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Regression and Time Series Analysis

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The data in the table below represent the annual revenues(in billion of dollars)
of McDonald's Corporation over the 31-year period from 1975 t0 2005.

Year Coded Yr Revenues
1975 0 1
1976 1 1.2
1977 2 1.4
1978 3 1.7
1979 4 1.9
1980 5 2.2
1981 6 2.5
1982 7 2.8
1983 8 3.1
1984 9 3.4
1985 10 3.8
1986 11 4.2
1987 12 4.9
1988 13 5.6
1989 14 6.1
1990 15 6.8
1991 16 6.7
1992 17 7.1
1993 18 7.4
1994 19 8.3
1995 20 9.8
1996 21 10.7
1997 22 11.4
1998 23 12.4
1999 24 13.3
2000 25 14.2
2001 26 14.9
2002 27 15.4
2003 28 17.1
2004 29 19
2005 30 20.5

a) Calculate a three-year moving average to the data (add a column to the table)

b) Using a smoothing coefficient of W = 0.75, exponentially smooth the series
(add a column to the table, use data analysis to smooth)

c) Plot the results from a) and b) with the time series.

d) Compute a quadratic trend forecasting equation and plot the predicted result with the data against the coded years.

e) Compute an exponential trend forcasting equation and plot the predicted results with the data against the coded years.

f) Compute a second -order autoregressive model, test for the significance of the second-order
autoregressive parameter.

g) Compute a first-order autoregressive model, test for the significance of the first-oder
autoregressive parameter, and plot the predicted results with the data against the coded years.

Pr2 Simple Linear Regression.

The owner of a chain of ice cream store would like to study
the effect of atmosperic temperature on sales during the summer season.
A sample of 21 consecutive days is selected, with the results stored in the table below

Temperature, in degrees Sales, in thousand of $
63 1.52
70 1.68
73 1.8
75 2.05
80 2.36
82 2.25
85 2.68
88 2.9
90 3.14
91 3.06
92 3.24
75 1.92
98 3.4
100 3.28
92 3.17
87 2.83
84 2.58
88 2.86
80 2.26
82 2.14
76 1.98

a) Construct a scatter diagram

b) Using Data Analysis/Regression, find the regression coefficients b0 and b1.

c) Graph the regression line on the scatter diagram

d) Interpret the meaning of b0 and b1 in this problem.

e) Predict the sales per store for a day in which the temperature is 83F.

f) determine the coefficient of determination and interpret its meaning.

g) Graph the residuals in this model and determine
the adequacy of the model (linearity).

h) At the 0.05 level of significance, is there evidence
of a linear relationship between sales and temperature?
(based on t-test or p-value)

Pr3. Anova one factor

In order to test the strength of four brands of trash bags,
one-pound weights were placed into bags, one at a time
until the bag broke.
A total of 40 bags, 10 of each brand, were used. The data in the table below
gives the weight required to break the trash bag.

34 32 33 26
30 42 34 18
40 34 32 20
38 36 40 15
36 32 40 20
30 40 34 20
30 36 36 17
42 43 34 18
36 30 32 19
38 38 34 20

a) Using Data analysis/Anova, test at the level of significance 0.05 for the evidence
of the differences in the mean strength of the four brands of trash bags .

b) If appropriate, determine which brands differ in mean strength

Use Turkey-Kramer procedure.

In Word:

1. The names of the coefficients in the regression equation are __________________________.
2. In the time series the independent variable is __________________________________________
3. The exponential equation in autoregression model is transformed into____________________ by using ____________function.
4. The average of odd number of values calculated for each consecutive cell is called ______________________________.
5. In the autoregressive models the independent variables are________________________________________
6. The multiple regression equation has one _________________ and several _____________________.
7. The regression line fits the points on the scatter diagram that means __________________________________________________________________________________________________
8. A column of the t-table is picked according to the __________________.
9. The symbol used for order of the autoregressive model is ________.
10. The goal of the ANOVA method is to test if ____________ among the________________ are significant.
11. The explained variation is LARGE / SMALL when coefficient of determination is close to one.
12. Periodic pattern on the time series with time period of 2 to10 years is called____________________.
13. In the autoregressive model of the third-order the number of intercepts in the equation is__________________________
14. In the simple regression method the coefficient of determination r^2 is calculated as _________________________________________________.
15. The multiple regression models can be improved by adding _____________________ term.
16. The Tukey-Kramer procedure calculates the _________________________ _____ and compares them with ______________________________________in Q tets.
17. In the ANOVA method the number of factors may be __________________________________.
18. The zero in the confidence interval for slope indicates that the relationship is STRONG / WEAK.
19. No pattern on the plot of the residuals means that the regression model is appropriate to use. TRUE /FALSE.
20. The contribution of a term to a regression model is significant if __________________ of the term is ____________________.

