These multicollinearity issues cannot be resolved by just

= λ , . . . , λ T −1 ′ , H = Z, , Z = ⎡ ⎢ ⎢ ⎣ z .. . z T −1 ⎤ ⎥ ⎥ ⎦= ⎡ ⎢ ⎢ ⎣ 1 y l, y u, · · · y l, 1−p y u, 1−p .. . .. . .. . .. . .. . 1 y l,T −1 y u,T −1 · · · y l,T −p y u,T −p ⎤ ⎥ ⎥ ⎦ . The least-squares estimators of the parameters γ l and γ u are γ l = H ′ H −1 H ′ y l , γ u = H ′ H −1 H ′ y u . 3.11 The next theorem establishes the asymptotic properties of the two-step estimators γ l and γ u . Theorem 2. Consistency and asymptotic normality of the second-step OLS estimator Under the following assumptions: i plim T →∞ H ′ HT = B −1 , which is nonsingular; ii H ′ Jβ ∗ T converges uniformly in probability to the matrix function Qβ ∗ ; J ′ β ∗ Jβ ∗ T is bounded uniformly in probability at least in a neighborhood of true value β ∗ ; iii E|h t −1,i v lt | 2 ∞, E|h t −1,i v ut | 2 ∞, and E|j t −1,i v lt | 2 ∞ for all t and i = 1, . . . , 2p + 2; iv l,T ≡ varT −12 H ′ v l p − → l and u,T ≡ varT −12 H ′ v u p − → u , and l , u are finite and positive definite. Then, the two-step estimators γ l and γ u a converge to their true values in probability, b with asymptotic normal distributions, that is, √ T γ l − γ l d − → N0, B l B ′ , and √ T γ u − γ u d − → N0, B u B ′ , where B ≡ plim T →∞ H ′ HT −1 = plim T →∞ H ′ HT −1 , and l = l + C 2 l Q ′ S Q + M l + M ′ l , 3.12 u = u + C 2 u Q ′ S Q + M u + M ′ u , 3.13 with Q = plim T →∞ H ′ Jβ ∗ T , S = Jβ ∗ V β ∗ Jβ ∗ ′ , M l = plim T →∞ E H ′ v l − ′ HC l T M u = plim T →∞ E H ′ v u − ′ HC u T . In Equations 3.12 and 3.13 , the first terms l and u are the variance–covariance matrices of the errors v lt and v ut , respectively, if were observable. The second term Q ′ S Q captures the uncertainty induced by the estimates of . The last two terms, M l and M u , capture the covariances between the error terms v lt and v ut with . Although v lt and v ut are mar- tingale difference sequences, they are correlated with λ t +i for i = 0, 1, . . . , T − t. This is a further difference with Heckman’s two-step estimator. In Heckman’s covariance matrix, the matrix M is zero because in a cross-sectional setting the error v is uncorrelated with the inverse of the Mills ratio. Since the asymp- totic variance–covariance matrices in 3.12 and 3.13 cap- ture the heteroscedasticity induced by the observability restric- tion together with the time dependence induced by , Newey and West’s 1994 HAC variance–covariance matrix estimator should suffice to estimate B l B and B u B consistently. We also estimate the unconditional variances σ 2 l and σ 2 u of the re- spective errors ε lt and ε ut and their correlation coefficient ρ by implementing a simple method of moments based on the results of Nath 1972 see the online Appendix. 3.5 Two-Step Estimation: Implementation Issues The implementation of the two-step estimator may be subject to multicollinearity, and consequently the parameters γ l and γ u in the second step, Equation 3.10 , may not be precisely estimated or, in extreme cases, they may not be identified at all. There are two reasons for multicollinearity. First, the functional form 3.3 of the inverse of the Mills ratio λ· is nearly linear over a wide range of its argument y t −1 , β σ m so that the estimated regressor is almost collinear with the regressors in

