Consider the following data for the average cost of various fuels and electricit

Question

Consider the following data for the average cost of various fuels and electricity for a twelve year period. The data measure the following: 6. Electricity Natural Gas Fuel Oil Gasoline Residential rate per kilowatt hour Residential natural gas per 1000 cubic feet Residential fuel oil per gallon Regular gasoline per gallon Your model will attempt to predict residential electricity costs using the cost of the other fuels a) State whether you would expect a positive or negative relationship between electricity costs and each of the independent variables and explain why Use EXCEL, to create a correlation using all independent variables provided. Discuss all aspects of the correlation matrix, as it compares to your expectations in part (a) and what it tells you about the potential results of a regression analysis b) c) Use EXCEL to run a multiple regression to estimate electricity costs using all independent d) Test for the significance of each slope coefficient at the 5% level. (incorporate all tests into ONE e) Based on your correlation matrix and the results of your regression do you suspect f) Explain how you test for multicollinearity but do not test. variables provided. Test if the overall regression is significant at the 5% level five step answer as in the slides.) multicollinearity? If yes, which independent variable(s) to you think could be responsible? How could you rerun the regression taking into account the issues in the first regression? Which variable or variables would you keep as an explanatory variable(s), and why?

Explanation / Answer

Here independent variable is Electricity and dependent variables are gas, gasoline and oil.

We have to find correlation between electricity and each of the independent variable.

We can find correlation between two variables by using EXCEL.

syntax is,

=CORREL(array1, array2)

where array1 is select range of electricity.

array2 is select range of gas or gasoline or oil.

Correlation between Electricity and gas is 0.9641.

There is high positive correlation between Electricity and gas.

Correlation between electricity and gasoline is 0.6876.

There is positive correlation between electricity and gasoline.

Correlation between electricity and oil is 0.498.

There is positive correlation between electricity and oil.

Multiple regression in MINITAB :

steps :

STAT --> Regression --> Regression --> Response : electricity --> Predictors : natural gas, gasoline and fuel oil --> --> options : select on VIF --> Results : select second option --> ok --> ok

Output is,

Regression Analysis: electricity versus natural gas, gasoline, fuel oil

The regression equation is
electricity = 1.20 + 0.881 natural gas + 3.92 gasoline - 5.44 fuel oil

Predictor Coef SE Coef T P VIF
Constant 1.1984 0.4474 2.68 0.028
natural gas 0.88063 0.09966 8.84 0.000 2.2
gasoline 3.921 1.413 2.77 0.024 11.6
fuel oil -5.437 1.857 -2.93 0.019 8.8

S = 0.379873 R-Sq = 96.6% R-Sq(adj) = 95.3%

Analysis of Variance

Source DF SS MS F P
Regression 3 32.910 10.970 76.02 0.000
Residual Error 8 1.154 0.144
Total 11 34.064

Here we have to test the hypothesis that,
H0 : Bj = 0 Vs H1 :Bj 0

where Bj is the population slope of jth independent variable.

Assume alpha = 0.05

Test statistic F = 76.02

P-value = 0.000

P-value < alpha

Reject H0 at 5% level of significance.

Conclusion : Atleast one population slope is differ than 0.

Test for significance of each slope :

Here we have to test the hypothesis that

H0 : B = 0 Vs H1 : B 0

where B is slope of independent variable.

We see that all the P-values (natural gas, gasolineand fuel oil) are < alpha

Reject H0 at 5% level of significance.

Conclusion : Population slope for all the three variables are differ than 0.

Here problem of multicollinearity occur because we see in the first part of answer that there is correlation between electricity and all the three independent variables.

Here we see that VIF for three variables are 2.2, 11.6 and 8.8.

VIF for last two variables is 11.6 and 8.8 which are > 5.

So last two factors are indicate multicollinearity.

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