Deck 4: Hypothesis Testing

Question 1

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

Answer 1

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.

Answer 2

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

Answer 3

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.

Question 5

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

Question 6

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

Question 7

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

Question 8

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

Question 9

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

Question 10

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

Question 11

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

Question 12

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

Question 13

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

Question 14

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

Question 15

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

Question 16

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

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.

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.

Question 19

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

Question 20

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

Question 21

If the u_i's are binomially distributed, then $\hat { \beta } _ { 1 }$ will be biased.

Question 22

$\hat { \beta } _ { 1 }$ is BLUE in the presence of perfect multicollinearity.

Question 23

A violation of assumption 2 of the CLRM carries more severe consequences than a violation of assumption 4.

Question 24

Multicollinearity can lead to unexpected signs on regression coefficients.

Question 25

Serial correlation results in inefficient estimates of the structural parameters.