proiect modelare

1. Download data for at least 4 economic variables that you think are related.

Make sure they have the same frequency (daily, weekly, monthly or yearly data) and the

same number of observations- you need contemporaneous values for all variables. The

sample should include at least 30 observations. Compute means, standard deviations,

skewness and kurtosis for all the variables and built histograms to characterize their

distributions. Describe and graph the data. Show your sources.

MonthY(Retail and food services sales) mil. $

X ( Total

disponible personal Income) bil. $

X (Consumar price

index - CPI)

X (Interest

rate) %2007-Jan. 329,736 11,640.70 202.416 8.25Feb. 324,287 11,713.80 203.499 8.25Mar. 375,294 11,788.20 205.352 8.25Apr. 359,619 11,815.80 206.686 8.25May 392,640 11,843.00 207.949 8.25Jun. 377,695 11,858.10 208.352 8.25Jul. 373,405 11,906.90 208.299 8.25Aug. 388,846 11,931.90 207.917 8.25Sept. 354,721 12,024.50 208.49 7.75Oct. 369,434 12,065.10 208.936 7.5Nov. 378,619 12,132.00 210.177 7.5Dec. 427,225 12,227.20 210.036 7.252008-Jan. 343,616 12,258.00 211.08 6Feb. 345,010 12,294.00 211.693 6Mar. 374,208 12,349.20 213.528 5.25Apr. 370,352 12,336.50 214.823 5May 399,773 12,522.10 216.632 5Jun. 380,389 12,524.10 218.815 5Jul. 385,985 12,409.70 219.964 5Aug. 385,211 12,462.60 219.086 5Sept. 353,128 12,468.90 218.783 5Oct. 352,847 12,435.00 216.573 4Nov. 338,774 12,376.10 212.425 4Dec. 388,025 12,257.70 210.228 3.252009-Jan. 313,864 12,160.20 211.143 3.25Feb. 303,504 12,072.20 212.193 3.25Mar. 333,230 12,047.30 212.709 3.25Apr. 334,767 12,110.50 213.24 3.25May 353,263 12,310.80 213.856 3.25Jun. 349,960 12,189.00 215.693 3.25Jul. 353,617 12,148.30 215.351 3.25Aug. 359,221 12,173.80 215.834 3.25

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Sept. 330,260 12,169.70 215.969 3.25Oct. 344,716 12,178.70 216.177 3.25Nov. 345,700 12,237.40 216.33 3.25Dec. 408,576 12,300.70 215.949 3.252010-Jan. 321,550 12,324.30 216.687 3.25Feb. 317,961 12,337.20 216.741 3.25Mar. 369,339 12,389.40 217.631 3.25Apr. 366,002 12,478.50 218.009 3.25May 375,699 12,532.80 218.178 3.25Jun. 369,031 12,540.00 217.965 3.25Jul. 372,451 12,559.80 218.011 3.25Aug. 373,373 12,622.10 218.312 3.25Sept. 355,549 12,619.30 218.439 3.25

Sources: www.census.gov (retail and food services sales), www.bea.gov (personal income), www.bls.gov (CPI), www.bankofcanada.ca (interest rate).

Means:

=

= 360454.9333 – The mean of retail and food services sales in USA beginning with January 2007 to September 2010 is 360454.9333.

= 12225.40222 – The mean of personal income in USA beginning with January 2007 to September 2010 is 12225.40222.

= 213.4701333 – The mean of the consumer price index in USA beginning with January 2007 to September 2010 is 213.4701333.

= 4.95 – The mean of the interest rate in USA beginning with January 2007 to September 2010 is 4.95.

Standard deviation:

= =

= 26036.6843 – The degree of dispersion of the retail and food services sales in USA from the mean value is 250.7240547.

= 250.7240547 – The degree of dispersion of personal income in USA from the mean value is 250.7240547.

= 4.54299426 - The degree of dispersion of consumer price index in USA from the mean value is 4.54299426.

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http://www.bankofcanada.ca/

http://www.bls.gov/

http://www.bea.gov/

http://www.census.gov/

= 2.027929979 - The degree of dispersion of interest rate in USA from the mean value is 2.027929979.

