Usage: |
z = rqfit(x,y{,tau,ci,alpha,iid,interp,tcrit})
|
Input: |
| x | n x p design matrix of explanatory variables.
|
| y | n x 1 vector, dependent variable.
|
| tau | desired quantile, default value = 0.5.
If tau is inside <0,1>, a single quantile solution
is computed and returned.
If tau is outside of <0,1>, solutions for all quantiles
are sought and the program computes the whole quantile
regression solution as a process in tau. The resulting
arrays containing the primal and dual solutions and
betahat(tau) are called sol and dsol.
It should be emphasized that this form of the solution
can be both memory and cpu quite intensive. On typical
machines it is not recommended for problems with n > 10,000.
|
| ci | logical flag for confidence intervals (nonzero = TRUE),
default value = 0. If ci = 0, only the estimated coefficients
are returned. If ci != 0, confidence intervals for the
parameters are computed using the rank inversion method of
Koenker (1994). Note that for large problems the option
ci != 0 can be rather slow. Note also that rank inversion
only works for p > 1, an error message is printed in the case
that ci != 0 and p = 1.
|
| alpha | the nominal coverage probability for the confidence
intervals, i.e., aplha/2 gives the level of significance
for confidence intervals, default value = 0.1.
|
| iid | logical flag for iid errors (nonzero = TRUE),
default value = 1.
If iid != 0, then the rank inversion (see
parameter ci) is based on an assumption of iid error model
and the original version of the rank inversion intervals is
used (as in Koenker, 1994).
If iid = 0, then it is based on the heterogeneity error
assumption. See Koenker and Machado (1999) for further details.
|
| interp | logical flag for smoothed confidence intervals (nonzero = TRUE),
default value = 1.
As with typical order statistic type confidence intervals
the test statistic is discrete, so it is reasonable to consider
intervals that interpolate between values of the parameter
just below the specified cutoff and values just above the
specified cutoff.
If interp != 0, this function returns a single interval based
on linear interpolation of the two intervals.
If interp = 0, then the 2 "exact" values above
and below on which the interpolation would be based are returned.
Moreover, in this case c.values and p.values which give
the critical values and p.values of the upper and lower intervals
are returned.
|
| tcrit | logical flag for finite sample adjustment using t-statistics
(nonzero = TRUE), default value = 1.
If tcrit != 0, Student t critical values are used,
while for tcrit = 0 normal ones are employed.
|
Output: |
| z.coefs | p x 1 or p x m matrix.
If tau is in <0,1>, the only column (p x 1) contains estimated
coefficients.
If tau is outside <0,1>, then p x m matrix contains
estimated coefficients for all quantiles = sol[4:(p+3),],
see sol description. |
| z.intervals | nothing, p x 2, or p x 4 matrix containing confidence intervals.
If ci = 0, then no confidence intervals are computed.
If ci != 0 and interp != 0, then variable intervals has 2 columns,
interpolated "lower bound" and "upper bound".
If ci != 0 and interp = 0, then variable intervals contains
"lower bound", "Lower Bound", "upper bound", "Upper Bound".
See description of ci and interp parameters for further information. |
| z.res | n x 1 vector of residuals.
Not supplied if tau is not inside <0,1>. |
| z.sol | The primal solution array. This is a (p+3) by J matrix whose
first row contains the 'breakpoints' tau_1,tau_2,...tau_J,
of the quantile function, i.e. the values in [0,1] at which the
solution changes, row two contains the corresponding quantiles
evaluated at the mean design point, i.e. the inner product of
xbar and b(tau_i), the third row contains the value of the objective
function evaluated at the corresponding tau_j, and the last p rows
of the matrix give b(tau_i). The solution b(tau_i) prevails from
tau_i to tau_i+1. Portnoy (1991) shows that J=O_p(n log n). |
| z.dsol | The dual solution array. This is an by J matrix containing the
dual solution corresponding to sol, the ij-th entry is 1 if
y_i > x_i b(tau_j), is 0 if y_i < x_i b(tau_j), and is between
0 and 1 otherwise, i.e. if the residual is zero. See Gutenbrunner and
Jureckova(1991) for a detailed discussion of the statistical
interpretation of dsol. The use of dsol in inference is described
in Gutenbrunner, Jureckova, Koenker, and Portnoy (1994). |
| z.cval | c-values, see the description of interp parameter for further information.
Not supplied if tau is not in <0,1> or ci == 0. |
| z.pval | p-values, see the description of interp parameter for further information.
Not supplied if tau is not in <0,1> or ci == 0. |