Session 4

Applied Regression -- Prof. Juran2 Outline for Session 4 Summary Measures for the Full Model –Top Section of the Output –Interval Estimation More Multiple Regression –Movers Nonlinear Regression –Insurance

Applied Regression -- Prof. Juran3 Top Section: Summary Statistics

Applied Regression -- Prof. Juran4

5

6 Top Section: Summary Statistics

Applied Regression -- Prof. Juran7

8 As stated earlier R 2 is closely related to the correlation between X and Y, indeed Furthermore, R 2, and thus r X,Y, is closely related to the slope of the regression line via Thus, testing the significance of the slope, testing the significance of R 2 and testing the significance of r X,Y are essentially equivalent.

Applied Regression -- Prof. Juran9

10

Applied Regression -- Prof. Juran11

Applied Regression -- Prof. Juran12

Applied Regression -- Prof. Juran13

Applied Regression -- Prof. Juran14

Applied Regression -- Prof. Juran15 Interval Estimation

Applied Regression -- Prof. Juran16 An Image of the Residuals xixi yiyi (x i, y i ) X Y The observed values: The fitted values: The residuals: Recall: The regression line passes through the data so that the sum of squared residuals is as small as possible. (x i, y i )

Applied Regression -- Prof. Juran17 Regression and Prediction Regression lines are frequently used for predicting future values of Y given future, conjectural or speculative values of X. Suppose we posit a future value of X, say x 0. The predicted value,, is

Applied Regression -- Prof. Juran18 Under our assumptions this is an unbiased estimate of Y given that x=x 0, regardless of the value of x 0. Let  0 = E ( Y(x 0 )) and thus, since the estimate is unbiased,  0 = b 0 + b 1 x 0. However, be alert to the fact that this estimate (prediction) of a future value has a standard error of Furthermore, the standard error of the prediction of the expected (mean) value of Y given x = x 0 is

Applied Regression -- Prof. Juran19 From these facts it follows that a 2-sided “confidence” interval on the expected value of Y given x= x 0,   , is given by

Applied Regression -- Prof. Juran20 A 2-sided “prediction”interval on future individual values of Y given x = x 0, y  , is given by

Applied Regression -- Prof. Juran21 Confidence Interval on E ( Y ( x 0 )) Prediction Interval on Y ( x 0 )

Applied Regression -- Prof. Juran22 Note that both of these intervals are parabolic functions in x 0, have their minimum interval width at x 0 =, and their widths depend on and on S xx The sum of squared x term appears so often in regression equations that it is useful to use the abbreviation S xx. Note that S xx can easily be obtained from the variance as computed in most spreadsheets or statistics packages.

Applied Regression -- Prof. Juran23 An Image of the Prediction and Confidence Intervals

Applied Regression -- Prof. Juran24

Applied Regression -- Prof. Juran25

Applied Regression -- Prof. Juran26

Applied Regression -- Prof. Juran27 All-Around Movers The management question here is whether historical data can be used to create a cost estimation model for intra-Manhattan apartment moves. The dependent variable is the number of labor hours used, which is a proxy for total cost in the moving business. There are two potential independent variables: volume (in cubic feet) and the number of rooms in the apartment being vacated.

Applied Regression -- Prof. Juran28 Summary Statistics

Applied Regression -- Prof. Juran29

Applied Regression -- Prof. Juran30

Applied Regression -- Prof. Juran31

Applied Regression -- Prof. Juran32

Applied Regression -- Prof. Juran33 The Most Obvious Simple Regression

Applied Regression -- Prof. Juran34

Applied Regression -- Prof. Juran35 An Alternative Simple Regression Model

Applied Regression -- Prof. Juran36

Applied Regression -- Prof. Juran37 A Multiple Regression Model

Applied Regression -- Prof. Juran38

Applied Regression -- Prof. Juran39 Volume is the best single predictor, but perhaps not useful if customers are to be expected to collect these data and enter them on a web site. Rooms is a pretty good predictor (not as good as Volume), and may be more useful on a practical basis. Preliminary Observations

Applied Regression -- Prof. Juran40 The multiple regression model makes better predictions, but not much better than either of the simple regression models. The multiple regression model has problems with multicollinearity. Notice the lack of significance for the Rooms variable (and the strange coefficient). Preliminary Observations

Applied Regression -- Prof. Juran41 Prediction intervals, corresponding to the estimated number of hours for one specific move, given one specific value for the number of rooms. Confidence intervals, corresponding to the estimated population average number of hours over a large number of moves, all with the same number of rooms.