Skewness:

= 1/n*

= 0.034228706 – The degree of asymmetry of the sales distribution around its mean is 0.034228706. In this case, the skewness is positive and that indicates a distribution with an asymmetric tail extending towards more positive values.

= -0.491795728 - The degree of asymmetry of the income distribution around its mean is -0.491795728. In this case, the skewness is negative and that indicates a distribution with an asymmetric tail extending towards more negative values.

= -0.629726058 - The degree of asymmetry of the CPI distribution around its mean is -0.629726058. In this case, the skewness is negative and that indicates a distribution with an asymmetric tail extending towards more negative values.

= 0.705329021 - The degree of asymmetry of the interest rate distribution around its mean is 0.705329021. In this case, the skewness is positive and that indicates a distribution with an asymmetric tail extending towards more positive values.

Kurtosis:

k = 0.016183697 – Positive kurtosis indicates a relatively peaked distribution of sales.

k = -0.426401663, k = -0.505730192, k = -1.19736224 – Negative kurtosis indicates a relatively flat distribution of this three economic indicators: personal income, CPI and interest rate.

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Histogram for Y

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303504 324124.1667 344744.3333 365364.5 385984.6667 406604.8333 More

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2. Select the dependent variable and build a multiple regression model that makes

economic sense. Run a battery of regressions of the dependent variable on all

combinations of one, two and three other variables. Create the ANOVA table in each

case. What do you observe? Comment on the values you obtained for the coefcients.

Compute the regression statistics in Excel by minimizing the sum of squared residuals

(using Solver), and using Regression in the Data Analysis tool-pack. Verify that you

obtained the same values for the coefficients irrespective of the method used. Create a

summary table of the results and interpret it.

Combinations of one:

Model 1.1. Dependent variable: y=Sales

Independent variable: x=Personal Income

y=114575.8743(b ) + 20.11214474(b )*x For a personal income of 0, the sales will be around 115000. But from an economic point

of view the coefficient b has no relevance.

The coefficient b tells us that each additional unit of personal income adds an average of

about 20 to the sales.

Model 1.2. Dependent variable: y=Sales

Independent variable: x= Consumer price index

y=15.38986188 + 0.016202352x

For a CPI=0, the value of sales will be around 15. But from an economic point of view the coefficient b has no relevance.The coefficient b tells us that each additional unit of CPI adds an average of about

0.02% to the value of sales.

Model 1.3. Dependent variable: y= Sales

Independent variable: x= Interest rate

y= 342427.657 + 3641.873998x

For an interest rate=0 the value of sales will be around 343000. The coefficient b tells us that if the interest rate increases with 1 unit, it adds an average

of about 3642 to the value of sales.

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We have compared the value of Significance F for the three models of regression

and we have observed that the Model 1.2 is the best model of one combination because it

is the single one with Significance F <0.05.

Combinations of two

Model 2.1. Dependent variable: y= sales

Independent variable: x =personal income; x = CPI;

y= 156310.0338+ 64.04961164 x -2711.795584 x

For a personal income of 0, the sales will be around 157 000. The value of 64 for b means that if personal income increases by one unit while CPI remains constant, sales will increase by aprox.64. The value of -2712 for b means that if CPI increases by one unit while personal income remains constant, sales will decrease by aprox. 2712.


Independent variable: x =CPI; x = interest rate;

y= -652741.257+ 4479.369565 x + 11512.03473 x

For a CPI of 0, the sales will be around -652742. The value of 4480 for b means that if CPI increases by one unit while interest rate remains constant, sales will increase by aprox. 4480. The value of 11512 for b means that if interest rate increases by one unit while CPI remains constant, sales will increase by aprox. 11512.


Independent variable: x =personal income; x = interest rate;

y= -614274.5472+ 10040.83217x +75.66437034 x

For a personal income of 0, the sales will be around -614280. The value of 10041 for b means that if personal income increases by one unit while interest rate remains constant, sales will increase by aprox. 10041. The value of 75 for b means that if interest rate increases by one unit while personal income remains constant, sales will increase by aprox. 75.

