Here, _γ_ is equivalent to 2 _α_ , and must be positive for convergence to hold. Under the hypothesis of convergence, _γ_ is more than 0, so the dispersion of productivity levels falls as a function of time. Because of the penalty term 2 log (log _t_ ), the t-statistic will convergence to _−∞_ where the hypothesis of convergence is rejected. A one-sided t-test of _α ≥_ 0 will assess the hypothesis for a given sample. In addition to the significance of the coefficient, the magnitude of _γ_ shows the degree of convergence in effect. For 0 _< γ <_ 2, convergence in growth rates but not levels will occur. For _γ >_ 2, convergence in levels will hold. The above convergence test is appropriate to test for convergence within a particular group but must be combined with an additional algorithm to test for club convergence among multiple potential clubs of economies. Phillips Sul propose the following procedure to establish the presence of “club” convergence

- Order economies according to income in the final period (or average in last half).

- Choose a core group _Gk_ of _k_ economies and compute the test statistic _tk_ = _t_ ( _Gk_ ). Choose group size k to maximize the test statistic _tk s.t._ min _tk > −_ 1 _._ 65 (the 5% critical value for the t-test). Where the minimum test statistic is not met with _k_ = 2, drop the first economy and proceed to maximize _tk_ .

- Add one country at a time to group _k_ and include in the group if the t-statistic exceeds the criterion _c_<sup>_∗_</sup> .

- Form a new group for those countries not included in the initial three steps. If the remaining countries have _tk >_ 1 _._ 65 then there are two groups. Otherwise, repeat steps 1-3. It may be the case that the remaining economies are divergent and that there is no additional club.

- If _c∗_ = 0, then the requirement to be added to an existing group will be very conservative (relative to _−_ 1 _._ 655% critical value).

# **Converging in levels vs growth rates**

Applying the PS routine to our market-exchange rate adjusted data produces 5 convergence clubs ( **Table 3** ). The parameter _γ_ = 2 _β_ is between 0 and 1 in the majority of cases. This suggests that the convergence clubs uncovered show a tendency for relative convergence, or the reduction of the size of the gap in relative productivity levels over time, but not necessarily full convergence to the same level of output per worker. However, this does not rule out a substantial closure of productivity gaps over time—instead, it implies that over time productivity growth rates will align, alongside smaller productivity gaps between members of clubs. Full convergence in levels is a very strict condition not met even by advanced economies, where productivity gaps have declined considerably over the past 50 years, but persistent smaller gaps remain. Applying the PS log-t test to advanced economies yields a _γ_ of just 0.15, in line with the results for many of the convergence clubs identified for in Table 3. Estimated over the period 1970-2000, there are considerably fewer members of the two highest convergence clubs, which as noted in the main text is a result of fewer EMDEs displaying fast-convergence characteristics.

# **Determinants of club membership**

As noted in the main text, initial productivity, governance quality, education, and economic complexity are all statistically significant drivers of club membership in a multinomial logit regression. Further details of these regressors are provided in **Table 4** . To preserve degrees of freedom, 34