The hard problems: the Delhi School set

Questions in other exams ask for a number from a formula. The hard ones here ask what must be true, so the work is to find the right object and read off the answer. Almost all of them fall into seven families, and this lesson takes them in turn. Four habits run through all of them.

1. Rewrite, do not re-estimate. Most questions can be settled by writing one regression in terms of another. You need no data.
2. Find the quantity that is invariant. A fit does not change when the regressors are replaced by combinations of themselves; R², t ratios and F do not change when units change; the coefficients do.
3. Name the object the table gives you. A table of coefficients and standard errors hands you t ratios, adjusted R², standardised effects and, with a sample size, F.
4. Check the claim against the assumptions. “Which statement is false?” is usually a statement that is true only under an assumption the question has removed (an intercept, a constant error variance, uncorrelated regressors).

The earlier lessons taught the ideas. This one is about recognising them under a different cover, so each part begins with the disguise, then shows one question worked out. The right answer to each of the 28 questions is, in the end, a line or two.

The disguise: a statement about residuals, fitted values or a re-parametrised regression, with no numbers at all. The tools: the residuals are orthogonal to every regressor and sum to zero (with an intercept); Y = Ŷ + e splits Y into two uncorrelated pieces; the coefficients of a regression on combinations of the regressors are the same fit in different coordinates; and Var(b_j) = σ² ÷ [SST_j(1 − R_j²)].

The disguise: a regression printed with standard errors, an R², a sample size, and a question that needs one more step. The tools: t = b ÷ se against about 1.98 at 100 degrees of freedom; adjusted R² = 1 − (1 − R²)(n − 1) ÷ (n − k − 1); the F test for adding a variable from two residual sums of squares; a standardised effect b × s_x ÷ s_y; and the explained sum of squares of a dummy regression from the group sizes and the gap.

Other questions in this set: the home-sales table of DSE 2019 asks about both the significance of two coefficients (t = 23.4 ÷ 10.76 = 2.17 and 35.6 ÷ 10.5 = 3.39, both beyond 1.98: significant) and the adjusted R² (1 − 0.28 × 106 ÷ 100 = 0.7032). DSE 2016 Q12 asks for a standardised effect: 2.3 × 5 ÷ 500 = 0.023 standard deviations of birth weight. Lessons 3 and 4 teach each step.

The disguise: a gender gap, a dummy coded differently, two complementary dummies and no constant, a product of dummies, or a threshold in income. The tools: a dummy regression fits group means; a dummy’s coefficient is a gap that only means something given the rest of the equation; recoding a dummy is a change of variable you can substitute through; and the precision of a gap depends on the group sizes.

Also here: DSE 2014 Q47, the two complementary dummies without a constant, which gives the two group means and is the same as separate regressions (the “trap” version, with a constant as well, fails); DSE 2015 Q42, where the gap is most precise with an equal split because Var = σ²(1 ÷ n₁ + 1 ÷ n₂) is smallest at n₁ = n₂; DSE 2011 Q38, where one dummy tests a gender gap after controls; and DSE 2016 Q06, the kinked Engel curve, whose false option is the one that calls it a standard common-intercept formulation, when the term (X − X*)D makes the two segments meet at X*. Lesson 5 covers each.

The disguise: a fitted quadratic, or a variable “measured in decades instead of years”. The tools: the slope of a quadratic is b + 2cx, so with c < 0 the effect falls throughout, stays positive until −b ÷ 2c and then turns; and a variable in bigger units has a proportionally bigger coefficient.

In the home-sales question (DSE 2019 Q39), age in decades is age ÷ 10, so its coefficient is ten times larger: 0.090 × 10 = 0.900. The official key printed with the paper marks 0.009, which has the units backwards: a larger unit gets a larger coefficient. The platform stores 0.900 and flags the conflict. Lesson 6 has the rule, and it is a good habit to check any such answer against a one-line example.

The disguise: a researcher leaves a variable out (or puts a useless one in) and you are asked about bias, variance or what two coefficients reveal. The tools: the short-regression slope equals the long one plus β₂δ; an irrelevant variable costs variance, not bias; and a wrong functional form is an omitted term.

The same identity decides DSE 2011 Q39 (an omitted complement, a₂ > 0 and positive δ: an upward bias) and DSE 2014 Q50 (the wrong model has a larger error variance estimate in expectation, E[σ̂²_v] ≥ σ²_u). DSE 2015 Q45 and DSE 2018 Q28 test the other direction: an irrelevant variable leaves the estimates unbiased and raises their variances. DSE 2016 Q09 fits a line to a quadratic cost curve: the marginal cost is biased, and the residuals follow a pattern (a U-shape against output) that looks like autocorrelation if the data are ordered by output. Lesson 7 has the details.

The disguise: an equation whose regressors include a combination of the others; or a regression with a high R², a significant F and no significant t. The tools: perfect collinearity means only some combinations of coefficients are identified; near collinearity inflates variances but never biases; and low pairwise correlations do not rule out collinearity among three or more variables.

DSE 2012 Q10 asks which is not an indicator of multicollinearity: bias in the coefficients, since collinearity inflates variances and leaves the estimates unbiased. DSE 2016 Q08 has four low pairwise correlations and still a collinearity pattern (high R², significant F, no significant t); the false statement says the coefficients are biased because the correlations are small, when neither the small correlations nor the collinearity creates bias. See lesson 8.

The disguise: an error variance that depends on a variable, and a question on what OLS does or how to test. The tools: unbiasedness needs only E(u | x) = 0, so the coefficients stay unbiased; the usual standard errors assume constant variance, so they are wrong; White’s auxiliary regression takes the squared residuals on the regressors, their squares and cross-products.

The hardest step in the exam is the first: recognising which technique a question is inviting. Seven situations, pick the opening move.

Seven sets, one for each technique above, with all 28 DSE econometrics questions above the easy level. They are open to you whether or not you have worked through the lessons, and each is graded at the end. The easy DSE questions sit in the lessons that teach them.