Statistics and Econometrics Questions and Answers

Answer: D

In a multiple regression, the individual regression coefficients are tested with the t test, each coefficient being divided by its standard error. The F test is used to test the joint significance of all the coefficients together.

The Z test needs a known variance or a large sample, and the chi-square test is used for goodness of fit or for variances, not for individual coefficients. So the t test is the one used for each coefficient in this situation.

Answer: A

In the standard regression model, Y has a probability distribution because it depends on the random disturbance. X is treated as fixed in repeated samples, so it does not have a distribution.

The disturbance term is assumed to be uncorrelated with X, which excludes the third option. The residuals are not zero for all points in a sample, so the last option is wrong too. Only the first statement is true.

Answer: C

Statements (i), (ii) and (iii) hold for a triangular or recursive system. OLS can be applied to each equation separately and gives valid estimates. The explanatory variables may be correlated with the errors of other equations, but not with the error of the equation in which they appear.

Applying 2SLS to such a system gives unbiased but inefficient estimates, so (iii) is true. Statement (iv) is false because correlation with the error of its own equation would make OLS inconsistent.

Answer: C

If heteroscedasticity is ignored, the OLS estimator remains unbiased and consistent but is no longer efficient, because it no longer has the minimum variance among linear unbiased estimators, and the usual standard errors are wrong.

The options saying that OLS will be biased, or that it will be inconsistent, are therefore false. 'All of the above' includes those two, so it is not correct either, and only the statement about inefficiency is true.

Answer: C

The pairs are the unit root test with stationarity, the contingency table with the chi-square test, the regression coefficient with the Student's t-test, and autocorrelation with the Durbin-Watson test. This is a-3, b-4, c-2 and d-1.

A unit root test checks whether a series is stationary, a contingency table is tested with chi-square for association, a regression coefficient is tested with the t-test, and the Durbin-Watson statistic tests for serial correlation. The other codes pair the Durbin-Watson test with the unit root test.

Answer: A

The first equation is unidentified by the order condition. The condition needs the number of excluded exogenous variables to be at least the number of included endogenous variables minus one.

The first equation has three endogenous variables (Y1, Y2 and Y3) and two exogenous variables (X1 and X2), while X3 is excluded. Since 1 is less than 2, the order condition fails, so the equation is unidentified. It is not just identified or over-identified, and no reduced form is needed to apply the order condition.

Answer: A

In a binomial distribution the mean is np and the variance npq. Given mean + variance = 15 and mean x variance = 54, the two are the roots of x² - 15x + 54 = 0, that is 9 and 6. Since the variance is less than the mean, the mean is 9 and the variance 6. Then q = 6/9 = 2/3 and p = 1/3, so n = 9/(1/3) = 27. The answer is (A). The values 30, 24 and 33 do not give a mean of 9 with p = 1/3.

Answer: D

The correlation coefficient is the geometric mean of the two regression coefficients: r = \(\sqrt{b_{xy} \times b_{yx}}\), with the sign of the coefficients. Option (A) is their product, which equals r squared and not r. Option (B) takes a ratio, and (C) has a form that does not occur. So (D) is the answer.

Answer: B

The chi-square test is used to test the association between qualitative variables (attributes), with the observed frequencies in a contingency table compared with the frequencies expected if the variables were independent. Analysis of variance uses the F test and the difference between two means uses the Z or t test, so (A), (C) and (D) are wrong. The answer is (B). The test compares observed with expected frequencies, and the larger the difference, the stronger the evidence against independence.

Answer: C

The correct matching is I-3, II-4, III-2, IV-1. The coefficient of variation is the standard deviation divided by the mean, \(\sigma/\bar{x}\). The coefficient of correlation is \(\sum xy / n\sigma_{x}\sigma_{y}\), written as in the list. The coefficient of skewness is \(\mu_{3}/\sigma^{3}\) and the coefficient of kurtosis is \(\mu_{4}/\sigma^{4}\). The code 3 4 2 1 is option (C). The other options swap skewness and kurtosis or the first two.