Measuring intragenerational mobility using aggregate data

Abstract

We propose a new method to approximate income distribution dynamics at the micro level using only macro data on aggregate moments of the income distribution. Under the assumption that individual incomes follow a lognormal autoregressive process, we show that the evolution of the mean and standard deviation of log income across individuals provides sufficient information to bound the degree of mobility. We estimate mobility bounds for 46 countries, using time series data on aggregate moments of the income distribution available in the World Inequality Database and the World Bank’s PovcalNet database. This new data allows us to study the correlates of mobility, and to document churning in the top and bottom of the income distribution, in a much larger set of countries than was previously possible.

Appendices

Appendix

1.1 Appendix A: Proofs

In the main text we assumed for notational convenience that survey data are available in consecutive periods $t$ and $t-1$. In reality however, survey data often are available at irregular frequencies that differ over time and across countries, and in the empirical part of the paper we work with such irregularly-spaced data. In this appendix we provide proofs for the case of irregularly-spaced data, i.e. for two surveys available in periods $t$ and $t-k$. The propositions as stated in the main text obtain for the special case of $k=1$.

1.2 Preliminaries

Iterating the data generating process in Eq. 1 backwards for $k$ periods results in:

$${y}_{it}={\rho }^{k}{y}_{it-k}+\left(\frac{1-{\rho }^{k}}{1-\rho }\right){\lambda }_{i}+{\tilde{\delta }}_{t}+ {\tilde{\varepsilon }}_{it}$$

(9)

where ${\tilde{\delta }}_{t}\equiv \sum_{s=0}^{k-1}{\rho }^{s}{\delta }_{t-s}$ and ${\tilde{\varepsilon }}_{it}\equiv \sum_{s=0}^{k-1}{\rho }^{s}{\varepsilon }_{it-s}$. Evaluating this expression at $t-k$, iterating back $t-k$ periods to the initial period zero, and using the assumption on initial income in Assumption A1, Eq. (9) becomes:

$$ \begin{aligned} y_{it - k} = & \rho^{t - k} \left( {\frac{{\lambda_{i} }}{1 - \rho } + \delta_{0} + \varepsilon_{i0} } \right) + \left( {\frac{{1 - \rho^{t - k} }}{1 - \rho }} \right)\lambda_{i} + \tilde{\delta }_{t - k} + \tilde{\varepsilon }_{it - k} \\ = & \frac{{\lambda_{i} }}{1 - \rho } + \rho^{t - k} \left( {\delta_{0} + \varepsilon_{i0} } \right) + \tilde{\delta }_{t - k} + \tilde{\varepsilon }_{it - k} \\ \end{aligned} $$

(10)

1.3 Proof of Proposition 1

Given Eq. 10 which states that ${y}_{it-k}$ is a linear combination of ${\varepsilon }_{i0}, \dots ,{\varepsilon }_{it-k}$ and ${\lambda }_{i}$, and Assumption A1 that these shocks are jointly normally distributed, it follows that ${y}_{it-k}$ is normally distributed for all $t-k\ge 0$. Equation 10 also implies that $COV\left[{y}_{it-k},{\lambda }_{i}\right]=\frac{{\sigma }_{\lambda }^{2}}{1-\rho }$. To complete the proof of Proposition 1 we need to find $COV\left[{y}_{it},{y}_{it-k}\right]$. Using Eq. 9 we have:

$$ \begin{aligned} COV\left[ {y_{it} ,y_{it - k} } \right] = & \rho^{k} \sigma_{t - k}^{2} + \left( {\frac{{1 - \rho^{k} }}{1 - \rho }} \right)COV\left[ {y_{it - k} ,\lambda_{i} } \right] \\ = & \left( {\rho^{k} + \frac{{1 - \rho^{k} }}{{\left( {1 - \rho } \right)^{2} }}\frac{{\sigma_{\lambda }^{2} }}{{\sigma_{t - k}^{2} }}} \right)\sigma_{t - k}^{2} \\ = & \beta_{t - k,k} \sigma_{t - k}^{2} \\ \end{aligned} $$

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Setting $k=1$ and adopting the more compact notational convention that ${\beta }_{t-\mathrm{1,1}}={\beta }_{t-1}$ gives ${\beta }_{t-1}=\rho +\frac{{\sigma }_{\lambda }^{2}}{1-\rho }\frac{1}{{\sigma }_{t-1}^{2}}$ as in the main text.

