Chapter 2 Nonlinear econometrics for finance
MAXIMUM LIKELIHOOD
The likelihood principle says “pick the parameters that make the sample more likely to have occurred”.
Assume the sample is a draw from T normal random variables. Write
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x ∼ N(μ,Σ).
Hence,xisaT vector,μisaT vector,andΣisaT×T matrix. Thelikelihood
function is nothing but the joint probability distribution of the observations. Thus, 1 1 ⊤−1
L({x},μ,Σ)=(2π)T/2|Σ|1/2exp −2(x−μ) Σ (x−μ) . Now, take logs
11 logL({x},μ,Σ) = log (2π)T/2 |Σ|1/2 exp(−2(x−μ)⊤Σ−1(x−μ))
= −T2 log(2π) − 12 log |Σ| − 21(x − μ)⊤Σ−1(x − μ). Maximum likelihood provides estimates of μ and Σ given the sample, i.e., x.
Maximum likelihood in a time-series context
Consider a sample of observations xt t = 1,…,T. Assume dependence in the ob- servations. In other words, p(xt+1, xt) ̸= p(xt+1)p(xt). In general, p(xt+1, xt) = p(xt+1 |xt )p(xt ).
Chapter 2 Nonlinear econometrics for finance
L({x} , θ) = p(xT |xT −1, xT −2, …, θ)p(xT −1|xT −2, xT −3, …, θ)…p(x1, θ). The log-likelihood becomes
l({x} , θ) = log L({x} , θ) = log p(xt|xt−1, …, θ).
The maximum likelihood (ML) estimator
1 T T t=1
1 T ∂ log p(xt|xt−1, …, θT ) T t=1 ∂θ
θT = arg max [QT (θ)] = arg max θθ
The term ∂ log p(xt|xt−1,…,θ0) (the derivative of the log transition density with respect ∂θ
log p(xt|xt−1, …, θ)
to θ0) is usually called “the score”. Some properties:
p(xt|xt−1, …, θ0)dxt = 1 ⇒
∂p(xt|xt−1, …, θ0)
∂θ dxt = 0
∂ log p(xt|xt−1, …, θ0)p(xt|xt−1, …, θ0)dxt = 0, ∂θ
∂ log p(xt|xt−1, …, θ0)
Chapter 2 Nonlinear econometrics for finance
• The score is unforecastable. Its conditional (and unconditional) expectation are equal to zero (at the parameter value).
2.3 Consistency and asymptotic normality of the ML estimator
• We assume dependence in the data from the start.
• By Taylor’s expansion, stopped at the first order, around θ0:
∂ Q ( θ ) ∂ Q ( θ ) ∂ 2 Q ( θ )
TT−T0=T0θ−θ.
⊤T0
• Now, notice that ∂QT (θT ) ≈ 0 since we are minimizing the criterion with respect to ∂θ
θ to find θT .
• We begin with consistency. Write
∂2QT(θ0)−1 ∂QT(θ0) θT − θ0 = − ⊤ .
We will show that the first term, i.e., ∂2QT (θ0) converges in probability as the number ∂θ∂θ⊤
of observations grows without bound. As for the second term, by the WLLN, we have
d×1 vector
∂QT (θ0) ∂θ
= 1 T ∂ log p(xt|xt−1, …, θ0) T t=1 ∂θ
p ∂ log p(xt|xt−1, …, θ0) →E ∂θ .
Chapter 2 Nonlinear econometrics for finance
However, because of the properties of the score E
∂ log p(xt |xt−1 ,…,θ0 ) ∂θ
= 0. Thus,
• We now turn to asymptotic normality. After standardizing (by
θ T →p θ 0 . Hence, the ML estimator is consistent.
∂2QT (θ0)−1√
∂QT (θ0) ∂θ
θ T − θ 0 →p 0
T ), write
• Let us focus on (a) first. What is √T ∂QT (θ0)? ∂θ
T √ ∂QT (θ0) √ 1 ∂ log p(xt|xt−1, …, θ0)
T ∂θ = T Tt=1 Yd N (0, Ω0 ),
∂ log p(xt |xt−1 ,…,θ0 )
where,again,E ∂θ =0andΩ0 =E
∂ log p(xt |xt−1 ,…,θ0 ) ∂ log p(xt |xt−1 ,…,θ0 ) ∂θ ∂θ⊤
(why? because the autocorrelation terms are zero – the score is not forecastable). • Now, we need to consider the additional term (b), namely ∂2QT (θ0). Write
∂θ∂θ⊤ ∂2QT(θ0) = 1 T ∂2 logp(xt|xt−1,…,θ0).
• This term converges to B0 = E
T t=1 ∂θ∂θ⊤ ∂2 logp(xt|xt−1,…,θ0)
∂θ∂θ⊤ by an application of the WLLN: ∂2QT(θ0)→p B. (1)
Chapter 2 Nonlinear econometrics for finance • We can now put everything together and obtain:
∂2QT (θ0)−1 √ ∂QT (θ0)
=− ∂θ∂θ⊤ T ∂θ
Yd NB−1ΩB−1. 000
• Thus, the ML estimator is asymptotically normal.
