For a Poisson random variable with parameter λ, what are its mean and variance?

The mean is λ and the variance is also λ

The mean is λ and the variance is λ²

The mean is np and the variance is np(1 − p)

The mean is λ/2 and the variance is λ

The Central Limit Theorem is most accurately described by which statement?

The sampling distribution of the mean approaches normality with standard error σ/√n as n grows, for any parent distribution of finite variance

The raw observations themselves become normally distributed as the sample size grows large, whatever the shape of the population from which they are drawn

The sampling distribution of the mean approaches normality with standard error σ/√n as n grows, for any parent distribution of finite variance

The population variance shrinks to zero as the sample becomes very large

Any statistic computed from a large sample is unbiased for its population parameter

At the 5% significance level, a researcher tests H₀: μ = 50 against the two-sided alternative H₁: μ ≠ 50. What does the 5% significance level represent here?

The probability of rejecting H₀ when it is in fact true

The probability that H₀ is true given the data

The probability of failing to reject H₀ when it is false

The power of the test against the alternative

Under the Gauss–Markov assumptions, the OLS estimator is described by which property?

It is the best (minimum-variance) linear unbiased estimator, with no normality assumption required

It is the maximum-likelihood estimator whatever the distribution of the regression errors, so normality is irrelevant

It is the best (minimum-variance) linear unbiased estimator, with no normality assumption required

It is biased but has the smallest possible variance among all estimators

It is unbiased only when the regressors are perfectly uncorrelated with each other

Two economically unrelated variables are each a random walk, and one is regressed on the other by OLS. The regression reports a high R² and a highly significant slope, yet the Durbin–Watson statistic is very low. This pattern is best explained as:

Spurious regression, because both series are non-stationary I(1) processes

A genuine long-run causal relationship that the researcher has correctly uncovered

Perfect multicollinearity between the two variables

Spurious regression, because both series are non-stationary I(1) processes

Heteroscedasticity that has inflated the coefficient estimate

A consumer maximises utility subject to a budget constraint using a Lagrangian. The optimal value of the Lagrange multiplier λ is best interpreted as:

The marginal utility of income — the rise in maximised utility per unit relaxation of the budget

The total utility the consumer enjoys at the chosen optimum bundle, expressed in the same units as the objective function

The marginal utility of income — the rise in maximised utility per unit relaxation of the budget

The ratio of the two goods' prices at the optimum

The quantity of the cheaper good purchased at the optimum

Consider the consequences of heteroscedasticity in an otherwise correctly specified OLS regression. Which of the following statements are correct?

The OLS coefficient estimates remain unbiased

The usual OLS standard errors become invalid, so t and F tests can mislead

OLS is no longer efficient — it is not the minimum-variance linear unbiased estimator

The OLS coefficient estimates remain unbiased

The usual OLS standard errors become invalid, so t and F tests can mislead

OLS is no longer efficient — it is not the minimum-variance linear unbiased estimator

The OLS coefficient estimates become biased and inconsistent

Regarding the Laspeyres, Paasche and Fisher price indices, which of the following statements are correct?

The Laspeyres index uses base-year quantities as weights

Fisher's ideal index is the geometric mean of the Laspeyres and Paasche indices

Fisher's index satisfies both the time-reversal and factor-reversal tests

The Laspeyres index uses base-year quantities as weights

Fisher's ideal index is the geometric mean of the Laspeyres and Paasche indices

Fisher's index satisfies both the time-reversal and factor-reversal tests

The Paasche index uses base-year quantities as weights

Two series are each integrated of order one, I(1). Which of the following statements are correct?

They are cointegrated if some linear combination of them is stationary, I(0)

If they are cointegrated, an error-correction model links short-run changes to the lagged equilibrium error

Differencing each series once produces a stationary series

They are cointegrated if some linear combination of them is stationary, I(0)

If they are cointegrated, an error-correction model links short-run changes to the lagged equilibrium error

Differencing each series once produces a stationary series

Any regression of one I(1) series on the other is guaranteed to be free of spurious results

In an exactly identified structural equation, the order condition for identification is best stated as:

The number of excluded exogenous variables equals the number of included endogenous variables minus one

The number of structural equations in the model equals the total number of exogenous variables appearing anywhere in the system

