<p>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)</p>
<p>ln_gdp_growth = ln_gdp_80 - ln_gdp_65</p>
<p><a href="https://deepnote.com/@carlos-mendez/R-Augmented-Solow-Model-d90f7550-909c-407d-8295-9ba49e81764f" rel="nofollow noreferrer">https://deepnote.com/@carlos-mendez/R-Augmented-Solow-Model-d90f7550-909c-407d-8295-9ba49e81764f</a></p>
<p>According to the equation that Tony quoted, It should be calculated like this:
ln_gdp_growth = (ln_gdp_80 - ln_gdp_65)/25</p>

CC BY-SA 4.0
2023-12-12T00:24:54.157Z
2023-12-12T00:36:17.097Z
2023-12-12T00:36:17.097Z
kostas2323
<p>The typical equation for testing absolute convergence (even if in the practice this specification has serious shortcomings) is:</p>
<p><span class="math-container">\begin{equation}
\frac{1}{T} \left[ ln(y_{it}) - ln(y_{i,t-T}) \right]= \beta_0 + \beta_1 ln(y_{i,t-T}) + u_{it}
\end{equation}</span>
where the left hand side of this equation is just the geometric average growth rate of GDP per capita from time <span class="math-container">$t-T$</span> to <span class="math-container">$t$</span>. For instance, if <span class="math-container">$T=10$</span>, the left hand side variable denotes the geometric average growth rate from <span class="math-container">$t-10$</span> to <span class="math-container">$t$</span>, i.e. in 10 years.</p>
<p>If I write,</p>
<p><span class="math-container">\begin{equation}
ln(y_{it}) - ln(y_{i,t-1}) = \beta_0 + \beta_1 ln(y_{i,t-1}) + u_{it}
\end{equation}</span></p>
<p>I'm considering the annual growth rate of GDP per capita as left hand side variable</p>

CC BY-SA 4.0
2023-12-12T17:32:25.940Z
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

\begin{equation}
\frac{1}{20} \left( \ln(gdp_{2015}) - \ln(gdp_{1995}) \right)
\end{equation}


or calculate the annual GDP growth rate with this formula

\begin{equation}
   \left (\ln(gdp_{2015}) - \ln(gdp_{1995} \right) ) 
\end{equation}
kostas2323
CC BY-SA 4.0
2023-12-12T17:59:32.647Z
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
CC BY-SA 4.0
2023-12-12T19:01:05.500Z
So, is it correct to use a model in this form to test the convergence hypothesis?
\begin{equation}
    \ln\left(\frac{\text{gdp}_{2015}}{\text{gdp}_{1995}}\right) = b_0 - b_1\ln(\text{gdp}_{1995}) + u_{it}

\end{equation}
kostas2323
CC BY-SA 4.0
2023-12-12T19:23:54.340Z
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
CC BY-SA 4.0
2023-12-12T19:38:53.590Z
Great, thank you very much for your help and for your time.
kostas2323
CC BY-SA 4.0
2023-12-11T23:36:37.963Z
2023-12-11T23:36:37.963Z
Tony
<p>I have noticed that some researchers in order to test convergence hypothesis apply a model in this form:</p>
<p><span class="math-container">$(1/T) \ln(y_{it}/y_{i,t-1}) = b_0 - b_1\ln(y_{i,t-1}) + u_{it} $</span></p>
<p>(Barro &amp; Sala-I-Martin, 2004)</p>
<p>While others apply this form:</p>
<p><span class="math-container">$\ln(y_{it}/y_{i,t-1}) = b_0 - b_1\ln(y_{i,t-1}) + u_{it} $</span></p>
<p>(Mankiw et al., 1992)</p>
<p>What's the difference between these two models?</p>

<macroeconomics><econometrics><economic-growth><applied-econometrics>
Absolute Convergence
kostas2323
# Absolute Convergence

(Asked by: username_0 on 2023-12-11)

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?

**Tags:** macroeconomics, econometrics, economic-growth, applied-econometrics

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## 2 answers

### Answer by username_1 on 2023-12-11. Score: 4 | Accepted answer

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

\begin{equation}
\frac{1}{T} \left[ ln(y\_{it}) - ln(y\_{i,t-T}) \right]= \beta\_0 + \beta\_1 ln(y\_{i,t-T}) + u\_{it}
\end{equation}
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,

\begin{equation}
ln(y\_{it}) - ln(y\_{i,t-1}) = \beta\_0 + \beta\_1 ln(y\_{i,t-1}) + u\_{it}
\end{equation}

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

> **Comments:**
> **username_0**: 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  \begin{equation} \frac{1}{20} \left( \ln(gdp_{2015}) - \ln(gdp_{1995}) \right) \end{equation}   or calculate the annual GDP growth rate with this formula  \begin{equation}    \left (\ln(gdp_{2015}) - \ln(gdp_{1995} \right) )  \end{equation}  
> **username_1**: 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.  
> **username_0**: So, is it correct to use a model in this form to test the convergence hypothesis? \begin{equation}     \ln\left(\frac{\text{gdp}_{2015}}{\text{gdp}_{1995}}\right) = b_0 - b_1\ln(\text{gdp}_{1995}) + u_{it}  \end{equation}  
> **username_1**: 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.  
> **username_0**: Great, thank you very much for your help and for your time.  

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### Answer by username_0 on 2023-12-12. Score: 1

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


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