Day 5: Equation Balance to Wrap Up Week 1

Slides

• The Main reveal presentation

• A reveal presentation on Variance models

• A reveal presentation on Switching models

• A worked example on ARCH

Equation Balance

The article by Pickup and Kellstedt forms the basis for summary remarks on time series and a useful transition into the study of multiple time series formally for week two.

The lingering issue that does not get adequate treatment owing to time is fractional integration methods. Indeed, the article by Lebo and Grant (2016) summarising the issues in the 2016 special issue is worth digesting.

A Note on ARCH and GARCH

ARCH and GARCH Models

• Models of the conditional volatility (variance) in the errors
• Widely used in financial econometrics with volatility in an assets being taken as an indicator of risk

ARCH Model of order \(q\)

\[ e_t = \sigma_t Z_t \] \(Z_t\) = white noise \(\sigma_t\) = standard deviation

\[ \sigma^{2}_t = a_0 + a_1e^{2}_{t-1} + \ldots + a_{q}e^{2}_{t-q} \]

GARCH Model

• \(p\) is order of the GARCH terms \(\sigma^2\)

• \(q\) is order of ARCH terms \(e^2\)

\[\sigma^2_{t} = w + \underbrace{\Sigma_{i=1}^{q} a_i e^{2}_{t-i}}_{(ARCH components)} + \underbrace{\Sigma_{i=1}^{p} B_i \sigma^{2}_{t-i}}_{(GARCH components)} \] Here’s a consolidated reference table. The rugarch package covers the widest range of these models in R — I’ve noted where fGarch or other packages offer cleaner alternatives, and flagged the few models with no direct R equivalent.

Computing: Variance Model Types

Stata Keyword	Description	R Model Specification (rugarch::ugarchspec)	Alt. Package
arch	ARCH	variance.model = list(model = "sGARCH", garchOrder = c(q, 0))	fGarch::garchFit(~garch(q,0))
garch	GARCH	variance.model = list(model = "sGARCH", garchOrder = c(p, q))	fGarch::garchFit(~garch(p,q)); tseries::garch()
saarch	Simple asymmetric ARCH	variance.model = list(model = "fGARCH", submodel = "AVGARCH")	—
tarch	Threshold ARCH	variance.model = list(model = "gjrGARCH")	—
aarch	Asymmetric ARCH	variance.model = list(model = "apARCH") with power fixed at 2	fGarch::garchFit(~aparch(p,q))
narch	Nonlinear ARCH	variance.model = list(model = "fGARCH", submodel = "NGARCH")	—
narchk	Nonlinear ARCH with single shift	variance.model = list(model = "fGARCH", submodel = "NAGARCH")	—
abarch	Absolute value ARCH	variance.model = list(model = "fGARCH", submodel = "AVGARCH")	—
atarch	Absolute threshold ARCH	variance.model = list(model = "fGARCH", submodel = "TGARCH")	—
sdgarch	Lags of \(\sigma_t\)	variance.model = list(model = "fGARCH", submodel = "TGARCH")	—
earch	News terms — Nelson (1991) EGARCH	variance.model = list(model = "eGARCH")	—
egarch	Lags of \(\ln(\sigma_t^2)\)	variance.model = list(model = "eGARCH")	—
parch	Power ARCH	variance.model = list(model = "apARCH") with η = 0 (no asymmetry)	fGarch::garchFit(~aparch(p,q))
tparch	Threshold power ARCH	variance.model = list(model = "apARCH") with GJR-type asymmetry	—
aparch	Asymmetric power ARCH	variance.model = list(model = "apARCH")	fGarch::garchFit(~aparch(p,q))
nparch	Nonlinear power ARCH	No direct equivalent; closest: fGARCH submodel "NGARCH"	—
nparchk	Nonlinear power ARCH with single shift	No direct equivalent in standard R packages	—
pgarch	Power GARCH	variance.model = list(model = "apARCH") (includes lagged power variance terms)	—

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Model Options

Stata Option	Description	R Equivalent
archm	ARCH-in-mean term in mean equation	mean.model = list(archm = TRUE, archpow = 2) — set archpow = 1 for \(\sigma_t\) instead of \(\sigma_t^2\)
arima	ARIMA(p,d,q) for mean equation	mean.model = list(armaOrder = c(p, q)) for the ARMA part; integrate the series manually beforehand with diff() for the d component
het	Include regressors in conditional variance	variance.model = list(external.regressors = X) where X is a T × k matrix

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A note on workflow. In R, ugarchspec() builds the model specification and ugarchfit() estimates it:

library(rugarch)

spec <- ugarchspec(
  variance.model = list(model = "gjrGARCH", garchOrder = c(1, 1)),
  mean.model     = list(armaOrder = c(1, 0), archm = FALSE),
  distribution.model = "norm"
)

fit <- ugarchfit(spec, data = y)

The key difference from Stata is that Stata’s arch command lets you mix components additively (e.g. combine garch and tarch terms in one call), whereas rugarch requires you to select a single named model family that bundles the equivalent terms together.