Every variable in the system appears in every equation

The error terms of all equations are mutually uncorrelated

Statistics, Econometrics and Mathematical Economics

This chapter answers Section XH1.3 of the Humanities and Social Sciences (XH) paper, "Statistics, Econometrics and Mathematical Economics" — part of Section XH1 Economics, one of the six optional sections, chosen at application. It builds the quantitative toolkit an economist uses: probability and its distributions and the Central Limit Theorem; index numbers and the construction of price indices; sampling, estimation and hypothesis testing; the classical linear regression model, the Gauss–Markov theorem and the three classical breakdowns (heteroscedasticity, multicollinearity, autocorrelation), dummy variables, simultaneous equations and identification; time-series ideas from AR, MA and ARMA to non-stationarity, spurious regression, cointegration, error correction, causality and VAR; and the mathematical economics of differential calculus, matrices and Cramer's rule, static optimisation, the input–output model, linear programming and difference and differential equations. Worked numericals run through index numbers, a regression slope and t-ratio, Cramer's rule, a linear-programming optimum and a Leontief output.

1. Probability, distributions and the Central Limit Theorem

Probability measures how likely an event is, on a scale from 0 to 1. The classical view counts equally likely outcomes, the frequentist view is a long-run relative frequency, and the subjective view is a degree of belief. Key rules: the addition rule P(A ∪ B) = P(A) + P(B) − P(A ∩ B); for independent events P(A ∩ B) = P(A)P(B); conditional probability P(A | B) = P(A ∩ B)/P(B); and Bayes' theorem, which reverses a conditional, P(A | B) = P(B | A)P(A)/P(B). A random variable attaches a number to each outcome; its expectation E(X) is the probability-weighted mean and its variance Var(X) = E(X²) − [E(X)]² the spread.

Standard distributions and their mean and variance
Distribution	Use	Mean	Variance
Binomial (n, p)	Number of successes in n independent trials	np	np(1 − p)
Poisson (λ)	Rare events in a fixed interval	λ	λ
Normal (μ, σ²)	Continuous, bell-shaped, symmetric	μ	σ²
Uniform on [a, b]	Every value equally likely	(a + b)/2	(b − a)²/12

🎯 Why the Central Limit Theorem matters

The Central Limit Theorem (CLT) says that the sample mean of n independent draws from almost any distribution with finite variance is approximately normal for large n, with mean μ and variance σ²/n, so its standard deviation (the standard error) is σ/√n. This is why normal-based t and z tests work even when the data are not normal: it is the mean, not the raw data, that becomes normal. It also explains why quadrupling the sample size only halves the standard error — precision grows with √n, not n.

2. Index numbers and the construction of price indices

An index number tracks how a group of related quantities — prices, output, cost of living — moves relative to a base period set at 100. A price index aggregates many prices with quantity weights. Two conventions differ by whose basket they weight. The Laspeyres index uses base-year quantities q₀: L = (Σp₁q₀ / Σp₀q₀) × 100. The Paasche index uses current-year quantities q₁: P = (Σp₁q₁ / Σp₀q₁) × 100. Fisher's ideal index is their geometric mean, F = √(L × P). Laspeyres tends to overstate and Paasche to understate the rise in the cost of living, because consumers substitute away from goods that become dearer.

Time-reversal test: a good index should satisfy P₀₁ × P₁₀ = 1 (reversing the dates inverts the index). Laspeyres and Paasche fail it; Fisher passes.
Factor-reversal test: the price index times the quantity index should equal the value ratio Σp₁q₁/Σp₀q₀. Of the three indices only Fisher passes, which is why it is called ideal.
Common series: the Consumer Price Index (CPI) for retail cost of living, the Wholesale Price Index (WPI) for bulk transactions, and GDP deflator = (nominal GDP/real GDP) × 100, which is an implicit Paasche-type index covering all output.

⚠️ State the base year and whose weights

A price-index number is meaningless without saying which year is the base (= 100) and whether base-year or current-year quantities weight it. Laspeyres and Paasche use the same prices but different weights and give different numbers, so a question must name the index it wants.

