1. ESTIMATING & FORECASTING DEMAND Chapter 4 slide 1 Regression Analysis estimates the equation that best fits the data and measures whether the relationship is statistically significant. An airline seeks to estimate the demand relationship between seats sold (Q) and average fare (P): Q = a + bP, based on 16 past observations of P & Q. # Seats sold per flight Average Fare Here, the best linear equation turns out to be: Q = 479 – 1.63P. Best-fit Line

2. REGRESSION ANALYSIS 4.2 More generally, Multiple Regression allows for multiple explanatory variables: Q = a + bP + cP  + dY. The power of Multiple Regression: Even with multiple variables that simultaneously influence sales, it ’ s able to estimate the separate variable influences (i.e. coefficients). Important Regression Statistics 1. R 2 (ranging between 0 and 1) measures the proportion of variation in Q explained by the right-hand side variables. 2. F = R 2 /(k - 1) (1 - R 2 )/(N - k) The F stat (larger F better) indicates the statistical significance (or lack thereof) of the relationship.

3. REGRESSION STATISTICS 4.3 For the estimated OLS equation: Q = 28.8 - 2.12P + 1.03P  + 3.09Y, (.34) (.47) (1.00) R 2 =.78 (78% of Q ’ s variation explained) and F = 13.86 (well above 95% significance threshold). Standard Errors and t-stats Each coefficient ’ s standard error (its standard deviation) measures the uncertainty around its estimate. The coefficient ’ s 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.

4. The standard error of the regression measures the uncertainty around the forecast of Q. A 95% confidence interval around the forecast is: ± 2 standard errors. REGRESSION ISSUES 4.4 1. 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?

5. 4.5 CHOOSING A REGRESSION EQUATION 1. 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?

6. 4.6 FORECASTING DEMAND Time-Series Models indentify 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, Q t = a + bt 2. Quadratic, Q t = a + bt + ct 2 3. Exponential, Q t = br t, estimated as log(Q t ) = log(b) + log(r)t

7. 4.7 FORECASTING DEMAND Barometric Models indentify patterns among different variables over time. Movements in “ Leading Indicators ” predict future changes in economic activity. The Index of Leading Indicators 1.Weekly manufacturing hours 2. Manufacturer ’ s New Orders 3. Changes in Unfilled Orders 4. Plant and Equipment Orders 5.Number of Housing Building Permits 6. Changes in Sensitive Materials Prices 7. Percentage of firms receiving Slower Deliveries 8. Growth in the Money Supply 9. Index of Consumer Confidence 10. The S&P 500 Index 11. Weekly Claims for Unemployment Insurance