Question 1

In a regression model, if variance of the dependent variable, y, conditional on an explanatory variable, x, or Var(y|x), is not constant, _____.&#10;A)the t statistics are invalid and confidence intervals are valid for small sample sizes&#10;B)the t statistics are valid and confidence intervals are invalid for small sample sizes&#10;C) the t statistics and the confidence intervals are valid no matter how large the sample size is&#10;D) the t statistics and the confidence intervals are both invalid no matter how large the sample size is

Accepted Answer

If variance of the dependent variable conditional on an explanatory variable is not a constant the usual t statistics and the confidence intervals are both invalid no matter how large the sample size is.

Question 2

Which of the following statements is true?&#10;A)In large samples there are not many discrepancies between the outcomes of the F test and the LM test.&#10;B)Degrees of freedom of the unrestricted model are necessary for using the LM test.&#10;C) The LM test can be used to test hypotheses with single restrictions only and provides inefficient results for multiple restrictions.&#10;D) The LM statistic is derived on the basis of the normality assumption.

Accepted Answer

In large samples there are not many discrepancies between the F test and the LM test because asymptotically the two statistics have the same probability of a Type 1 error.

Question 3

Which of the following statements is true?&#10;A)The standard error of a regression,   , is not an unbiased estimator for   , the standard deviation of the error, u, in a multiple regression model.&#10;B)In time series regressions, OLS estimators are always unbiased.&#10;C) Almost all economists agree that unbiasedness is a minimal requirement for an estimator in regression analysis.&#10;D) All estimators in a regression model that are consistent are also unbiased.

Accepted Answer

The standard error of a regression is not an unbiased estimator for the standard deviation of the error in a multiple regression model.

Question 4

When the error term is not normally distributed, then   &#8203; is sometimes called the:&#10;A)&#8203;asymptotic standard error.&#10;B)&#8203;asymptotic t statistic.&#10;C) &#8203;asymptotic confidence interval.&#10;D) &#8203;asymptotic normality.

Accepted Answer

When the error term is not normally distributed, the standard error can still be estimated using the maximum likelihood method, but it is no longer a true standard error. Instead, it is known as the asymptotic standard error because it is only approximately valid for large sample sizes. When the error term is not normally distributed, then 11eab06d_250f_12c1_beed_6f9f5d630e62_TB2133_11 u200bis sometimes called the asymptotic standard error.

Question 5

In a multiple regression model, the OLS estimator is consistent if:&#10;A)there is no correlation between the dependent variables and the error term.&#10;B)there is a perfect correlation between the dependent variables and the error term.&#10;C) the sample size is less than the number of parameters in the model.&#10;D) there is no correlation between the independent variables and the error term.

Accepted Answer

The consistency of the OLS estimator relies on the absence of correlation between the independent variables and the error term. In a multiple regression model, the OLS estimator is consistent if there is no correlation between the explanatory variables and the error term.

Question 6

The Cauchy-Schwartz inequality implies that the asymptotic variance of &#8203;   is:&#10;A)&#8203;greater than   .&#10;B)&#8203;less than or equal to   .&#10;C) &#8203;equal to&#10;  .&#10;D) &#8203;less than&#10;  .

Accepted Answer

The Cauchy-Schwartz inequality states that for any two random variables X and Y, the following inequality holds:

|Cov(X,Y)| ≤ √(Var(X)Var(Y))

In this case, X is 11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11 and Y is 1. We know that the variance of Y is 0, since it is a constant. Therefore, we can simplify the inequality to:

|Cov(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11,1)| ≤ √(Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11) * 0)

The covariance between two independent random variables is 0, so we can further simplify to:

|Corr(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11,1)| ≤ √Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11)

The correlation between 11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11 and 1 is 1, since 1 is a constant. Therefore, we can simplify further to:

1 ≤ √Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11)

Squaring both sides gives:

1 ≤ Var(11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11)

Therefore, the asymptotic variance of 11eab06d_250f_af02_beed_87b0fdfcd9e6_TB2133_11 is less than or equal to 1, and the best choice is B. The Cauchy-Schwartz inequality implies that the asymptotic variance of 11eab06d_2510_2437_beed_9f56e623ae51_TB2133_11 is less than or equal to 11eab06d_2510_4b48_beed_4f018129c482_TB2133_11 .

Question 7

If the error term is correlated with any of the independent variables, the OLS estimators are:&#10;A)biased and consistent.&#10;B)unbiased and inconsistent.&#10;C) biased and inconsistent.&#10;D) unbiased and consistent.

