Stata 15 help for arch

[TS] arch -- Autoregressive conditional heteroskedasticity (ARCH) family of estimators


arch depvar [indepvars] [if] [in] [weight] [, options]

options Description ------------------------------------------------------------------------- Model noconstant suppress constant term arch(numlist) ARCH terms garch(numlist) GARCH terms saarch(numlist) simple asymmetric ARCH terms tarch(numlist) threshold ARCH terms aarch(numlist) asymmetric ARCH terms narch(numlist) nonlinear ARCH terms narchk(numlist) nonlinear ARCH terms with single shift abarch(numlist) absolute value ARCH terms atarch(numlist) absolute threshold ARCH terms sdgarch(numlist) lags of s_t earch(numlist) new terms in Nelson's EGARCH model egarch(numlist) lags of ln(s_t^2) parch(numlist) power ARCH terms tparch(numlist) threshold power ARCH terms aparch(numlist) asymmetric power ARCH terms nparch(numlist) nonlinear power ARCH terms nparchk(numlist) nonlinear power ARCH terms with single shift pgarch(numlist) power GARCH terms constraints(constraints) apply specified linear constraints collinear keep collinear variables

Model 2 archm include ARCH-in-mean term in the mean-equation specification archmlags(numlist) include specified lags of conditional variance in mean equation archmexp(exp) apply transformation in exp to any ARCH-in-mean terms arima(#p, #d, #q) specify ARIMA(p,d,q) model for dependent variable ar(numlist) autoregressive terms of the structural model disturbance ma(numlist) moving-average terms of the structural model disturbances

Model 3 distribution(dist [#]) use dist distribution for errors (may be gaussian, normal, t, or ged; default is gaussian) het(varlist) include varlist in the specification of the conditional variance savespace conserve memory during estimation

Priming arch0(xb) compute priming values on the basis of the expected unconditional variance; the default arch0(xb0) compute priming values on the basis of the estimated variance of the residuals from OLS arch0(xbwt) compute priming values on the basis of the weighted sum of squares from OLS residuals arch0(xb0wt) compute priming values on the basis of the weighted sum of squares from OLS residuals, with more weight at earlier times arch0(zero) set priming values of ARCH terms to zero arch0(#) set priming values of ARCH terms to # arma0(zero) set all priming values of ARMA terms to zero; the default arma0(p) begin estimation after observation p, where p is the maximum AR lag in model arma0(q) begin estimation after observation q, where q is the maximum MA lag in model arma0(pq) begin estimation after observation (p + q) arma0(#) set priming values of ARMA terms to # condobs(#) set conditioning observations at the start of the sample to #

SE/Robust vce(vcetype) vcetype may be opg, robust, or oim

Reporting level(#) set confidence level; default is level(95) detail report list of gaps in time series nocnsreport do not display constraints display_options control columns and column formats, row spacing, and line width

Maximization maximize_options control the maximization process; seldom used

coeflegend display legend instead of statistics ------------------------------------------------------------------------- You must tsset your data before using arch; see [TS] tsset. depvar and varlist may contain time-series operators; see tsvarlist. by, fp, rolling, statsby, and xi are allowed; see prefix. iweights are allowed; see weight. coeflegend does not appear in the dialog box. See [TS] arch postestimation for features available after estimation.

To fit an ARCH(#m) model with Gaussian errors, type

. arch depvar ..., arch(1/#m)

To fit a GARCH(#m,#k) model assuming that the errors follow Student's t distribution with 7 degrees of freedom, type

. arch depvar ..., arch(1/#m) garch(1/#k) distribution(t 7)

You can also fit many other models.



Statistics > Time series > ARCH/GARCH > ARCH and GARCH models


Statistics > Time series > ARCH/GARCH > Nelson's EGARCH model


Statistics > Time series > ARCH/GARCH > Threshold ARCH model


Statistics > Time series > ARCH/GARCH > GJR form of threshold ARCH model


Statistics > Time series > ARCH/GARCH > Simple asymmetric ARCH model


Statistics > Time series > ARCH/GARCH > Power ARCH model


Statistics > Time series > ARCH/GARCH > Nonlinear ARCH model


Statistics > Time series > ARCH/GARCH > Nonlinear ARCH model with one shift


Statistics > Time series > ARCH/GARCH > Asymmetric power ARCH model


Statistics > Time series > ARCH/GARCH > Nonlinear power ARCH model


arch fits regression models in which the volatility of a series varies through time. Usually, periods of high and low volatility are grouped together. ARCH models estimate future volatility as a function of prior volatility. To accomplish this, arch fits models of autoregressive conditional heteroskedasticity (ARCH) by using conditional maximum likelihood. In addition to ARCH terms, models may include multiplicative heteroskedasticity. Gaussian (normal), Student's t, and generalized error distributions are supported.

