# Absolute Convergence

I have noticed that some researchers in order to test convergence hypothesis apply a model in this form:

$$(1/T) \ln(y_{it}/y_{i,t-1}) = b_0 - b_1\ln(y_{i,t-1}) + u_{it}$$

(Barro & Sala-I-Martin, 2004)

While others apply this form:

$$\ln(y_{it}/y_{i,t-1}) = b_0 - b_1\ln(y_{i,t-1}) + u_{it}$$

(Mankiw et al., 1992)

What's the difference between these two models?

The typical equation for testing absolute convergence (even if in the practice this specification has serious shortcomings) is:

$$$$\frac{1}{T} \left[ ln(y_{it}) - ln(y_{i,t-T}) \right]= \beta_0 + \beta_1 ln(y_{i,t-T}) + u_{it}$$$$ where the left hand side of this equation is just the geometric average growth rate of GDP per capita from time $$t-T$$ to $$t$$. For instance, if $$T=10$$, the left hand side variable denotes the geometric average growth rate from $$t-10$$ to $$t$$, i.e. in 10 years.

If I write,

$$$$ln(y_{it}) - ln(y_{i,t-1}) = \beta_0 + \beta_1 ln(y_{i,t-1}) + u_{it}$$$$

I'm considering the annual growth rate of GDP per capita as left hand side variable

• Well, if I have GDP per capita data for the year 1995 and 2015, I can either calculate the geometric mean growth rate using this formula $$\frac{1}{20} \left( \ln(gdp_{2015}) - \ln(gdp_{1995}) \right)$$ or calculate the annual GDP growth rate with this formula $$\left (\ln(gdp_{2015}) - \ln(gdp_{1995} \right) )$$ Commented Dec 12, 2023 at 17:32
• No, the second formula is incorrect when $T>1$, as it does not provide the annual growth rate. It roughly computes the total growth occurring from 1995 to 2015.
– Tony
Commented Dec 12, 2023 at 17:59
• So, is it correct to use a model in this form to test the convergence hypothesis? $$\ln\left(\frac{\text{gdp}_{2015}}{\text{gdp}_{1995}}\right) = b_0 - b_1\ln(\text{gdp}_{1995}) + u_{it}$$ Commented Dec 12, 2023 at 19:01
• It is correct (particularly for an assignment or passing the Macro I exam. However, if you aim to write a paper discussing absolute convergence testing, it may need further refinement. Refer to Acemoglu and Molina, 2021 for more insights on this aspect). Keep in mind that the variable on the left-hand side roughly represents the total growth occurring from 1995 to 2015.
– Tony
Commented Dec 12, 2023 at 19:23
• Great, thank you very much for your help and for your time. Commented Dec 12, 2023 at 19:38

You are right and I have found this formula in many articles. But on this page they suggest this method of calculating Growth rate of GDP per capita.(This page shows step by step replication of Augmented Solow model)

ln_gdp_growth = ln_gdp_80 - ln_gdp_65

https://deepnote.com/@carlos-mendez/R-Augmented-Solow-Model-d90f7550-909c-407d-8295-9ba49e81764f

According to the equation that Tony quoted, It should be calculated like this: ln_gdp_growth = (ln_gdp_80 - ln_gdp_65)/25