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."

Accepted Answer

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.

Accepted Answer

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 
 $\Sigma \left( Y _ { i } - \bar { Y } \right) = 0$

Accepted Answer

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.

Accepted Answer

The coefficient r² = 0.77 represents the square of the correlation coefficient (R^2) in a regression analysis, which indicates that 77% of the variation in Y is explained by the variable in the regression, leaving 23% of the variation in Y explained by variables other than the one in the regression, not 33%.

Question 5

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

Accepted Answer

A regression model with a small r-squared value (like 0.002) can still be considered a good fit in certain contexts, especially when dealing with complex data sets where even small amounts of explained variance are meaningful, or in fields where prediction accuracy is inherently limited.

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.

Accepted Answer

If all observations do not lie on a straight line, there will always be some deviation (errors) from the fitted line, making it impossible for the sum of squared errors to be zero.

Question 7

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

Accepted Answer

regression analysis and ordinary least squares (OLS) are not the same method. Regression analysis is a statistical process for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables. 
On the other hand, Ordinary Least Squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function.
So, while OLS is a method used in linear regression analysis, they are not the same thing. Regression analysis is a broader concept, and OLS is a specific technique used within that framework.

Question 8

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

Accepted Answer

This expression represents the partial derivative of the sum of squared residuals with respect to $\hat{\beta}_0$ in the context of linear regression, and the derivative correctly simplifies to $2 \Sigma (Y_i - \hat{\beta}_0 - \hat{\beta}_1X_i)(-1)$, indicating the rate of change of the sum of squared residuals with respect to $\hat{\beta}_0$.

Question 9

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

Accepted Answer

The formula provided is the covariance formula, which calculates the covariance between two variables X and Y. The formula is:Cov(X, Y) = Σ((Xi - X̄)(Yi - Ȳ)) / (n-1)where:Xi is the ith value of XX̄ is the mean of XYi is the ith value of YȲ is the mean of Yn is the number of observationsThe formula provided is a simplified version of the covariance formula, which is used when the mean of both X and Y is 0. In this case, the formula simplifies to:Cov(X, Y) = Σ(XiYi)which is equivalent to the formula provided.

Question 10

Equation 4 below is for the SER².

Accepted Answer

The equation provided is for the SER², which stands for the squared error of the residuals.

Question 11

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

Accepted Answer

The statement is ambiguous without specifying what "Equation 2" refers to, making it impossible to accurately determine its correctness in relation to the equation for $\hat { \beta } _ { 1 }$ without additional context.

Question 12

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

Accepted Answer

The standard error of $\hat { \beta } _ { 1 }$ typically refers to the standard deviation of the sampling distribution of the estimator $\hat { \beta } _ { 1 }$ in regression analysis, which is not denoted as "Equation 3" without context. The actual formula for the standard error of $\hat { \beta } _ { 1 }$ depends on the specific statistical model and estimation method being used.

Question 13

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

Accepted Answer

The provided text does not contain an equation, making it impossible to determine its correctness or similarity to any other equation.

Quiz 1: The Nature of Econometrics

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

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

Unless all the observations lie on a straight line, it is impossible to fit a line such that ∑ei2\sum e _ { i } ^ { 2 }∑ei2​ = 0.

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

Σ(Xi−Xˉ)(Yi−Yˉ)=ΣYiXi−YˉΣXi\Sigma \left( X _ { i } - \bar { X } \right) \left( Y _ { i } - \bar { Y } \right) = \Sigma Y _ { i } X _ { i } - \bar { Y } \Sigma X _ { i }Σ(Xi​−Xˉ)(Yi​−Yˉ)=ΣYi​Xi​−YˉΣXi​

Equation 4 below is for the SER2.

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

Equation 3 below is the standard error of β^1\hat { \beta } _ { 1 }β^​1​

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