Concerning the regression equation itself, models may also contain ARCH-in-mean and ARMA terms.

The following are commonly fitted models:

Common term Options to specify --------------------------------------------------------------------- ARCH arch()

GARCH arch() garch()

ARCH-in-mean archm arch() [garch()]

GARCH with ARMA terms arch() garch() ar() ma()

EGARCH earch() egarch()

TARCH, threshold ARCH abarch() atarch() sdgarch()

GJR, form of threshold ARCH arch() tarch() [garch()]

SAARCH, simple asymmetric ARCH arch() saarch() [garch()]

PARCH, power ARCH parch() [pgarch()]

NARCH, nonlinear ARCH narch() [garch()]

NARCHK, NARCH with one shift narchk() [garch()]

A-PARCH, asymmetric power ARCH aparch() [pgarch()]

NPARCH, nonlinear power ARCH nparch() [pgarch()] ---------------------------------------------------------------------


+-------+ ----+ Model +------------------------------------------------------------

noconstant; see [R] estimation options.

arch(numlist) specifies the ARCH terms (lags of e_t^2).

Specify arch(1) to include first-order terms, arch(1/2) to specify first- and second-order terms, arch(1/3) to specify first-, second-, and third-order terms, etc. Terms may be omitted. Specify arch(1/3 5) to specify terms with lags 1, 2, 3, and 5. All the options work this way.

arch() may not be specified with aarch(), narch(), narchk(), nparchk(), or nparch(), as this would result in collinear terms.

garch(numlist) specifies the GARCH terms (lags of s_t^2).

saarch(numlist) specifies the simple asymmetric ARCH terms. Adding these terms is one way to make the standard ARCH and GARCH models respond asymmetrically to positive and negative innovations. Specifying saarch() with arch() and garch() corresponds to the SAARCH model of Engle (1990).

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

tarch(numlist) specifies the threshold ARCH terms. Adding these is another way to make the standard ARCH and GARCH models respond asymmetrically to positive and negative innovations. Specifying tarch() with arch() and garch() corresponds to one form of the GJR model (Glosten, Jagannathan, and Runkle 1993).

tarch() may not be specified with tparch() or aarch(), as this would result in collinear terms.

aarch(numlist) specifies the lags of the two-parameter term a(|e_t|+g*e_t)^2. This term provides the same underlying form of asymmetry as including arch() and tarch(), but it is expressed in a different way.

aarch() may not be specified with arch() or tarch(), as this would result in collinear terms.

narch(numlist) specifies lags of the two-parameter term a(e_t-ki)^2. This term allows the minimum conditional variance to occur at a value of lagged innovations other than zero. For any term specified at lag L, the minimum contribution to conditional variance of that lag occurs when the squared innovations at that lag are equal to the estimated constant k_L.

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

narchk(numlist) specifies lags of the two-parameter term a(e_t-k)^2; this is a variation of narch() with k held constant for all lags.

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

abarch(numlist) specifies lags of the term |e_t|.

atarch(numlist) specifies lags of |e_t|(e_t > 0), where (e_t > 0) represents the indicator function returning 1 when true and 0 when false. Like the TARCH terms, these ATARCH terms allow the effect of unanticipated innovations to be asymmetric about zero.

sdgarch(numlist) specifies lags of s_t. Combining atarch(), abarch(), and sdgarch() produces the model by Zakoian (1994) that the author called the TARCH model. The acronym TARCH, however, refers to any model using thresholding to obtain asymmetry.

earch(numlist) specifies lags of the two-parameter term a*z_t+g*(|z_t|- sqrt(2/pi)). These terms represent the influence of news -- lagged innovations -- in Nelson's (1991) EGARCH model. For these terms, z_t=e_t/s_t, and arch assumes z_t ~ N(0,1). Nelson derived the general form of an EGARCH model for any assumed distribution and performed estimation assuming a generalized error distribution (GED). See Hamilton (1994) for a derivation where z_t is assumed normal. The z_t terms can be parameterized in either of these two equivalent ways. arch uses Nelson's original parameterization; see Hamilton (1994) for an equivalent alternative.

egarch(numlist) specifies lags of ln(s_t^2).

