Solution. (a) \(F = \bigl ((150 - 120)/2\bigr )/\bigl (120/56\bigr ) = 15/2.143 = 7.0\), against the \(5\%\) point \(F_{2,56} = 3.16\): the restrictions are rejected.

(b) By Theorem 9.3.1, \(W = 60(30)/120 = 15.0\), \(\text {LR} = 60\log (150/120) = 13.39\) and \(\text {LM} = 60(30)/150 = 12.0\), so \(W \geq \text {LR} \geq \text {LM}\). All exceed the \(\chi ^2_2\) point \(5.99\). Also \(W = \frac {n}{n-k}qF = \frac {60}{56}(2)(7.0) = 15.0\).

Solution. (a) \(R^2 > d\), the classic sign of a spurious regression (Theorem 12.2.5). The residual ADF statistic \(-1.8\) is far above the Engle–Granger \(5\%\) critical value \(-3.34\), so the null of no cointegration is not rejected: the residuals are \(I(1)\), the \(t\) ratio of \(34\) is meaningless, and the regression says nothing about the relation between money and output.

(b) Since the series are \(I(1)\) but not cointegrated, model the relation in differences (growth rates), for instance in a VAR in differences, with tests of Granger causality; or look for omitted variables (prices, interest rates) with which the two might be cointegrated, and test with Johansen’s method.

Solution. (a) \(41.2\) far exceeds the \(\chi ^2_5\) point \(11.07\), so homoscedasticity is rejected. Heteroscedasticity does not bias least squares (Theorem 9.4.1); it invalidates the conventional standard errors.

(b) With the robust standard error, \(t = 0.045/0.021 = 2.14\), significant at \(5\%\) but much less so than the conventional \(t = 3.75\) suggests.

(c) Measurement error in \(q\) attenuates its coefficient (Theorem 13.1.1) and, because cash flow is correlated with true \(q\), biases the cash flow coefficient too, typically upwards, since cash flow picks up the part of investment opportunities that mismeasured \(q\) misses. The remedy is IV: instrument \(q\) with lagged values or an alternative measure, and test with Hausman’s statistic.

Solution. (a) Demand now excludes no exogenous variable, so it fails the order condition and is underidentified. Supply excludes income, which appears in demand, so it is exactly identified (Theorem 14.2.1).

(b) Regress \(q\) on \(p\) and \(w\) in the supply equation, with instruments the constant, \(w\) and income. Exactly identified, it equals the IV estimator and ILS: income shifts demand along the supply curve.

Solution. (a) \(H = (0.081 - 0.064)^2/(0.012^2 - 0.007^2) = 0.000289/0.000095 = 3.04\), below the \(\chi ^2_1\) point \(3.84\): random effects is not rejected at \(5\%\), though the evidence is borderline (at \(10\%\), \(2.71\), it would be).

(b) Random effects is lower, which suggests that union members have, if anything, lower unobserved earning power; unions may organise workers who would otherwise earn less, so the between comparison understates the union effect.

(c) The fixed effects estimate is identified only by switchers, whose changes may be few and partly due to misclassification, which attenuates it strongly in first differences or within data (Section 16.2). Its large standard error reflects the small within variation.

Solution. (a) \(\Phi (0.4) = 0.655\), and the marginal effect is \(\phi (0.4)(0.6) = 0.368 \times 0.6 = 0.221\) per decade: another ten years of education would raise the probability by \(22\) points, or about \(2.2\) points per year at this point on the curve.

(b) The number of young children, or non-labour income such as a spouse’s earnings, affects the decision to work but plausibly not the wage offered. Without such a variable the selection term \(\lambda (z'\gamma )\) is nearly collinear with the wage regressors, and the correction is identified only by the functional form (Section 17.3).

Solution. (a) The impact elasticity is \(0.4\) and the long-run elasticity \(0.9\).

(b) A quarter of the gap closes each period, so \(\phi = 0.75\), and the half-life is \(\log 0.5/\log 0.75 = 2.4\) periods.

(c) From Theorem 12.5.1, \(1 - \phi = 0.25\), \(\beta _0 = 0.4\) and \(\theta = (\beta _0 + \beta _1)/(1 - \phi ) = 0.9\), so \(\beta _1 = 0.225 - 0.4 = -0.175\): \(c_t = 0.75c_{t-1} + 0.4y_t - 0.175y_{t-1} + \varepsilon _t\). The common factor restriction requires \(\beta _1 = -\phi \beta _0 = -0.3\); it fails, consistently with the impact and long-run elasticities being different.

Solution. (a) Standard errors only, since advertising is exogenous: HAC standard errors or FGLS (Section 11); also consider a dynamic model. (b) Bias from sample selection: Heckman’s model with an exclusion restriction (Section 17.3); the wage is also endogenous, which calls for IV. (c) Bias from an omitted variable correlated with the regressor: farm fixed effects with panel data, or an instrument for fertiliser such as its price (Sections 16 and 13). (d) Standard errors only: cluster by school, and note the Moulton factor (Section 16.5). (e) A censored outcome: Tobit or a two-part model (Section 17.2).

Solution. The reliability of \(x\) is \(36/45 = 0.8\), so \(\plim b = 0.8 \times 0.8 = 0.64\) by Theorem 13.1.1. The error \(n\) in \(y\) only adds to the disturbance: it is uncorrelated with \(x\), so it does not affect the probability limit, but it increases the variance of \(b\).

Solution. (a) The first has \(6 - 2 = 4\) degrees of freedom, with \(5\%\) point \(9.49\): \(J = 9.1\) does not quite reject. The second has \(8\), with \(5\%\) point \(15.51\): \(11.0\) does not reject.

(b) With \(10\) instruments the weighting matrix \(\hat S\) is a \(10 \times 10\) matrix estimated from the data, and with many moments its inverse is poorly estimated, which weakens the \(J\) test and biases two-step GMM (Section 15.5). A \(J\) statistic that grows less than its degrees of freedom when instruments are added is a common sign of this loss of power, not evidence that the model fits better.