Applied Regression -- Prof. Juran42 Validity of the Rooms Model

Applied Regression -- Prof. Juran43 Analysis of the Residuals

Applied Regression -- Prof. Juran44

Applied Regression -- Prof. Juran45 Comments on the Rooms Model Good explanatory power Statistically Significant Points fit the line well But… Small apartments tend to be over-estimated Large apartments tend to be badly estimated, especially on the high side Maybe could use more data Maybe nonlinear

Applied Regression -- Prof. Juran46 A Non-linear Model? Note: If Ae B , then ln ( A ) = B.

Applied Regression -- Prof. Juran47

Applied Regression -- Prof. Juran48

Applied Regression -- Prof. Juran49

Applied Regression -- Prof. Juran50 Residual Analysis Histogram of Residuals Residual Error Frequency Histogram of Residuals Residual Error Frequency Linear Model Exponential Model

Applied Regression -- Prof. Juran51 Residual Errors vs. Predictions Predicted Hours Errors (Hours) Linear Model Residual Errors vs. Predictions Predicted Hours Errors (Hours) Exponential Model

Applied Regression -- Prof. Juran52 Residual Errors vs. Rooms Rooms Errors (Hours) Linear Model Residual Errors vs. Rooms Rooms Errors (Hours) Exponential Model

Applied Regression -- Prof. Juran53 Conclusions Regression analysis is technically easy Creating a reliable model is subject to creativity and judgment The Rooms model (either linear or otherwise) is reasonably useful for this managerial application The most serious estimation problem is when we try to make predictions for large apartments. What about a separate model for very large apartments?

Applied Regression -- Prof. Juran54

Insurance Case Applied Regression -- Prof. Juran55

Insurance Case Applied Regression -- Prof. Juran56

Insurance Case Applied Regression -- Prof. Juran57 The regression with exponential equation has a higher R 2. One "real world" explanation: companies that generate very high ROAEs will be rewarded with higher valuation multiples The relationship might be exponential as opposed to linear because an investment will compound at this higher ROAE. The primary driver for this is that Duck is an outlier in both dimensions – it has a VERY high P/B and ROAE.

Insurance Case Applied Regression -- Prof. Juran58

Applied Regression -- Prof. Juran59

Insurance Case Applied Regression -- Prof. Juran60 What is the implied P/B multiple and implied total value of Circle? Using the following equation to calculate the implied P/B multiple: Plugging in 14.2 for x, we get y = The implied book value is $2.5 billion times P/B multiple of = an estimated total value of $ billion.

Insurance Case Applied Regression -- Prof. Juran61 3. Abe has announced that it will be making an acquisition. It is trying to decide whether to pay in stock or in cash. a. If Abe pays with stock, the pro-forma ROAE of the combined company will be 12.2% and the pro-forma book value will be $16.5 billion. What is the implied P/B multiple and implied total value of the pro-forma company? b. If Abe pays with cash, the pro-forma ROAE of the combined company will be 15.5% and the pro-forma book value will be $11.5 billion. What is the implied P/B multiple and implied total value of the pro-forma company? c. If the goal is to maximize the pro-forma total value of the new company, how should Abe pay for the acquisition?

Insurance Case Applied Regression -- Prof. Juran62 Depending on which version of the equation we use, there are several possible results for the estimate P/B of the new company: Abe should pay in cash, since the total value would be $ billion higher than if Abe paid in stock.

Insurance Case Applied Regression -- Prof. Juran63 4. Assume that before the acquisition, Abe has a book value of $11.5 billion and an ROAE of 12.8%. Abe will either issue $5 billion in stock or use $5 billion in cash to complete the acquisition. What incremental value, if any, is created in both the stock and cash scenarios described above?

Insurance Case Applied Regression -- Prof. Juran64 Abe's total value before the acquisition is determined by taking its ROAE of 12.8% and applying the regression equation, to get an implied P/B multiple of x. Applying that to total book value of $11.5 billion, we would get an implied total value of $ billion. Adding in the $5 billion cost of the proposed acquisition, we would get an adjusted value for Abe of $ billion. In both the scenarios described in question 3 (stock and cash), the pro- forma total value would be LESS than $ billion. Thus, NO incremental value is created. (The exact result will vary depending on which model you use.)

Insurance Case Applied Regression -- Prof. Juran65

Insurance Case Applied Regression -- Prof. Juran66

Insurance Case Applied Regression -- Prof. Juran67

Applied Regression -- Prof. Juran68 Summary Summary Measures for the Full Model –Top Section of the Output –Interval Estimation More Multiple Regression –Movers Nonlinear Regression –Insurance

Applied Regression -- Prof. Juran69 For Session 5 Cigarettes –Do a full multiple regression model of the cigarette data, and answer the questions: Cars –Do a multiple regression model of the cars data –Just quantitative independent variables; we’ll talk next time about the qualitative ones