Statistics and Econometrics Questions and Answers

Answer: B

It is the formula of Laspeyre's Price Index.

The numerator values base-year quantities (\(Q_{o}\)) at current prices (\(P_{t}\)), and the denominator values the same base-year quantities at base-year prices (\(P_{o}\)). Using base-year quantities as weights is the defining feature of the Laspeyres index:

\( P_{L} = \frac{\sum P_{t}Q_{o}}{\sum P_{o}Q_{o}} \times 100 \)

The Paasche index uses current-year quantities, \(\frac{\sum P_{t}Q_{t}}{\sum P_{o}Q_{t}}\). The GDP deflator is a Paasche-type index, and the CPI is a particular application rather than a formula.

Answer: D

The one that is not a test of adequacy is the Bowley Test.

Irving Fisher proposed tests that a good index number should satisfy: the time reversal test, the factor reversal test and the circular test (along with the unit test). Fisher's ideal index satisfies the first two.

Bowley is known for an index number formula (the arithmetic mean of the Laspeyres and Paasche indices) and for his measure of skewness, not for a test of adequacy.

Answer: D

(i), (ii) and (iii) are true.

• (i) The geometric mean is the right average for rates of growth, such as average percentage increase in sales or production over several years.
• (ii) It is used in constructing index numbers, notably Fisher's ideal index, which is the geometric mean of the Laspeyres and Paasche indices.
• (iii) It gives more weight to small items and less to large ones than the arithmetic mean does, so textbooks describe it as suitable where that weighting is wanted.

Since all three statements hold, the options keeping only some of them, or calling them all false, are wrong.

Answer: A

The relationship is Mode = 3 Median - 2 Mean.

For moderately skewed distributions, Karl Pearson observed that the median lies about one-third of the way from the mean to the mode, which gives the empirical formula:

\[ \text{Mode} = 3\,\text{Median} - 2\,\text{Mean} \]

A check: in a symmetric distribution all three are equal, and \(3m - 2m = m\). The other combinations fail that check (for example \(2m - 3m = -m\)).

Answer: C

There are 216 exhaustive cases.

Each die can show any of 6 faces, independently of the others, so by the multiplication principle:

\[ 6 \times 6 \times 6 = 6^{3} = 216 \]

18 adds the faces instead of multiplying (\(6 + 6 + 6\)), 108 is half of 216, and 72 corresponds to no standard count. Exhaustive cases list every possible outcome, so all \(6^{3}\) combinations count.

Answer: C

The one that is not non-parametric (non-probability) sampling is Stratified sampling.

Stratified sampling is a probability method: the population is divided into strata and a random sample is drawn from each, so every unit has a known chance of selection.

Judgmental (purposive), convenience and quota sampling choose units by the researcher's judgment or by ease of access, without random selection, so they are non-probability methods. Quota sampling looks like stratified sampling but fills the quotas non-randomly.

Answer: C

The one not used to test stationarity is the Engle-Granger Test.

The Engle-Granger procedure is a test for cointegration: whether a linear combination of non-stationary series is stationary, that is, whether they share a long-run relationship.

The Dickey-Fuller and Phillips-Perron tests are unit root tests, the standard tests of stationarity, and the autocorrelation function (correlogram) is also used to judge stationarity: a slowly decaying ACF suggests a non-stationary series.

Answer: D

2SLS is most appropriate for an Over Identified Equation.

In simultaneous-equation models, an over-identified equation has more excluded exogenous variables than needed, so indirect least squares gives more than one estimate of the structural parameters. Two-stage least squares combines all the instruments into one: it first regresses the endogenous regressors on all exogenous variables, then uses the fitted values in the second stage.

An exactly identified equation can be estimated by indirect least squares (2SLS gives the same result), and an under-identified (unidentified) equation cannot be estimated by any method.

Answer: D

\(r_{12.3}\) measures the partial correlation between Y and X₂.

With Y labelled 1, X₂ labelled 2 and X₃ labelled 3, the subscripts before the dot name the two variables being correlated, and those after the dot are held constant. So \(r_{12.3}\) is the correlation between Y and X₂ after removing the linear influence of X₃ from both.

The simple correlation between Y and X₂ is \(r_{12}\), without the dot. The partial correlation between Y and X₃ would be \(r_{13.2}\), and X₂ with X₃ would be \(r_{23}\).

Answer: A

It means the value of r is not at all significant.

The probable error of the correlation coefficient is \(PE = 0.6745 \times \frac{1 - r^{2}}{\sqrt{n}}\). The usual rules of interpretation are:

• If \(r\) is less than PE, the correlation is not significant at all.
• If \(r\) is more than six times PE, it is definitely significant.
• In between, the result is not conclusive.

So a value of \(r\) below its probable error cannot be called significant, moderately or highly.