MAE256 Analytical Methods In Economics And Finance

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MAE256 Analytical Methods In Economics And Finance

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MAE256 Analytical Methods In Economics And Finance

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Course Code: MAE256
University: Deakin University

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Country: Australia

Questions:
(i)Present the descriptive statistics of the variables Sale Price, Length and Weight. Comment on the means and measures of dispersion of the variables. (ii)Estimate the following simple regression model of sale price on length. 
SalePrice=β0+ β1length+u
Write down the sample regression function and interpret the coefficient estimates.  (iii)Now estimate the following simple regression model with a log-log specification,
log(SalePrice)=β0+ β1log(length)+uReport your regression results in a sample regression function. Interpret the estimated coefficient of log(length). What did you expect this coefficient to be before the estimation and is the sign of this estimate what you expect it to be?(iv)A model that relates the car sale price to the car length and car weight is:
SalePrice=β0+ β1length+ β1weight+uReport your results in a sample regression function. What can you conclude regarding comparison of the goodness of fit of this regression model versus the regression model in part (ii)?

Answer:
The regression model to be estimated is given as 
The obtained regression result is given below

Regression Statistics

Multiple R

0.73

R Square

0.53

Adjusted R Square

0.52

Standard Error

0.31

Observations

251

 

ANOVA

 

df

SS

MS

F

Significance F

Regression

2

26.51383593

13.25692

138.071

5.0158E-41

Residual

248

23.81177806

0.096015

 

 

Total

250

50.32561399

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

1.1679

1.8535

0.6301

0.5292

-2.4827

4.8185

log(length)

-0.6118

0.3793

-1.6128

0.1080

-1.3590

0.1353

log (weight)

1.7832

0.1313

13.5817

0.0000

1.5246

2.0418

The regression equation as estimated from the regression result is
The estimated elasticity of sales price with respect to weight is 1.7832. In order to test the statistical significance of the variable weight, t test needs to be conducted. The computed t value is
The critical t value corresponding to 1% level of significance with 249 degrees of freedom is 2.5957. As the computed t value is greater than the critical t value, it thus falls in the critical or rejection region. The null hypothesis of no significant relation between weight and sales price is thus rejected (Chatterjee and Hadi 2015). The variable thus is statistically significant at 1% level. The result is further supported by the estimate of p value. The obtained p value for the co-efficient is 0.0000. The p value less than the significance level again indicates weight is statistically significant at the given level of significance. 
The estimated coefficient for length is -0.6118. The negative value of the coefficient indicates that length has a negative effect on the sales price. The computed t value is (-0.6118/0.3793) = -1.6128. The critical t value at 1% level of significance and given degrees of freedom is 2.5957. As the computed t value is less than critical t value, the null hypothesis of no significant relation between length and sales price is accepted (Draper and Smith 2014). The p value for the coefficient length equals 0.1080. As the p value is greater than significance level of 0.01, the variable length is statistically insignificant.   
Result of the regression output is produced below

Regression Statistics

Multiple R

0.90

R Square

0.80

Adjusted R Square

0.80

Standard Error

0.20

Observations

251

 

ANOVA

 

df

SS

MS

F

Significance F

Regression

4

40.39446274

10.0986157

250.14818

2.04037E-85

Residual

246

9.931151252

0.04037053

 

 

Total

250

50.32561399

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

7.9273

1.2830

6.1786

0.0000

5.4002

10.4545

log(length)

-0.9598

0.2487

-3.8584

0.0001

-1.4497

-0.4698

log (weight)

1.0414

0.1170

8.9028

0.0000

0.8110

1.2718

Horsepower

0.0045

0.0003

16.5861

0.0000

0.0040

0.0051

Luggage Size

-0.0038

0.0013

-2.8199

0.0052

-0.0064

-0.0011

The t value computed for each of the independent variable is given as
The computed t values for all the independent variables are lower than critical t value of 2.5957. This implies that the four independent variables are statistically significant. The conclusion further gains support from the obtained P values of the coefficient estimates. The p values are less than significant value of 0.01. This in turn means rejection of null hypothesis of no significant relation between Sales Price and the chosen independent variables (Darlington and Hayes 2016).
For testing overall significance of the model, F test has to be conducted.
Hypothesis
Null Hypothesis (H0): 
Alternative Hypothesis (H1): At least one of the coefficient is not zero
Test Statistics  
Using the regression result, the F value is computed as 
Decision rule
The null hypothesis is rejected if the computed F value exceeds the tabulated F value at the chosen level of significance (Wooldridge 2015).
Conclusion
At 5% significance level, the tabulated F value is 2.4083. As the tabulated F value is less than computed F value the null hypothesis stating the model does not have statistical significance is rejected. The model is therefore statistically significant as a whole. The corresponding p value for the F statistics is 0.0000, which is less than 0.05. P value less than the significance level again implies the model is overall significant. 
The critical F value corresponding to 1% level of significance is obtained as 3.3962. The computed F value exceeds the critical F value. This implies, the model has an overall significance at 1% level of significance. 
In order to test whether luxury cars are more expensive the regression model in (v) needs to be re-estimated using the variable ‘luxury’ as an independent variable. The new model thus becomes 
The regression result is summarized below

Regression Statistics

Multiple R

0.85

R Square

0.72

Adjusted R Square

0.71

Standard Error

0.24

Observations

251

 

ANOVA

 

df

SS

MS

F

Significance F

Regression

3

36.06300056

12.021

208.1797

2.51517E-67

Residual

247

14.26261344

0.0577434

 

 

Total

250

50.32561399

 

 

 

 

 

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

0.9070

1.4375

0.6309

0.5287

-1.9243

3.7383

log(length)

-0.0694

0.2972

-0.2335

0.8156

-0.6547

0.5160

log (weight)

1.3458

0.1073

12.5371

0.0000

1.1344

1.5573

Luxury

0.4530

0.0352

12.8597

0.0000

0.3836

0.5224

The estimated regression equation is
The statistical significance of the variable luxury needs to be tested to decide whether luxury cars are more expensive or not. The coefficient of luxury is positive having a value of 0.4530. The computed t statistics is (0.4530/0.0352) = 12.8597. The critical t value is 1.9697. The computed t value exceeds the tabulated t value implying rejection of null hypothesis of no significant relation between sales price and luxury. The associated p value for the coefficient is 0.0000. The p value lower than the significance level implies that the variable is statistically significant at 5% level of significance. It can therefore be concluded that luxury cars are more expensive relative to other types of cars.
References list 
Chatterjee, S. and Hadi, A.S., 2015. Regression analysis by example. John Wiley & Sons.
Darlington, R.B. and Hayes, A.F., 2016. Regression analysis and linear models: Concepts, applications, and implementation. Guilford Publications.
Draper, N.R. and Smith, H., 2014. Applied regression analysis(Vol. 326). John Wiley & Sons.
Wooldridge, J.M., 2015. Introductory econometrics: A modern approach. Nelson Education.

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