Question 1

What is a Monte Carlo study? In addition, give examples. Why are these studies useful?

Accepted Answer

A Monte Carlo study is a statistics study that draws repeated samples from a known population and then analyzes the characteristics of the samples. For example, one may draw repeated samples of 3 from a deck of 9 playing cards (all the hearts from 2 through 10). The average value of the 9 cards is 6. The mean value of a typical sample draw will approach 6 as well as the number of draws approaches infinity. This Monte Carlo experiment shows that the mean value of a sample of 3 cards is an unbiased estimator of the population mean. Another example would be a computer simulation where a sample of 40 observations on X and Y is considered the population. A regression is run to ascertain the values of $\hat { \beta } _ { 0 }$ and $\hat { \beta } _ { 1 }$ . Then 10,000 samples of n=20 can be obtained and $\hat { \beta } _ { 0 }$ and $\hat { \beta } _ { 1 }$ calculated for each. The average of these 10,000 $\hat { \beta } _ { 0 }$ s and $\hat { \beta } _ { 1 }$ s should equal the population values of $\beta _ { 0 }$ and $\beta _ { 1 }$ if the estimation technique is unbiased. Monte Carlo studies are useful because they can assess the properties of statistical estimators and tests.

Question 2

If a regression is in the incorrect functional form, explain why it is unlikely that E[u_i_│X_i] = 0.

Accepted Answer

Here we have applied a straight line when an alternate functional form is appropriate. Notice E[ $U _ { i }$ |X_i] ≠ 0. When the value of X (PMILK) is low, u_i is likely to be positive, not 0. Similarly, when the value of X (PMILK) > 2.5, u_i is likely to be positive. When 1.0 < PMILK < 2.5, u_i is likely to be negative. Thus, E[u_i] depends on Xi and is not expected to equal to zero.

Question 3

Explain how this: $$\begin{array} { l } 
E \left[ w _ { 1 } { } ^ { 2 } u _ { 1 } { } ^ { 2 } + w _ { 2 } { } ^ { 2 } u _ { 2 } { } ^ { 2 } + \ldots + w _ { n } { } ^ { 2 } u _ { n } ^ { 2 } + 2 w _ { 1 } u _ { 1 } w _ { 2 } u _ { 2 } + 2 w _ { 1 } u _ { 1 } w _ { 3 } u _ { 3 } + \ldots \right. \
\left. + 2 w _ { 2 } u _ { 2 } w _ { 3 } u _ { 3 } + 2 w _ { 2 } u _ { 2 } w _ { 4 } u _ { 4 } + \ldots + 2 w _ { n - 1 } u _ { n - 1 } w _ { n } u _ { n } \right]
\end{array}$$

Accepted Answer

collapses to this: $\sigma ^ { 2 } \Sigma w _ { i } ^ { 2 }$
A ssume that $\mathrm { E } \left[ \mathrm { u } _ { i } \mathrm { u } _ { j } \right] = 0$ for all $\mathrm { i } \neq \mathrm { j }$, then the lengthy expression becomes:
$$E \left[ w _ { 1 } { } ^ { 2 } u _ { 1 } { } ^ { 2 } + w _ { 2 } { } ^ { 2 } u _ { 2 } { } ^ { 2 } + \ldots + w _ { n } { } ^ { 2 } u _ { n } ^ { 2 } \right]$$
Next, assume $\mathrm { E } \left[ \mathrm { u } _ { \mathbf { i } } ^ { 2 } \right] = \mathrm { E } \left[ \mathrm { u } _ { \mathbf { j } } ^ { 2 } \right] = \sigma ^ { 2 }$ and we have:
$$E \left[ w _ { 1 } { } ^ { 2 } u _ { 1 } { } ^ { 2 } + w _ { 2 } { } ^ { 2 } u _ { 2 } { } ^ { 2 } + \ldots + w _ { n } { } ^ { 2 } u _ { n } { } ^ { 2 } \right] = \sigma ^ { 2 } E \left[ w _ { 1 } { } ^ { 2 } + w _ { 2 } { } ^ { 2 } + \ldots + w _ { n } { } ^ { 2 } \right] = \sigma ^ { 2 } \Sigma w _ { i } { } ^ { 2 }$$

Question 4

What is data mining? Explain why hypothesis tests, such as a test of significance, are invalid when data have been mined. Defend data mining as an econometric technique.

Accepted Answer

The answer of What is data mining? Explain why hypothesis...

