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I am trying to replicate Islam(1995) Convergence model for EU countries. I know that LSDV estimator is biased,however i have applied it in order to compare the results with other methods. My question is, what diagnostic tests should I run on such a model?

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  • $\begingroup$ Are you able to detect the sign and source of the bias? $\endgroup$
    – Tony
    Jan 6 at 12:26
  • $\begingroup$ I think the bias comes from the fact that the time lag of the dependent variable (initial gdp) that I set as independent in order to test the convergence hypothesis is correlated with the error term. Where according to them Barro, R. J. (2015) and Nickell(1981, p. 1422) the coefficient of the variable (initial gdp) is upward biased. $\endgroup$
    – kostas2323
    Jan 6 at 15:44

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A generic representation of the cross-country growth regression is given by \begin{equation} \Delta y_{it} = \beta y_{i,t-1} + x_{i,t} \psi + \alpha_i + \delta_t + \varepsilon_{i,t} \label{baseline_1} \end{equation} where $y_{i,t}$ denotes the Log GDP per capita of country $i$ at time $t$, $x_{i,t}$ is a row vector of size $K$ including the correlates of economic growth, and $\psi$ is the corresponding $K$-dimensional column vector of coefficients. Additionally, $\alpha_i$ denotes the unobservable country-specific effects, $\delta_t$ denotes the unobservable time effects, and $\varepsilon_{i,t}$ denotes the stochastic disturbance term.

Estimating the equation above using OLS is not viable, as it would yield biased and inconsistent estimates. This issue arises because OLS ignores the presence of $\alpha_i$ and $\delta_t$, which jointly go into the composite error term, $\nu_{i,t} = \alpha_i + \delta_t + \varepsilon_{i,t}$. To see this, \begin{equation} \mathbb{E}[y_{i,t-1}\alpha_i] = \mathbb{E}[(\beta y_{i,t-2} + x_{i,t-1} \psi + \alpha_i + \delta_{t-1}+ \varepsilon_{i,t-1}) \alpha_i ] \ne 0 \label{bias1} \end{equation} The inequality in the equation arises from the assurance that at least $\mathbb{E}(\alpha^2_i) \ne 0$. Given that $\mathbb{E}(\alpha^2_i) = \mathrm{Var}(\alpha_i)$ unequivocally, it follows that $\mathrm{Var}(\alpha_i) > 0$. Consequently, pooling data and employing the least squares estimator would introduce an upward bias to $\hat{\beta}$. This bias stems from the positive correlation between the lagged dependent variable and the composite error term.

Estimating the first equation using the within estimator is not feasible. In this scenario, the within transformation eliminates $\alpha_i$, but the term $(y_{i,t-1} - \bar{y}_{i,- 1})$, where $\bar{y}_{i,- 1} = \sum_{t=2}^{T} \frac{y_{i,t-1}}{T-1}$, remains correlated with $(\varepsilon_{i,t}-\bar{\varepsilon}_{i})$. The correlation arises because $y_{i,t-1}$ is correlated with $\bar{\varepsilon}_{i}$ by construction, as the average includes $\varepsilon_{i,t-1}$, which is unambiguously correlated with $y_{i,t-1}$. Additionally, $\varepsilon_{i,t}$ is correlated with $\bar{y}_{i,- 1}$ because the latter contains $y_{i,t}$, which by construction varies with $\varepsilon_{i,t}$. These are the leading terms causing the correlation and they are both of order $T-1$. Nickell (1981) showed that the within estimator is biased of $\mathcal{O}(1/T)$. In other words, the within estimator is downward biased and the bias reduces as $T$ increases. However, it has been shown that the bias in the within estimator can be sizeable even when $T$ is not small.

Due to the assumed data-generating process, a consistent estimate of your estimator is likely to fall between the and Within Estimate (Fixed effects), $\hat{\beta}_{\text{FE}}$ and OLS estimate, $\hat{\beta}_{\text{OLS}}$.

You can employ either the two-step GMM estimator or the System-GMM estimator (Bundell and Bond, 1998) and check if it falls within the aforementioned interval.

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