Methodology

Stance

The study is quantitative and explanatory, organized around a single methodological commitment: a null result under one functional form is not evidence of absence, it is evidence about that functional form. The design is therefore comparative across specifications rather than confirmatory within one.

Design

Repeated cross-section over four annual waves, drawn from a government DT training programme. The data are administrative in origin rather than researcher-solicited, which strengthens coverage and weakens control over instrumentation — a trade-off that shapes the study’s central open problem.

Analytical strategy

Three nested specifications estimated in sequence, so that each addition is tested rather than assumed:

Step 1 (M_main):  Y = b0 + b1*T + b2*E + b3*O + b4*EDU + year_dummies + e
Step 2 (M_m7):    + b5*(T x E) + b6*(T x O) + b7*(E x O)
Step 3 (M_rsm):   + b8*T^2 + b9*E^2 + b10*O^2          <- main model

Incremental F-tests between steps establish whether each layer of complexity earns its place. The second-order step is required not because it maximizes fit but because surface interpretation is undefined without full curvature terms.

Triangulation

Five methods are applied to the same substantive question so that convergence or divergence between them is itself evidence:

Method	Approach	What it identifies
Paired comparison	pre versus post	within-respondent change
Propensity score matching	matched comparison	selection into training
Staggered difference-in-differences	pre-post by cohort	within-firm change
Cross-lagged panel model	t1 to t2 cross-lag	temporal ordering
Response surface analysis	second-order surface	functional form

Known design limitation under revision

Two of these five methods assume genuine panel structure, and most firms appear in only one wave. The pending revision restricts those two methods to the subset of firms observed in multiple years. Year fixed effects do not substitute for firm fixed effects on single-wave cohorts.

Inference

Heteroskedasticity-consistent (HC3) standard errors throughout; bootstrap confidence intervals for derived quantities that have no closed-form standard error, including the location of the surface stationary point.

Causal claims

The response surface analysis is cross-sectional and supports associational claims only. Six alternative designs that would establish time precedence have been drafted — lagged, difference-based, cross-lagged, time-varying, growth-curve, and dynamic-optimum specifications. The first of these is the intended addition to the robustness section; the last is a potential methodological paper in its own right.