Regional and Sector-specific Determinants of Industry Dynamics and the Displacement–replacement Effects

Abstract

In this paper, we empirically assess the importance of regional and sector-specific determinants of industry dynamics. To this aim we test three hypotheses (originally proposed by Shapiro and Khemani (1987, Int J Indust Organ 5:15–26)) for the relationship between the entry and exit of firms: independence, symmetry and simultaneity. Estimates from a panel data system of equations seem to confirm the simultaneity hypothesis for Spain, i.e. we find evidence of a displacement (replacement) effect between the gross rate of entry (exit) and the gross rate of exit (entry). Also, our results show that, irrespective of the hypothesis we use, both sectorial and regional variables affect entry and exit.

Appendix: Estimation methods

The econometric framework is given by a system of M equations (m = 1, … ,M):

$$ y_{m}= X_{m}\beta_{m}+u_{m} $$

(6)

and an error component structure:

$$ u_{m}= Z_{\mu}\mu_{m}+Z_{\lambda}\lambda _{m}+ Z_{\eta}\eta_{m}+\varepsilon_{m} $$

(7)

in which Z _μ = I _N  ⊗ e _T  ⊗ e _Q, Z _λ = e _N  ⊗ I _T   ⊗ e _Q ,Zη = e _T  ⊗ e _N   ⊗ I _Q ; e _N , e _T and e _Q are vectors of ones and I _N ,I _T and I _Q are identity matrices of dimension N, T and Q, respectively. ɛ _m is an idiosyncratic shock with classical properties and μ′ = (μ₁,μ₂, … ,μ _n ),λ′ = (λ₁,λ₂, … ,λ _t ) and η′ = (η₁,η₂, … ,η _q ). Also, y _m is a vector (NTQ) × 1. X _m is a matrix of explanatory variables whose dimension is (NTQ) ×  (k _m  + 1) and β _m is the vector (k _m  + 1) of model coefficients. In the application in this paper, M = 2 (entry and exit), N = 17 (regions), T = 15 (1980–1994) and Q = 11 (sectors), so that NTQ = 2,805.

To determine the most suitable method for estimating the parameters of Eq. (3) and systems (4) and (5), we must take into account the underlying assumptions in the various hypotheses regarding the stochastic behaviour of the variables and the error terms. Under the independence hypothesis, we used OLS and Random Effects estimators (see Table 3). The algebra of these estimators is omitted because they are so widely used—see e.g. Baltagi (2001) for details. Under the simultaneity hypothesis, we are dealing with a simultaneous equations model (SEM), while under the symmetry hypothesis the analytical reference corresponds to the particular case that defines a system of seemingly unrelated regressions (SUR). These are less familiar estimation techniques, so they probably need the following short descriptions.

1.1 Symmetry hypothesis SUR

From (6) and (7), we assume, without loss of generality, that the latent variables are random and independent vectors of the form ${\mu\sim (0,\Sigma_{\mu}\otimes I_{N}),\lambda \sim (0,\Sigma_{\lambda}\otimes I_{T}),\eta \sim (0,\Sigma _{\eta}\otimes I_{Q})}$ and ɛ ∼ (0,Σ_ɛ ⊗ I _NTQ ), where ${\Sigma _\mu=\left[ {\sigma_{\mu _{\rm ml}}^2}\right],\Sigma_\eta =\left[{\sigma_{\eta_{\rm ml}}^2}\right],\Sigma_\eta =\left[{\sigma_{\eta_{\rm ml}}^2} \right]}$ and ${\Sigma _\varepsilon =\left[{\sigma_{\varepsilon _{\rm ml}}^2}\right]}$ are matrices of dimension M × M. Also, the matrix of variances and covariances of the system Ω = [Ω_ml] will be (Wansbeek and Kapteyn 1982):

$$\Omega =\sum\limits_{s=1}^5 {\xi_s \otimes V_s}$$

(8)

in which ξ₁ = Σ_ɛ, ξ₂ = TQΣ_μ + Σ_ɛ, ξ₃ = NQΣ_λ + Σ_ɛ, ξ₄ =  NTΣ_η + Σ_ɛ and ξ␣₅ = TQΣ_μ + NQΣ_λ + NTΣ_η + Σ_ɛ are the characteristic roots of Ω. Moreover, ${V_{1}=P,V_2=E_N \otimes \bar{J}_T \otimes \bar{J}_Q,V_3 =\bar{J}_N \otimes E_T \otimes \bar{J}_Q,V_4 =\bar{J}_N \otimes\bar{J}_T \otimes E_Q,V_5 =\bar{J}_N \otimes \bar{J}_T \otimes \bar{J}_Q}$ are the corresponding matrices of eigenprojectors, in which ${E_{N} = I_{N} -\bar{J}_{N}, E_{T}= I_{T} -\bar{J}_{T}}$ and ${E_{Q} = I_{Q} -\bar{J}_{Q}}$ . Given that, for every scalar r, it can be demonstrated that ${\Omega^r=\sum\limits_{s=1}^5 {\xi _s^r \otimes V_s}}$ , from (8) the vector of parameters in (6) can be estimated by GLS. Further, to obtain feasible GLS we must first estimate the characteristic roots of Ω. One way is to use ANOVA estimates like ${\hat{\xi}=u'V_{s}u/tr(V_{s}),s = 1,2,3,4}$ and substitute the vector u with the residuals from the OLS (Avery 1977) or fixed-effects (Baltagi 1980) estimates. Both techniques provide asymptotically efficient estimates of the model coefficients. These are reported in Table 4.

1.2 Simultaneity hypothesis SEM

In this case the model is analogous to that from expressions (6), (7) and (8), except that there are endogenous variables on the right-hand side of the equation. Of the various methods in the literature for estimating SEM with panel data, the properties and simplicity of the one proposed by Baltagi (1981) make it best suited to our application (see Baltagi and Li 1992). The estimation methods are based on two-stage least squares (2SLS) with limited information and three-stage least squares (3SLS) with complete information. The identification condition is simply that the number of exogenous variables not included in the corresponding equation is greater than or equal to the number of endogenous variables.

Let the model given by (6) be rewritten in this case in compact form. A transformation matrix A is applied such that y ^* = Ay, Z ^* = AZ and u ^* = Au. If the matrix of instruments used is W, the vector of coefficients will be given by ${\beta_{W}=(Z ^{\ast \prime} P_{W}Z^{\ast}){^{-1}}Z^{\ast \prime} P_{W}Y^{\ast}}$ , where ${P_{W}=W(W ^\prime W)^{-1}W ^\prime}$ is the projection matrix of the instruments. In particular, if we define the transformation matrix in terms of the elements of the main diagonal of the matrix of variances and covariances of each equation ( ${A =\Omega _{\rm mm}^{-1/2}}$),and apply 2SLS to the transformed model, we obtain the error component two-stage least squares (EC2SLS) estimator (Cornwell et al. 1992). Similarly, if we use the complete matrix (A = Ω^−1/2) and 3SLS, we obtain the error component three-stage least squares (EC3SLS) estimator. Both GLS estimates are consistent and, in their feasible version, they are based on the residuals from an initial 2SLS estimation. These estimates are reported in Table 5.¹