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How do I understand EI standard errors?

${\mathfrak{E}I}$ standard errors (and other uncertainty estimates, such as confidence intervals, etc.) are logically very similar to, and can be interpreted analogously to, standard errors for least squares regression (LS) coefficients:

1. In LS, the variance of predicted values as estimates of $y_i$, $V(\hat y_i+\epsilon)=V(x_ib)+\sigma^2$ do not go to zero as $n$ gets larger. In ${\mathfrak{E}I}$, the variances of the precinct-level parameters ( $V(\beta^b_i)$ and $V(\beta^w_i)$) do not go to zero as $n$ gets larger.

2. In LS, $V(\hat y)$, i.e. the ``sample'' variance of $\hat y_i$ over $i$, does not go to zero as $n$ gets larger. In ${\mathfrak{E}I}$, $V(\beta^b)$ and $V(\beta^w)$, i.e. the ``sample'' variance of $\beta^b_i$ and $\beta^w_i$ over $i$, does not go to zero as $n$ gets larger.

3. In LS, $V(b)$ goes to zero as $n$ gets larger (where $b$ is essentially any scalar function of $\hat y_i$, for all $i$, such as a LS coefficient). In ${\mathfrak{E}I}$, $V(B)$ goes to zero as $n$ gets larger (where B is the weighted average of the $\beta_i$'s over all precincts).

4. In LS, the number of explanatory variables we include tends in common practice to be an increasing function of the number of observations we have, and so we don't see very small standard errors unless there's a mistake. In ${\mathfrak{E}I}$, covariates are only sometimes used to correct for some types of aggregation bias and the number included is, in practice, independent of the number of observations and the number of quantities that may be of interest.

5. In LS, the variance of a prediction is a function of estimation uncertainty (sampling error, $V(b)$, where $b$ is the regression coefficient) and fundamental uncertainty ( $V(\epsilon_i)$). In ${\mathfrak{E}I}$, $V(\beta^b_i)$ and $V(\beta^w_i)$ are functions of estimation uncertainty ($V(\psi)$, where psi are the 5 parameters of the truncated bivariate normal), and fundamental uncertainty ($\Sigma$, which is composed of three elements of $\psi$).

6. In LS, it is standard practice is to include in the formal computation of the standard error only estimation and fundamental variability and to exclude uncertainty due to the possibility of omitted variable bias, endogeneity, measurement error, selection bias, etc. It is possible to include these other possible problems in the computation of the standard error, but few computer programs allow it. In ${\mathfrak{E}I}$, the standard errors include only estimation and fundamental uncertainty, and exclude possible violations of the 3 model assumptions that the model in an application may not be robust to (i.e., serious violations of no aggregation bias, truncated bivariate normality, and spatial independence). Only by using the diagnostics and bringing in additional qualitative information, not represented in the aggregate data, can one add assessments of uncertainty to the formal measures given by the program. Of course, many of the standard problems of regression do not affect ecological inferences.