Question 1

The formula for the standard error of the regression coefficient, when moving from one explanatory variable to two explanatory variables,
A)stays the same.
B)changes, unless the second explanatory variable is a binary variable.
C)changes.
D)changes, unless you test for a null hypothesis that the addition regression coefficient is zero.

Accepted Answer

The addition of a second explanatory variable changes the formula for the standard error of the regression coefficient. Specifically, it involves adding a term that takes into account the relationship between the two explanatory variables. This is true regardless of whether the second variable is binary or not, and regardless of whether a null hypothesis test is performed. Therefore, options A, B, and D are incorrect.

Question 2

When testing joint hypothesis, you should
A)use t-statistics for each hypothesis and reject the null hypothesis is all of the restrictions fail.
B)use the F-statistic and reject all the hypothesis if the statistic exceeds the critical value.
C)use t-statistics for each hypothesis and reject the null hypothesis once the statistic exceeds the critical value for a single hypothesis.
D)use the F-statistics and reject at least one of the hypothesis if the statistic exceeds the critical value.

Accepted Answer

When testing joint hypotheses, the appropriate approach is to use the F-statistic. If the calculated F-statistic exceeds the critical value, it indicates that at least one of the hypotheses in the joint test is rejected. This method is used because it considers the overall model and the interdependencies between variables, unlike t-tests which evaluate individual coefficients independently.

Question 3

In the multiple regression model, the t-statistic for testing that the slope is significantly different from zero is calculated A)by dividing the estimate by its standard error. B)from the square root of the F-statistic. C)by multiplying the p-value by 1.96. D)using the adjusted R² and the confidence interval.

Accepted Answer

The t-statistic for testing that the slope is significantly different from zero in a multiple regression model is calculated by dividing the estimate by its standard error. This gives us the number of standard errors away from the null hypothesis (β=0), and we can compare it to the t-distribution to calculate the p-value and assess statistical significance.

Question 4

To test joint linear hypotheses in the multiple regression model, you need to A)compare the sums of squared residuals from the restricted and unrestricted model. B)use the heteroskedasticity-robust F-statistic. C)use several t-statistics and perform tests using the standard normal distribution. D)compare the adjusted R² for the model which imposes the restrictions, and the unrestricted model.

Accepted Answer

The answer of To test joint linear hypotheses in the...

Question 5

The overall regression F-statistic tests the null hypothesis that
A)all slope coefficients are zero.
B)all slope coefficients and the intercept are zero.
C)the intercept in the regression and at least one, but not all, of the slope coefficients is zero.
D)the slope coefficient of the variable of interest is zero, but that the other slope coefficients are not.

Accepted Answer

The overall regression F-statistic tests the null hypothesis that all slope coefficients are zero.

Question 6

If you reject a joint null hypothesis using the F-test in a multiple hypothesis setting, then
A)a series of t-tests may or may not give you the same conclusion.
B)the regression is always significant.
C)all of the hypotheses are always simultaneously rejected.
D)the F-statistic must be negative.

Accepted Answer

When rejecting a joint null hypothesis using the F-test in a multiple hypothesis setting, it means that at least one of the individual null hypotheses is false. However, it does not necessarily mean that all of them are false. Therefore, a series of t-tests on each individual hypothesis may give different conclusions since each one is being tested separately.

Question 7

The homoskedasticity-only F-statistic is given by the following formula
A) 
B) $F = \frac { \left( S S R _ { \text {restricted } } - S S R _ { \text {unrestricted } } \right) / q } { S S R _ { \text {restricted } } / \left( n - k _ { \text {unrestricted } } - 1 \right) }$
C) $F = \frac { \left( S S R _ { \text {restricted } } - S S R _ { \text {unrestricted} } \right) / q } { S S R _ { \text {unrestricted } } / \left( n - k _ { \text {unrestricted } } - 1 \right) }$
D)

Accepted Answer

The answer of The homoskedasticity-only F-statistic is given by the...