• The formula for the asymptotic variance simplifies dramatically. Again, write
∂p(xt|xt−1, …, θ0) p(xt|xt−1, …, θ0)dxt = 1 ⇒ ∂θ dxt = 0
∂ log p(xt|xt−1, …, θ0)p(xt|xt−1, …, θ0)dxt = 0 ∂θ
∂2 log p(xt|xt−1, …, θ0)p(x |x , …, θ )dx ∂θ∂θ⊤ t t−1 0 t
∂ log p(xt|xt−1, …, θ0) ∂p(xt|xt−1, …, θ0)
+ ∂θ ∂θ⊤ dxt
∂2 log p(xt|xt−1, …, θ0)
∂θ∂θ⊤ p(xt|xt−1, …, θ0)dxt
∂ log p(xt|xt−1, …, θ0) ∂ log p(xt|xt−1, …, θ0)
∂θ ∂θ⊤ p(xt|xt−1, …, θ0)dxt
• This relation implies that
Nonlinear econometrics for finance
T θT−θ0 N(0, −B−1)
= N 0,E − ∂θ∂θ⊤
∂2 log p(xt|xt−1, …, θ0)−1
∂ log p(xt|xt−1, …, θ0) ∂ log p(xt|xt−1, …, θ0)−1 =N0,E∂θ∂θ⊤ .
• Notice that the precision of our estimates depends on the curvature of the likelihood function at the true parameter value.
• The ML estimator is asymptotically efficient (no other estimator can have a smaller asymptotic variance).
• The asymptotic variance of the maximum likelihood estimator is typically called “Information matrix”.
• Variance-covariance estimation. Use sample analogues: 1T ∂2logp(x|x ,…,θ) p
t t−1 T →B T t=1 ∂θ∂θ⊤ 0
1 T ∂ log p(xt|xt−1, …, θT ) ∂ log p(xt|xt−1, …, θT ) p
T t=1 ∂θ ∂θ⊤ • Note: MLE is GMM on the score. We find θ so that
1 T ∂ log p(xt|xt−1, …, θ) = 0.
Chapter 2 Nonlinear econometrics for finance
This is nothing but a sample counterpart to
∂ log p(xt|xt−1, …, θ0)
E ∂θ =0, i.e., the expectation of the score is equal to zero!
In light of the last two bullet points, we could have derived all of the asymptotic properties of the maximum likelihood estimator by using the (more general) GMM formulae.
Testing asset pricing models (when the factors are returns)
Consider a one-factor model, for notational simplicity (the CAPM, for example). Write
Rt =α+βft+εt ft =μf+ut,
Now, we need conditions on the error terms. Write
ut 0 0 σu2
The likelihood becomes
L({v} , θ) =
= p(vT |θ)p(vT −1|θ)…p(v1|θ)
= p(εT |θ)p(εT −1|θ)…p(ε1|θ)p(uT |θ)p(uT −1|θ)…p(u1|θ) 7
ε 0σ20 vt= t ∼iidN , ε .
p(vT |vT −1, vT −2, …, θ)p(vT −1|vT −2, vT −3, …, θ)…p(v1, θ)
Chapter 2 Nonlinear econometrics for finance 1 1 ⊤ 2 −1
• Estimates:
= (2π)T/2 (σε2)T/2 exp −2ε (σεI) ε
1 1 ⊤ 2 −1
×(2π)T/2 (σu2)T/2 exp −2u (σuI) u . T T 1T
t=1 T T 1T
• The log-likelihood is:
l({v},θ) = −2 log(2π)− 2 logσε2 − 2σε2
(Rt −α−βft)2
−2 log(2π)− 2 logσu2 − 2σu2
or, better,
Tl({v},θ) = −2log(2π)−2logσε2 −2Tσε2
= σε2T (Rt−α−βft)=0 t=1
(Rt−α−βft)ft=0 ∂l 1 11T
∂σε2 = −2σε2+2(σ2)2T (Rt−α−βft)2=0
(Rt −α−βft)2
t=1 1 1 1T
−2log(2π)−2logσu2 −2Tσu2 ∂l 11T
ε t=1 ∂μ = 2Tσ2 (ft−μf)=0
∂l 2T f u t=1
∂σu2 = − 2 σ u2 + 2 ( σ 2 ) 2 T ( f t − μ f ) 2 = 0
Chapter 2 Nonlinear econometrics for finance • If we now solve this system of 5 equations in 5 unknowns, we find
(Rt − R)(ft − f)
t=1 μf = f,
α = R−βf,
(ft −f)2 t=1
σu = T (ft−μf). t=1
σ ε = ( R t − α − β f t ) , T
• Hence, θT = (β,α,σε,μf,σu).
• To do statistical inference, we now have to compute standard errors. What is the asymptotic variance-covariance matrix? We need to compute the second derivative of the likelihood function with respect to the parameters.
