arch.hlp

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  reason.  As the number of parameters and the flexibility of the
  specification increase, larger amounts of data are required to estimate the
  parameters of the conditional heteroskedasticity.

{pmore}
   Note that when g goes to 1, the full term goes to zero for many
   observations and, at this point, can be numerically unstable.

{phang}
{opth nparch(numlist)} specifies lags of the two-parameter
   term a|e_t-ki|^p.

{pmore}
{cmd:nparch()} may not be specified with {cmd:arch()},
   {cmd:saarch()}, {cmd:narch()}, {cmd:narchk()}, or {cmd:nparchk()}, as this
   would result in collinear terms.

{phang}
{opth nparchk(numlist)} specifies lags of the two-parameter
   term a|e_t-k|^p; note that this is a variation on {opt nparch()} with k
   held constant for all lags.  This is a direct analog of {opt narchk()},
   except for the power of p.
   {opt nparchk()} corresponds to an extended form of the model of Higgins and
   Bera as presented by Bollerslev, Engle, and Nelson.
   {opt nparchk()} would typically be combined with the option {opt pgarch()}.

{pmore}
   {opt nparchk()} may not be specified with {opt arch()}, {opt saarch()},
   {opt narch()}, {opt narchk()}, or {opt nparch()}, as this would result in
   collinear terms.

{phang}
{opth pgarch(numlist)} specifies lags of s_t^p.

{phang}
{opt constraints(constraints)}; see {help estimation options##constraints():estimation options}.

{dlgtab:Model 2}

{phang}
{opt archm} specifies that an ARCH-in-mean term be included in the
   specification of the mean equation.  This term allows the expected value of
   {depvar} to depend on the conditional variance.  ARCH-in-mean is most
   commonly used in evaluating financial time series when a theory supports a
   trade-off between asset risk and return.  By default, no ARCH-in-mean terms
   are included in the model.

{pmore}
   {opt archm} specifies that the contemporaneous expected conditional
   variance be included in the mean equation.

{phang}
{opth archmlags(numlist)} is an expansion of {opt archm} that includes lags of
   the conditional variance s_t^2 in the mean equation.  To specify a
   contemporaneous and once-lagged variance, specify either
   {bf:{cmd:archm archmlags(1)}} or {cmd:archmlags(0/1)}.

{phang}
{opth archmexp(exp)} applies the transformation in {it:exp} to any
ARCH-in-mean terms in the model.  The expression should contain an {cmd:X}
wherever a value of the conditional variance is to enter the expression.  This
option can be used to produce the commonly used ARCH-in-mean of the
conditional standard deviation.

{phang}
{opt arima(#p,#d,#q)} is an alternative,
shorthand notation for specifying autoregressive models in the
dependent variable.  The dependent variable and any independent variables are
differenced {it:#d} times, 1 through {it:#p} lags of autocorrelations are
included, and 1 through {it:#q} lags of moving averages are included.
For example, the specification

{pin2}
{cmd:. arch y, arima(2,1,3)}

{pmore}
   is equivalent to

{pin2}
{cmd:. arch D.y, ar(1/2) ma(1/3)}

{pmore}
   The former is easier to write for "classic" ARIMA models of the mean
   equation, but it is not nearly as expressive as the latter.  If gaps in the
   AR or MA lags are to be modeled, or if different operators are to be
   applied to the independent variables, the latter syntax is required.

{phang}
{opth ar(numlist)} specifies the autoregressive terms of the structural model
   disturbance to be included in the model.  For example, {cmd:ar(1/3)}
   specifies that lags 1, 2, and 3 of the structural disturbance be included
   in the model.  {cmd:ar(1/4)} specifies that lags 1 and 4 be included,
   possibly to account for quarterly effects.

{pmore}
   If the model does not contain regressors, these terms can also be
   considered autoregressive terms for the dependent variable; see
   {helpb arima}.

{phang}
{opth ma(numlist)} specifies the moving-average terms to be
   included in the model.  These are the terms for the lagged innovations or
   white-noise disturbances.

{dlgtab:Model 3}

{phang}
{opth het(varlist)} specifies that {it:varlist} be included in the
   specification of the conditional variance.  {it:varlist} may contain
   time-series operators.  This varlist enters the variance specification
   collectively as multiplicative heteroskedasticity.

{phang}
{opt savespace} conserves memory by retaining only those variables required
   for estimation.  The original dataset is restored after estimation.  This
   option is rarely used and should be specified only if there is insufficient
   memory to fit a model without the option.  Note that {opt arch} requires
   considerably more temporary storage during estimation than most estimation
   commands in Stata.

{dlgtab:Priming}

{phang}
{opt arch0(cond_method)} is a rarely used option that specifies
how to compute the conditioning (presample or priming) values for s_t^2 and
e_t^2.  In the presample period, it is assumed that s_t^2 = e_t^2 and that
this value is constant.  If {opt arch0()} is not specified, the priming values
are computed as the expected unconditional variance given the current
estimates of the b coefficients and any ARMA parameters.  See 
{bf:[TS] arch} for details.

{phang}
{opt arma0(cond_method)} is a rarely used option that specifies
how the e_t values are initialized at the beginning of the sample for the ARMA
component, if the model has one.  This option has an effect only when AR or MA
terms are included in the model (options {opt ar()}, {opt ma()}, or
{opt arima()} specified).  See 
{bf:[TS] arch} for details.

{phang}
{opt condobs(#)} is a rarely used option that specifies a fixed
   number of conditioning observations at the start of the sample.
   Over these priming observations, the recursions necessary to generate
   predicted disturbances are performed, but only to initialize pre-estimation
   values of e_t, e_t^2, and s_t^2.  Any required lags of e_t before the
   initialization period are taken to be their expected value of 0 (or the
   value specified in {opt arma0()}), and required values of e_t^2 and s_t^2
   assume the values specified by {opt arch0()}.  {opt condobs} can be used if
   conditioning observations are desired for the lags in the ARCH terms of the
   model.  If {opt arma()} is also specified, the maximum number of
   conditioning observations required by {opt arma()} and {opt condobs(#)} is
   used.