For the following options, the model is parameterized in terms of h(e_t)^p and s_t^p. One p is estimated, even when more than one option is specified.

parch(numlist) specifies lags of |e_t|^p. parch() combined with pgarch() corresponds to the class of nonlinear models of conditional variance suggested by Higgins and Bera (1992).

tparch(numlist) specifies lags of (e_t>0)|e_t|^p, where (e_t > 0) represents the indicator function returning 1 when true and 0 when false. As with tarch(), tparch() specifies terms that allow for a differential impact of "good" (positive innovations) and "bad" (negative innovations) news for lags specified by numlist.

tparch() may not be specified with tarch(), as this would result in collinear terms.

aparch(numlist) specifies lags of the two-parameter term a(|e_t|+g*e_t)^p. This asymmetric power ARCH model, A-PARCH, was proposed by Ding, Granger, and Engle (1993) and corresponds to a Box-Cox function in the lagged innovations. The authors fit the original A-PARCH model on more than 16,000 daily observations of the Standard and Poor's 500, and for good reason. As the number of parameters and the flexibility of the specification increase, more data are required to estimate the parameters of the conditional heteroskedasticity. See Ding, Granger, and Engle (1993) for a discussion of how seven popular ARCH models nest within the A-PARCH model.

When g goes to 1, the full term goes to zero for many observations and can then be numerically unstable.

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

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

nparchk(numlist) specifies lags of the two-parameter term a|e_t-k|^p; this is a variation of nparch() with k held constant for all lags. This is a direct analog of narchk(), except for the power of p. nparchk() corresponds to an extended form of the model of Higgins and Bera (1992) as presented by Bollerslev, Engle, and Nelson (1994). nparchk() would typically be combined with the pgarch() option.

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

pgarch(numlist) specifies lags of (s_t)^p.

constraints(constraints), collinear; see [R] estimation options.

+---------+ ----+ Model 2 +----------------------------------------------------------

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 tradeoff between asset risk and return. By default, no ARCH-in-mean terms are included in the model.

archm specifies that the contemporaneous expected conditional variance be included in the mean equation.

archmlags(numlist) is an expansion of 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 archm archmlags(1) or archmlags(0/1).

archmexp(exp) applies the transformation in exp to any ARCH-in-mean terms in the model. The expression should contain an 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.

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 #d times, 1 through #p lags of autocorrelations are included, and 1 through #q lags of moving averages are included. For example, the specification

. arch y, arima(2,1,3)

is equivalent to

. arch D.y, ar(1/2) ma(1/3)

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 independent variables, the latter syntax is required.

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

If the model does not contain regressors, these terms can also be considered autoregressive terms for the dependent variable; see [TS] arima.

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.

+---------+ ----+ Model 3 +----------------------------------------------------------

distribution(dist [#]) specifies the distribution to assume for the error term. dist may be gaussian, normal, t, or ged. gaussian and normal are synonyms, and # cannot be specified with them.

If distribution(t) is specified, arch assumes that the errors follow Student's t distribution, and the degree-of-freedom parameter is estimated along with the other parameters of the model. If distribution(t #) is specified, then arch uses Student's t distribution with # degrees of freedom. # must be greater than 2.

If distribution(ged) is specified, arch assumes that the errors have a generalized error distribution, and the shape parameter is estimated along with the other parameters of the model. If distribution(ged #) is specified, then arch uses the generalized error distribution with shape parameter #. # must be positive. The generalized error distribution is identical to the normal distribution when the shape parameter equals 2.

het(varlist) specifies that varlist be included in the specification of the conditional variance. varlist may contain time-series operators. This varlist enters the variance specification collectively as multiplicative heteroskedasticity; see Judge et al. (1985). If het() is not specified, the model will not contain multiplicative heteroskedasticity.

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. arch requires considerably more temporary storage during estimation than most estimation commands in Stata.

+---------+ ----+ Priming +----------------------------------------------------------

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 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 [TS] arch for details.

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 (the ar(), ma(), or arima() option specified). See [TS] arch for details.