Question 8

For a single restriction (q = 1), the F-statistic
A)is the square root of the t-statistic.
B)has a critical value of 1.96.
C)will be negative.
D)is the square of the t-statistic.

Accepted Answer

When q = 1, the F-statistic is equal to the square of the t-statistic. This is because the F-statistic is calculated as the ratio of the explained variance to the unexplained variance, and when there is only one restriction, the explained variance is equal to the variance explained by the single regression coefficient (which is what the t-statistic tests), while the unexplained variance is the residual variance (which is equal to the total variance minus the explained variance). Therefore, the F-statistic is equal to the square of the t-statistic.

Question 9

When your multiple regression function includes a single omitted variable regressor, then
A)use a two-sided alternative hypothesis to check the influence of all included variables.
B)the estimator for your included regressors will be biased if at least one of the included variables is correlated with the omitted variable.
C)the estimator for your included regressors will always be biased.
D)lower the critical value to 1.645 from 1.96 in a two-sided alternative hypothesis to test the significance of the coefficients of the included variables.

Accepted Answer

When a single omitted variable is included in a multiple regression function, the estimator for the included regressors will be biased if at least one of the included variables is correlated with the omitted variable. This is known as the omitted variable bias. Therefore, option B is the correct choice. The other options are incorrect. We do not need to use a two-sided alternative hypothesis to check the influence of all included variables (option A). The estimator for the included regressors will not always be biased (option C). Finally, we do not need to lower the critical value to 1.645 from 1.96 in a two-sided alternative hypothesis to test the significance of the coefficients of the included variables (option D).

Question 10

If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution you can
A)safely assume that your regression results are significant.
B)reject the null hypothesis.
C)reject the assumption that the error terms are homoskedastic.
D)conclude that most of the actual values are very close to the regression line.

Accepted Answer

The critical value from the standard normal distribution represents the level of significance chosen for the hypothesis test. If the absolute value of the calculated t-statistic exceeds this critical value, it indicates that the calculated result is unlikely to happen by chance alone, and we can reject the null hypothesis, which usually states that there is no significant relationship between the independent variable(s) and the dependent variable. Therefore, option B is the best choice. The other options are not directly related or do not follow from the t-statistic being significant.

Question 11

The homoskedasticity-only F-statistic is given by the following formula
A) 
B) 
C) $F = \frac { \left( R _ { \text {unrestricted } } ^ { 2 } - R _ { \text {restricted } } ^ { 2 } \right) / q } { \left( 1 - R _ { \text {unrestricted } } ^ { 2 } \right) / \left( n - k _ { \text {restricted } } - 1 \right) }$
D)

Accepted Answer

The answer of The homoskedasticity-only F-statistic is given by the...

Question 12

All of the following are correct formulae for the homoskedasticity-only F-statistic, with the exception of
A) $F = \frac { \left( S S R _ { \text {restricted } } - S S R _ { \text {unrestricted } } \right) / q } { S S R _ { \text {unrestricted} } / \left( n - k _ { \text {unrestricted } } - 1 \right) }$
B) $F = \frac { \left( S S R _ { \text {unrestricted } } - S S R _ { \text {restictad } } \right) / q } { S S R _ { \text {restricted} } / \left( n - k _ { \text {restricted } } - 1 \right) }$
C) 
D) $F = \left( \frac { S S R _ { \text {restricted } } } { S S R _ { \text {unrestricted } } } - 1 \right) \times \frac { \left( n - k _ { \text {unrestricted} } - 1 \right) } { q }$

Accepted Answer

The answer of All of the following are correct formulae...

Question 13

Let R²_unrestricted and R²_restricted be 0.4366 and 0.4149 respectively. The difference between the unrestricted and the restricted model is that you have imposed two restrictions. There are 420 observations. The F-statistic in this case is A)4.61 B)8.01 C)10.34 D)7.71

Accepted Answer

To calculate the F-statistic, we first need to find the sum of squared residuals (SSR) for both models. Let SSR_unrestricted and SSR_restricted be the sum of squared residuals for the unrestricted and restricted models, respectively.

