Practical Exercises Multiple Regression Model

Practical Exercises Multiple Regression Model

PGDBA 2014-2015

1) To explain what determines the price of air conditioners, one researcher obtained the following regression results based on a sample of 19 air conditioners.

EMBED Equation.3 + 19.729X3i + 7.653 X4i

se = (.005) (8.992) (3.082)

R2 = 0.84

Where, Y= the price, in dollars

X2= the energy rating of AC

X3= the energy efficiency ratio

X4= the number of settings

se= standard errors

a) Interpret the regression results

R2 = 0.84, this means that 84% of the relation is explain by the model.Se for X2 = 0.05, therefore, T value of X2= (0.023-0)/0.005 = 4.6

Se for X3 = 8.992, therefore, T value of X3 = (19.729-0)/8.992 = 2.19

Se for X4 = 3.082, therefore, T value of X4 = (7.653-0)/3.082 = 2.48

From the T values above, we can say that we are above 95% confident that variables Energy Ratings of AC (X2), Energy Efficiency Ratio (X3) and Number of Settings (X4) influence (positively) the Price (Y).

b) Do the results make economic sense?From the above equation denotes that for every unit rise in Energy Rating AC, there is an increase of .023 units in Price (if energy efficiency ratio and numbers of settings are constant).

For every unit raise in Energy Efficiency Ratio, there is a 19.729 units raise in Price (if Energy Rating AC and Number of Settings are constant).

For every unit raise in Number of Settings, there is a 7.653 units raise in Price (if Energy Rating AC and Energy Efficiency Ratio are constant.

Therefore we can say that rise in X2, X3 and X4 will lead to rise in Price (Y)

c) At =5%, test the hypothesis (using t test) that the energy rating has no effect on the price of an AC vs. that it has a positive effect.

H0: Coefficient of X2 = 0If Coefficient of X2 = 0,

Then from the above equation,

Se for X2 = 0.05, therefore, T value of X2= (0.023-0)/0.005 = 4.6

This shows that we are ~99.99% confident that the coefficient is no equal to 0

Therefore, we reject H0.

d) Would you accept the null hypothesis that the 3 explanatory variables together do not explain a substantial variation in the prices of AC? Show clearly all your calculations. [Hint: Use F test]H0: All 3 variables (X2, X3, and X4) do not explain the Y variableF = (R2/(doff.))/((1-R2)/(doff.)) = = (R2/(k-1))/((1-R2)/(n-k))

Given

n=19k= 4

F = (0.84/2) / ((1-8.4)/15) = 0.000003225This means it is ~99.9999% significant.

Therefore, we reject H0, which means that X2, X3 and X4 explain the variable Y.2. U.S. defense budget outlays, 1962-1981. In order to explain the US defense budget, you are asked to consider the following model:

Where = defense budget-outlay for year t, $ billions

= GNP for year t, $ billion

= US military sales/assistance in year t, $ billions

= aerospace industry sales, $ billions

To test this model, you are given the data in the following Table.

YearDefense budget outlays, YGNP, X2US military sales/assistance, X3Aerospace industry sales, X4Conflicts 100,000+, X5

196251.1560.30.616.00

196352.3590.50.916.40

196453.6632.41.116.70

196549.6684.91.417.01

196656.8749.91.620.21

196770.1793.91.023.41

196880.5865.00.825.61

196981.2931.41.524.61

197080.3992.71.024.81

197177.71077.61.521.71

197278.31185.92.9521.51

197374.51326.44.824.30

197477.81434.210.326.80

197585.61549.216.029.50

197689.41718.014.730.40

197797.51918.38.333.30

1978105.22163.911.038.00

1979117.72417.813.046.20

1980135.92633.115.357.60

1981162.12937.718.068.90

a. Estimate the parameters of this model and their standard errors and obtain , and .

Using Multiple Regression Analysis,

R2 = 0.9710

= 0.9656SE for GNP, X2 = 0.0070SE for US Military Sales/assistance, X3 = 0.4539SE for Aerospace Industry Sales, X4 = 0.2776

The Model is, Y = 22.7751 + 0.01670 X2 - 0.6961 X3 + 1.4677 X4b. Comment on the results, taking into account any prior expectations you have about the relationship between Y and the various X variables.H0: X2, X3, X4 does not have any impact on Y

P Value for GNP, X2 = 0.03007

P Value for US Military Sales/assistance, X3 = 0.1446

P Value for Aerospace Industry Sales, X4 = 0.000073

From the above result we are above 95% confident that the variables GNP (X2), Aerospace Industry Sales (X4) are significant and impact the value Defense Budget Outlays (Y) (also t-stat is greater than 2).

However, since we are less than 95% confident that the variable US Military Sales/assistance (X3) is significant (also t-stat value is less than 2), therefor it we are not confident that it will have an impact on Y. Reject H0, since X2 and X4 impact the value of Y.c. What other variable(s) might you want to include in the model and why?

3. The demand for cable. The table gives data used by a telephone cable manufacturer to predict sales to a major customer for the period 1968-1983.