Accepted Answer

If the error term is correlated with any of the independent variables, the OLS estimator will be biased and inconsistent. This is known as endogeneity. The bias arises because the OLS estimator is no longer the Best Linear Unbiased Estimator (BLUE) and the inconsistency arises because the estimator will not converge to the true population parameter as the sample size increases. Therefore, option C ("biased and inconsistent") is the correct answer. If the error term is correlated with any of the independent variables, then the OLS estimators are biased and inconsistent.

Question 8

In the multiple regression model , if x₁ is correlated with u but the other independent variables are uncorrelated with u, then all of the OLS estimators are generally consistent.

Accepted Answer

In the multiple regression model 11eab06d_2512_953c_beed_73facb132d40_TB2133_11 , if x₁ is correlated with u but the other independent variables are uncorrelated with u, then all of the OLS estimators are generally inconsistent.u200b

Question 9

The LM statistic follows a:&#10;A)t distribution.&#10;B)f distribution.&#10;C)&#10;  distribution.&#10;D) binomial distribution.

Accepted Answer

The LM statistic follows a 11eab06d_250e_9d8f_beed_f389f6e61c08_TB2133_11 distribution.

Question 10

If _j, an unbiased estimator of _j, is also a consistent estimator of _j, then when the sample size tends to infinity: A)the distribution of _j collapses to a single value of zero. B)the distribution of _j diverges away from a single value of zero. C) the distribution of _j collapses to the single point _j. D) the distribution of _j diverges away from _j.

Accepted Answer

This question is using notation and terminology from statistical inference. The symbol 11eab06d_2507_9890_beed_f5bfb5255f5c_TB2133_11 _j represents a parameter of the population, which is unknown and fixed. An estimator is a random variable that takes on different values depending on the sample that is used to compute it. An estimator is unbiased if its expected value is equal to the parameter it is estimating. An estimator is consistent if it approaches the true parameter value as the sample size increases. The question tells us that an unbiased estimator of the parameter 11eab06d_2507_9890_beed_f5bfb5255f5c_TB2133_11 _j is also a consistent estimator of the parameter 11eab06d_2507_9891_beed_a707c99d9773_TB2133_11 _j. This means that as the sample size tends to infinity, the estimator approaches the true parameter value of both parameters. Since the two parameters are different, it's possible that their true values are different, and the estimator might approach different values for the two parameters. The question asks what happens to the distribution of another parameter, 11eab06d_2507_e6b4_beed_bd4d616fa060_TB2133_11 _j, as the sample size tends to infinity. It gives us two options: either the distribution collapses to a single point (option C), or the distribution diverges away from a single point (option D). To answer this question, we need to remember that the estimator is a random variable, which means that it has a distribution. As the sample size tends to infinity, the estimator approaches the true parameter value, which means that its distribution becomes more and more concentrated around the true value. Option C says that the distribution collapses to a single point, which means that the estimator becomes more and more precise as the sample size increases, until it is almost certain to take on a specific value (the true parameter value). Option D says that the distribution diverges away from a single value, which doesn't make sense in this context - if the estimator is consistent, it approaches the true parameter value, not some other value. Therefore, the best choice is C, and the explanation is that as the sample size tends to infinity, the distribution of 11eab06d_2507_e6b4_beed_bd4d616fa060_TB2133_11 _j collapses to the single point 11eab06d_2507_e6b5_beed_d5548108e8c2_TB2133_11 _j, because the estimator is consistent and becomes more and more precise as the sample size increases. If 11eab06d_2508_34d8_beed_37d62d2463c0_TB2133_11 _j, an unbiased estimator of 11eab06d_2508_34d9_beed_f94e5c7a6a91_TB2133_11 _j, is also a consistent estimator of 11eab06d_2508_34da_beed_f9bce1f3dfd4_TB2133_11 _j, then when the sample size tends to infinity the distribution of 11eab06d_2508_5beb_beed_b3c070dea107_TB2133_11 _j collapses to the single point 11eab06d_2508_5bec_beed_b3e5173b9fce_TB2133_11 _j.

Question 11

If _j is an OLS estimator of a regression coefficient associated with one of the explanatory variables, such that j = 1, 2, …., n, asymptotic standard error of _j will refer to the: A)estimated variance of _j when the error term is normally distributed. B)estimated variance of a given coefficient when the error term is not normally distributed. C) square root of the estimated variance of _j when the error term is normally distributed. D) square root of the estimated variance of _j when the error term is not normally distributed.

Accepted Answer

The answer of If _j is an OLS...

Question 12

A useful rule of thumb is that standard errors are expected to shrink at a rate that is the inverse of the:&#10;A)square root of the sample size.&#10;B)product of the sample size and the number of parameters in the model.&#10;C) square of the sample size.&#10;D) sum of the sample size and the number of parameters in the model.