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 preestimation 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 arma0()), and required values of e_t^2 and s_t^2 assume the values specified by arch0(). condobs() can be used if conditioning observations are desired for the lags in the ARCH terms of the model. If arma() is also specified, the maximum number of conditioning observations required by arma() and condobs(#) is used.

+-----------+ ----+ SE/Robust +--------------------------------------------------------

vce(vcetype) specifies the type of standard error reported, which includes types that are robust to some kinds of misspecification (robust) and that are derived from asymptotic theory (oim, opg); see [R] vce_option.

For ARCH models, the robust or quasi-maximum likelihood estimates (QMLE) of variance are robust to symmetric nonnormality in the disturbances. The robust variance estimates generally are not robust to functional misspecification of the mean equation; see Bollerslev and Wooldridge (1992).

The robust variance estimates computed by arch are based on the full Huber/White/sandwich formulation, as discussed in [P] _robust. Many other 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.

+-----------+ ----+ Reporting +--------------------------------------------------------

level(#); see [R] estimation options.

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.

nocnsreport; see [R] estimation options.

display_options: noci, nopvalues, vsquish, cformat(%fmt), pformat(%fmt), sformat(%fmt), and nolstretch; see [R] estimation options.

+--------------+ ----+ Maximization +-----------------------------------------------------

maximize_options: difficult, technique(algorithm_spec), iterate(#), [no]log, trace, gradient, showstep, hessian, showtolerance, tolerance(#), ltolerance(#), gtolerance(#), nrtolerance(#), nonrtolerance, and from(init_specs); see [R] maximize for all options except gtolerance(), and see below for information on gtolerance().

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

Setting technique() to something other than the default or BHHH changes the vcetype to vce(oim).

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

gtolerance(#) specifies the tolerance for the gradient relative to the coefficients. When |g_i*b_i| < gtolerance() for all parameters b_i and the corresponding elements of the gradient g_i, the gradient tolerance criterion is met. The default gradient tolerance for arch is gtolerance(.05).

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

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

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 maximums. The default starting values are obtained via a series of regressions, producing results that, on the basis of 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 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.

The standard syntax for from() accepts a matrix, a list of values, or coefficient name value pairs; see [R] maximize. arch also allows the following:

from(archb0) sets the starting value for all the ARCH/GARCH/... parameters in the conditional-variance equation to 0.

from(armab0) sets the starting value for all ARMA parameters in the model to 0.

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

The following option is available with arch but is not shown in the dialog box:

coeflegend; see [R] estimation options.


--------------------------------------------------------------------------- Setup . webuse wpi1

ARCH model with three lags . arch D.ln_wpi, arch(1/3)

GARCH(2,1) model . arch D.ln_wpi, arch(1/2) garch(1)

Same as above, but assuming that errors follow the generalized error distribution . arch D.ln_wpi, arch(1/2) garch(1) distribution(ged)

--------------------------------------------------------------------------- Setup . webuse urates

GARCH(1,1) model with covariates . arch illinois indiana kentucky, arch(1) garch(1)

Same as above, but assuming that errors follow Student's t distribution with 6 degrees of freedom . arch illinois indiana kentucky, arch(1) garch(1) distribution(t 6)

--------------------------------------------------------------------------- Setup . webuse wpi1

GARCH(1,1) model with ARMA disturbances . arch D.ln_wpi, ar(1) ma(1 4) arch(1) garch(1)

EGARCH model with ARMA disturbances . arch D.ln_wpi, ar(1) ma(1 4) earch(1) egarch(1)

Setup . constraint 1 (3/4)*[ARCH]l1.arch = [ARCH]l2.arch . constraint 2 (2/4)*[ARCH]l1.arch = [ARCH]l3.arch . constraint 3 (1/4)*[ARCH]l1.arch = [ARCH]l4.arch

ARCH model with constraints . arch D.ln_wpi, ar(1) ma(1 4) arch(1/4) constraint(1/3)

--------------------------------------------------------------------------- Setup . webuse dow1

Threshold ARCH model . arch D.ln_dow, tarch(1) ---------------------------------------------------------------------------

Stored results

arch stores the following in e():

