I am using the following regression:

$$\text{Test score} = \beta_0+\beta_1\text{Mother's employment}+\beta_2\text{Mother's education}$$

where "Mother's employment" is a set of dummy variables indicating whether the mother works more than 35 hours a week, is unemployed or is absent, and "Mother's education" is also a set of dummy variables indicating if the mother has a high school diploma, a college degree or a PhD.

If the mother is absent, then "Mother's education" is not applicable, i.e. there is no answer. How do I deal with this in Stata? Mean imputation? How do I do that with dummy variables?

I will get to the 0.632 estimator, but it'll be a somewhat long development:

Suppose we want to predict $Y$ with $X$ using the function $f$, where $f$ may depend on some parameters that are estimated using the data $(\mathbf{Y}, \mathbf{X})$, e.g. $f(\mathbf{X}) = \mathbf{X}\mathbf{\beta}$ 

A naïve estimate of prediction error is $$\overline{err} = \dfrac{1}{N}\sum_{i=1}^N L(y_i,f(x_i))$$ where $L$ is some loss function, e.g. squared error loss. This is often called training error. Efron et al. calls it apparent error rate or resubstitution rate. It's not very good since we use our data $(x_i,y_i)$ to fit $f$. This results in $\overline{err}$ being downward biased. You want to know how well your model $f$ does in predicting new values. 

Often we use cross-validation as a simple way to estimate the expected extra-sample prediction error (how well does our model do on data not in our training set?). $$Err = \text{E}\left[ L(Y, f(X))\right]$$ 

A popular way to do this is to do $K$-fold cross-validation. Split your data into $K$ groups (e.g. 10). For each group $k$, fit your model on the remaining $K-1$ groups and test it on the $k$th group. Our cross-validated extra-sample prediction error is just the average $$Err_{CV} = \dfrac{1}{N}\sum_{i=1}^N L(y_i, f_{-\kappa(i)}(x_i))$$ where $\kappa$ is some index function that indicates the partition to which observation $i$ is allocated and $f_{-\kappa(i)}(x_i)$ is the predicted value of $x_i$ using data not in the $\kappa(i)$th set. 

This estimator is approximately unbiased for the true prediction error when $K=N$ and has larger variance and is more computationally expensive for larger $K$. So once again we see the bias–variance trade-off at play.

Instead of cross-validation we could use the bootstrap to estimate the extra-sample prediction error. Bootstrap resampling can be used to estimate the sampling distribution of any statistic. If our training data is $\mathbf{X} = (x_1,\ldots,x_N)$, then we can think of taking $B$ bootstrap samples (with replacement) from this set $\mathbf{Z}_1,\ldots,\mathbf{Z}_B$ where each $\mathbf{Z}_i$ is a set of $N$ samples. Now we can use our bootstrap samples to estimate extra-sample prediction error: $$Err_{boot} = \dfrac{1}{B}\sum_{b=1}^B\dfrac{1}{N}\sum_{i=1}^N L(y_i, f_b(x_i))$$ where $f_b(x_i)$ is the predicted value at $x_i$ from the model fit to the $b$th bootstrap dataset. Unfortunately, this is not a particularly good estimator because bootstrap samples used to produce $f_b(x_i)$ may have contained $x_i$. The leave-one-out bootstrap estimator offers an improvement by mimicking cross-validation and is defined as: $$Err_{boot(1)} = \dfrac{1}{N}\sum_{i=1}^N\dfrac{1}{|C^{-i}|}\sum_{b\in C^{-i}}L(y_i,f_b(x_i))$$ where $C^{-i}$ is the set of indices for the bootstrap samples that do not contain observation $i$, and $|C^{-i}|$ is the number of such samples. $Err_{boot(1)}$ solves the overfitting problem, but is still biased (this one is upward biased). The bias is due to non-distinct observations in the bootstrap samples that result from sampling with replacement. The average number of distinct observations in each sample is about $0.632N$ (see this answer for an explanation of why Why on average does each bootstrap sample contain roughly two thirds of observations?). To solve the bias problem, Efron and Tibshirani proposed the 0.632 estimator:
$$ Err_{.632} = 0.368\overline{err} + 0.632Err_{boot(1)}$$ where $$\overline{err} = \dfrac{1}{N}\sum_{i=1}^N L(y_i,f(x_i))$$ is the naïve estimate of prediction error often called training error. The idea is to average a downward biased estimate and an upward biased estimate. 

However, if we have a highly overfit prediction function (i.e. $\overline{err}=0$) then even the .632 estimator will be downward biased. The .632+ estimator is designed to be a less-biased compromise between $\overline{err}$ and $Err_{boot(1)}$. 
$$
Err_{.632+} = (1 - w) \overline{err} + w Err_{boot(1)}
$$
with 
$$w = \dfrac{0.632}{1 - 0.368R} \quad\text{and}\quad R = \dfrac{Err_{boot(1)} - \overline{err}}{\gamma - \overline{err}}
$$
where $\gamma$ is the no-information error rate, estimated by evaluating the prediction model on all possible combinations of targets $y_i$ and predictors $x_i$. 

$$\gamma = \dfrac{1}{N^2}\sum_{i=1}^N\sum_{j=1}^N L(y_i, f(x_j))$$.

Here $R$ measures the relative overfitting rate. If there is no overfitting (R=0, when the $Err_{boot(1)} = \overline{err}$) this is equal to the .632 estimator. 