3. Sampling, estimation and hypothesis testing

A sample stands in for a population. Probability sampling — simple random, systematic, stratified, cluster — lets us quantify error; non-probability sampling (convenience, quota) cannot. A statistic computed from the sample, such as the mean, has a sampling distribution; the CLT gives its shape. An estimator should be unbiased (its expectation equals the parameter), efficient (least variance) and consistent (it converges to the parameter as n grows). A confidence interval such as x̄ ± 1.96·(σ/√n) gives a 95% interval: over repeated samples, 95% of such intervals cover the true mean.

Hypothesis testing pits a null hypothesis H₀ against an alternative H₁. A test statistic is compared with a critical value at a chosen significance level α (the probability of rejecting a true H₀, a Type I error). A Type II error is failing to reject a false H₀, with probability β; power = 1 − β. A two-tailed test splits α between both tails; a one-tailed test puts it all in one. Standard tools: the z test when σ is known, the t test when σ is estimated, the chi-square test for goodness of fit and independence, and the F test for comparing variances and for overall regression significance.

⚠️ A p-value is not the probability H₀ is true

The p-value is the probability of a result at least as extreme as the one observed, assuming H₀ is true — not the probability that H₀ holds. Reject H₀ when p < α. A smaller α makes rejection harder (fewer Type I errors) but raises Type II error, so the two cannot both be cut at a fixed sample size.

4. Regression, Gauss–Markov and the classical breakdowns

The simple linear regression model is Yᵢ = β₀ + β₁Xᵢ + uᵢ. Ordinary least squares (OLS) chooses β̂₀, β̂₁ to minimise the sum of squared residuals, giving slope β̂₁ = Σ(Xᵢ − X̄)(Yᵢ − Ȳ)/Σ(Xᵢ − X̄)² and intercept β̂₀ = Ȳ − β̂₁X̄. R² is the share of the variation in Y explained. The Gauss–Markov theorem states that, under its assumptions — linearity in parameters, zero mean error E(u)=0, homoscedasticity (constant error variance), no autocorrelation, and regressors uncorrelated with the error (plus no perfect multicollinearity in multiple regression) — OLS is BLUE: the Best (minimum-variance) Linear Unbiased Estimator. Normality of the errors is not needed for BLUE, only for the exact t and F tests in small samples.

The three classical breakdowns
Problem	What breaks	Consequence for OLS and usual remedy	A detection test
Heteroscedasticity	Error variance not constant	Still unbiased, no longer efficient; SEs wrong. Remedy: robust (White) SEs, or WLS/GLS	Breusch–Pagan, White
Autocorrelation	Errors correlated across observations	Still unbiased, inefficient; SEs wrong. Remedy: Newey–West (HAC) SEs, or GLS such as Cochrane–Orcutt	Durbin–Watson, Breusch–Godfrey
Multicollinearity	Regressors strongly correlated	Still BLUE but SEs inflated, signs unstable. Remedy: more data, or drop or combine collinear regressors (beware specification bias)	High VIF, high R² with few significant t

Dummy (indicator) variables encode a qualitative category as 0 or 1 — say D = 1 for urban, 0 for rural. A single dummy shifts the intercept; a dummy interacted with X shifts the slope. With a category of k levels, include k − 1 dummies and let the omitted level be the base, or the model cannot be estimated — the dummy-variable trap is perfect multicollinearity from including all k plus an intercept.

5. Simultaneous equations, identification and time series

In a simultaneous-equation model variables are jointly determined — price and quantity by demand and supply together — so a regressor is correlated with the error and OLS is biased (simultaneity bias). A recursive system has a triangular structure (with errors uncorrelated across equations) and can be estimated by OLS one equation at a time; a non-recursive system cannot, because OLS is then biased. The identification problem asks whether an equation's parameters can be recovered at all. The order condition (necessary, not sufficient): an equation can be identified only if the number of excluded exogenous variables is at least the number of included endogenous variables (counting the dependent variable) minus one — exactly equal gives exact identification, more gives over-identification, fewer gives under-identification. The rank condition (necessary and sufficient) requires that the matrix of coefficients, in the other equations, of the variables excluded from this equation has rank equal to the number of equations minus one. Identified equations are estimated by instrumental variables such as 2SLS.

