I am trying to run the following regression with panel data. I first construct country-level beta_i of z_it on x_t. Then, at the country level, I regress Y_i on those beta_i, with population weights.
I want to calculate bootstrap standard errors using the SD of the bootstrap estimates of the 2nd stage, but I'm not sure what procedure would be appropriate. Is there a standard one? I tried a few ways, but the bootstrap mean is always smaller in magnitude than the statistic — maybe this is just because of measurement error/attenuation bias?
The approaches I tried:
1. Weighted resampling at the country level, then within each country, run a single random sample with replacement of the year-level data; use that to calculate beta_i. Run the cross-sectional regression without weights.
2. Weighted panel-level resampling of the z_it and associated x_t, compute beta_i, run cross-sectional regression without weights.
3. Resample at the year level without weights. Compute beta_i and run cross-sectional regression with weights.
4. The same as #1, except all countries always appear in the cross-sectional regression. i.e., in each country, run a simple random sample with replacement of the year-level data; use that to calculate beta_i. Run cross-sectional regression without weights.
I thought #1 was the most sensible, since it seemed like the only way to cluster sampling at the country level while simulating variability in the generated regressor.
Any help would be appreciated.
I want to calculate bootstrap standard errors using the SD of the bootstrap estimates of the 2nd stage, but I'm not sure what procedure would be appropriate. Is there a standard one? I tried a few ways, but the bootstrap mean is always smaller in magnitude than the statistic — maybe this is just because of measurement error/attenuation bias?
The approaches I tried:
1. Weighted resampling at the country level, then within each country, run a single random sample with replacement of the year-level data; use that to calculate beta_i. Run the cross-sectional regression without weights.
2. Weighted panel-level resampling of the z_it and associated x_t, compute beta_i, run cross-sectional regression without weights.
3. Resample at the year level without weights. Compute beta_i and run cross-sectional regression with weights.
4. The same as #1, except all countries always appear in the cross-sectional regression. i.e., in each country, run a simple random sample with replacement of the year-level data; use that to calculate beta_i. Run cross-sectional regression without weights.
I thought #1 was the most sensible, since it seemed like the only way to cluster sampling at the country level while simulating variability in the generated regressor.
Any help would be appreciated.