The variables in the table are defined as follows:

Y = annual sales in MPF, million paired feet

X2 = gross national product (GNP), $, billions

X3 = housing starts, thousand of units

X4 = unemployment rate, %

X5 = prime rate lagged 6 months

X6 = customer line gains, %

YearX2, GNPX3, housing startsX4, unemployment, %X5, prime rate lag, 6 mos.X6, customer line gains, %Y, total plastic purchases (MPF)

19681051.81503.63.65.85.95873

19691078.81486.73.56.74.57852

19701075.31434.85.08.44.28189

19711107.52035.66.06.24.27497

19721171.12360.85.65.44.98534

19731235.02043.94.95.95.08688

19741217.81331.95.69.44.17270

19751202.31160.08.59.43.45020

19761271.01535.07.77.24.26035

19771332.71961.87.06.64.57425

19781399.22009.36.07.63.99400

19791431.61721.96.010.64.49350

19801480.71298.07.214.93.96540

19811510.31100.07.616.63.17675

19821492.21039.09.217.50.67419

19831535.41200.08.816.01.57923

You are to consider the following model:

a. Estimate the preceding regression

Using Multiple Regression Analysis we got the following output.

From the table we got the following output,R2 = 0.8227 (i.e. 82.27% of the relation is explained by this model)

= 0.7341

SE for X2 = 2.5125SE for X3 = 0.8435

SE for X4 = 187.7072

SE for X5 = 147.0496

SE for X6 = 292.1447

B1 = 5962.6555

B2 = 4.8836

B3 = 2.3639

B4 = -819.1287

B5 = 12.0104

B6 = -851.3926

F value = 0.0016 (~99.99% significant)

T stat value of X2 = 1.9437 (less than 95% confident that the variable impacts output)T stat value of X3 = 2.8023 (more than 95% confident that the variable impacts output)T stat value of X4 = -4.3638 (more than 95% confident that the variable impacts output)T stat value of X5 = 0.0816 (less than 95% confident that the variable impacts output)T stat value of X6 = -2.9142 (more than 95% confident that the variable impacts output)

The Model

Yi = 5962.6555 + 4.8836X2 + 2.3639X3 - 819.1287X4 + 12.0104X5 - 851.3926X6b. What are the expected signs of the coefficients of this model?

According to the original model provided, it was expected that all the coefficients should be positive.c. Are the empirical results in accordance with prior expectations?

However, after using Multiple Regression Analysis, we find out that

The Model

Yi = 5962.6555 + 4.8836X2 + 2.3639X3 - 819.1287X4 + 12.0104X5 - 851.3926X6Coefficients

B1 = 5962.6555

B2 = 4.8836

B3 = 2.3639

B4 = -819.1287

B5 = 12.0104

B6 = -851.3926

This means that X4 and X6 have negative impact on the output.

d. Are the estimated partial regression coefficients individually statistically significant at the 5 percent level of significance?

From the output, at 5 percent level of significance, we find out that the X2 and X5 for not statistically significant

T stat value of X2 = 1.9437 (less than 95% confident that the variable impacts output)

T stat value of X3 = 2.8023 (more than 95% confident that the variable impacts output)

T stat value of X4 = -4.3638 (more than 95% confident that the variable impacts output)

T stat value of X5 = 0.0816 (less than 95% confident that the variable impacts output)

T stat value of X6 = -2.9142 (more than 95% confident that the variable impacts output)

e. Suppose you first regress Y on X2, X3 and X4 only then decide to add the variables X5 and X6. How would you find out if it is worth adding the variables X5 and X6? Which test do you use? Show the necessary calculations.

Using Multiple regression by regressing Y on X2,X3 and X4, we get the following output.

R2 = 0.6012 (i.e. 60.12% of the relation is explained by this model)

= 0.5015

F value = 0.0095

By adding the variable X5 and X6, we get the following output.

R2 = 0.8227 (i.e. 82.27% of the relation is explained by this model)

= 0.7341

F value = 0.0016This shows that the model is better represented by adding X5 and X6 as it has higher value of R2 (better explained) and high value of F (more significant).

4) The following data set provides information about performance of public sector in India. Run a multiple regression of output on other variables and interpret the results.YearOutputCapitalRaw MaterialLabor

1985443.34461.17373.041214722

1986379.05536.1259.881607128

1987413.52509.88322.811545389

1988466.2677.5380.491664926

1989473.05610.72381.381593680

1990544.1598.71424.611609015

1991508.71601.06381.931626806

1992468.36585.623471543411

1993451.3429.62313.531568153

1994413.1527.87260.361565944

1995391.15468.95244.061530029

1996445.07451.34296.751477010

1997406.75446.3283.871468938

1998388.31446.51239.241467611

1999266.69509.39127.871434503

2000307.4488.33153.81371988

Output = Nominal Output deflated by WPI (base 1981-82 = 100) Measured in Rupee crore

Capital = Capital employed deflated by WPI (base 1981-82 = 100), Measured in Rupee crore

Raw material = Raw material used is deflated by WPI (base 1981-82 = 100), Measured in Rupee crore

Labor = Number of Labor employed

Using Multiple Regression Analysis we get the following output.

ModelOutput = 73.6 0.16 Capital + 0.80 Raw Material + 0.0001 Labor

From the model we can find that,

For every unit rise in Capital, there is -0.16 units decrease in Output.

For every unit rise in Material, there is 0.80 units increase in Output.

For every unit rise in Capital, there is 0.0001 units increase in Output.

R2 = 0.9431 (94.31% if the relation is explained by this model)Adjusted R2 = 0.9289

SE for Capital = 0.0908

SE for Raw material = 0.0672

SE for Labor = 0.00005

F value = ~99.99%, which means the variance is significant within the model.T-stat value of Capital = -1.7640 (less than 95% confident that the variable impacts output)

T-stat value of Material = 12.0058 (more than 95% confident that the variable impacts output)

T-stat value of labor = 12.0058 (more than 95% confident that the variable impacts output)

There for Capital is not statistically significant to impact the output._1250318485.unknown

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