A time series is stationary if its mean, variance and autocovariances do not change over time. A random walk Yₜ = Yₜ₋₁ + εₜ is non-stationary (a unit root); differencing it once yields a stationary series, so it is integrated of order 1, I(1). Regressing one unrelated I(1) series on another typically gives a high R² and significant t with no real relationship — a spurious regression — which the low Durbin–Watson statistic betrays. The Dickey–Fuller / ADF test checks for a unit root (H₀: a unit root is present).

AR, MA, ARMA: an AR(p) model regresses Yₜ on its own past values; an MA(q) model on past error terms; ARMA(p, q) combines both, and ARIMA adds d rounds of differencing for an I(d) series. An AR(1) is stationary only if |φ| < 1 (in general, the roots of the AR lag polynomial must lie outside the unit circle); a finite MA is always stationary.
Cointegration and error correction: two I(1) series are cointegrated if a linear combination of them is I(0) — a genuine long-run equilibrium. The Engle–Granger / error-correction model (ECM) then ties short-run changes to the lagged deviation from that equilibrium, with a negative adjustment coefficient pulling the system back.
Causality and VAR: Granger causality says X Granger-causes Y if past X helps predict Y beyond Y's own past — prediction, not true cause. A vector autoregression (VAR) models several series together, each on lags of all, used for forecasting and impulse-response analysis.

6. Mathematical economics: calculus, matrices, optimisation and dynamics

Differential calculus underlies marginal analysis: marginal cost is dC/dq, marginal utility is dU/dx, and elasticity is a logarithmic derivative (dln y/dln x). An unconstrained optimum needs the first derivative zero (first-order condition) and the second derivative negative for a maximum, positive for a minimum (second-order condition). A constrained optimum uses a Lagrangian L = f(x) + λ[c − g(x)] for the constraint g(x) = c, where the multiplier λ is the shadow price — the gain in the objective per unit relaxation of the constraint. Euler's theorem says a function homogeneous of degree one exhausts its output among factors paid their marginal products.

Linear algebra solves systems of equations. For AX = B with a square A, Cramer's rule gives each unknown as xᵢ = det(Aᵢ)/det(A), where Aᵢ replaces column i of A by B; it needs det(A) ≠ 0. The input–output (Leontief) model writes each sector's output as X = AX + d, where A is the technical-coefficient matrix and d final demand, solved as X = (I − A)⁻¹d; the Leontief inverse (I − A)⁻¹ shows the total (direct plus indirect) output each unit of final demand requires. The Hawkins–Simon condition ensures a non-negative viable solution.

Linear programming (LP) maximises or minimises a linear objective subject to linear inequality constraints. Its feasible region is a convex polygon, and an optimum — if one exists — sits at a vertex (corner point), so evaluating the objective at each vertex finds it. Every LP has a dual, and when the primal has an optimum so does the dual, with equal optimal value (strong duality); the dual variables are shadow prices. Dynamics: a difference equation yₜ = a·yₜ₋₁ + b has a stable path converging to the steady state y* = b/(1 − a) when |a| < 1; a differential equation dy/dt = a·y + b is stable when a < 0. Difference equations describe the cobweb model and discrete-time debt dynamics; differential equations describe continuous growth paths.

Key takeaways

Expectation is the probability-weighted mean and Var(X) = E(X²) − [E(X)]²; binomial mean np, Poisson mean and variance both λ. The CLT makes the sample mean approximately normal with standard error σ/√n, so precision grows with √n.
Laspeyres weights by base-year quantities, Paasche by current-year; Fisher is their geometric mean and alone passes the time- and factor-reversal tests. Always state the base year and whose weights.
In hypothesis testing α is the Type I error rate, power is 1 − β, and a two-tailed test splits α between both tails. A p-value is the chance of so extreme a result if H₀ is true, not the probability H₀ holds.
Under the Gauss–Markov assumptions OLS is BLUE. Heteroscedasticity and autocorrelation keep OLS unbiased but make it inefficient with wrong standard errors; multicollinearity inflates standard errors. Use k − 1 dummies for k categories.
A random walk is I(1); regressing unrelated I(1) series gives spurious regression. Cointegrated I(1) series share a long-run equilibrium captured by an error-correction model; Granger causality is predictive, and the order condition is only a necessary check for identification (the rank condition settles it).
Cramer's rule gives xᵢ = det(Aᵢ)/det(A); the Leontief model solves X = (I − A)⁻¹d; a Lagrange multiplier is a shadow price; and a linear-programming optimum sits at a vertex of the feasible region.

