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Econ 399

Winter 2015

Midterm Exam

Question 1 (60 marks)

Suppose we estimated a model to study the factors that affect the sales of a particular restaurant. The sample contains data on 33 branches for the restaurant, and we used the following variables:

Sales: Revenues to the restaurant measured in dollars.

Competition: number of competitors within the same location of the restaurant.

Population: number of people living in the restaurant`s surrounding area.

Income: average income of the population living in the restaurant`s area measured in dollars

The dependant variable and all the explanatory variables are expressed in the natural logarithmic form.

1.   Report the results of model 1 in a full reporting form. ( 4 marks)

2.   Interpret the meaning of the estimated coefficients      and      in Model 1. ( 4 marks)

3.   Briefly  explain the  meanings  of the  calculated value  of        and        in Model  1 .  Of the

two, which do you recommend typically using? Why? (4 marks)

4.   Suppose we change the unit of measurement of the dependant variable sales by multiplying the original variable sales by 10. Report the result of the new model in a full reporting form. ( 4 marks)

5.   Develop  and  test  appropriate  hypothesis  about  the  individual  slope  coefficient        at  the  5%

significance level. ( 4 marks)

6.   Develop and conduct an overall significance test for model 1 at the 5% significance level. ( 4 marks)

7.   Construct a 95% confidence interval estimate of the true      coefficient and interpret its meaning. ( 4

marks)

8.   Based on the results of Model 1, which of the three explanatory variables has more effect on the dependant variable? and why? ( 4 marks)

9.   Suppose that all the Classical Assumptions hold.  Does this mean that the true (marginal) effect of the competition on sales is -0.377 (i.e. the population parameter is -0.377)? ( 4 marks)

10. What are the consequences of adding an irrelevant variable to model 1? ( 4 marks)

11. Suppose we omitted a relevant explanatory variable like population from Model 1. And suppose the correlation between population  and  income  is negative, while there  is  zero  correlation between population and competition. What are the consequences of omitting population from our model? (4 marks).

12. Suppose now we estimated model 1 but without population and Income and we get model 2. Test the restriction of excluding both population and Income from model 1. ( 6 marks)

13. Which of the two models: model 1 or model 2 do you prefer and why? ( 10 marks)

Model 1

. reg sales competition population Income

Source

SS     df     MS

Model Residual

703457412

.(.)294463032

3  .234485804 29  .010153898

Total

.997920444   32  .031185014

Number of obs =     33

F( 3,   29) =  23.09

Prob > F     = 0.0000

R-squared    = 0.7049

Adj R-squared = 0.6744

Root MSE     =  .10077

sales

Coef.

Std.

Err.

t

P>|t|

[95% Conf.

Interval]

competition

-.3779502

.064

9484

-5.82

0.000

-.5107846

-.2451157

population

.352044

.055

9798

6.29

0.000

.2375525

.4665355

Income

.1590619

.084

5355

1.88

0.070

-.0138326

.3319564

cons

6.657826

.772

1623

8.62

0.000

5.078577

8.237076

Model 2

. reg sales competition

Source

SS     df     MS

Model Residual

017734189

.(.)980186255

1 .017734189 31 .031618911

Total

.997920444  32 .031185014

Number of obs =    33

F( 1,  31) =  0.56

Prob > F    = 0.4596

R-squared   = 0.0178

Adj R-squared = -0.0139

Root MSE    = .17782

sales     Coef.  Std. Err.    t  P>|t|   [95% Conf. Interval]

competition

cons

-.052729  .0704072

11.799  .1023621

-0.75

115.27

0.460

0.000

-.1963254

11.59023

0908675

1(.)2.00777

Question 2 ( 15 marks)

Under the classical linear model (CLM) assumptions, the OLS estimators have stronger properties than what they would have under the Gauss-Markov assumptions.

1)  State the Classical Linear Model (CLM) assumptions. ( 10 marks)

2)  Briefly explain the properties of the OLS estimators if these assumptions are satisfied. ( 5 marks)

Question 3 (25 marks)

Consider the following sample regression model without an intercept:                        and the corresponding

true population regression model is                       . Assume that the standard assumptions about this model

apply.

1)  Derive the ordinary least squares (OLS) estimator of

2)  Derive the expected value of the OLS estimator     . Is      an unbiased estimator of    ?