1.4 Proof of Proposition 2

Taking unconditional expectations of both sides of Eq. 9 gives the following irregularly-spaced analog of Eq. 3.

$${\mu }_{t}={\rho }^{k}{\mu }_{t-k}+{\tilde{\delta }}_{t}$$

(12)

Taking unconditional variances of both sides of Eq. 9 gives the following irregularly-spaced analog of Eq. 4:

$$ \begin{aligned} \sigma_{t}^{2} = & \rho^{2k} \sigma_{t - k}^{2} + \left( {\frac{{1 - \rho^{k} }}{1 - \rho }} \right)^{2} \sigma_{\lambda }^{2} + 2\rho^{k} \left( {\frac{{1 - \rho^{k} }}{1 - \rho }} \right)COV\left( {y_{it - k} ,\lambda_{i} } \right) + \mathop \sum \limits_{s = 0}^{k - 1} \rho^{2s} \sigma_{\varepsilon t - s}^{2} \\ = & \rho^{2k} \sigma_{t - k}^{2} + \left( {\frac{{1 - \rho^{k} }}{1 - \rho }} \right)^{2} \sigma_{\lambda }^{2} + 2\rho^{k} \left( {\frac{{1 - \rho^{k} }}{1 - \rho }} \right)\frac{{\sigma_{\lambda }^{2} }}{1 - \rho } + \mathop \sum \limits_{s = 0}^{k - 1} \rho^{2s} \sigma_{\varepsilon t - s}^{2} \\ = & \rho^{2k} \sigma_{t - k}^{2} + \frac{{1 - \rho^{2k} }}{{\left( {1 - \rho } \right)^{2} }}\sigma_{\lambda }^{2} + \tilde{\sigma }_{\varepsilon t}^{2} \\ \end{aligned} $$

(13)

where ${\tilde{\sigma }}_{\varepsilon t}^{2}\equiv \sum_{s=0}^{k-1}{\rho }^{s}{\sigma }_{\varepsilon t-s}^{2}$. Setting $k=1$ retrieves the result in the main text.

1.4.1 Proof of Proposition 3

To prove Proposition 3 we first show that the anonymous growth rate of group average incomes is a weighted average of the anonymous growth incidence curves within that group:

$$ \begin{aligned} g_{t,t - k}^{A} \left( {p,q} \right) \equiv & \frac{{Y_{t} \left( {p,q} \right) - Y_{t - k} \left( {p,q} \right)}}{{Y_{t - 1} \left( {p,q} \right)}} \\ = & \frac{1}{q - p}\mathop \int_{p}^{q} \frac{{Y_{t} \left( s \right) - Y_{t - k} \left( s \right)}}{{Y_{t - k} \left( s \right)}}\frac{{Y_{t - k} \left( s \right)}}{{Y_{t - k} \left( {p,q} \right)}}ds \\ \approx & \mathop \int_{p}^{q} \left( {y_{t} \left( s \right) - y_{t - k} \left( s \right)} \right)w_{t - k} \left( s \right)ds \\ = & \mathop \int_ {p}^{q} g_{t,t - k}^{A} \left( s \right)w_{t - k} \left( s \right)ds \\ \end{aligned} $$

(14)

where ${Y}_{t}\left(p,q\right)$ is mean income between the ${p}^{th}$ and ${q}^{th}$ percentile of the income distribution for $p\le q$, and ${w}_{t-k}(s)\equiv \frac{1}{q-p}\frac{{Y}_{t-k}\left(s\right)}{{Y}_{t-k}\left(p,q\right)}$ is the share of percentile $s$ in the total income of the group at time $t-k$. Similarly, the non-anonymous growth rate of group average incomes is the same weighted average of the non-anonymous growth incidence curves:

$$ \begin{aligned} g_{t,t - k}^{NA} \left( {p,q} \right) \equiv & \frac{{E\left[ {Y_{t|t - k} \left( {p,q} \right)} \right] - Y_{t - k} \left( {p,q} \right)}}{{Y_{t - k} \left( {p,q} \right)}} \\ = & \frac{1}{q - p}\mathop \int _{p}^{q} E\left[ {\frac{{e^{{y_{t|t - k} \left( s \right)}} - e^{{y_{t - k} \left( s \right)}} }}{{e^{{y_{t - k} \left( s \right)}} }}} \right] \frac{{e^{{y_{t - k} \left( s \right)}} }}{{Y_{t - k} \left( {p,q} \right)}}ds \\ \approx & \mathop \int _{p}^{q} \left( {E\left[ {y_{t|t - k} \left( s \right)} \right] - y_{t - k} \left( s \right)} \right)w_{t - k} \left( s \right)ds \\ = & \mathop \int_{p}^{q} g_{t,t - k}^{NA} \left( s \right)w_{t - k} \left( s \right)ds \\ \end{aligned} $$

(15)

The expression for the difference between the anonymous and non-anonymous group average growth rates in the main text follows from subtracting Eqs. 14 and (15).