Practice questions (17)

Attempt each one before opening the answer. Every explanation names the tempting wrong option as well as the right one, because that is where marks are lost.

Take 2015 as the base year (index = 100). Two goods have base-year prices and quantities and current-year prices as follows. Good A: base price 8, base quantity 10, current price 10. Good B: base price 5, base quantity 4, current price 7. Compute the Laspeyres price index for the current year (base-year quantities as weights). Give the number to the nearest integer.
Numerical answer — type the value.
Show answer
Answer: 128
Laspeyres L = (Σp₁q₀ / Σp₀q₀) × 100. Σp₁q₀ = 10×10 + 7×4 = 100 + 28 = 128. Σp₀q₀ = 8×10 + 5×4 = 80 + 20 = 100. L = (128/100) × 100 = 128. Using current-year quantities instead would give Paasche, a different number, which is the standard slip.
Five observations give X = 1, 2, 3, 4, 5 and Y = 2, 4, 5, 4, 5. Fit the OLS regression of Y on X, Yᵢ = β₀ + β₁Xᵢ + uᵢ. Compute the estimated slope β̂₁ to one decimal place.
Numerical answer — type the value.
Show answer
Answer: 0.6
X̄ = 3, Ȳ = 4. β̂₁ = Σ(X − X̄)(Y − Ȳ)/Σ(X − X̄)². Numerator = (−2)(−2) + (−1)(0) + (0)(1) + (1)(0) + (2)(1) = 4 + 0 + 0 + 0 + 2 = 6. Denominator = 4 + 1 + 0 + 1 + 4 = 10. β̂₁ = 6/10 = 0.6. (The intercept would be Ȳ − β̂₁X̄ = 4 − 1.8 = 2.2, but only the slope is asked.)
In a regression, an estimated coefficient is 2.4 with an estimated standard error of 0.8. Compute the t-statistic for testing the null hypothesis that the true coefficient equals zero.
Numerical answer — type the value.
Show answer
Answer: 3
The t-statistic for H₀: β = 0 is the coefficient divided by its standard error, t = β̂/SE(β̂) = 2.4/0.8 = 3. A |t| of 3 exceeds the usual 5% two-tailed critical value near 1.96–2.0, so the coefficient is significant at conventional levels.
Solve the system 2x + y = 8 and x + 3y = 9 using Cramer's rule. Give the value of x.
Numerical answer — type the value.
Show answer
Answer: 3
The coefficient determinant D = |2 1; 1 3| = (2)(3) − (1)(1) = 5. Replacing the x-column by the right-hand side gives Dₓ = |8 1; 9 3| = (8)(3) − (1)(9) = 15. So x = Dₓ/D = 15/5 = 3 (and y = 2, since 2(3) + 2 = 8).
Maximise Z = 3x + 2y subject to x + y ≤ 4, x ≤ 3, y ≤ 3 and x, y ≥ 0. Give the maximum value of Z.
Numerical answer — type the value.
Show answer
Answer: 11
The optimum of a linear program sits at a vertex of the feasible region. Vertices: (0,0) gives 0; (3,0) gives 9; (3,1) gives 11; (1,3) gives 9; (0,3) gives 6. The corner (3,1) is where x = 3 meets x + y = 4. The maximum is Z = 11 at (3, 1).
A two-sector economy has technical-coefficient matrix A with a₁₁ = 0.2, a₁₂ = 0.4, a₂₁ = 0.3, a₂₂ = 0.1 (aᵢⱼ is input from sector i per unit of sector j's output). Final demand is 24 for sector 1 and 12 for sector 2. Using X = (I − A)⁻¹d, compute the total output required from sector 1.
Numerical answer — type the value.
Show answer
Answer: 44
I − A = [0.8, −0.4; −0.3, 0.9], with determinant (0.8)(0.9) − (−0.4)(−0.3) = 0.72 − 0.12 = 0.60. Solving (I − A)X = d: 0.8X₁ − 0.4X₂ = 24 and −0.3X₁ + 0.9X₂ = 12. The inverse gives X₁ = (0.9·24 + 0.4·12)/0.60 = (21.6 + 4.8)/0.60 = 26.4/0.60 = 44 (and X₂ = 28). Total output of sector 1 is 44.
For a Poisson random variable with parameter λ, what are its mean and variance?
1. The mean is λ and the variance is also λ
2. The mean is λ and the variance is λ²
3. The mean is np and the variance is np(1 − p)
4. The mean is λ/2 and the variance is λ
Show answer
Answer: A — The mean is λ and the variance is also λ
A defining feature of the Poisson distribution is that its mean and variance are equal, both λ. The mean np with variance np(1 − p) belongs to the binomial, which the Poisson approximates when n is large and p small with np = λ.
The Central Limit Theorem is most accurately described by which statement?
1. The raw observations themselves become normally distributed as the sample size grows large, whatever the shape of the population from which they are drawn
2. The sampling distribution of the mean approaches normality with standard error σ/√n as n grows, for any parent distribution of finite variance
3. The population variance shrinks to zero as the sample becomes very large
4. Any statistic computed from a large sample is unbiased for its population parameter
Show answer
