SEASONAL PATTERN DATE D1D2D3  1990 Q  1990 Q  1990 Q  1990 Q

SEASONAL PATTERN MODELLED 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 VariableCoefficientStd. Errort-StatisticProb. C 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)

Graphical Evaluation of Fit and Error Terms NOT RANDOM

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 VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms RANDOM

SEASONAL PART OF THE MODEL DYNAMIC PART OF THE MODEL COLLEGE TUITION PRICE INDEX GENERALIZED AUTOREGRESSIVE TREND ESTIMATION Dependent Variable: LFEE Method: Least Squares Sample(adjusted): 1987:1 1995:4 Included observations: 36 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. C LFEE(-1) LFEE(-2) LFEE(-3) LFEE(-4) 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)

GAP SALES FORECAST

SIMPLE AUTOREGRESSIVE REGRESSION MODEL Dependent Variable: LSALES Method: Least Squares Sample(adjusted): 1985:2 1999:4 Included observations: 59 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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

Graphical Evaluation of Fit and Error Terms NOT RANDOM

AUTOREGRESSIVE REGRESSION MODEL WITH SEASONAL DUMMIES Dependent Variable: LSALES Method: Least Squares Sample(adjusted): 1985:3 1999:4 Included observations: 58 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms RANDOM

ALTERNATIVE SEASONAL MODELLING  FOR NONSEASONAL DATA, THE AUTOREGRESSIVE MODEL CAN BE WRITTEN AS  IF THE LENGTH OF THE SEASONALITY IS S, THE SESONAL AUTOREGRESSIVE MODEL CAN BE WRITTEN AS

SEASONAL LAGGED AUTOREGRESSIVE REGRESSION MODEL Dependent Variable: LSALES Method: Least Squares Sample(adjusted): 1986:1 1999:4 Included observations: 56 after adjusting endpoints VariableCoefficientStd. Errort-StatisticProb. 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)

Graphical Evaluation of Fit and Error Terms