Then, we can use the formula for the F-statistic:

F = ((SSR_restricted - SSR_unrestricted) / q) / (SSR_unrestricted / (n - k))

where q is the number of restrictions (in this case, q = 2), n is the sample size (420), and k is the number of parameters estimated in the unrestricted model (13).

From the information given in the question, we have:

SSR_unrestricted = (1 - 0.4366) * 420 = 235.812
SSR_restricted = (1 - 0.4149) * 420 = 243.912

Plugging these values into the formula, we get:

F = ((243.912 - 235.812) / 2) / (235.812 / (420 - 13))
  ≈ 8.01

Therefore, the answer is B.

Question 14

When there are two coefficients, the resulting confidence sets are
A)rectangles.
B)ellipses.
C)squares.
D)trapezoids.

Accepted Answer

When there are two coefficients, the resulting confidence sets are represented by ellipses. This is because the confidence sets are determined by the joint distribution of the two coefficients, which is typically a bivariate normal distribution. The level of confidence determines the size and shape of the ellipse.

Question 15

When testing the null hypothesis that two regression slopes are zero simultaneously, then you cannot reject the null hypothesis at the 5% level, if the ellipse contains the point A)(-1.96, 1.96). B) $| ( 0,1.96 ) |$ C)(0,0). D)(1.96², 1.96²).

Accepted Answer

The point (0,0) represents the scenario where both regression slopes are exactly zero, which is consistent with the null hypothesis that the slopes are zero. If the confidence ellipse contains this point, it indicates that the data does not provide sufficient evidence to reject the null hypothesis at the 5% level.

Question 16

The confidence interval for a single coefficient in a multiple regression A)makes little sense because the population parameter is unknown. B)should not be computed because there are other coefficients present in the model. C)contains information from a large number of hypothesis tests. D)should only be calculated if the regression R² is identical to the adjusted R².

Accepted Answer

The confidence interval for a single coefficient in a multiple regression contains information from a large number of hypothesis tests, reflecting the uncertainty around the estimate of that coefficient. It is a useful measure for understanding the precision of the estimate and does not depend on the population parameter being known, the presence of other coefficients, or specific values of model fit statistics like R-squared.

Question 17

A 95% confidence set for two or more coefficients is a set that contains
A)the sample values of these coefficients in 95% of randomly drawn samples.
B)integer values only.
C)the same values as the 95% confidence intervals constructed for the coefficients.
D)the population values of these coefficients in 95% of randomly drawn samples.

Accepted Answer

A 95% confidence set for coefficients is designed to contain the true population values of these coefficients in 95% of all possible samples, not just the sample values or specific integer values. It's about the population parameters, not the sample statistics or their specific numerical characteristics.

Question 18

All of the following are examples of joint hypotheses on multiple regression coefficients, with the exception of A)H₀ : ?₁ + ?₂= 1 B)H₀ : $\frac { \beta _ { 3 } } { \beta _ { 2 } }$ = ?₁and ?₄= 0 C)H₀ : ?₂ = 0 and ?₃= 0 D)H₀ : ?₁ = -?₂and ?₁ + ?₂= 1

Accepted Answer

A joint hypothesis involves more than one restriction on the coefficients. Option A is a single restriction, not a joint hypothesis.

Question 19

The following linear hypothesis can be tested using the F-test with the exception of A)β₂ = 1 and β₃= β₄/β₅. B)β₂=0. C)β₁ + β₂ = 1 and β₃ = -2β₄. D)β₀ = β₁ and β₁= 0.

Accepted Answer

The F-test can be used to test hypotheses that involve linear combinations of the regression coefficients. Option A includes such linear combinations and can therefore be tested using the F-test. Options B, C, and D do not involve linear combinations and cannot be tested using the F-test.