Z. These multicollinearity issues cannot be resolved by just

dropping some of the regressors because the inclusion of is necessary to guarantee the consistency of the estimators ˆ β l and ˆ β u . The second reason pertains to those cases in which the ob- servability condition is not binding. When the observability con- dition is not binding, the population value of λ· is zero. Within a sample, we will observe values close to zero and very small variance in ˆλ t . The direct consequence is that C l and C u are not identifiable. In the simulation Section, we will discuss cases in which this problem is severe. For these two reasons, we propose a modified second-step es- timator that overcomes the identification problem of C l and C u , and in addition, provides a direct identification of the uncondi- tional variances σ 2 l and σ 2 u of the respective structural errors ε lt and ε ut and their correlation coefficient ρ. 3.6 Two-Step Estimation: A Modified Two-Step Estimator The first step of the estimation is identical to that explained in Section 3.3 , from which we obtain the estimates and σ m . In the second step, we exploit the relationships among C l , C u , σ 2 u , and σ 2 l , that is, C u + C l = σ 2 u − σ 2 l σ m and C u − C l = σ m . 3.14 If σ 2 l , σ 2 u , and σ m were known, the system of Equation 3.14 would have a unique solution, and C l and C u will be uniquely identified. By writing σ 2 u and σ 2 l as functions of C l and C u , that is σ 2 u C u and σ 2 l C l , we propose the following minimum distance estimator, which permits identifying C l and C u , C l , C u = arg min C l ,C u C u + C l − σ 2 u C u − σ 2 l C l σ m 2 , such that C u − C l = σ m . 3.15 Our first task is to find σ 2 u C u and σ 2 l C l . To do so, observe that the unconditional variance σ 2 u and σ 2 l of the error terms ε ut Downloaded by [Universitas Maritim Raja Ali Haji] at 22:18 11 January 2016 and ε lt can be written as σ 2 l = varε lt = varEε lt |ε t ≥ y t −1 ; β + Evarε l,t |ε t ≥ y t −1 ; β = C 2 l varλ t −1 + Evarv lt |y t −1 . 3.16 Similarly, σ 2 u = C 2 u varλ t −1 + Evarv ut |y t −1 , and σ 2 m = varε t = σ 2 m varλ t −1 + Evarv t |y t −1 , 3.17 with v t = v ut − v lt = y + z t −1 β − σ m λ t −1 by subtracting 3.5 and 3.6 , and β defined by 3.4 . From 3.17 , we have varλ t −1 = 1 − E[varv t |y t −1 ]σ 2 m , so that we need consistent estimators for the population mo- ments Evarv lt |y t −1 , Evarv ut |y t −1 , and Evarv t |y t −1 to obtain σ 2 u C u and σ 2 l C l as functions of sample information. Proposition 1 guarantees that this is the case. First, let us call v t = y t + z t −1 β − σ m λ t −1 , 3.18 u lt = y lt − z t −1 β l C l − C l λ t −1 , 3.19 u ut = y ut − z t −1 β u C u − C u λ t −1 , 3.20 where β and λ t −1 are the estimates from the first step, and β l C l and β u C u are the concentrated OLS estimates of β in 3.10 , that is β l C l = Z ′ Z −1 Z ′ Y l − C l , β u C u = Z ′ Z −1 Z ′ Y u − C u . 3.21 Proposition 1. Under Assumptions 1–5 and for φ- or α -mixing sequences v lt and v ut with at least finite second moments, we have that T t =1 v 2 t T p − → Evarv t |y t −1 , T t =1 u 2 lt T p − → Evarv lt |y t −1 , T t =1 u 2 ut T p − → Evarv ut | y t −1 , and therefore, σ 2 l C l ≡ C 2 l 1 − T t =1 v 2 t T σ 2 m + T t =1 u 2 lt T p − → σ 2 l and σ 2 u C u ≡ C 2 u 1 − T t =1 v 2 t T σ 2 m + T t =1 u 2 ut T p − → σ 2 u . The implementation of the minimum distance estimator in 3.15 is described in Figure 2 . We proceed as follows: 1. pick any point C ∗ l , C ∗ u on the line C u = ˆσ m + C l ; 2. compute the corresponding concentrated β l C ∗ l and β u C ∗ u as in 3.21 ; 3. compute the corresponding residuals u lt , u ut , and v t as in 3.19 , 3.20 , and 3.18 , respectively; 4. calculate the intercept σ 2 u C ∗ u − σ 2 l C ∗ l σ m to obtain the point C ∗ l , C ∗∗ u on the line C u = [σ 2 u C ∗ u − σ 2 l C ∗ l ] σ m + C l ; 5. assess the distance C ∗ u − C ∗∗ u 2 ; 6. go back to 1. Repeat until the distance function 3.15 is minimized by the minimizer C l , C u . Given the optimal solution C l , C u , the estimators of the parameters β of the original model are readily available as well as the variance–covariance matrix of the errors ε lt and ε ut , Figure 2. Minimum distance estimator. that is β l = β l C l = Z ′ Z −1 Z ′ Y l − C l , β u = β u C u = Z ′ Z −1 Z ′ Y u − C u , σ 2 l = σ 2 l C l , σ 2 u = σ 2 u C u , ρ = σ 2 m − σ 2 l − σ 2 u −2 σ l σ u . 3.22 Theorem 3. Consistency of Modified Two-Step Estimator The modified two-step estimator C l , C u and those defined in 3.22 converge in probability to the true values of the parame- ters. To prove Theorem 3, which states the consistency of estimates C l and C u in 3.15 , we only need to verify the assumptions stated in Theorem 7.3.2 in Mittelhammer, Judge, and Miller 2000 that guarantee the consistency of extremum estimators. 3 Proposition 1 shows that the restricted objective function in 3.15 converges in probability to that provided in 3.14 . In addition, since the system of equations 3.14 has a unique solu- tion and the restricted objective function 3.15 is a continuous and convex function in C l and C u , it is uniquely minimized at the true values of C l and C u . 4. SIMULATION We perform Monte Carlo simulations to assess the finite sam- ple performance of the two proposed estimation strategies: the two-step and modified two-step estimators; and compare these 3 See Newey and MacFadden 1994 , pp. 2133–2134 for the proof. The four assumptions are a mθ,

Y, X converges uniformly in probability to a function