To complete the proof we need to evaluate and sign the integral

$$\int_{p}^{q}{\Phi }^{-1}(s){w}_{t-k}\left(s\right)ds=\frac{1}{{Y}_{t-k}\left(p,q\right)}\frac{1}{q-p}\int_{p}^{q}{\Phi }^{-1}\left(s\right){e}^{{\mu }_{t-k}+{\sigma }_{t-k}{\Phi }^{-1}\left(s\right)}ds$$

(16)

To evaluate this remaining integral, recall that ${Y}_{t-k}\left(p,q\right)\equiv \frac{1}{q-p}\underset{p}{\overset{q}{\int }}{e}^{{y}_{t-k}\left(s\right)}ds=\frac{1}{q-p}\underset{p}{\overset{q}{\int }}{e}^{{\mu }_{t-k}+{\sigma }_{t-k}{\Phi }^{-1}\left(s\right)}ds$. Differentiating ${Y}_{t-k}\left(p,q\right)$ with respect to ${\sigma }_{t-k}$ yields:

$$\frac{\partial {Y}_{t-k}\left(p,q\right)}{\partial {\sigma }_{t-k}}=\frac{1}{q-p}\int_{p}^{q}{\Phi }^{-1}\left(s\right){e}^{{\mu }_{t-k}+{\sigma }_{t-k}{\Phi }^{-1}\left(s\right)}ds$$

(17)

This means that we can find the integral we need simply by differentiating group average income with respect to ${\sigma }_{t-k}$. Combining Eqs. 16 and 17 and using the property of the truncated lognormal distribution that ${Y}_{t-k}\left(p,q\right)={e}^{{\mu }_{t-k}+\frac{{\sigma }_{t-k}^{2}}{2}}\left(\Phi \left({\Phi }^{-1}\left(q\right)-{\sigma }_{t-k}\right)-\Phi \left({\Phi }^{-1}\left(p\right)-{\sigma }_{t-k}\right)\right)/(q-p)$, we obtain:

$$ \begin{aligned} \mathop \int_{p}^{q} {\Phi }^{ - 1} \left( s \right)w_{t - k} \left( s \right)ds = &\frac{{\partial Y_{t - k} \left( {p,q} \right)}}{{\partial \sigma_{t - k} }}\frac{1}{{Y_{t - k} \left( {p,q} \right)}} \\ = &\frac{\partial }{{\partial \sigma_{t - k} }}\left( {e^{{\mu_{t - k} + \frac{{\sigma_{t - k}^{2} }}{2}}} \left( {{\Phi }\left( {{\Phi }^{ - 1} \left( q \right) - \sigma_{t - k} } \right) - {\Phi }\left( {{\Phi }^{ - 1} \left( p \right) - \sigma_{t - k} } \right)} \right)} \right)\frac{1}{{Y_{t - k} \left( {p,q} \right)}} \\ = & \left( {\sigma_{t - k} - \frac{{\phi \left( {{\Phi }^{ - 1} \left( q \right) - \sigma_{t - k} } \right) - \phi \left( {{\Phi }^{ - 1} \left( p \right) - \sigma_{t - k} } \right)}}{{{\Phi }\left( {{\Phi }^{ - 1} \left( q \right) - \sigma_{t - k} } \right) - {\Phi }\left( {{\Phi }^{ - 1} \left( p \right) - \sigma_{t - k} } \right)}}} \right) \\ \end{aligned} $$

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For the particular case of average incomes in the bottom $q$ percent this expression simplifies to

$${\int }_{0}^{q}{\Phi }^{-1}\left(s\right){w}_{t-k}\left(s\right)ds=\left({\sigma }_{t-k}-\frac{\phi \left({\Phi }^{-1}\left(q\right)-{\sigma }_{t-k}\right)}{\Phi \left({\Phi }^{-1}\left(q\right)-{\sigma }_{t-k}\right)}\right)$$

(19)

while for the particular case of average incomes in the top $p$ percent this expression simplifies to

$${\int }_{p}^{1}{\Phi }^{-1}\left(s\right){w}_{t-k}\left(s\right)ds=\left({\sigma }_{t-k}+\frac{\phi \left({\Phi }^{-1}\left(p\right)-{\sigma }_{t-k}\right)}{1-\Phi \left({\Phi }^{-1}\left(p\right)-{\sigma }_{t-k}\right)}\right)$$

(20)

Finally, note that ${g}_{t}^{i}\left(p,q\right)$, $i=A,NA$, can also be represented as: ${g}_{t}^{i}\left(p,q\right)={g}_{t}^{i}\left(z(p,q,{\sigma }_{t-1})\right)$, where $z(p,q,\sigma )$ satisfies: $p\le z\left(p,q,\sigma \right)\le q$. This follows directly from the fact that ${g}_{t}^{i}\left(z\right)$ is a monotonic function of z.