Scalars e(N) number of observations e(N_gaps) number of gaps e(condobs) number of conditioning observations e(k) number of parameters e(k_eq) number of equations in e(b) e(k_eq_model) number of equations in overall model test e(k_dv) number of dependent variables e(k_aux) number of auxiliary parameters e(df_m) model degrees of freedom e(ll) log likelihood e(chi2) chi-squared e(p) p-value for model test e(archi) sigma_0^2=epsilon_0^2, priming values e(archany) 1 if model contains ARCH terms, 0 otherwise e(tdf) degrees of freedom for Student's t distribution e(shape) shape parameter for generalized error distribution e(tmin) minimum time e(tmax) maximum time e(power) varphi for power ARCH terms e(rank) rank of e(V) e(ic) number of iterations e(rc) return code e(converged) 1 if converged, 0 otherwise

Macros e(cmd) arch e(cmdline) command as typed e(depvar) name of dependent variable e(covariates) list of covariates e(eqnames) names of equations e(wtype) weight type e(wexp) weight expression e(title) title in estimation output e(tmins) formatted minimum time e(tmaxs) formatted maximum time e(dist) distribution for error term: gaussian, t, or ged e(mhet) 1 if multiplicative heteroskedasticity e(dfopt) yes if degrees of freedom for t distribution or shape parameter for GED distribution was estimated, no otherwise e(chi2type) Wald; type of model chi-squared test e(vce) vcetype specified in vce() e(vcetype) title used to label Std. Err. e(ma) lags for moving-average terms e(ar) lags for autoregressive terms e(arch) lags for ARCH terms e(archm) ARCH-in-mean lags e(archmexp) ARCH-in-mean exp e(earch) lags for EARCH terms e(egarch) lags for EGARCH terms e(aarch) lags for AARCH terms e(narch) lags for NARCH terms e(aparch) lags for A-PARCH terms e(nparch) lags for NPARCH terms e(saarch) lags for SAARCH terms e(parch) lags for PARCH terms e(tparch) lags for TPARCH terms e(abarch) lags for ABARCH terms e(tarch) lags for TARCH terms e(atarch) lags for ATARCH terms e(sdgarch) lags for SDGARCH terms e(pgarch) lags for PGARCH terms e(garch) lags for GARCH terms e(opt) type of optimization e(ml_method) type of ml method e(user) name of likelihood-evaluator program e(technique) maximization technique e(tech) maximization technique, including number of iterations e(tech_steps) number of iterations performed before switching techniques e(properties) b V e(estat_cmd) program used to implement estat e(predict) program used to implement predict e(marginsok) predictions allowed by margins e(marginsnotok) predictions disallowed by margins

Matrices e(b) coefficient vector e(Cns) constraints matrix e(ilog) iteration log (up to 20 iterations) e(gradient) gradient vector e(V) variance-covariance matrix of the estimators e(V_modelbased) model-based variance

Functions e(sample) marks estimation sample


Bollerslev, T., R. F. Engle, and D. B. Nelson. 1994. ARCH models. In Handbook of Econometrics, Volume IV, ed. R. F. Engle and D. L. McFadden. New York: Elsevier.

Bollerslev, T., and J. M. Wooldridge. 1992. Quasi-maximum likelihood estimation and inference in dynamic models with time-varying covariances. Econometric Reviews 11: 143-172.

Ding, Z., C. W. J. Granger, and R. F. Engle. 1993. A long memory property of stock market returns and a new model. Journal of Empirical Finance 1: 83-106.

Engle, R. F. 1990. Discussion: Stock volatility and the crash of '87. Review of Financial Studies 3: 103-106.

Glosten, L. R., R. Jagannathan, and D. E. Runkle. 1993. On the relation between the expected value and the volatility of the nominal excess return on stocks. Journal of Finance 48: 1779-1801.

Hamilton, J. D. 1994. Time Series Analysis. Princeton: Princeton University Press.

Higgins, M. L., and A. K. Bera. 1992. A class of nonlinear ARCH models. International Economic Review 33: 137-158.

Judge, G. G., W. E. Griffiths, R. C. Hill, H. L├╝tkepohl, and T.-C. Lee. 1985. The Theory and Practice of Econometrics. 2nd ed. New York: Wiley.

Nelson, D. B. 1991. Conditional heteroskedasticity in asset returns: A new approach. Econometrica 59: 347-370.

Zakoian, J. M. 1994. Threshold heteroskedastic models. Journal of Economic Dynamics and Control 18: 931-955.

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