Answer: B — The sampling distribution of the mean approaches normality with standard error σ/√n as n grows, for any parent distribution of finite variance
The CLT is about the distribution of the sample mean, not of the raw data: for large n the mean is approximately normal with standard error σ/√n whatever the parent shape. The raw observations keep their own distribution; the population variance σ² is a fixed parameter; and large samples do not make every statistic unbiased.
At the 5% significance level, a researcher tests H₀: μ = 50 against the two-sided alternative H₁: μ ≠ 50. What does the 5% significance level represent here?
1. The probability of rejecting H₀ when it is in fact true
2. The probability that H₀ is true given the data
3. The probability of failing to reject H₀ when it is false
4. The power of the test against the alternative
Show answer
Answer: A — The probability of rejecting H₀ when it is in fact true
The significance level α is the probability of a Type I error — rejecting a true null — set by the researcher before the test. The probability of H₀ given the data is a Bayesian posterior, not α; failing to reject a false null is the Type II error β; and power is 1 − β.
Under the Gauss–Markov assumptions, the OLS estimator is described by which property?
1. It is the maximum-likelihood estimator whatever the distribution of the regression errors, so normality is irrelevant
2. It is the best (minimum-variance) linear unbiased estimator, with no normality assumption required
3. It is biased but has the smallest possible variance among all estimators
4. It is unbiased only when the regressors are perfectly uncorrelated with each other
Show answer
Answer: B — It is the best (minimum-variance) linear unbiased estimator, with no normality assumption required
Gauss–Markov makes OLS BLUE — the minimum-variance estimator within the class of linear unbiased estimators — and normality is not among its assumptions (normality is needed only for exact small-sample t and F inference). OLS is unbiased, not biased; it coincides with maximum likelihood only if the errors happen to be normal, which Gauss–Markov does not assume; and ordinary correlation among regressors (short of perfect collinearity) does not destroy unbiasedness.
Two economically unrelated variables are each a random walk, and one is regressed on the other by OLS. The regression reports a high R² and a highly significant slope, yet the Durbin–Watson statistic is very low. This pattern is best explained as:
1. A genuine long-run causal relationship that the researcher has correctly uncovered
2. Perfect multicollinearity between the two variables
3. Spurious regression, because both series are non-stationary I(1) processes
4. Heteroscedasticity that has inflated the coefficient estimate
Show answer
Answer: C — Spurious regression, because both series are non-stationary I(1) processes
A high R² with a very low Durbin–Watson from regressing one non-stationary (unit-root, I(1)) series on another unrelated one is the classic signature of spurious regression: the apparent fit is an artefact of shared trending, not a relationship. Multicollinearity concerns correlation among regressors in one equation, and heteroscedasticity affects efficiency and standard errors, not this trend-driven false fit.
A consumer maximises utility subject to a budget constraint using a Lagrangian. The optimal value of the Lagrange multiplier λ is best interpreted as:
1. The total utility the consumer enjoys at the chosen optimum bundle, expressed in the same units as the objective function
2. The marginal utility of income — the rise in maximised utility per unit relaxation of the budget
3. The ratio of the two goods' prices at the optimum
4. The quantity of the cheaper good purchased at the optimum
Show answer
Answer: B — The marginal utility of income — the rise in maximised utility per unit relaxation of the budget
The multiplier is the shadow price of the constraint: in consumer choice it measures how much maximised utility rises if the budget is loosened by one unit, i.e. the marginal utility of income. It is not total utility, not the price ratio (that equals the ratio of marginal utilities at the optimum), and not a quantity.
Consider the consequences of heteroscedasticity in an otherwise correctly specified OLS regression. Which of the following statements are correct?
1. The OLS coefficient estimates remain unbiased