Accepted Answer

The answer of A useful rule of thumb is that...

Question 13

The n-R-squared statistic also refers to the:&#10;A)F statistic.&#10;B)t statistic.&#10;C) z statistic.&#10;D) LM statistic.

Accepted Answer

The answer of The n-R-squared statistic also refers to the:&#10;A)F...

Question 14

Which of the following statements is true under the Gauss-Markov assumptions?&#10;A)Among a certain class of estimators, OLS estimators are best linear unbiased, but are asymptotically inefficient.&#10;B)Among a certain class of estimators, OLS estimators are biased but asymptotically efficient.&#10;C) Among a certain class of estimators, OLS estimators are best linear unbiased and asymptotically efficient.&#10;D) The LM test is independent of the Gauss-Markov assumptions.

Accepted Answer

The answer of Which of the following statements is true...

Question 15

If OLS estimators satisfy asymptotic normality, it implies that: A)they are approximately normally distributed in large enough sample sizes. B)they are approximately normally distributed in samples with less than 10 observations. C) they have a constant mean equal to zero and variance equal to ². D) they have a constant mean equal to one and variance equal to .

Accepted Answer

The answer of If OLS estimators satisfy asymptotic normality, it...

Question 16

If the model satisfies the first four Gauss-Markov assumptions, then v has: A)a zero mean and is correlated with only x₁. B)a zero mean and is correlated with x₁ and x₂. C) a zero mean and is correlated with only x₂. D) a zero mean and is uncorrelated with x₁ and x₂.

Accepted Answer

The answer of If the model   &#8203; satisfies...

Question 17

If variance of an independent variable in a regression model, say x₁, is greater than 0, or Var(x₁) > 0, the inconsistency in ₁ (estimator associated with x₁) is negative, if x₁ and the error term are positively related.

Accepted Answer

The answer of If variance of an independent variable in...

Question 18

If _j, an unbiased estimator of _j, is consistent, then the: A)distribution of _j becomes more and more loosely distributed around _j as the sample size grows. B)distribution of _j becomes more and more tightly distributed around _j as the sample size grows. C) distribution of _j tends toward a standard normal distribution as the sample size grows. D) distribution of _j remains unaffected as the sample size grows.

Accepted Answer

The answer of If _j, an unbiased estimator...

Question 19

An auxiliary regression refers to a regression that is used:&#10;A)when the dependent variables are qualitative in nature.&#10;B)when the independent variables are qualitative in nature.&#10;C) to compute a test statistic but whose coefficients are not of direct interest.&#10;D) to compute coefficients which are of direct interest in the analysis.

Accepted Answer

The answer of An auxiliary regression refers to a regression...

Question 20

If ₁ = Cov(x₁,x₂) / Var(x₁) where x₁ and x₂ are two independent variables in a regression equation, which of the following statements is true? A)If x₂ has a positive partial effect on the dependent variable, and ₁ > 0, then the inconsistency in the simple regression slope estimator associated with x₁ is negative. B)If x₂ has a positive partial effect on the dependent variable, and ₁ > 0, then the inconsistency in the simple regression slope estimator associated with x₁ is positive. C) If x₁ has a positive partial effect on the dependent variable, and ₁ > 0, then the inconsistency in the simple regression slope estimator associated with x₁ is negative. D) If x₁ has a positive partial effect on the dependent variable, and ₁ > 0, then the inconsistency in the simple regression slope estimator associated with x₁ is positive.

Accepted Answer

The answer of If ₁ = Cov(x₁,x₂) /...

Question 21

A normally distributed random variable is symmetrically distributed about its mean, it can take on any positive or negative value (but with zero probability), and more than 95% of the area under the distribution is within two standard deviations.

Accepted Answer

The answer of A normally distributed random variable is symmetrically...

Question 22

If Cov(z,x) &#8800; 0, then z and x are correlated.&#8203;

Accepted Answer

The answer of If Cov(z,x) &#8800; 0, then z and...

Question 23

The F statistic is also referred to as the score statistic.

Accepted Answer

The answer of The F statistic is also referred to...

Question 24

Even if the error terms in a regression equation, u₁, u₂, …, u_n, are not normally distributed, the estimated coefficients can be normally distributed.

Accepted Answer

The answer of Even if the error terms in a...

Question 25

The LM statistic requires estimation of the unrestricted model only.

Accepted Answer

The answer of The LM statistic requires estimation of the...

Deck 5: Multiple Regression Analysis: Ols Asymptotics