Download 5 - NYU Stern

C22.0103 SPRING 2012 Homework Set 5
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1. The value of a certain asset is currently at $50,000. The proportional weekly changes
in value have been modeled as lognormal with  = 0.0024 and  = 0.0044. (This means
P
that log t is assumed to be normally distributed with mean  = 0.0024 and  = 0.0044.)
Pt 1
(a)
(b)
(c)
(d)
Find the probability that this asset will be worth more than $55,000 after one year
(52 weeks). The logarithms here are base-e, as are all logarithms that follow. In
anticipation of parts (b) through (d), it would help to solve this first in terms of
general  and . If you’re facile with Excel, this gets much easier.
Repeat the calculation with  = 0.0010 and  = 0.0044.
Repeat the calculation with  = 0.0024 and  = 0.0080.
Repeat the calculation with  = 0.0010 and  = 0.0080.
2. Consider a set of (x, y) data in which
n = 100
 xi = 500
 xi2 = 2,628
 yi = 4,400
 yi2 = 196,744
 xi yi = 22,396
Find the fitted equation resulting from the regression of Y on x. Then obtain s, the
estimate of the noise standard deviation. (The symbol s is also used for this purpose.)
This will be the only time that you’ll be asked to do the regression work from summary
statistics.
3. The file M\HEATING.MTP deals with the heating bill for dwelling units of various
numbers of rooms.
(a)
(b)
(c)
(d)
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What are the names of the variables in this data set?
Obtain a scatterplot of the two variables. Which variable should be on the
horizontal axis?
Find the linear regression equation resulting from the regression of FUELBILL on
ROOMS.
Examine the residual-fitted-plot from this regression. Is there a tendency for
larger dwelling units to have more variable heating bills? In Minitab, this plot is
available in Stat  Regression  Regression; use Graphs  Residuals
versus fits.
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C22.0103 SPRING 2012 Homework Set 5
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4. The expanding residuals seen in the residual-versus-fitted plot of the previous problem
suggest a failure of the regression model. The usual corrective action is to replace the
dependent variable, here FUELBILL, with its logarithm. Take this action and repeat the
regression. Note the revised fitted model equation and give the new residual-versusfitted plot. Finally, compare the value of R2 for the two versions.
NOTE: You can use any base for logarithm, but base e is recommended. In
Minitab, you would use Calc  Calculator, and the function name is LOGE.
5. MBS11, problem 10.16, page 573. It might help to examine a scatterplot. The data
file can be obtained at www.stern.nyu.edu/~gsimon/statdata; look for the McClaveBenson-Sincich set and click on Exercises and then on DIAMONDS. (You can do a
right-click and then Save Link As to put a link on the Windows desktop.) A residual
versus fitted plot for these data will be requested in the next problem.
6. The analysis of the previous data set will not be very satisfying. Please address the
following issues.
(a)
Show the residual versus fitted plot for the DIAMOND data.
(b)
The residuals are more spread at the right, and this suggests that PRICE should
be replaced by LnPRICE. Use Minitab to create the variable LnPRICE, do the
regression of LnPRICE on CARAT and give the residual versus fitted plot.
Note the fitted regression line and the values of R2 and s .
NOTE: You can create new variables (columns) in Minitab through the
command Calc  Calculator. The Minitab function for natural
logarithm is LOGE.
(c)
You are suspicious that there might be a little additional curvature. Use Minitab
to create the variable C2 = CARAT  CARAT. Then do the regression of
LnPRICE on (CARAT, C2). Note the fitted equation, R2, s , and give the
residual versus fitted plot.
(d)
Which do you prefer, the results of (c) or the results of (d)?
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Why?
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