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Solution Summary

The detailed solution is provided for testing of interceptor, slope in regression analysis. Lag (r) test statistics value is calculated in order to decide the existence of auto-correlation and also moving averages, exponential smoothing and other time series techniques are used in this solution. Attached in Word and Excel.

See Also This Related BrainMass Solution

Multiple Regression Analysis, Time Series Analysis

See attached data file.


One day, after reporting the performance of the company to the shareholders, the CEO of A. Fictitious & Co. decided that he would like to quantify the impact of the company's expenditures has on how much sales it generates. In other words, he would like to know if the company increases the amount spent on marketing by one dollar, how large of an increase (or decrease) in sales would be expected? The major categories of expenditures and how much was spent in each are known for the company, as is the total sales generated per quarter for the last five years. He has the data file with the relevant data sent to you, and asks you to do the multiple-regression analysis to find out the answer to his questions. Oh, and he also asks you to do a time-series analysis on the total sales per quarter and forecast the amount of sales expected in the future.

Part I. Multiple Regression:

1. Look over the expenditure categories that the CEO gave you. Check to see if there interaction between the category Capital Equipment and Materials, and the category Salary and Benefits. Namely, do a multiple regression model with quarterly sales as the y-variable and the four expenditure categories given in the Data set as the x-variables. Do a second multiple regression model with four expenditure categories plus an interaction term between the category Capital Equipment and Materials, and the category Salary and Benefits. After comparing the two models, which is the better model? Can you conclude whether the two categories are independent of each other?

2. Based on your analysis in Question 1, write down the best-fit multiple regression equation for this problem with quarterly sales as the y-variable and the expenditure categories as the x-variables; do not forget the interaction term if there is one. Define each variable in the equation.

3. Answer the CEO's question. Namely, for each the four categories of expenditure (marketing, R&D, equipment and supplies, and salaries and benefits), if the CEO increases spending in one category by one million dollars (holding the others fixed), how much increase in sales should he expect? If you found an interaction term, explain its effects as well.

4. Being a very cautious person, you decide to also give the CEO the confidence intervals for the rates of increase your calculated in Question 3. Calculate the 95% confidence intervals for the slopes you calculated in Question 2, including the interaction term if you found one.

5. Being even more cautious - you are reporting the results to the CEO, after all - you decide to do a residual error analysis by applying the F-Test on the entire regression model. Do so, and interpret the results.

Part II. Time-series Analysis

The CEO noticed that he has five years of quarterly sales data in hand, and they form a time series. He decided to also ask you to perform time-series analysis on it, and use it to forecast what future sales are expected to be at the end of 1Q 2009.

6. Plot the quarterly sales as a function of time in your Excel data spreadsheet. From the shape of these graphs, and any analysis that you think is needed, determine what type of trend model is best suitable for this data. Write down the equation for the trend model, and define and explain each of the variables as it applies to this problem.

7. Do a regression analysis on the data for the trend model you decided on in Question 6, and determine the parameters for the model.

8. Answer the CEO's question. Tell him how much sales are expected to be at the end of 1Q 2009. Be careful, and also include the 95% confidence interval for this number.

Part III. Conclusions

9. Write a report to the CEO of your findings. Which expenditure has the largest impact on sales, and which one has the least impact on sales? How fast do you expect sales to increase in time?

10. When tasked to do this analysis, the CEO made a number of assumptions about the data, and what can be extrapolated from it. List down the assumptions he made, and criticize each assumption. Criticize also the conclusions that you drew from your analysis for your report to the CEO. As a starting point, remember that the data covered a period of five years.

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