# Forecasting (moving average/ exponential smoothing) problems

Q17.26 (pp661).

The dean of a law school has developed a regression equation for

estimating the starting salary (thousands of dollars) of a new graduate on the basis of two independent variables:

x1 = the score on the Law School Admission Test (LSAT) at the time of

the application, and

x2 = whether the new graduate's position is in the private sector (coded

as 1) or the public sector (coded as 0).

(a) For the estimation equation, ŷ = 25 + 0.1 x1 + 30 x2, interpret

the partial regression coefficients.

(b) Estimate the starting salary for a new graduate with an LSAT

score of 160 who is entering the private sector.

Q17.32(p664).

Following data transformation, regression analysis results in the

estimation equation: log ŷ = 3.15 + 0.473 log x1 + 30 x2,

Transform this equation into to the equivalent multiplicative model

with estimation equation: ŷ =

[ The base of the logarithms is 10. ]

Q18.4 (p691).

The trend equation ŷ = 1200 + 35x has been fitted to a time series for industry worker days lost due to job related injuries. If x=1 for 1991, estimate the number of worker days lost during 2008.

Q18.9 The following data show residential and commercial natural gas consumption (quadrillion BTU) from 1985 through 2000

Year Consumption

Year Consumption

1985 27.6 1993 30.6

1986 26.9 1994 30.8

1987 27.6 1995 31.6

1988 28.9 1996 33.0

1989 29.4 1997 33.0

1990 29.2 1998 32.8

1991 30.1 1999 35.8

1992 29.6 2000 37.4

(a)Put a three-year centered moving average over the original series.

Year Consumption

Three-Year Moving

Total Three-Year Moving

Average

1985 27.6

1986 26.9

1987 27.6

1988 28.9

1989 29.4

1990 29.2

1991 30.1

1992 29.6

1993 30.6

1994 30.8

1995 31.6

1996 33.0

1997 33.0

1998 32.8

1999 35.8

2000 37.4

(b) Put a five-year centered moving average over the original series.

Year Consumption

Five-Year Moving

Total Five-Year Moving

Average

1985 27.6

1986 26.9

1987 27.6

1988 28.9

1989 29.4

1990 29.2

1991 30.1

1992 29.6

1993 30.6

1994 30.8

1995 31.6

1996 33.0

1997 33.0

1998 32.8

1999 35.8

2000 37.4

(c) Why is the moving average "smoother" when N =5?

18.30 When exponential smoothing is used in fitting a curve to a time series, the approach is slightly different from its application to forecasting. Compare the appropriate formulas and point out how they differ.

18.38 The following data are the wellhead prices for domestically produced natural gas, in dollars per thousand cubic feet, from 1987 through 1994. Given these data and the trend equations shown here, use the MAD criterion to determine which equation is the better fit. Repeat the evaluation using the MSE criterion.

Year x = Year Code y = Price

1987 1 $1.67

1988 2 $1.69

1989 3 $1.69

1990 4 $1.71

1991 5 $1.64

1992 6 $1.80

1993 7 $2.09

1994 8 $2.27

18.42 When autocorrelation of the residuals is present, what effect can this have on interval estimation and significance tests regarding the regression model involved?

18.43 What is the Durbin-Watson test for autocorrelation, and how can it be useful in evaluating the relevance of a given regression model that has been fitted to a set of time series data?

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#### Solution Preview

Q17.26 (pp661).

The dean of a law school has developed a regression equation for

estimating the starting salary (thousands of dollars) of a new graduate on the basis of two independent variables:

x1 = the score on the Law School Admission Test (LSAT) at the time of

the application, and

x2 = whether the new graduate's position is in the private sector (coded

as 1) or the public sector (coded as 0).

(a) For the estimation equation, ŷ = 25 + 0.1 x1 + 30 x2, interpret

the partial regression coefficients.

The partial regression coefficient for LSAT score (X1) of 0.1 implies that, on average, with a unit increase in the LSAT exam score leads to an increase of $0.1 (thousand of dollars) in starting salary.

The partial regression coefficient for graduate's position (X2) of 30 implies that, on average, if a graduate's position is 1 (i.e. private sector), the starting salary is $30 (thousands of dollars) higher than graduates with position 0 (i.e. public sector).

(b) Estimate the starting salary for a new graduate with an LSAT

score of 160 who is entering the private sector.

Input X1= 160, and X2= 1 in the equation:

= 25 + 0.1*160 + 30*1

= $71 (thousands of dollars)

Q17.32(p664).

Following data transformation, regression analysis results in the

estimation equation: log ŷ = 3.15 + 0.473 log x1 + 30 x2,

Transform this equation into to the equivalent multiplicative model

with estimation equation: ŷ =

[ The base of the logarithms is 10. ]

We know that -- (i) can also be expressed as log = log b0 + b1 log x1 + b2 log x2 -- (ii).

Therefore, by comparing (ii) with the given equation log = 3.15 + 0.473 log x1 + 0.354 log x2 , we know

log b0 = 3.15 => b0 = ...

#### Solution Summary

The expert examines the moving average/ exponential smoothing calculations on various problems were performed.