<p>Let me clarify the concepts of absolute and conditional convergence:</p>
<p>The hypothesis that poor economies tend to grow faster per capita than rich ones without conditioning on any other characteristics of economies is referred to as absolute convergence (or unconditional convergence). This implies convergence in income per capita levels.
The most elementary absolute convergence test (also known as the unconditional <span class="math-container">$\beta$</span>-convergence test) reads
<span class="math-container">\begin{equation}
\frac{1}{T}[log(y_{it})-log(y_{i,t-T})] = \alpha + \beta log (y_{i,t-T})  + u_{it}
\end{equation}</span></p>
<p>The test for absolute converge reads:</p>
<p><span class="math-container">\begin{equation}
H_0: \beta=0 \\
H_1: \beta&lt;0
\end{equation}</span>
If you can reject <span class="math-container">$H_0$</span>, you have evidence of absolute convergence.
In this equation, we are investigating how country’s gross domestic product (GDP) per capita today depends on its GDP per capita in the past (unconditional convergence).</p>
<p>When studying conditional convergence, we investigate whether the relationship between the growth rate in GDP per capita and the initial level of GDP per capita is different when conditioning on
various determinants or “correlates” of growth. Correlates of growth can be human capital, the % of investments over GDP, and so on.
Conditioning on the correlates of growth means to augment the initial regression equation as</p>
<p><span class="math-container">\begin{equation}
\frac{1}{T}[log(y_{it})-log(y_{i,t-T})] = \alpha + \beta log (y_{i,t-T}) + x_{it} \gamma + u_{it}
\end{equation}</span></p>
<p>where <span class="math-container">$\gamma$</span> is a <span class="math-container">$K$</span>-dimensional column vector of parameters and
<span class="math-container">$x$</span> is a <span class="math-container">$K$</span>-dimensional row vector of conditioning variables that
determine the steady-state value of GDP per capita variables such as rates of physical and human capital accumulation.
In this case, countries are assumed to have different steady states only because of the microeconomic variation controlled for by the inclusion of <span class="math-container">$x$</span> and a negative estimated value of <span class="math-container">$\beta$</span> is taken as evidence that
each is converging to its particular steady state. Such tests are called tests of “conditional convergence” to distinguish them from tests of “absolute convergence” based on the first equation.</p>
<p>To answer your question, if in your equation you get a negative and statistically significant <span class="math-container">$\hat{\beta}$</span>, you have found evidence of conditional convergence. Even if the other predictors have non statistically significant coefficients, you don't care as long as  the adjusted R-squared has improved and the coefficients on the other variables are jointly statistically significant (you need an F-test).</p>

CC BY-SA 4.0
2023-12-19T23:34:40.693Z
Thank you, your comment helped me a lot!!!
kostas2323
CC BY-SA 4.0
2023-12-18T22:10:17.830Z
2023-12-18T22:10:17.830Z
Tony
<p>Lets assume that we have a theoretical model in this specification to test conditional convergence:</p>
<p><span class="math-container">\begin{equation}
\frac{1}{T} \ln\left(\frac{y_{it}}{y_{it-1}}\right) = a - (1 - e^{-\beta T})\ln(y_{it-1}) + (1 - e^{-\beta T})\ln(s_{it}) - (1 - e^{-\beta T})\ln(n_{it} + g + \delta) + \varepsilon_{it}
\end{equation}</span></p>
<p>I know that the least squares method is not the best method to estimate the above model. However, I nevertheless use it to compare the results with other methods.</p>
<p>With this estimation method, I get a statistically significant (and negative) coefficient of the initial GDP per capita variable, but the coefficients of all other variables are not statistically significant. In this case, would you say that <em>conditional convergence</em> as a hypothesis is rejected by using this method?</p>

CC BY-SA 4.0
2023-12-17T15:48:41.943Z
This could be a more valuable question (and might encourage others to provide answers) if you provided a definition of what you mean by conditional convergence.
BKay
<macroeconomics><econometrics><economic-growth><applied-econometrics>
Conditional Convergence
kostas2323
# Conditional Convergence

(Asked by: username_0 on 2023-12-15)

Lets assume that we have a theoretical model in this specification to test conditional convergence:

\begin{equation}
\frac{1}{T} \ln\left(\frac{y\_{it}}{y\_{it-1}}\right) = a - (1 - e^{-\beta T})\ln(y\_{it-1}) + (1 - e^{-\beta T})\ln(s\_{it}) - (1 - e^{-\beta T})\ln(n\_{it} + g + \delta) + \varepsilon\_{it}
\end{equation}

I know that the least squares method is not the best method to estimate the above model. However, I nevertheless use it to compare the results with other methods.

With this estimation method, I get a statistically significant (and negative) coefficient of the initial GDP per capita variable, but the coefficients of all other variables are not statistically significant. In this case, would you say that *conditional convergence* as a hypothesis is rejected by using this method?

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

> **Comments:**
> **username_1**: This could be a more valuable question (and might encourage others to provide answers) if you provided a definition of what you mean by conditional convergence.  

---

## 1 answers

### Answer by username_2 on 2023-12-18. Score: 7 | Accepted answer

Let me clarify the concepts of absolute and conditional convergence:

The hypothesis that poor economies tend to grow faster per capita than rich ones without conditioning on any other characteristics of economies is referred to as absolute convergence (or unconditional convergence). This implies convergence in income per capita levels.
The most elementary absolute convergence test (also known as the unconditional $\beta$-convergence test) reads
\begin{equation}
\frac{1}{T}[log(y\_{it})-log(y\_{i,t-T})] = \alpha + \beta log (y\_{i,t-T}) + u\_{it}
\end{equation}

The test for absolute converge reads:

\begin{equation}
H\_0: \beta=0 \\
H\_1: \beta<0
\end{equation}
If you can reject $H\_0$, you have evidence of absolute convergence.
In this equation, we are investigating how country’s gross domestic product (GDP) per capita today depends on its GDP per capita in the past (unconditional convergence).

When studying conditional convergence, we investigate whether the relationship between the growth rate in GDP per capita and the initial level of GDP per capita is different when conditioning on
various determinants or “correlates” of growth. Correlates of growth can be human capital, the % of investments over GDP, and so on.
Conditioning on the correlates of growth means to augment the initial regression equation as

\begin{equation}
\frac{1}{T}[log(y\_{it})-log(y\_{i,t-T})] = \alpha + \beta log (y\_{i,t-T}) + x\_{it} \gamma + u\_{it}
\end{equation}

where $\gamma$ is a $K$-dimensional column vector of parameters and
$x$ is a $K$-dimensional row vector of conditioning variables that
determine the steady-state value of GDP per capita variables such as rates of physical and human capital accumulation.
In this case, countries are assumed to have different steady states only because of the microeconomic variation controlled for by the inclusion of $x$ and a negative estimated value of $\beta$ is taken as evidence that
each is converging to its particular steady state. Such tests are called tests of “conditional convergence” to distinguish them from tests of “absolute convergence” based on the first equation.

To answer your question, if in your equation you get a negative and statistically significant $\hat{\beta}$, you have found evidence of conditional convergence. Even if the other predictors have non statistically significant coefficients, you don't care as long as the adjusted R-squared has improved and the coefficients on the other variables are jointly statistically significant (you need an F-test).

> **Comments:**
> **username_0**: Thank you, your comment helped me a lot!!!  

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