CAPÍTULO 2 PROPUESTA DE SOLUCIÓN
2.1 A RQUITECTURA DE MOCIC
4.1 Empirical framework
Following Baumol (1986), Barro (1991a) and Barro and Sala-i-Martin (1991, 1992 and 1995), convergence across a sample of regions or provinces within a country is conventionally estimated using cross-sectional regressions as follows:
1
𝑇𝑙𝑜𝑔 (𝑦𝑦𝑖,𝑡
𝑖,𝑡−𝑇) = 𝑥 − [(1−𝑒𝑇𝛽𝑇)] . 𝑙𝑜𝑔(𝑦𝑖.𝑡−𝑇) + [(1−𝑒𝑇𝛽𝑇)] . 𝑙𝑜𝑔(𝑦̂𝑖)∗+ 𝜇𝑖,𝑡 [1]
Where 𝑖 represents a region/province, 𝑡 is time6, 𝑦𝑖,𝑡 is per capita GDP (equivalent to per capita income), (𝑦̂𝑖)∗ is steady state per capita GDP, 𝑥 is the steady state growth rate of GDP per capita, T is the length of the observation interval, the coefficient β is the rate of convergence, and μi,t is an error term which is a distributed lag of disturbances between t-T and t. Thus the convergence coefficient, β, represents the rate at which 𝑦𝑖,𝑡 approaches(𝑦̂𝑖)∗.
By transformation, assuming simple relation between GDP per unit of effective labour and GDP per capita, equation [1] can be written as follows:
1
𝑇𝑙𝑜𝑔 (𝑦𝑦𝑖,𝑡
𝑖,𝑡−𝑇) = 𝛽1. 𝑙𝑜𝑔(𝑦𝑖.𝑡−𝑇) + 𝜇𝑖,𝑡 [2]
6 This is a single point in time, not a continuous time variable
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The estimation of a significant negative β implies the existence of unconditional beta convergence (Barro and Sala-i-Martin, 1995).
Unconditional beta convergence means that economies, which in this case are provinces, are converging to a common steady state.
According to Solow (1956), the neoclassical growth model assumes the existence of unconditional convergence, when economies have similar technology, preferences and institutions. However, the structure of many economies is different in many respects – economically, demographically and politically (Sakikawa, 2012). Therefore, each economy may have a different steady state path, and thus conditional beta convergence is more appropriate than unconditional beta convergence. Conditional beta convergence holds when an economy grows faster the further it is from its steady state level. By adding variables to control for the different steady states, the following equation is derived:
1
𝑇𝑙𝑜𝑔 (𝑦𝑦𝑖,𝑡
𝑖,𝑡−𝑇) = 𝛽1. 𝑙𝑜𝑔(𝑦𝑖.𝑡−𝑇) + 𝛾𝑋𝑖,𝑡+ 𝜇𝑖,𝑡 [3]
Where 𝑋𝑖,𝑡 is a set of variables that serve to control for differences in the steady states. The estimation of a significant negative β implies the existence of conditional beta convergence. That is, the poorer the economy, the faster it converges to its own steady state level of GDP per capita.
The conventional cross-sectional analysis of convergence received criticism (Quah, 1993; Bernard and Durlauf, 1996). The criticism mainly lies in the failure of cross-sectional regressions to control for unobserved heterogeneities across economies. As shown in Casselli (1996), if those heterogeneities are not controlled for, this may result in omitted variable bias, which may lead to wrong conclusions about the beta coefficient. For instance, a lack of a strong negative correlation between initial income and its subsequent growth rate can be interpreted as a result of slow convergence.
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Moreover, in the context of South Africa, the cross-sectional regression approach is limited for making sufficiently valid inferences about convergence, because of the fewer cross-sections. South Africa has only nine provinces, which makes the number of observations in a cross-sectional regression to be equal to nine, i.e. N=9.