Question 5

Explain why calculating VIFs is a more thorough test for multicollinearity than considering correlation coefficients.

Accepted Answer

The answer of Explain why calculating VIFs is a more...

Question 6

Prove that the following equation is undefined in the presence of perfect multicollinearity.

Accepted Answer

The answer of Prove that the following equation is undefined...

Question 7

Assumption 1 of the CLRM (Classical Linear Regression Model) is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Accepted Answer

Assumption 1 of the CLRM states that the relationship between the independent and dependent variables is linear in parameters, which is necessary for $\hat{\beta}_1$ to be linear, but other assumptions are also needed to ensure linearity in estimation.

Question 8

Assumption 1 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Accepted Answer

Assumption 1 of the Classical Linear Regression Model (CLRM), which states that the relationship between the dependent and independent variables is linear, is necessary to ensure that the Ordinary Least Squares (OLS) estimator $\hat { \beta } _ { 1 }$ is unbiased and consistent. However, it is not sufficient on its own to prove that $\hat { \beta } _ { 1 }$ is the Best Linear Unbiased Estimator (BLUE); additional assumptions are required for the Gauss-Markov theorem to hold, which establishes the BLUE property.

Question 9

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Accepted Answer

Assumption 2 of the Classical Linear Regression Model (CLRM) refers to the assumption of no perfect multicollinearity, not directly to the linearity of the estimator $\hat { \beta } _ { 1 }$. The linearity of $\hat { \beta } _ { 1 }$ in the parameters and in the regressors is a fundamental property of the linear regression model, not specifically tied to Assumption 2.

Question 10

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is unbiased.

Accepted Answer

Assumption 2 of the Classical Linear Regression Model (CLRM), which states that the explanatory variable is not perfectly collinear, is necessary to ensure that the OLS estimator can be computed. However, to prove that $\hat { \beta } _ { 1 }$ is unbiased, we also need other assumptions such as the expectation of the error term being zero conditional on the explanatory variables.

Question 11

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Accepted Answer

Assumption 2 of the Classical Linear Regression Model (CLRM) states that the expected value of the error term is zero, which is necessary for unbiasedness but not sufficient alone to ensure that $\hat{\beta}_1$ is the Best Linear Unbiased Estimator (BLUE). Other assumptions are also needed for it to be BLUE.

Question 12

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Accepted Answer

Assumption 3 of the Classical Linear Regression Model (CLRM) pertains to the zero conditional mean assumption, which states that the expected value of the error term, given any value of the independent variable, is zero. This assumption is crucial for unbiasedness of the OLS estimators but does not directly pertain to proving the linearity of $\hat { \beta } _ { 1 }$. Linearity of the estimator is a property ensured by the linear formulation of the model itself and the method of estimation (OLS), rather than by this specific assumption.

Question 13

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ isunbiased.

Accepted Answer

The answer of Assumption 3 of the CLRM is necessary...

Question 14

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Accepted Answer

Assumption 3 of the Classical Linear Regression Model (CLRM), which is the assumption of zero conditional mean (E[ε|X] = 0), is necessary to ensure that the OLS estimator $\hat { \beta } _ { 1 }$ is unbiased. However, being unbiased alone does not guarantee that it is the best (in the sense of having the smallest variance among all unbiased estimators), which requires additional assumptions like homoscedasticity and no autocorrelation of errors.

Question 15

Assumption 4 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Accepted Answer

Assumption 4 of the Classical Linear Regression Model (CLRM) pertains to the zero conditional mean assumption, which states that the expected value of the error term, given any value of the independent variable, is zero. This assumption is crucial for unbiasedness of the estimators, not directly for proving linearity of $\hat { \beta } _ { 1 }$. Linearity of the estimator is a property derived from the linear specification of the model itself, not from this assumption.

Question 16

Assumption 4 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is unbiased.

Accepted Answer

The answer of Assumption 4 of the CLRM is necessary...

Question 17

u_i ~ N(0, $\sigma$²) indicates that the true error terms are normally distributed with an expected value of 0 and that each true error term has a variance equal to some constant, $\sigma$².

Accepted Answer

The notation "~ N(0, ?²)" implies that the subject (in this case, the true error terms) follows a normal distribution with a mean (expected value) of 0 and a variance denoted by "?²," indicating that each error term has a constant variance, which is a fundamental assumption in many statistical models, including linear regression.

Question 18

If the u_i's are more likely to be positive when one of the explanatory variables is at higher values, then all of the estimators will be biased.

