Statistics and Econometrics Questions and Answers

Answer: C

The outcomes are written as \({}^{3}C_{2}\).

Getting exactly two heads with three coins means choosing which 2 of the 3 coins show heads:

\[ {}^{3}C_{2} = \frac{3!}{2!\,1!} = 3 \]

The three outcomes are HHT, HTH and THH. \({}^{3}C_{3}\) counts three heads, \({}^{3}C_{0}\) no heads, and \({}^{3}C_{1}\) one head (which also equals 3 numerically, but it counts the choice of the single head, not the two heads the question asks about).

Answer: B

The Bernoulli distribution is associated with the Binomial Distribution.

A Bernoulli trial has two outcomes, success with probability \(p\) and failure with probability \(1 - p\). The binomial distribution gives the number of successes in \(n\) independent Bernoulli trials, so a binomial variable is the sum of \(n\) Bernoulli variables, and the Bernoulli distribution is the binomial with \(n = 1\).

The multinomial extends the binomial to more than two outcomes, the hypergeometric covers sampling without replacement, and the Poisson is a limiting case of the binomial for rare events. None is the distribution built directly from Bernoulli trials in this way.

Answer: D

The suitable models are ARCH and GARCH.

Financial returns often show volatility clustering: calm periods and turbulent periods, with the variance changing over time (heteroscedasticity). Engle's ARCH model makes the conditional variance depend on past squared errors, and Bollerslev's GARCH adds past variances as well, so they model this time-varying volatility directly.

AR, MA and ARMA models describe the conditional mean of a series and assume a constant error variance, so they cannot capture changing volatility.

Answer: D

Direct OLS leads to biased and inconsistent estimation.

In a simultaneous equations model, an endogenous variable that appears as an explanatory variable is correlated with the error term of its equation, because it is jointly determined with the dependent variable. This violates a key OLS assumption, and the result is simultaneity bias: the estimates are biased, and the bias does not disappear even in large samples, so they are also inconsistent.

That is why methods such as indirect least squares and two-stage least squares are used for such models.

Answer: B

A Type I error is the probability of rejecting the true hypothesis.

A Type I error occurs when the null hypothesis is true but the test rejects it. Its probability is the level of significance, \(\alpha\).

Accepting (failing to reject) a false null hypothesis is a Type II error, with probability \(\beta\). Rejecting a false hypothesis and accepting a true one are correct decisions, not errors.

Answer: D

The Dorbish and Bowley index is \(\frac{L + P}{2}\).

Laspeyres' index uses base-year quantities as weights and tends to overstate price change, while Paasche's uses current-year quantities and tends to understate it. Dorbish and Bowley proposed taking the arithmetic mean of the two to balance these biases.

Fisher's ideal index is the geometric mean, \(\sqrt{L \times P}\), not \(L \times P\) itself. The sum \(L + P\) and the ratio \(\frac{L}{P}\) are not index formulas.

Answer: A

The coefficient of determination is \(1 - \frac{\Sigma \hat{u}_{i}^{2}}{\Sigma y_{i}^{2}}\).

Here \(y_{i}\) are deviations of the dependent variable from its mean, so \(\Sigma y_{i}^{2}\) is the total sum of squares (TSS), and \(\Sigma \hat{u}_{i}^{2}\) is the residual sum of squares (RSS). Since TSS = ESS + RSS,

\[ R^{2} = \frac{ESS}{TSS} = 1 - \frac{RSS}{TSS} \]

\(R^{2}\) is the share of the variation in \(Y\) explained by the regression. The ratio \(\frac{\Sigma \hat{u}_{i}^{2}}{\Sigma y_{i}^{2}}\) alone is the unexplained share, \(1 - R^{2}\), and the options with \(\Sigma x_{i}^{2}\) or \(\Sigma \hat{x}_{i}^{2}\) use the wrong sums of squares.

Answer: D

For rates, proportions and ratios the geometric mean is the most appropriate average, since it multiplies the values and takes the root, which deals correctly with growth rates and ratios. The key is (D). The arithmetic mean is suited to absolute amounts and would overstate the average of ratios. The median is a positional average that ignores the size of the items. The harmonic mean is used for rates of the type of speed over fixed distances and is not the key for this question.

Answer: A

The official key is (A). In the regression Y = a + bX the constant 'a' is the value of Y when X is zero, so it shows the autonomous part of Y, the part that does not depend on X, and the reason gives this as the value that holds when X = 0. The wording that 'a' is the 'minimum value of Y' is loose, since Y could be lower if b were negative, but the key accepts the reason as the explanation. Options (B), (C) and (D) call the reason an incorrect or insufficient explanation.

Answer: C

The theory of probability deals with uncertain events (a-2), the mean is a single value that represents a distribution (b-1), the normal distribution is symmetrical with almost the whole area within the mean plus or minus three sigma (c-4), and the sample is a miniature replica of the population (d-3). The code is 2, 1, 4, 3, option (C). Options (A), (B) and (D) pair the theory of probability with a single value or with a symmetrical distribution, and so are wrong.