observations:

(y_{t}, x_{t}), t = 1, …, n
y_{t} ≥ 0, y_{t} = 1 × 1, x_{t} = k × 1

model: censored regression model or “Tobit”

y_{t} = max{0, x′_{t}β_{0} + u_{t}}, t = 1, …, n.

problem: estimate the population parameter β using the LAD estimator β̂. (Powell, 1984) Since the median of median of y_{t} is

m(y_{t}) = max{0, x′_{t}β_{0}}

the censored LAD estimator is obtained as the solution to the problem:

min_{β} ∑_{t=1}^{n} |y_{t} − max{0, x′_{t}β}|.

FOC: alternatively, the estimator β̂ can be obtained as the root of

S_{x}(β) = n^{− \frac{1}{2}} ∑_{t=1}^{n} I(x′_{t}β > 0)\{I(y_{t} − x′_{t}β ≤ 0) − \frac{1}{2}\}x_{t}

Since E{S_{x}(β_{0})} = 0, S_{x}(β) can serve as an estimating equation. Resampling Method: A new resampling method has been proposed by Parzen, Wei and Ying (Biometrika, 1994) recently, when the random vector S_{x}(β_{0}) is exactly or asymptotically pivotal. That is, the exact or limiting distribution is completely known or can be estimated consistently.

We apply this resampling method to the censored LAD estimator proposed by Powell (1984). In this case, the conditional distibution (given x) of S_{x}(β_{0}) can be exactly generated by a weighted sum of independent and centered Bernoulli variables with probability of “success” \frac{1}{2}:

U = n^{− \frac{1}{2}} ∑_{t=1}^{n} I(x′_{t}β_{0} > 0)\{ξ_{t} − \frac{1}{2}\}x_{t}.

To explain in more detail note that, when I(x′_{t}β_{0} > 0) = 1, it follows that x′_{t}β = max{0, x′_{t}β_{0}} = m(y_{t}|x_{t}) and therefore I(y_{t} − x′_{t}β_{0} ≤ 0) is a Bernoulli variate. On the other hand, when I(x′_{t}β_{0} > 0) = 0 the corresponding summand of S_{x}(β_{0}) is zero. Thus, the summation in U runs over the observations for which x′_{t}β_{0} > 0 holds:

U = n^{− \frac{1}{2}} ∑_{t:x′_{t}β_{0}>0}\{ξ_{t} − \frac{1}{2}\}x_{t}.

The distribution of U can be consistently estimated by

Û = n^{− \frac{1}{2}} ∑_{t=1}^{n} I(x′_{t}β̂ > 0)\{ξ_{t} − \frac{1}{2}\}x_{t}.

Under conditions, for large n, the distribution of (β̂ − β_{0}) can be approximated by the conditional distribution of (β̃ − β_{U}). To generate the distribution of β_{U} empirically we need to be able to solve the equation

S_{x}(β) = û

for a given realization of û of Û. To this end, create an extra data point (y_{n+1}, x_{n+1}) with y_{n+1} extremely large so that I(y_{n+1} − x′_{t+1}β ≤ 0) becomes always 0. For a given û, we generate x_{n+1} = 2n^{\frac{1}{2}} û. Now, note that

n^{− \frac{1}{2}} ∑_{t=1}^{n+1} I(x′_{t}β > 0)\{I(y_{t} − x′_{t}β ≤ 0) − \frac{1}{2}\}x_{t} = 0

is equivalent to

n^{− \frac{1}{2}} ∑_{t=1}^{n} I(x′_{t}β > 0)\{I(y_{t} − x′_{t}β ≤ 0) − \frac{1}{2}\}x_{t} = I(x′_{n+1}β > 0)û

while we would like the right-hand side of last equality to be û. Therefore

S(β) = û

is equivalent to

n^{− \frac{1}{2}} ∑_{t=1}^{n} I(x′_{t}β > 0)\{I(y_{t} − x′_{t}β ≤ 0)−\frac{1}{2}\}x_{t} + n^{− \frac{1}{2}} \{I(y_{n+1} − x′_{n+1}β ≤ 0)−\frac{1}{2}\}x_{n+1} = 0

which in practice can be accomplished by minimizing the objective function:

min_{β} {∑_{t=1}^{n} |y_{t} − max{0, x′_{t}β}| + |y_{n+1} − x′_{n+1}β|}

Hopefully, this optimization program can be carried out with the algorithm for nonlinear regression quantiles developed recently by Koenker and Park (J. of Econometrics, 1995).