2. The usual OLS standard errors become invalid, so t and F tests can mislead
3. OLS is no longer efficient — it is not the minimum-variance linear unbiased estimator
4. The OLS coefficient estimates become biased and inconsistent
Show answer
Answer: A — The OLS coefficient estimates remain unbiased; B — The usual OLS standard errors become invalid, so t and F tests can mislead; C — OLS is no longer efficient — it is not the minimum-variance linear unbiased estimator
Heteroscedasticity violates the constant-variance assumption but leaves the slope estimates unbiased and consistent; what it destroys is efficiency and the validity of the usual standard errors, so inference is unreliable (the remedy is robust standard errors or GLS). It does not bias the coefficients, so that claim is wrong.
Regarding the Laspeyres, Paasche and Fisher price indices, which of the following statements are correct?
1. The Laspeyres index uses base-year quantities as weights
2. Fisher's ideal index is the geometric mean of the Laspeyres and Paasche indices
3. Fisher's index satisfies both the time-reversal and factor-reversal tests
4. The Paasche index uses base-year quantities as weights
Show answer
Answer: A — The Laspeyres index uses base-year quantities as weights; B — Fisher's ideal index is the geometric mean of the Laspeyres and Paasche indices; C — Fisher's index satisfies both the time-reversal and factor-reversal tests
Laspeyres weights by base-year quantities q₀, Fisher is √(L × P) and, of the three indices, alone passes both reversal tests. The claim that Paasche weights by base-year quantities is wrong: Paasche uses current-year quantities q₁, which is precisely what distinguishes it from Laspeyres.
Two series are each integrated of order one, I(1). Which of the following statements are correct?
1. They are cointegrated if some linear combination of them is stationary, I(0)
2. If they are cointegrated, an error-correction model links short-run changes to the lagged equilibrium error
3. Differencing each series once produces a stationary series
4. Any regression of one I(1) series on the other is guaranteed to be free of spurious results
Show answer
Answer: A — They are cointegrated if some linear combination of them is stationary, I(0); B — If they are cointegrated, an error-correction model links short-run changes to the lagged equilibrium error; C — Differencing each series once produces a stationary series
Cointegration is exactly a stationary linear combination of I(1) series, and it licenses an error-correction representation; differencing an I(1) series once gives an I(0) series by definition. The last claim is false: without cointegration, regressing one I(1) series on another is precisely the setting that produces spurious regression.
In an exactly identified structural equation, the order condition for identification is best stated as:
1. The number of excluded exogenous variables equals the number of included endogenous variables minus one
2. The number of structural equations in the model equals the total number of exogenous variables appearing anywhere in the system
3. Every variable in the system appears in every equation
4. The error terms of all equations are mutually uncorrelated
Show answer
Answer: A — The number of excluded exogenous variables equals the number of included endogenous variables minus one
The order condition (necessary, not sufficient) counts excluded exogenous variables against included endogenous ones: exact identification is equality (excluded exogenous = included endogenous − 1), more gives over-identification. The other statements describe neither the order condition nor identification; mutual uncorrelatedness of errors concerns a recursive system's estimability, a separate matter.
Which sampling method guarantees that every unit of the population has a known, non-zero probability of selection?
1. Convenience sampling
2. Quota sampling
3. Purposive (judgement) sampling
4. Simple random sampling
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Answer: D — Simple random sampling
Simple random sampling is a probability method: every unit has a known, equal, non-zero chance of selection, which is what lets sampling error be quantified. Convenience, quota and purposive sampling are non-probability methods in which selection probabilities are unknown and the sample can be systematically biased.