DSCI 5340: Predictive Modeling and Business Forecasting Spring 2013 – Dr. Nick Evangelopoulos

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DSCI 5340: Predictive Modeling and Business Forecasting

Spring 2013 – Dr. Nick Evangelopoulos

Lecture 3: Time Series Regression (Ch. 6)

Material based on:Bowerman-O’Connell-Koehler, Brooks/Cole

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DSCI 5340FORECASTING

Review of Homework in Textbook

Ex 4.20 Page 210

Ex 4.21 Page 212

Ex 5.5 Page 266

EX 5.10 Page 268

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Insurance Innovation Data Ex 4.20 Page 210

4.20 part a. Since there are two parallel lines – one for Mutual and one for Stock, a dummy variable can show the difference in the intercepts of the models.

Y = 0 + 1X + 2DS +

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4.20 part bY = 0 + 1X + 2DS +

a,m = 0 + 1a + 2(0) for X = a assets, and m= Mutual type of company (D = 0)

a,s = 0 + 1a + 2(1) for X = a assets, and m= Stock type of company (D = 1)

a,m - a,s = 2

2 is the difference between the mean of number of months pasted (y) for a mutual type of company and a stock type of company.

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Since the p-value for 2 is 3.74187E-05, reject H0: 2 = 0, and conclude that 2 is not equal to 0.

95% CI for 2 is 4.98 to 11.13, which does not include 0. Type of Firm is a significant variable in predicting Number of months elapsed at both a significance level of 5% and 1%.

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Ex 4.20 part d - Interaction

Since the data indicate that the lines for the two type of firms are parallel.

A p-value of .9821 is less than any reasonable alpha level. So the beta coefficient for xDs cannot be assumed to be nonzero.

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Ex 4.21 page 212

a. Y = 0 + 1DM + 2DT +

For Bottom Shelf, DM = 0 and DT = 0 which implies:

B = 0 + 1(0) + 2(0) = 0.

For Middle Shelf, DM = 1 and DT = 0 which implies:

M = 0 + 1(1) + 2(0) = 0+ 1.For Top Shelf, DM = 0 and DT = 1 which implies: T = 0 + 1(0) + 2(1) = 0+ 2.

M could be used to represent 1

T could be used to represent 2

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Ex 4.21 part b

4.21 part b. Note if M and T are equal to zero, then:

B = 0, M = 0+ M = 0+ 0, and T = 0 + 2 = 0+ 0.

Thus, H0:M =0 and T = 0 implies H0:B = M = T .

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Ex 4.21 part c

Since B = 0, M = 0+ M, and T = 0+ T, we can solve for 0, M, and T .

Therefore, M=M -B, T = T -B,

and M-T = M - T .

Note that t(.025, df=15) = 2.131 (see table on page 593).

95% CI for M is 21.4 +/– 2.131*1.433, which is 18.35 to 24.45.

95% CI for T is -4.30 +/– 2.131*1.433, which is -7.35 to -1.25.

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Ex 4.21 part d page 213

Note that the Fit is 77.2 which corresponds to the mean of the Middle Shelf sales. Thus the output at the bottom of the Analysis of Variance is for a 95% CI and 95% PI for mean sales when using a middle display height.

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Ex 4.21 part e.

Note that in part c, we were not able to get a confidence interval on M - T since it was equal to M-T . However, if the following model is used: Y = 0 + 1DB + 2DM + then

M - T is equal to M since the Top Shelf is now the reference group.

Note that t(.025, df=15) = 2.131 (same as before).

95% CI for M - T (note equal to M)is 25.7 +/– 2.131*1.433, which is 22.65 to 28.75.

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Page 266 Ex 5.5 - outliers

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EX 5.10 part a, Page 268

Y* = 0 + 1X + where Y* = ln(Y).

Prediction point for 7 desktop computers and 95% PI for Y* is 5.0206 and 4.3402 to 5.7010.

Prediction point for 7 desktop computers and 95% PI for Y is exp(5.0206) = 151.5 and exp(4.3402) = 76.72 to exp(5.7010) = 299.166.

Note that putting a “.” for Y* in the data with an X = 7 will provide a prediction interval and predicted value for this value in SAS.

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EX 5.10 part b, Page 268

There are a couple of small residuals at -.59979.

It may be possible to remove one of these residuals at a time or to try adding a square term to the model.

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Chapter 6 Polynomial Fits

Use higher order terms when curvature exists in graph of y and x. Typically, x is time and square and cubic terms are added to increase the R square.

Interactions can also be formed with higher order terms.

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Requirements for Fitting a pth-Order Polynomial Regression Model

1. The number of levels of x must be greater than or equal to (p + 1).

2. The sample size n must be greater than (p + 1) to allow sufficient degrees of freedom for estimating 2.

ppxxxy ...2

210

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Count the number of times that a curve changes directions.

A polynomial fit would have the highest order term be equal to one minus the number of times the curve changes directions. What degree polynomial would you use here?

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The use of a model outside its range is dangerous (although sometimes unavoidable).

GN

P (

y)

Inflation Rate (x,%)

Extrapolation

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xy 10

Line Tending Upward: 1 > 0Curve Tending Downward: 1 < 0

Trend and coefficient sign

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2210 xxy

Holds Water: 2 > 0

Does Not Hold Water: 2 < 0

Curvature and coefficient sign

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x

y1

10

Curve Tending Upward: 1 < 0

Curve Tending Downward: 1 > 0

Inverse relationship

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xey 10

Curve Tending Upward: 1 < 0Curve Tending Downward: 1 > 0

Exponential curve

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S Curve

y = exp(0 + 1(1/x) + )

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Line Tending Upward: b1 > 0Curve Tending Downward: b1 < 0

xy 10)log(

Logarithmic transformation for Y

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)log(10 xy

Curve Tending Upward: b1 > 0Curve Tending Downward: b1 < 0

Logarithmic transformation for X

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Examples of autocorrelation in residuals

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Detecting Autocorrelation

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Detecting Positive Autocorrelation

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Detecting Negative Autocorrelation

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Rules of thumb for DW

If DW is close to 2 then there is no autocorrelation.

If DW is close to 0 then there is positive autocorrelation.

If DW is close to 4 then there is negative autocorrelation.

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Modeling Seasonal Factor with Dummy Variables

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Trigonometric Models

Model two is for increasing variation cyclically.

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Autoregressive errors

Use Proc ARIMA for a First Order Autoregressive Process for the Error Term

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Prediction Intervals for Autoregressive Models

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Page 318 Ex 6.3Page 318 Ex 6.4

Homework in Textbook

Documents

DSCI 5340: Predictive Modeling and Business Forecasting Spring 2013 – Dr. Nick Evangelopoulos