2014年-世界发展银行全球_Estimation_of_Normal_Mixtures_in_a_Nested_Error_Model_with_an_Application_to_Small_Area_Estimation_of_Poverty_and_Inequality_33页_950kb

Summary of "Estimation of Normal Mixtures in a Nested Error Model with an Application to Small Area Estimation of Poverty and Inequality"

Core Content

This paper presents a method for estimating distribution functions in a nested error model, particularly for small area estimation of poverty and inequality. The approach is based on Empirical Bayes (EB) estimation and uses normal mixtures to model the error distributions, allowing for flexibility without assuming a specific functional form.

The main objective is to improve the accuracy of estimating nonlinear functions of the dependent variable, such as poverty and inequality measures, by accounting for the nested error structure in the linear mixed model. This structure introduces a convolution problem, where the area-level error $ u_a $ is contaminated by the average of the household-level errors $ \varepsilon_{ah} $, and thus, ignoring this contamination can lead to significant bias in the estimates.

Main Views

• The standard linear mixed model for log income is given by:
  $$
  y_{ah} = x_{ah}^T \beta + u_a + \varepsilon_{ah}
  $$
  where $ u_a $ and $ \varepsilon_{ah} $ are zero-mean, independent errors.

• The non-normality of errors can introduce substantial bias in estimates of poverty and inequality, especially when the poverty rate is between 20% and 30%. Simulations show this bias can be as high as 2–3%.

• The Empirical Bayes approach is more accurate than the non-EB method used by Elbers, Lanjouw, and Lanjouw (2003), but requires a proper estimation of the conditional distribution of the area-level error $ u_a $ given the sample data.

• The paper proposes a non-parametric method using normal mixtures to estimate the error distributions, which allows for the flexibility of modeling non-normal errors while still accommodating EB estimation.

• The conditional distribution of $ u_a $ given $ \bar{e}a $ (the average of the total residuals) is also a normal mixture, and its parameters can be derived from the parameters of the unconditional normal mixtures of $ u_a $ and $ \varepsilon{ah} $.

• The paper demonstrates that the standard normal assumption is a special case of the normal mixture model, where the number of components is one.

• The proposed estimator is more accurate than the ELL estimator and is computationally feasible, as it uses a modified EM algorithm to estimate the parameters of the normal mixtures.

Key Information

• Application: The method is applied to small area estimation of poverty and inequality, using income surveys combined with census data.

• Data setup:

  • $ x_{ah} $: Independent variables available for the entire population.
  • $ y_{ah} $: Income or expenditure data available only for a sample of households.
  • $ \bar{e}_a $: The average of the total residuals from the area $ a $, used to condition the error distributions.

• Welfare function $ W $: A nonlinear function of the dependent variable, such as the head-count poverty rate or the Gini index.

• Simulation-based estimation:

  • The estimator is based on simulating the distribution of $ y_{(a)} $ using the estimated parameters.
  • The final estimate for welfare is obtained by averaging over $ R $ simulated values:
    $$
    \hat{\mu} = \frac{1}{R} \sum_{r=1}^{R} W(\tilde{y}{(a)}^{(r)}, s{(a)})
    $$
  • Where $ \tilde{y}{ah}^{(r)} = x{ah}^T \tilde{\beta}^{(r)} + \tilde{u}a^{(r)} + \tilde{\varepsilon}{ah}^{(r)} $, and all parameters and errors are simulated from their estimated distributions.

• Comparison with existing methods:

  • ELL method: Uses non-parametric estimation of the error distribution, but ignores the conditional aspect, leading to potential bias.
  • MR method: Uses EB estimation but assumes normality of the errors, limiting flexibility.
  • The proposed method combines the flexibility of ELL with the efficiency of MR, by allowing non-normal error distributions and using EB estimation.

• Advantages:

  • Robustness: Handles non-normality of errors.
  • Accuracy: Reduces bias in the estimation of nonlinear functions of the dependent variable.
  • Applicability: Can be used for both survey-to-census and survey-to-survey predictions.

• Limitations:

  • Requires estimation of normal mixtures, which can be complex.
  • The convolution problem remains a challenge, as the area-level error is contaminated by the average of the household-level error.

Conclusion

The paper concludes that the proposed method using normal mixtures and Empirical Bayes estimation is a significant improvement over existing methods for estimating poverty and inequality in small areas. It provides a more accurate and flexible approach that accounts for the nested error structure and reduces the bias introduced by ignoring non-normality in the error distributions. The method is suitable for both developed and developing countries, with greater benefits in countries where income surveys cover a larger proportion of the population.