Accepted Answer

When the explanatory variables are at higher values, the u_i's are more likely to be positive. This means that the expected value of the u_i's is not zero, which will bias the estimators.

Question 19

TYPE I errors will be more likely in the presence of multicollinearity.

Accepted Answer

Multicollinearity primarily affects the precision of the estimates of the regression coefficients, making them less reliable, but it does not directly increase the likelihood of Type I errors, which are false positives in hypothesis testing.

Question 20

If E[ $\hat { \beta } _ { 1 } - \beta _ { 1 }$ ] = 0, then $\hat { \beta } _ { 1 }$ is unbiased.

Accepted Answer

If the expected value of the estimator minus the true parameter equals zero, it means that on average, the estimator hits the true parameter value, indicating that the estimator is unbiased.

Quiz 4: Hypothesis Testing

What is a Monte Carlo study? In addition, give examples. Why are these studies useful?

If a regression is in the incorrect functional form, explain why it is unlikely that E[u_i_│X_i] = 0.

What is data mining? Explain why hypothesis tests, such as a test of significance, are invalid when data have been mined. Defend data mining as an econometric technique.

Explain why calculating VIFs is a more thorough test for multicollinearity than considering correlation coefficients.

Prove that the following equation is undefined in the presence of perfect multicollinearity.

Assumption 1 of the CLRM (Classical Linear Regression Model) is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 1 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is unbiased.

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ isunbiased.

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Assumption 4 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 4 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is unbiased.

u_i ~ N(0, $\sigma$ ²) indicates that the true error terms are normally distributed with an expected value of 0 and that each true error term has a variance equal to some constant, $\sigma$ ².

If the u_i's are more likely to be positive when one of the explanatory variables is at higher values, then all of the estimators will be biased.

TYPE I errors will be more likely in the presence of multicollinearity.

If E[ $\hat { \beta } _ { 1 } - \beta _ { 1 }$ ] = 0, then $\hat { \beta } _ { 1 }$ is unbiased.

Quiz 4: Hypothesis Testing

What is a Monte Carlo study? In addition, give examples. Why are these studies useful?

If a regression is in the incorrect functional form, explain why it is unlikely that E[ui│Xi] = 0.

What is data mining? Explain why hypothesis tests, such as a test of significance, are invalid when data have been mined. Defend data mining as an econometric technique.

Explain why calculating VIFs is a more thorough test for multicollinearity than considering correlation coefficients.

Prove that the following equation is undefined in the presence of perfect multicollinearity.

Assumption 1 of the CLRM (Classical Linear Regression Model) is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is linear.

Assumption 1 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is best.

Assumption 2 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is linear.

Assumption 2 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is unbiased.

Assumption 2 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is best.

Assumption 3 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is linear.

Assumption 3 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ isunbiased.

Assumption 3 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is best.

Assumption 4 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is linear.

Assumption 4 of the CLRM is necessary but not sufficient to prove that β^1\hat { \beta } _ { 1 }β^​1​ is unbiased.

ui ~ N(0, σ\sigmaσ 2) indicates that the true error terms are normally distributed with an expected value of 0 and that each true error term has a variance equal to some constant, σ\sigmaσ 2.

If the ui's are more likely to be positive when one of the explanatory variables is at higher values, then all of the estimators will be biased.

TYPE I errors will be more likely in the presence of multicollinearity.

If E[ β^1−β1\hat { \beta } _ { 1 } - \beta _ { 1 }β^​1​−β1​ ] = 0, then β^1\hat { \beta } _ { 1 }β^​1​ is unbiased.

If a regression is in the incorrect functional form, explain why it is unlikely that E[u_i_│X_i] = 0.

Assumption 1 of the CLRM (Classical Linear Regression Model) is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 1 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is unbiased.

Assumption 2 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ isunbiased.

Assumption 3 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is best.

Assumption 4 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is linear.

Assumption 4 of the CLRM is necessary but not sufficient to prove that $\hat { \beta } _ { 1 }$ is unbiased.

u_i ~ N(0, $\sigma$ ²) indicates that the true error terms are normally distributed with an expected value of 0 and that each true error term has a variance equal to some constant, $\sigma$ ².

If the u_i's are more likely to be positive when one of the explanatory variables is at higher values, then all of the estimators will be biased.

If E[ $\hat { \beta } _ { 1 } - \beta _ { 1 }$ ] = 0, then $\hat { \beta } _ { 1 }$ is unbiased.