We briefly present our procedure for identifying a potential sample selection bias in the model for the skill structure of migration. It is a slight modification of Wooldridge (1995, 123-124), who proposes a method for testing for sample selection bias in panel data. It will become evident below that we impose very strong assumptions on the selection equation and the mechanism governing selection. These assumptions would often be inappropriate if we were to derive corrections for a sample selection bias in models with fixed effects. It turns out, however, that they do not pose a threat to the correct testing for a sample selection bias.
For further details on this, the reader is referred to Wooldridge (1995).
We start by rewriting the model for the skill structure of migration as:
yij = µi+ xijβ+ uij, j = 1, . . . , J, (H.1) where yij is the ij-specific log of the ratio of high-skilled migrations to low-skilled migrants, µi
is an unobserved country fixed effect, xij is a 1 × K vector of explanatory variables (including region dummies and interactions between region dummies and world region dummies), β is a K × 1 vector of parameters to be estimated, and uij is an independent and identically distributed error term. We explicitly allow for E(µi|xi1, . . . , xiJ) 6= E(µi). Since J is fixed, the asymptotic analysis is valid for I → ∞. Now suppose that (yij, xij) is sometimes unobserved, and that sij = (si1, . . . , siJ)0 is a vector of selection indicators with sij = 1 if (yij, xij) is observed and zero otherwise. Define xi ≡ (xi1, . . . , xiJ) and si ≡ (si1, . . . , siJ) and suppose that E(uij|µi, xi, si) = 0 ∀j, which implies that the selection process is strictly exogenous conditional on µi and xi. Then, our FE estimator employed in the main text is consistent and asymptotically normal even when selection arbitrarily depends on (µi, xi) (Wooldridge 1995, 118).
In our application, the explanatory variables xij are observed for all regions j = 1, . . . , J.
The variable yij is observed if sij = 1, but not otherwise. For each j = 1, . . . , J, define an
CHAPTER 6. NETWORKS AND SELECTION IN MIGRATION TO SPAIN 159
unobserved latent variable
h∗ij = δj0+ xi1δj1+ · · · + xiJδjJ + vij, (H.2) where vij is a stochastic term independent of (µi, xi), and δjpis a (K+1)×1 vector of unknown parameters, p = 1, 2, . . . , J.48 The binary selection indicator is defined as sij ≡ 1[h∗ij > 0].
Since si is a function of (xi, vi), where vi ≡ (vi1, . . . , viJ)0, a sufficient condition for the selection process to be strictly exogenous conditional on µi and xi is:
E(uij|µi, xi, vi) = 0, j = 1, . . . , J. (H.3) Under (H.3), there is no sample selection bias. An alternative that implies sample selection bias is:
E(uij|µi, xi, vi) = E(uij|vij) = ρvij, j = 1, . . . , J, (H.4) where ρ 6= 0 is some unknown scalar. Under the alternative (H.4) we have:
E(yij|µi, xi, si) = µi+ xijβ+ ρE(vij|µi, xi, si) = µi+ xijβ+ ρE(vij|xi, si). (H.5) Let E(vij|xi, si) = E(vij|xi, sij) and assume a standard uniform distribution for vij. Then,
E(vij|xi, sij = 1) = E(vij|xi, vij > −xiδj) = (1 + xiδj)/2. (H.6) and
E(yij|µi, xi, sij = 1) = ρ∗+ µi+ xijβ+ ρ∗xiδj, (H.7) where ρ∗ ≡ ρ/2 and xi now includes unity as its first element. The procedure to test for sample selection bias is as follows. We first obtain estimates of xiδj by estimating region-specific selection equations (where sij is the dependent variable) derived from equation (H.2), using linear probability models for the full data matrix. We then estimate equation (H.7) in an FE framework (within-transformed data), using only observations with sij = 1. We finally test H0 : ρ = 0, using the t-statistic for ρ∗.
48In the following, xij includes one element more than in equation (H.1), despite the fact that we use the same notation for convenience. We thus assume that there is exactly one exclusion restriction in equation (H.1). In the estimation, we use the log of the number of people holding country i’s nationality and migrating from region j in Spain to any other region k 6= j within or outside Spain over the period from January 1, 2006, to December 31, 2007, as an exclusion restriction.
160 CHAPTER 6. NETWORKS AND SELECTION IN MIGRATION TO SPAIN
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