We have compared the value of Significance F for the three models of regression

and we have observed that the Model 2.2 and Model 2.3. are relevant models of two

combinations because they both have Significance F <0.05.

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Combinations of three


Independent variable: x =personal income; x = CPI; x = interest rate;

y= -722949.6462+ 56.23933624 x + 1594.680709 x + 11199.87269 x

The value of 56 for b means that if personal income increases by one unit while CPI and interest rate remain constant, sales will increase by aprox. 56. The value of 1595 for b means that if CPI increases by one unit while personal income and interest rate remain constant, sales will increase by aprox. 1595. The value of 11120 for b means that if interest rate increases by one unit while personal income and CPI remain constant, sales will increase by aprox. 11120.

Because Significance F for this model is lower than 0.05 we can say this model is

a relevant one.

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Y(Retail and food servicessales)

All-period Average

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3 month MA

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ES alpha=0.2

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ES alpha=0.3

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X2(Consumer price index)

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ES alpha=0.3

Summary Table Regression

Model 1.1

Model 1.2

Model 1.3

Model 2.1

Model 2.2

Model 2.3

Model 3.1

Constant term

115000 15 343000 157 000 -652742 -614280 -7222950

Coefficient for

Personal Income

20 64 10041 56

Coefficient forCPI

0.02 -2712 4480 1595

Coefficient for

Interest rate

3642 11512 75 11120

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Summary Table (using Solver)

Model 1.1

Model 1.2

Model 1.3

Model 2.1

Model 2.2

Model 2.3

Model 3.1

Constant term 0,002454 -263970,1 -2345620 0,028035 26,00248 0,033276 0,018979

Coefficient for

Personal Income

28,32852 27,84473 28,30651 27,83183

Coefficient forCPI

-53687091 27,68454 1563,78 27,12872

Coefficient for

Interest rate

324285,6 1594,853 27,6845 31,8367

As we can see comparing the tables above, the values for the coefficients differ

from one method to another. We have obtained some values for the coefficients using

Regression and other values using Solver.

From the Summary tables we can observe that the values of the regression

coefficients associated with a given independent variable are differents for each model.

The values depend on what independent variables are included in the model.

We consider that Model 3.1. is the one we should rely on because it takes into

consideration the largest number of factors (independent variables) that can influence the

sales.

3. For two of the variables previously chosen remake the analysis we did in class:

1. Compute all-period average, 3 month Moving Average and Exponential Somoothing

with alpha = 0.2 and alpha = 0.3. 2. Decide on what method you could use for

forecasting using the precision coefficients. 3. Compute seasonality indexes and the

trend. 4. Use the Winter model to compute forcasts for 5 months into the future for the

two variables. Use the minimization of the sum of squared residuals to find the

exponential smoothing coefficients.

The two variables we have chosen are: Retail and food services sales and

Consumer Price Index.

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1) The values we have obtained for SALES in October using the three methods are

written in the following table:

All-period average 3 month MA ES with alpha=0.2 ES with alpha=0.3

October 360455 367124 362635.615 364451.6811

The values we have obtained for CPI in October using the three methods are written

in the following table:


October 213 218 217.7 218.05

2) SALES


MAD 21995.43 20694 21454.1 20509.2

MSE 738611259.2 753598800.7 727653365.6 703430711.2

MAPE 6.18% 5.78% 5.96% 5.71%

OCT. 360455 367124 362635.615 364451.6811

As we can see from the table above, the lowest values of the precision coefficients

are the ones obtained using the exponential smoothing with alpha=0.3 method. This

means that the value forcasted for October is the closest to the actual value (364451.6811

is the best value forcasted for October).

CPI


MAD 4.07 1.5 2.42 1.92

MSE 21.82 4.09 8.44 5.71

MAPE 1.89 % 0.7% 1.13% 0.9%

OCT. 213 218 217.7 218.05

As we can see from the table above, the lowest values of the precision coefficients

are the ones obtained using the three month Moving Average method. This means that

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the value forcasted for October is the closest to the actual value (218 is the best value

forcasted for October).

3)

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