# Time Series Problem

(See attached file for full problem description)

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Let {Z(s), s } is a stationary process. Define

Suppose (Z(s1),....,Z(sn)) is observed at know locations {s1,....,sn}. Please find an optimal linear prediction of Z(s) based on the observations and function. How would you estimate function from the observations?

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https://brainmass.com/statistics/multivariate-time-series-and-survival-analysis/time-series-problem-46183

#### Solution Summary

There is an attached solution of the estimation function from the observations. The solution is a step by step time series solution.

Statistics problems 14.5, 14.7, 14.25, 14.41, 14.43, 14.49, 16.7, 16.13, 16.15, 16.47

See attached word file for proper format of tables. See attached data files.

Multiple Regression & Time-Series Forecasting

14.5

A consumer organization wants to develop a regression model to predict mileage (as measured by miles per gallon) based on the horsepower of the car's engine and the weight of the car (in pounds). Data were collected from a sample of 50 recent car models, and the results are organized and stored in Auto.

A. State the multiple regression equation

B. Interpret the meaning of the slopes, b¹ and b² in this problem

C. Explain why the regression coefficient, bº, has no practical meaning in the context of this problem

D. Predict the mile per gallon for cars that have 60 horsepower and weigh 2,000 pounds

E. Construct a 95% confidence interval estimate for the mean miles per gallon for cars that have 60 horsepower and weigh 2,000 pounds

F. Construct a 95% prediction interval for the miles per gallon for an individual car that has 60 horsepower and weighs 2,000 pounds

14.7

The business problem facing the director of broadcasting operations for a television station was the issue of standby hours (i.e. hours in which unionized graphic artists at the station are paid but are not actually involved in any activity) and what factors were related to standby hours. The study included the following variables:

Standby hours (Y)-Total number of standby hours in a week

Total staff present (X¹)-Weekly total of people-days

Remote hours (X²)-Total number of hours worked by employees at locations away from the central plant

Data were collected for 26 weeks; these data are organized and stored in Standby.

A. State the multiple regression equation

B. Interpret the meaning of the slopes, b¹ and b², in this problem

C. Explain why the regression coefficient, bº , has no practical meaning in the context of this problem

D. Predict the standby hours for a week in which the total staff present have 310 people-days and the remote hours are 400

E. Construct a 95% confidence interval estimate for the mean standby hours for weeks in which the total staff present have 310 people-days and the remote hours are 400.

F. Construct a 95% prediction interval for the standby hours for a single week in which the total staff present have 310 people-days and the remote hours are 400

14.25

Use the following results:

Variable Coefficient Standard Error t Statistic p Value

INTERCEPT -0.02686 0.06905 -0.39 0.7034

FOREIMP 0.79116 0.06295 12.57 0.0000

MIDSOLE 0.60484 0.07174 8.43 0.0000

A. Construct a 95% confidence interval estimate of the population slope between durability and forefoot shock-absorbing capability

B. At the 0.05 level of significance, determine whether each independent variable make a significant contribution to the regression model. On the basis of these results, indicate the independent variables to include in this model.

14.41

The marketing manager of a large supermarket chain faced the business problem of determining the effect on the sales of pet food of shelf space and whether the product was placed at the front (=1) or back (=0) of the aisle. Data are collected from a random sample of equal-sized stores. The results are shown in the following table (and organized and stored in Petfood):

Store Shelf Space (Feet) Location Weekly Sales (Dolllars)

1 5 Back 160

2 5 Back 220

3 5 Back 140

4 10 Back 190

5 10 Back 240

6 10 Front 260

7 15 Back 230

8 15 Back 270

9 15 Front 280

10 20 Back 260

11 20 Back 290

12 20 Front 310

For (a) through (m), do not include an interaction term.

A. State the multiple regression equation that predicts sales based on shelf space and location.

B. Interpret the regression coefficients in (a).

C. Predict the weekly sales of pet food for a store with 8 feet of shelf space situated at the back of the aisle. Construct a 95% confidence interval estimate and a 95% prediction interval.

D. Perform a residual analysis on the results and determine whether the regression assumptions are valid.

E. Is there a significant relationship between sales and the two independent variables (shelf space and aisle position) at the 0.05 level of significance?

F. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data.

G. Construct and interpret 95% confidence interval estimates of the population slope for the relationship between sales and shelf space and between sales and aisle location.

H.

I. Compute and interpret the meaning of the coefficient of multiple determination, r

J. Compute and interpret the adjusted r²

K.

L. Compute the coefficients of partial determination and interpret their meaning

M. What assumption about the slop of shelf space with sales do you need to make in this problem?

N. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model

O. On the basis of the results of (f) and (n), which model is most appropriate? Explain

14.43

The owner of a moving company typically has his most experienced manager predict the total number of labor hours that will be required to complete an upcoming move. This approach has proved useful in the past, but the owner has the business objective of developing a more accurate method of predicting labor hours. In a preliminary effort to provide a more accurate method, the owner decided to use the number of cubic feet moved and whether there is an elevator in the apartment building as the independent variables and has collected data for 36 moves in which the origin and destination were within the borough of Manhattan in New York City and the travel time was an insignificant portion of the hours worked. The data are organized and stored in Moving. For (a) through (k), do not include an interaction term.

