 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. 

                         

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(C) MD*TECH Method and Data Technologies, 21.9.2000