{dlgtab:SE/Robust}

{phang}
{opt vce(vcetype)}; see {it:{help vce_option}}.

{phang}
{opt robust}; see {help estimation options##robust:estimation options}.

{pmore}
For ARCH models, the robust or quasi-maximum likelihood estimates (QMLE) of
variance are robust to symmetric non-normality in the disturbances.  The
robust variance estimates generally are not robust to functional
misspecification of the mean equation.

{pmore}
Note that the robust variance estimates computed by {opt arch} are based on
the full Huber/White/sandwich formulation, as discussed in {helpb _robust}.
In fact, many software packages report robust estimates that set some terms to
their expectations of zero (Bollerslev and Wooldridge 1992), which saves them
from calculating second derivatives of the log-likelihood function.

{dlgtab:Reporting}

{phang}
{opt level(#)}; see {help estimation options##level():estimation options}.

{phang}
{opt detail} specifies that a detailed list of any gaps in the series
be reported, including gaps due to missing observations or missing data for
the dependent variable or independent variables.

{marker maximize_options}{...}
{dlgtab:Max options}

{phang}
{it:maximize_options}:
{opt diff:icult},
{opt tech:nique(algorithm_spec)},
{opt iter:ate(#)},
[{cmd:{ul:no}}]{opt lo:g},
{opt tr:ace}, 
{opt grad:ient},
{opt showstep},
{opt hess:ian},
{opt shown:tolerance},
{opt tol:erance(#)},
{opt ltol:erance(#)},
{opt gtol:erance(#)},
{opt nrtol:erance(#)},
{opt nonrtol:erance},
{opt from(init_specs)}; see {help maximize}.

{pmore}
   These options are often more important for ARCH models than for other
   maximum likelihood models because of convergence problems associated with
   ARCH models{hline 2}ARCH model likelihoods are notoriously difficult to
   maximize.

{pmore}
   The following options are all related to maximization and are either
   particularly important in fitting ARCH models or not available for most
   other estimators.

{phang2}
{opt gtolerance(#)} specifies the threshold for the relative size of the
gradient; see {help maximize}.  The default for {opt arch} is
{cmd:gtolerance(0.5)}.

{pmore2}
{cmd:gtolerance(999)} may be specified to disable the gradient criterion.  If
the optimizer becomes stuck with repeated "(backed up)" messages, it is likely
that the gradient still contains substantial values, but an uphill direction
cannot be found for the likelihood.  With this option, results can often be
obtained, but is unclear whether the global maximum likelihood has been found.

{pmore2}
When the maximization is not going well, it is possible to set the maximum
number of iterations (see {help maximize}) to the point where the optimizer
appears to be stuck and to inspect the estimation results at that point.

{phang2}
{opt from(init_specs)} specifies the initial values of the 
coefficients.  
ARCH models may be sensitive to initial values and may have coefficient values
that correspond to local maxima.  The default starting values are obtained via
a series of regressions, producing results that, based on asymptotic theory,
are consistent for the b and ARMA parameters and generally reasonable for the
rest.  Nevertheless, these values may not always be feasible in that the
likelihood function cannot be evaluated at the initial values {opt arch} first
chooses.  In such cases, the estimation function is restarted with ARCH and
ARMA parameters initialized to zero.  It is possible, but unlikely, that even
these values will be infeasible and that you will have to supply initial
values yourself.

{pmore2}
The standard syntax for {opt from()} accepts a matrix, a list
of values, or coefficient name value pairs; see {help maximize}.  In
addition, {opt arch} allows the following:

{pmore2}
{cmd:from(archb0)},
sets the starting value for all the ARCH/GARCH/... parameters in the
conditional-variance equation to 0.

{pmore2}
{cmd:from(armab0)} sets the starting value for all ARMA parameters in the
model to 0.

{pmore2}
{cmd:from(archb0 armab0)} sets the starting values for all
ARCH/GARCH/... and ARMA parameters to 0.


{title:Examples}

{phang}{cmd:. arch interest, arch(1) garch(1)} {space 7} (GARCH model){p_end}
{phang}{cmd:. arch interest, arch(1/3) garch(1/2)} {space 3} (GARCH(3,2) model){p_end}
{phang}{cmd:. arch interest m1 m2, arch(1) garch(1)} {space 1} (GARCH model w/ covariates)

{phang}{cmd:. arch D.cpi, arch(1) garch(1)} {space 4} (GARCH model){p_end}
{phang}{cmd:. arch D.cpi, earch(1) egarch(1)} {space 2} (EGARCH model){p_end}
{phang}{cmd:. arch D.cpi, aparch(1) pgarch(1)} {space 1} (A-PARCH model)

{phang}{cmd:. arch D.cpi, ar(1) ma(1 4) arch(1) garch(1)} {space 1} (GARCH model w/ ARMA){p_end}
{phang}{cmd:. arch D.cpi, ar(1) ma(1 4) earch(1) egarch(1)}

{phang}{cmd:. arch D.cpi, ar(1) ma(1) het(oilprice govt) arch(1) garch(1)}{p_end}
{phang}{cmd:. arch D.cpi, ar(1) ma(1) het(oilprice govt) arch(1) garch(1) robust}


{title:Also see}

{psee}
Manual:  {bf:[TS] arch}

{psee}
Online:  {help arch postestimation};{break}
{helpb arima},
{helpb svar},
{helpb tsset},
{helpb var},
{helpb vec}
{p_end}