New ARCH options in TSP 4.4 (5/11/98)

A few new options have been recently added to the ARCH command in
TSP 4.4 to accomplish the following goals:
1. More standard methods for initializing presample values of h(t)
   and e(t)**2, which makes it easier to reproduce published results
   (HINIT= and E2INIT=)
2. Faster and more reliable iterations/convergence  (HITER=N)
3. Standard errors robust to non-normal errors  (HCOV=W)
4. Durbin-Watson autocorrelation and Jarque-Bera normality tests,
   based on the weighted residuals (@RES/sqrt(@HT)).

1. To initialize presample values of h(t), use:
   HINIT=ESTALL/OLS/[SSR]/STEADY/value :
         ESTALL:  estimate all required presample values of h(t) as
           additional "nuisance" parameters H(0), H(-1), ....
           When the sum of the alphas and betas (not counting alpha(0))
           is less than one, these presample values cannot be
           estimated consistently (i.e. they don't converge to a
           meaningful value if the sample size grows very large).
           This option was the default in previous versions of TSP.
           However, the estimates often were zero (the lower bound).
           This resulted in warnings about constrained parameters that 
           left many people concerned about whether the remainder of
           the coefficients were OK.  Probably the net effect in this
           case was to lower the first few in-sample values of h(t).
         OLS:  fix the presample values of h(t) at the ML OLS estimate,
           i.e. SSR(OLS)/T, where SSR(OLS) is the sum of squared
           residuals from a preliminary OLS regression of Y on the
           RHS variables.
         SSR:  set the presample values of h(t) to the current
           unconditional ML estimate, i.e. SSR(current)/T, where
           SSR(current) uses the parameter values from the current
           iteration.  This is the new default.  It is also described
           in Fiorentini, Calzolari and Panattoni, Journal of Applied
           Econometrics, 1996, p.401-403.  HINIT=SSR cannot be used
           in GARCH-M type models (when the MEAN or HEXP option is
           used), because then the SSR depends on h(0).
         STEADY:  set the presample values of h(t) to the steady-state
           value implied by the current estimates of the alphas and
           betas.  I.e.
           h(0) = alpha(0)/(1 - (alpha(1)+alpha(2)+...+alpha(NAR)
                                 + beta(1)+beta(2)+...+ beta(NMA)))
           To make sure this division can be performed, the parameters
           are restricted so that the denominator is positive.  This
           summation restriction of the alpha(i) and beta(j) is in
           addition to the usual non-negative restrictions and
           sum(beta(j)) < 1 .
           At present, any g(t) variables are assumed to have expectation
           zero.  Probably this should be modified to use their sample
           mean instead.
         value:  fix the presample values of h(t) at a value specified
           by the user.
  Note:  the old option UNCOND is now ignored.  It was similar to
    HINIT=SSR, but it used parameter values from the previous iteration,
    and did not guarantee improvement in the likelihood function.

  To initialize presample values of e(t)**2 (squared residuals), use:
  E2INIT=[HINIT]/INDATA/PREDATA:
         HINIT:  set presample values of e(t)**2 to the presample
           values of h(t), as given in the HINIT option above.
           This is the new default.
         INDATA:  Use the first NAR observations in the current sample
           to compute presample e(t).  Use the remaining NOB-NAR
           observations for residuals that enter the likelihood function
           directly.  This was the default in earlier versions of TSP.
         PREDATA:  Use NAR observations prior to the current sample
           to compute presample e(t).  If fewer than NAR such
           observations are available, take the remainder from the start
           of the current sample, and exclude those observations from
           entering the likelihood function directly.

   By using the new default HINIT and E2INIT values, TSP 4.4 can now
   closely reproduce some published results, such as Bollerslev
   and Ghysels, JBES, April 1996.  See this on the benchmarks web page:
       http://www-leland.stanford.edu/~clint/bench
   It may be somewhat surprising that the results may seem to vary
   quite a bit, depending on the initializations used, even though
   this example is a fairly large sample.

2. HITER=N  (new default) This is iteration with analytic second
   derivatives.  These derivatives are computed for all the available
   ARCH options, including MEAN, HEXP, GT, and all the above HINIT and
   E2INIT options.  Fiorentini, Calzolari, and Panattoni (1996)
   demonstrate that HITER=N results in much faster convergence, at
   least for the GARCH(1,1) model.
      A minor drawback at first with HITER=N was that when the
   (negative) Hessian was not positive definite, the generalized inverse
   had some zero rows and columns, and the indicated parameters
   did not change (even when their gradient was nonzero).  As a result,
   a "modified Cholesky factorization" has been employed for all models
   which use HITER=N.  It avoids this problem of the indefinite Hessian,
   and always makes sure the parameters are changed in a direction
   which improves the LogL.
      The second derivatives were extensively tested against the
   "discrete Hessian" (numerical differences of analytic first
   derivatives) with the new HESSCHEC option.  Note that HESSCHEC may
   indicate large "errors" at the default zero starting values in ARCH,
   so the checks were made with nonzero parameter values.

3. HCOV=W  (new default)  Also HCOV=N (new available option).
      Fiorentini, Calzolari, and Panattoni (1996) demonstrate that
   HCOV=W (they call it QML) provides the best parameter VCOV matrix
   (versus HCOV=B and HCOV=N) for the GARCH(1,1) model for both normal
   and (especially) non-normal disturbances, over a range of sample
   sizes.  They also showed that HCOV=W is very closely equivalent to
   the "BW" VCOV estimator studied by Bollerslev and Wooldridge,
   Econometric Reviews, 1992.  The BW matrix uses the expected Hessian
   (also called the Score or Information matrix), instead of the
   exact Hessian (N), in the same sandwich formula.
      A minor flaw in labelling occurs sometimes now when printing
   the parameter standard errors.  If the (negative) Hessian is
   not positive definite, corresponding standard errors for HCOV=N
   and HCOV=W will be set to zero.  This is fine, but instead of
   a warning message about singularity, the ARCH-specific warning
   message "some of the parameters are on the boundary of their
   constraints" is printed instead.  This problem will be fixed
   by passing additional information back from the constraint check
   to distinguish between these two sources of zeros in the VCOV.

4. ARCH now prints 2 sets of residual diagnostics -- weighted by
   the estimated variance (@HT) and unweighted ("original").
   The main purpose is to compute a more useful Durbin-Watson
   autocorrelation test and Jarque-Bera normality test, after
   correcting the residuals for the heteroskedasticity just
   estimated.