A. State the multiple regression equation for predicting labor hours, using the number of cubic feet moved and whether there is an elevator

B. Interpret the regression coefficients in (a)

C. Predict the labor hours for moving 500 cubic feet in an apartment building that has an elevator and construct a 95% confidence interval estimate and a 95% prediction interval

D. Perform a residual analysis on the results and determine whether the regression assumptions are valid

E. Is there significant relationship between labor hours and the two independent variables (cubic feet moved and whether there is an elevator in the apartment building) at the 0.05 level of significance?

F. At the 0.05 level of significance, determine whether each independent variable makes a contribution to the regression model. Indicate the most appropriate regression model for this set of data.

G. Construct a 95% confidence interval estimate of the population for the relationship between labor hours and cubic feet moved

H. Construct a 95% confidence interval estimate for the relationship between labor hours and the presence of an elevator

I. Compute and interpret the adjusted r²

J. Compute the coefficients of partial determination and interpret their meaning

K. What assumption do you need to make about the slope of labor hours with cubic feet moved?

L. Add an interaction term to the model and, at the 0.05 level of significance, determine whether it makes a significant contribution to the model

M. On the basis of the results of (f ) and (l), which model is most appropriate? Explain

14.49

The director of a training program for a large insurance company has the business objective of determining which training method is best for training underwriters. The three methods to be evaluated are traditional, CD-ROM based, and Web based. The 30 trainees are divided into three randomly assigned groups of 10. Before the start of the training, each trainee is given a proficiency exam that measures mathematics and computer skills. At the end of the training, all students take the same end-of-training exam. The results are organized and stored in Underwriting. Develop a multiple regression model to predict the score on the end-of-training exam, based on the score on the proficiency exam and the method of training used. For (a) through (k), do not include an interaction term.

A. State the multiple regression equation

B. Interpret the regression coefficient in (a).

C. Predict the end-of -training exam score for a student with a proficiency exam score of 100 who had Web-based training

D. Perform a residual analysis on your results and determine whether the regression assumptions are valid

E. Is there a significant relationship between the end-of-training exam score and the independent variables (proficiency score and training method) at the 0.05 level of significance

F. At the 0.05 level of significance, determine whether each independent variable make a contribution to the regression model for this set of data.

G. Construct and interpret 95% confidence interval estimate of the population slope for the relationship between end-of -training exam score and proficiency exam.

H. Construct and interpret 95% confidence interval estimate of the population slope for the relationship between end -of- training exam score and type of training method.

I. Compute and interpret the adjusted r²

J. Compute the coefficients of partial determination and interpret their meaning

K. What assumption about the slope of proficiency score with end-of-training exam score do you need to make in this problem?

L. Add interval terms to the model and, at the 0.05 level of significance, determine whether any interaction terms make a significant contribution to the model

M. On the basis of the results of (f ) and (l), which model is most appropriate? Explain

16.7

The following data (stored in Treasury) represent the three-month Treasury bill rates in the United States from 1991 to 2008:

Year Rate Year Rate

1991 5.38 2000 5.82

1992 3.43 2001 3.40

1993 3.00 2002 1.61

1994 4.25 2003 1.01

1995 5.49 2004 1.37

1996 5.01 2005 3.15

1997 5.06 2006 4.73

1998 4.78 2007 4.36

1999 4.64 2008 1.37

A. Plot the data

B. Fit a three-year moving average to the data and plot the results

C. Using a smoothing coefficient of W = 0.50, exponentially smooth the series and plot the results

D. What is your exponentially smoothed forecast for 2009?

E. Repeat (c) and (d), using a smoothing coefficient of W = 0.25

F. Compare the results of (d) and (e)

16.13

Gross domestic product (GDP) is a major indicator of a nation's overall economic activity. It consist of personal consumption expenditures, gross domestic investment, net experts of goods and services, and government consumption expenditures. The GDP (in billions of current dollars) for the United States from 1980 to 2008 is stored in GDP.

A. Plot the data

B. Compute a linear trend forecasting equation and plot the trend line

C. What are your forecasts for 2009 to 2010?

D. What conclusions can you reach concerning the trend in GDP?

16.15

The data in Strategic represent the amount of oil, in billions of barrels, held in the U.S. strategic oil reserve, from 1981 through 2008.

A. Plot the data

B. Compute a linear trend forecasting equation and plot the trend line.

C. Compute a quadratic trend forecasting equation and plot the results

D. Compute an exponential trend forecasting equation and plot the results

E. Which model is the most appropriate?

F. Using the most appropriate model, forecast the number of barrels, in billions, in 2009. Check how accurate your forecast is by locating the true value for 2009 on the Internet or in your library

16.47

The following data (stored in Credit) are monthly credit card charges (in millions of dollars) for a popular credit card issued by a large bank (the mane of which is not disclosed at its request):

Month 2007 2008 2009

January 31.9 39.4 45.0

February 27.0 36.2 39.6

March 31.3 40.5

April 31.0 44.6

May 39.4 46.8

June 40.7 44.7

July 42.3 52.2

August 49.5 54.0

September 45.0 48.8

October 50.0 55.8

November 50.9 58.7

December 58.5 63.4

A. Construct the time-series plot

B. Describe the monthly pattern that is evident in the data

C. In general, would you say that the overall dollar amounts charged on the bank's credit cards is increasing or decreasing? Explain

D. Note that December 2008 charges were more than $63 million, but those for February 2009 were less than $40 million. Was February's total close to what you would have expected?

E. Develop an exponential trend forecasting equation with monthly components.

F. Interpret the monthly compound growth rate.

G. Interpret the January multiplier

H. What is the predicted value for March 2009?

I. What is the predicted value for April 2009?

J. How can this type of time-series forecasting benefit the bank?