In order to correct the bias generated by heterogeneity, omitted variable bias and limited observation in the cross-sectional regressions, Islam (1995) suggested the use of panel data models which allow for differences in unobserved and unmeasurable parts of the differences, by modelling the region specific effects. Panel data models also add more observations, in that the cross-section and the time effect are considered. Accordingly, equation [3]
can be rewritten in a panel data setting as follows:
1
𝑇𝑙𝑜𝑔 (𝑦𝑦𝑖,𝑡
𝑖,𝑡−𝑇) = 𝛽1. 𝑙𝑜𝑔(𝑦𝑖.𝑡−𝑇) + 𝛾𝑋𝑖,𝑡+ 𝜂𝑖 + 𝜉𝑡 +𝜇𝑖,𝑡 [4]
Where 𝜂𝑖 is a country/region-specific fixed effect that indicates unobserved heterogeneities (or individual effect) and 𝜉𝑡 is a time-specific effect. Equation [4] represents a panel data model, which is estimated using either the fixed effect (FE) or random effect (RE) estimation. The FE and the RE estimations are the model framework from which the empirical analysis of convergence in this minor dissertation will be developed and tested. The choice between the FE and RE will be decided by the Hausman test.
Following Borys, Polgàr and Zlate (2008), equation [4] can be estimated using data for different time frequencies. However, although there is no consensus on the determination of the appropriate time intervals (Temple, 2000), the furthest one can go in shortening the time span is to take one year as a period (Dholakia, 2003) and equation [4] can be rewritten as:
𝑙𝑜𝑔 (𝑦𝑦𝑖,1997
𝑖,1996) = 𝛽1. 𝑙𝑜𝑔(𝑦𝑖.1996) + 𝛾𝑋𝑖,𝑡+ 𝜂𝑖 + 𝜉𝑡 +𝜇𝑖,𝑡 [5]
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Although the panel with annual observations generates a more comfortable number of observations (N*T = 9*18 = 162)7, the panel is vulnerable to cyclical demand related factors, which introduces extra “noise” into the regression (Borys et al., 2008). Moreover, the level of GDP per capita lagged by one year might be too recent to explain the real convergence process.
The use of averages is also common in many convergence studies (e.g.
Islam, 1995). According to Ding and Knight (2011), taking averages over several years tends to decrease the influence of short-term shocks and business cycles on economic growth, and reveals long-term relationships.
Islam (1995) used a five-yearly interval, but highlight that the time interval is dependent on the size of the cross-section and the time period covered.
As an example, for a 3-year interval time span (1996 to 1998), equation [3]
can be estimated as follows:
1
3𝑙𝑜𝑔 (𝑦𝑦𝑖,1998
𝑖,1996) = 𝛽1. 𝑙𝑜𝑔(𝑦𝑖.1996) + 𝜸𝑋𝒊,𝒕+ 𝜇𝑖,𝑡 [6]
Where, 𝑋𝒊,𝒕 is the average of each three-year interval for all steady state structural variables.
In addition to estimating the convergence coefficient, two parameters are also estimated (Barro and Sala-i-Martin, 1995 and Bonnefond, 2014). The first one is the implied speed of convergence, which is estimated as follows:
𝜆 =−ln (1 − 𝛽1𝑇) 𝑇
Implied speed of convergence 𝜆 represents the rate at which an economy is getting closer to its steady state level every year. The higher the speed of convergence, the faster an economy reaches its steady state level.
7 N stands for number of provinces and T is time period
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The second one is the half-life, which accounts for the time, in years, required for an economy to cover half the distance from its steady state level.
Following Barro and Sala-i-Martin (2003) and Sulaiman and Bryant (2010), half-life of convergence is computed as follows:
𝑒−𝛽𝑇 = 1
2⇒ 𝐻𝑎𝑙𝑓 𝑙𝑖𝑓𝑒(𝑇) = −ln (2) 𝛽
Another frequently used indicator for convergence measurement is the coefficient of variation of GDP per capita denoted by σ and calculated as follows:
𝜎 = √(1 𝑛⁄ ) ∑(𝑥𝑖− 𝑥̅)2⁄ 𝑥̅
Where 𝑥𝑖 is GDP per capita for the ith province, where there are n provinces, and 𝑥̅ is the sample mean for x. A scale of 0 to 1 is used to assess the extent of dispersion. Thus, higher values of σ indicate a more serious income disparity, and vice versa. A decline over time denotes sigma convergence.
In addition to beta convergence estimation, the speed of convergence, the half-life and sigma convergence are also estimated in this minor dissertation.