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Session 4. Outline for Session 4. Summary Measures for the Full Model Top Section of the Output Interval Estimation More Multiple Regression Movers Nonlinear Regression Insurance. Top Section: Summary Statistics. Top Section: Summary Statistics.

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outline for session 4
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. Juran

slide3

Top Section: Summary Statistics

Applied Regression -- Prof. Juran

slide6

Top Section: Summary Statistics

Applied Regression -- Prof. Juran

slide8
As stated earlier R2 is closely related to the correlation between X and Y, indeed

Furthermore, R2 , andthus rX,Y , is closely related to the slope of the regression line via

Thus, testing the significance of the slope, testing the significance of R2 and testing the significance of rX,Y are essentially equivalent.

Applied Regression -- Prof. Juran

interval estimation
Interval Estimation

Applied Regression -- Prof. Juran

an image of the residuals
An Image of the Residuals

Y

yi

(xi , yi)

X

xi

(xi , yi)

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.

Applied Regression -- Prof. Juran

regression and prediction
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 x0. The predicted value, , is

Applied Regression -- Prof. Juran

slide18
Under our assumptions this is an unbiased estimate of Y given that x=x0 ,regardless of the value of x0.

Let 0 = E(Y(x0)) and thus, since the estimate is unbiased, 0 = b0 + b1x0.

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 = x0 is

Applied Regression -- Prof. Juran

slide19
From these facts it follows that a 2-sided “confidence” interval on the expected value of Y given x= x0, 0, is given by

Applied Regression -- Prof. Juran

slide20
A 2-sided “prediction”interval on future individual values of Y given x = x0, y0, is given by

Applied Regression -- Prof. Juran

slide21

Confidence Interval on E(Y(x0))

Prediction Interval on Y(x0)

Applied Regression -- Prof. Juran

slide22
Note that both of these intervals are parabolic functions in x0, have their minimum interval width at x0 = , and their widths depend on and on Sxx

The sum of squared x term appears so often in regression equations that it is useful to use the abbreviation Sxx. Note that Sxx can easily be obtained from the variance as computed in most spreadsheets or statistics packages.

Applied Regression -- Prof. Juran

all around movers
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. Juran

slide28

Summary Statistics

Applied Regression -- Prof. Juran

slide33

The Most Obvious Simple Regression

Applied Regression -- Prof. Juran

slide35

An Alternative Simple Regression Model

Applied Regression -- Prof. Juran

slide37

A Multiple Regression Model

Applied Regression -- Prof. Juran

preliminary observations
Preliminary Observations
  • 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.

Applied Regression -- Prof. Juran

preliminary observations1
Preliminary Observations
  • 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).

Applied Regression -- Prof. Juran

slide41

, corresponding to the estimated number of hours for one

Prediction intervals

specific move, given one specific value for the number of rooms.

, corresponding to the estimated population average

Confidence intervals

number of hours over a large number of moves, all with the same number of

rooms.

Applied Regression -- Prof. Juran

slide42

Validity of the Rooms Model

Applied Regression -- Prof. Juran

analysis of the residuals
Analysis of the Residuals

Applied Regression -- Prof. Juran

comments on the rooms model
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. Juran

a non linear model

=

B

Note: If

, then

(A) = B.

ln

e

A

A Non-linear Model?

Applied Regression -- Prof. Juran

residual analysis

Histogram of Residuals

Histogram of Residuals

14

12

12

10

10

8

8

Frequency

Frequency

6

6

4

4

2

2

0

0

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

-35

-30

-25

-20

-15

-10

-5

0

5

10

15

20

25

30

35

Residual Error

Residual Error

Linear Model

Exponential Model

Residual Analysis

Applied Regression -- Prof. Juran

slide51

Residual Errors vs. Predictions

35

30

25

20

15

Errors (Hours)

10

5

0

-5

-10

-15

0

10

20

30

40

50

60

Predicted Hours

Linear Model

Residual Errors vs. Predictions

30

25

20

15

10

Errors (Hours)

5

0

-5

-10

-15

0

10

20

30

40

50

60

-20

Predicted Hours

Exponential Model

Applied Regression -- Prof. Juran

slide52

Residual Errors vs. Rooms

35

30

25

20

15

10

Errors (Hours)

5

0

-5

-10

-15

0

1

2

3

4

5

6

Rooms

Linear Model

Residual Errors vs. Rooms

30

25

20

15

10

5

Errors (Hours)

0

-5

-10

-15

0

1

2

3

4

5

6

-20

Rooms

Exponential Model

Applied Regression -- Prof. Juran

conclusions
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. Juran

insurance case
Insurance Case

Applied Regression -- Prof. Juran

insurance case1
Insurance Case

Applied Regression -- Prof. Juran

insurance case2
Insurance Case
  • The regression with exponential equation has a higher R2.
  • 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.

Applied Regression -- Prof. Juran

insurance case3
Insurance Case

Applied Regression -- Prof. Juran

insurance case4
Insurance Case

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 = 1.387481.

The implied book value is $2.5 billion times P/B multiple of 1.387481 = an estimated total value of $3.4687 billion.

Applied Regression -- Prof. Juran

insurance case5
Insurance Case
  • 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?

Applied Regression -- Prof. Juran

insurance case6
Insurance Case

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 $0.044527 billion higher than if Abe paid in stock.

Applied Regression -- Prof. Juran

insurance case7
Insurance Case

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?

Applied Regression -- Prof. Juran

insurance case8
Insurance Case

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 1.189957x.

Applying that to total book value of $11.5 billion, we would get an implied total value of $13.68451 billion. Adding in the $5 billion cost of the proposed acquisition, we would get an adjusted value for Abe of $18.68451 billion.

In both the scenarios described in question 3 (stock and cash), the pro-forma total value would be LESS than $18.68451 billion. Thus, NO incremental value is created.

(The exact result will vary depending on which model you use.)

Applied Regression -- Prof. Juran

insurance case9
Insurance Case

Applied Regression -- Prof. Juran

insurance case10
Insurance Case

Applied Regression -- Prof. Juran

insurance case11
Insurance Case

Applied Regression -- Prof. Juran

summary
Summary
  • Summary Measures for the Full Model
    • Top Section of the Output
    • Interval Estimation
  • More Multiple Regression
    • Movers
  • Nonlinear Regression
    • Insurance

Applied Regression -- Prof. Juran

for session 5
For Session 5
  • Cigarettes
    • Do a full multiple regression model of the cigarette data, and answer the questions:

www.ilr.cornell.edu/~hadi/RABE/Data/P081.txt

  • Cars
    • Do a multiple regression model of the cars data
    • Just quantitative independent variables; we’ll talk next time about the qualitative ones

Applied Regression -- Prof. Juran