Deck 1: The Nature of Econometrics

Question 1

Given $\begin{array} { c c } \frac { \mathrm { Yi } } { 5.5 } & \frac { \mathrm { Xi } } { 3.5 } \\14 & - 1.2 \\- 10 & - 2.3 \\11 & 7.8\end{array}$
Y_i = $\hat { \beta } _ { 0 }$ - $\hat { \beta } _ { 1 }$
X_i + e_i $\hat { \beta } _ { 0 }$ = 2.97
$\hat { \beta } _ { 1 }$ = 1.10
Calculate the value of the four e_i's.
Show your work making the details of your calculations apparent."

Answer 1

ei = Yi - $\hat { Y } _ { i }$ = Yi - ( $\hat { \beta } _ { 0 }$ + $\hat { \beta } _ { 1 }$ Xi) = 5.5 - (2.97 + (1.10 x 3.5)) = -1.33 (6.38)
ei = Yi - $\hat { Y } _ { i }$ = Yi - ( $\hat { \beta } _ { 0 }$ + $\hat { \beta } _ { 1 }$ Xi) = 14 - (2.97 + (-1.10 x -1.2)) = 12.35 (9.71)
ei = Yi - $\hat { Y } _ { i }$ = Yi - ( $\hat { \beta} _ { 0 }$ + $\hat { \beta } _ { 1 }$ Xi) = -10 - (2.97 + (-1.10 x -2.3)) = -10.44 (-15.50)
ei = Yi - $\hat { Y } _ { i }$ = Yi - ( $\hat { \beta } _ { 0 }$ + $\hat { \beta } _ { 1 }$ Xi) = 11 - (2.97 + (-1.10 x 7.8)) = -0.58 (16.61)

Question 2

$\begin{array} { |l| } \hline\mathrm { Y } _ { \mathrm { i } } = 45.33 - 16.20 \mathrm { X } _ { \mathrm { i } } + \mathrm { e } _ { \mathrm { i } } \\\text { (9.14) } ( 14.27 ) \leftarrow \text { standard errors } \\\text { SER } = 45.03 \quad \bar { Y } = 75.01 \quad \mathrm { r } ^ { 2 } = 0.77 \quad \mathrm { n } = 44 \\\hline\end{array}$

A) Interpret the structural parameters of the regression in the box above.
B) How many observations were used to estimate the structural parameters?
C) What is the estimated distance of an observation from the population regression line on average?
D) Interpret the standard error of the constant term.
E) If the structural parameters of this regression were re-calculated using a fresh set of data, would you be surprised if turned out to be -29.00? Explain.
F) According to the SER, is the fit of this regression line adequate? Explain.

Answer 2

A)If X=0, then Y is expected to = 45.33
If X↑ 1 unti,then Y is expected to↓ 16.20 units
B)44
C) 45.03
D) In repeated sampling $\hat { \beta } _ { 0 }$ typically varies by 9.14
E) No, since (-16.20 - 14.27) < -29.00
F)The fit is not adequate since SER > ½ Y-bar

Question 3

Answer any 5 of the following A through F:

A) Briefly explain in words how diplomatic parking fines can be used to predict the level of corruption in a nation. Also, write the econometric model used to make this prediction.
B) Define psychometrics.
C) Explain why minimizing the squared error terms is the premier method for fitting a line between observations considering that other options are available.
D) Give 3 distinct reasons why a student may not reduce their study time given the information in the box on the right. $\mathrm { GPA } _ { \mathrm { i } } = 3.1 + 0.02 \mathrm { ST } _ { \mathrm { i } } + \mathrm { e } _ { \mathrm { i } }$
(0.14) (0.08) $\leftarrow$ standard errors
where GPA is grade point average
ST is hours of study time on a typical day.
$$\mathrm { r } ^ { 2 } = 0.0002 \quad \mathrm { n } = 289$$
E) Prove
$\Sigma \left( X _ { i } - \bar { X } \right) ^ { 2 } = \Sigma X _ { i } { } ^ { 2 } - n \bar { X } ^ { 2 }$ F.
F. Prove
$\Sigma \left( Y _ { i } - \bar { Y } \right) = 0$

Answer 3

A) When diplomats rack up parking fines and don't pay them it may be a marker for the level of corruption in their home country. Diplomats from corrupt countries rack up more parking fines than those from less corrupt countries.
Level of Corruption_i = $\hat { \beta } _ { 0 }$ + $\hat { \beta } _ { 1 }$ unpaid parking fines per diplomat_i + e_i
B) The empirical determination of psychological laws.
C) This technique yields 1) a unique line that has 2) desirable statistical qualities.
D) 1) The variables may be poorly measured.
2) Other factors aside from study time may come into play.
3) These results are for typical students and I'm not typical.
E) $\Sigma \left( X _ { i } - \bar { X } \right) ^ { 2 } = \Sigma \left( X _ { i } - \bar { X } \right) \left( X _ { i } - \bar { X } \right) = \Sigma X _ { i } { } ^ { 2 } - \bar { X } \Sigma X _ { i } - \bar { X } \Sigma X _ { i } + n \bar { X } ^ { 2 } = \Sigma X _ { i } ^ { 2 } - \bar { X } \Sigma X _ { i } = \Sigma X _ { i } { } ^ { 2 } - n \bar { X } ^ { 2 }$ F) $\Sigma \left( Y _ { i } - \bar { Y } \right) = \Sigma Y _ { i } - \Sigma \bar { Y } = n \bar { Y } - n \bar { Y } = 0$

Question 4

If r² = 0.77, then 33% of the variation in Y is explained by variables other than the one in the regression.

Question 5

In some instances, a regression with r² = .002 can be considered a good fit.

Question 6

Unless all the observations lie on a straight line, it is impossible to fit a line such that $\sum e _ { i } ^ { 2 }$ = 0.

Question 7

Regression analysis and ordinary least-squares are the same method.

Question 8

$\frac { \partial \Sigma \left( Y _ { i } - \hat { \beta } _ { 0 } - \hat { \beta } _ { 1 } X _ { i } \right) ^ { 2 } } { \partial \hat { \beta } _ { 0 } }$ = $2 \Sigma \left( Y _ { i } - \hat { \beta } _ { 0 } - \hat { \beta } _ { 1 } X _ { i } \right) ( - 1 )$

Question 9

$\Sigma \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right) = \Sigma Y _ { i } X _ { i } - \bar { Y } \Sigma X _ { i }$

Question 10

Equation 4 below is for the SER².

Question 11

Equation 2 below is similar to the equation for $\hat { \beta } _ { 1 }$
, but not 100% correct.

Question 12

Equation 3 below is the standard error of $\hat { \beta } _ { 1 }$

Question 13

Equation 1 below is similar to the equation for r², but not 100% correct.