Appendix B: Details of OLS, bias-corrected, and MSE-minimizing estimates of ${\varvec{\rho}}$

This appendix describes our approach to estimating the autoregressive coefficient for log individual incomes using aggregate data. Let ${\widehat{\rho }}$ denote the OLS estimator of $\rho$ based on Eq. 3. Given that the available time series is short for many of the countries in our sample, this estimator will exhibit downwards finite-sample bias. We therefore also generate a corresponding bias-corrected estimator ${\widehat{\rho }}_{BC}$ using the procedure suggested in Andrews (1993). At the core of this procedure is the fact that the distribution of the OLS estimator is exclusively a function of the true autoregressive parameter and the sample size, and thus is independent of the parameters that describe the distribution of the error term and the time trend. We refer the interested reader to Andrews (1993) for a proof. We take advantage of this result, as suggested by Andrews (1993), by computing the median bias of the OLS estimator as a function of $\rho$ for each sample size separately (using numerical simulations), and then inverting this function to obtain a median-unbiased estimator for $\rho$.

Comparing the OLS and bias-corrected estimators highlights a tradeoff between bias and variance: while the OLS estimator is substantially downward biased, the bias-corrected estimator is much less precisely estimated. We address this tradeoff by taking a weighted average of the OLS and bias-corrected estimators with weight $\omega$ on the OLS estimator:

$$\widehat{\rho }=\omega {\widehat{\rho }}_{OLS}+(1-\omega ){\widehat{\rho }}_{BC}$$

(21)

The MSE of a weighted average of the two estimators is:

$$MSE\left[\omega {\widehat{\rho }}_{OLS}+\left(1-\omega \right){\widehat{\rho }}_{BC}\right]\equiv Bias{\left[\omega {\widehat{\rho }}_{OLS}+\left(1-\omega \right){\widehat{\rho }}_{BC}\right]}^{2} +V\left[\omega {\widehat{\rho }}_{OLS}+\left(1-\omega \right){\widehat{\rho }}_{BC}\right]$$

(22)

Setting the derivative of this expression with respect to $\omega$ equal to zero, and using the fact that $Bias\left[{\widehat{\rho }}_{BC}\right]=0$ results in this expression for the MSE-minimizing weight:

$${\omega }^{*}=\frac{V\left[{\widehat{\rho }}_{BC}\right]-COV\left[{\widehat{\rho }}_{OLS},{\widehat{\rho }}_{BC}\right]}{{Bias\left[{\widehat{\rho }}_{OLS}\right]}^{2}+V\left[{\widehat{\rho }}_{OLS}\right]-2COV\left[{\widehat{\rho }}_{OLS},{\widehat{\rho }}_{BC}\right]+V\left[{\widehat{\rho }}_{BC}\right]}$$

(23)

We approximate the bias of the OLS estimator as $Bias\left[{\widehat{\rho }}_{OLS}\right]\approx {\widehat{\rho }}_{OLS}-{\widehat{\rho }}_{BC}$. Since the bias-corrected estimator is a function of the OLS estimator, we can linearize to approximate $V\left[{\widehat{\rho }}_{BC}\right]\approx {\nabla }^{2}V\left[{\widehat{\rho }}_{OLS}\right]$ and $COV\left[{\widehat{\rho }}_{OLS},{\widehat{\rho }}_{BC}\right]\approx \nabla V\left[{\widehat{\rho }}_{OLS}\right]$, where $\nabla$ is the gradient of the bias corrected estimator as a function of the OLS estimator evaluated at the OLS estimate, that we compute numerically. Inserting these into Eq. (23) results in this MSE-minimizing weight:

$${\omega }^{*}=\frac{V\left[{\widehat{\rho }}_{OLS}\right]\nabla (\nabla -1)}{{\left({\widehat{\rho }}_{OLS}-{\widehat{\rho }}_{BC}\right)}^{2}+V\left[{\widehat{\rho }}_{OLS}\right]{\left(\nabla -1\right)}^{2}}$$

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See Tables

Table 7 Estimation details for WID sample

7 and

Table 8 Estimation details for PovcalNet sample

8