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ESTIMATING & FORECASTING DEMANDChapter 4 slide 1Regression Analysisestimates the equation thatbest fits the data and measureswhether the relationship isstatistically significant.An airline seeks to estimate thedemand relationship betweenseats sold (Q) and average fare (P): Q = a + bP,based on 16 past observationsof P & Q.# Seats sold per flightAverageFareHere, the best linear equationturns out to be: Q = 479 1.63P.Best-fit LineREGRESSION ANALYSIS4.2More generally, Multiple Regressionallows for multiple explanatory variables: Q = a + bP + cP + dY.The power of Multiple Regression:Even with multiple variables that simultaneously influence sales,its able to estimate the separate variable influences (i.e. coefficients).Important Regression Statistics1. R2 (ranging between 0 and 1) measures the proportion of variation in Q explained by the right-hand side variables. 2. F = R2/(k - 1)(1 - R2)/(N - k)The F stat (larger F better) indicates the statistical significance (or lack thereof) of the relationship.REGRESSION STATISTICS4.3For the estimated OLS equation: Q = 28.8 - 2.12P + 1.03P + 3.09Y, (.34) (.47) (1.00)R2 = .78 (78% of Qs variation explained) andF = 13.86 (well above 95% significance threshold).Standard Errors and t-statsEach coefficients standard error (its standard deviation) measures the uncertainty around its estimate.The coefficients t-stat = coefficient/SE,tests whether the coefficient is statistically significantly different t 0.Here, the t-stats are -6.24,2.20, and 3.09, indicating that all coefficients are statistically significant.The standard error of the regressionmeasures the uncertainty around the forecast of Q. A 95% confidence intervalaround the forecast is: 2 standard errors.REGRESSION ISSUES4.41. Which equation form? Linear? Polynomial, Multiplicative?2. Have explanatory variables been omitted?3. Are the explanatory variables multicolinear?4. Are the equation errors serially correlated?4.5CHOOSING A REGRESSION EQUATION1. Does the equation make economic sense? Are the right explanatory variables included?2. Are the signs and the magnitudes of the estimated coefficients reasonable? Do they make economic sense?3. Based on the regression statistics, does the equation have explanatory power? How well did it track the past data?4.6FORECASTING DEMANDTime-Series Modelsindentify patterns in a single variable over time.A Time Series can be decomposed into: 1. Trends 2. Business Cycles Three equations for estimating a time trend: 3. Seasonal Variation, and 4. Random Fluctuations.

1. Linear, Qt = a + bt2. Quadratic, Qt = a + bt + ct23. Exponential, Qt = brt, estimated as log(Qt) = log(b) + log(r)t 4.7FORECASTING DEMANDBarometric Modelsindentify patterns among different variables over time.Movements in Leading Indicatorspredict future changes in economic activity.The Index of Leading IndicatorsWeekly manufacturing hours Manufacturers New Orders Changes in Unfilled Orders Plant and Equipment OrdersNumber of Housing Building Permits6. Changes in Sensitive Materials Prices

7. Percentage of firms receiving Slower Deliveries 8. Growth in the Money Supply 9. Index of Consumer Confidence10. The S&P 500 Index11. Weekly Claims for Unemployment Insurance