.. index::
   single: Hildreth-Lu
   single: Autocorrelation
   single: []
   single: Generalized Least Squares

Hildreth-Lu Technique for Autocorrelation Correction
====================================================

**hl <rho1> <rho2> <incr> <y> = <x1>, [<x2> [, <x3> [, ... [, <xn>]]]]**
   In the Hildreth-Lu procedure, a guess of the autocorrelation coefficient of the errors, *rho*, is chosen, multiplied by the equation lagged once, and subtracted from the unlagged equation. The resulting equation then is estimated by ordinary regression.  Another value of the guess of *rho* then is chosen and the process repeated.  In this command, <rho1> is the starting guess of *rho*, <incr> is the amount by which it is incremented on each iteration, and <rho2> is an upper limit on the guess.  The <y> and <x1>, ..., <xn> values are as in the *r* command.  
      
   At the end of the process, you will get a table with this heading::

      RHO-HL    SEE 1 AHEAD  RHO-EST    SEE LONG-RUN

   RHO-HL shows the assumed *rho*, the SEE 1 AHEAD shows the standard error of estimate (SEE) of the estimated equation, RHO-EST shows the *rho* of the estimated equation, and SEE LONG-RUN shows the standard error using the fitted equation on the original, undifferenced data, without a knowledge of the true lagged value of the dependent variable, as must be done in forecasts of more than a few periods ahead.  If the *save* command is on, all of the estimated equations will placed in the .SAV file as undifferenced equations suitable for going into a model.  You must choose which one you want.

   For graphing the HL output, one should run the equation one last time with the chosen value of *rho* as both the starting and ending *rho*.  Then also run the equation with just the *r* command (no autocorrelation).  Finally, construct two graphs::

      gr depvar predp1 hlshort
      gr depvar predic hllong

   The first will compare the actual with two one-period ahead forecasts, the one from ordinary least squares (+'s) and one from Hildreth-Lu (x's). The second compares the actual with two forecasts not relying on the lagged value of the dependent variable.  Again, the uncorrected least squares fit is shown by +'s and the fit using the H-L correction is shown by x's. 

   The pauses at the end of each fit in the HL procedure can be eliminated by the *zip* command.  If you use *zip*, however, you must remember to do "zip off" before using the *(gr)aph* command. 
   
   *G7* does not provide the Corchrane-Orcutt correction, since Hildreth-Lu is better.