Question 20

If you wanted to test, using a 5% significance level, whether or not a specific slope coefficient is equal to one, then you should A)subtract 1 from the estimated coefficient, divide the difference by the standard error, and check if the resulting ratio is larger than 1.96. B)add and subtract 1.96 from the slope and check if that interval includes 1. C)see if the slope coefficient is between 0.95 and 1.05. D)check if the adjusted R² is close to 1.

Accepted Answer

To test whether a specific slope coefficient is equal to one, we need to calculate the t-statistic by subtracting 1 from the estimated coefficient and dividing the difference by the standard error. We then compare the resulting t-statistic to the critical value at a 5% significance level, which is 1.96 for a two-tailed test. Therefore, option A is the correct choice.

Quiz 7: Hypothesis Tests and Confidence Intervals in Multiple Regression

The formula for the standard error of the regression coefficient, when moving from one explanatory variable to two explanatory variables,

When testing joint hypothesis, you should

In the multiple regression model, the t-statistic for testing that the slope is significantly different from zero is calculated

To test joint linear hypotheses in the multiple regression model, you need to

The overall regression F-statistic tests the null hypothesis that

If you reject a joint null hypothesis using the F-test in a multiple hypothesis setting, then

The homoskedasticity-only F-statistic is given by the following formula

For a single restriction (q = 1) , the F-statistic

When your multiple regression function includes a single omitted variable regressor, then

If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution you can

The homoskedasticity-only F-statistic is given by the following formula

All of the following are correct formulae for the homoskedasticity-only F-statistic, with the exception of

Let R²_unrestricted and R²_restricted be 0.4366 and 0.4149 respectively. The difference between the unrestricted and the restricted model is that you have imposed two restrictions. There are 420 observations. The F-statistic in this case is

When there are two coefficients, the resulting confidence sets are

When testing the null hypothesis that two regression slopes are zero simultaneously, then you cannot reject the null hypothesis at the 5% level, if the ellipse contains the point

The confidence interval for a single coefficient in a multiple regression

A 95% confidence set for two or more coefficients is a set that contains

All of the following are examples of joint hypotheses on multiple regression coefficients, with the exception of

The following linear hypothesis can be tested using the F-test with the exception of

If you wanted to test, using a 5% significance level, whether or not a specific slope coefficient is equal to one, then you should

Quiz 7: Hypothesis Tests and Confidence Intervals in Multiple Regression

The formula for the standard error of the regression coefficient, when moving from one explanatory variable to two explanatory variables,

When testing joint hypothesis, you should

In the multiple regression model, the t-statistic for testing that the slope is significantly different from zero is calculated

To test joint linear hypotheses in the multiple regression model, you need to

The overall regression F-statistic tests the null hypothesis that

If you reject a joint null hypothesis using the F-test in a multiple hypothesis setting, then

The homoskedasticity-only F-statistic is given by the following formula

For a single restriction (q = 1) , the F-statistic

When your multiple regression function includes a single omitted variable regressor, then

If the absolute value of your calculated t-statistic exceeds the critical value from the standard normal distribution you can

The homoskedasticity-only F-statistic is given by the following formula

All of the following are correct formulae for the homoskedasticity-only F-statistic, with the exception of

Let R2unrestricted and R2restricted be 0.4366 and 0.4149 respectively. The difference between the unrestricted and the restricted model is that you have imposed two restrictions. There are 420 observations. The F-statistic in this case is

When there are two coefficients, the resulting confidence sets are

When testing the null hypothesis that two regression slopes are zero simultaneously, then you cannot reject the null hypothesis at the 5% level, if the ellipse contains the point

The confidence interval for a single coefficient in a multiple regression

A 95% confidence set for two or more coefficients is a set that contains

All of the following are examples of joint hypotheses on multiple regression coefficients, with the exception of

The following linear hypothesis can be tested using the F-test with the exception of

If you wanted to test, using a 5% significance level, whether or not a specific slope coefficient is equal to one, then you should

Let R²_unrestricted and R²_restricted be 0.4366 and 0.4149 respectively. The difference between the unrestricted and the restricted model is that you have imposed two restrictions. There are 420 observations. The F-statistic in this case is