• Let’s see …
∂l =−1, ∂α∂α σε2
∂l ∂ β ∂ β
1T ∂2logp(x|x ,…,θ) − tt−1T
Nonlinear econometrics for finance 1 f 000
•Recall:if • Thus,
a b −1 c d
d −c −b a
.SeeChapter0.
−b a 1 E(f) −1
f T ft t=1
2 2 000 =σε σε
0 0 … … …
0 0 … … …
0 0 … … …
1 E(f) 000 σε2 σε2
……….
E(f) E(f2) 0 0 0 22
σεσε →p0 0
……… 0 0 … … …
• To characterize the variance matrix of α and β, this matrix needs to be inverted.
One can show that the resulting limiting matrix is block-diagonal. Hence, the first block of the inverted matrix is just the inverted first-block (this is not true in general).
E(f) −1 σε2
1 E(f)−1 E(f) E(f2)
1 E(f2) −E(f) 2=.
E(f)E(f) V(f)−E(f) 1
Nonlinear econometrics for finance Hence, V(α) = 1σ2E(f2) = 1σ2V(f)+(E(f))2 = 1σ21+(E(f))2 and V(β) =
T ε V(f) T ε V(f) T ε V(f)
If we have N assets, then,
A famous (to test asset pricing models) chi-squared test can be obtained:
1 σε2 . T V(f)
V ( α ) = 1 + Σ ε .
E(f)2−1
T 1 + α Σ ε α 0 → χ N V ( f ) H : α = 0
It is derived from the fact that
√ dE(f)2
T ( α − α ) → N 0 , 1 + If we have N assets and k factors, then
⊤−1 −1⊤−1 d 2 T 1+E(f) Σf E(f) α Σε α 0→ χN.
The classical OLS case with pre-determined re- gressors: maximum likelihood
with E(ε) = 0 and E(εε⊤) = σ2IN . Assume the errors are normal.
Chapter 2 Nonlinear econometrics for finance • The log-likelihood is
l=−Nlog2π−Nlogσ2− 1 (Y−Xβ)⊤(Y−Xβ). 2 2 2σ2
• The first moment conditions:
∂l = 1X⊤(Y−Xβ)=0,
∂l =−N+1(Y−Xβ)⊤(Y−Xβ)=0.
∂σ2 2σ2 2σ4
• The second derivatives:
∂l = −X⊤X ∂β∂β⊤ σ2
∂l = N −(Y−Xβ)⊤(Y−Xβ) ∂σ2∂σ2 2σ4 σ6
β = (X⊤X)−1(X⊤Y ),
σ= (Y−Xβ)(Y−Xβ).
= −1X⊤(Y−Xβ). σ4
• What about the Fisher’s “information matrix”?
−1 N − E((Y −Xβ)⊤(Y −Xβ))
2σ4 σ2(X⊤X)−1
Chapter 2 Nonlinear econometrics for finance
The ML estimator of σ2 is biased (right?) but consistent and with the smallest possible variance.
Time-varying conditional variance
We have shown that the conditional first moment of stock returns can change over time and is predictable.
Does the conditional second moment of stock returns change over time too? In other words, is Vt(rt+1) a stochastic process? Let’s see …
DIFFPERCENT2
80 82 84 86 88 90 92 94 96 98 00 02 04 06
Figure 1: Squared percent S&P 500 returns.
Is the conditional first moment of excess stock market returns a function of the market returns’ conditional second moment? Differently put, is there a compensa- tion for time-varying variance risk at the market level, i.e., a risk-return trade-off? We will see …
Chapter 2 Nonlinear econometrics for finance
Autoregressive Conditional Heteroskedasticity (ARCH) – Engle (1982)
Consider a time series of generic returns {rt}Tt=1 . Model the conditional mean. For example, write
rt = μ + φrt−1 + εt, but other predictors could be used too, of course.
We know that the unconditional mean and variance are: E(rt) = σε2 .
μ and V ar(rt) = 1−φ
The conditional mean Et−1(rt) = μ + φrt−1 is time-varying, whereas the uncondi-
tional mean is not.
We wish to make the conditional variance time-varying too! Notice that V art−1(rt) = Et−1[rt − Et−1(rt)]2 = Et−1[ε2t ]. Hence, the key is time-variation in the squared residuals. Assume, for example, that
ε 2t = μ ∗ + φ ∗ ε 2t − 1 + π t , where πt is another white noise process with E(πt2) = σπ2 .
In other words, just like rt, ε2t follows an autoregressive process. We have:
Et−1[ε2t ] = μ∗ + φ∗ε2t−1. 14
Chapter 2 Nonlinear econometrics for finance
• This model is called “Autoregressive Conditional Heteroskedasticity of order 1” or ARCH(1).
• We can of course generalize the ARCH(1) model and write: ε2t = μ∗ + φ∗1ε2t−1 + … + φ∗pε2t−p + πt.
• This is an ARCH model of order p: ARCH(p). Some conditions:
• Note that the variances cannot go negative.
• In the ARCH(1) case, we might wish to require the following:
2. πt to be bounded from below by −μ∗ 3. |φ∗| < 1 for stationarity
An alternative expression of the model.
• The previous derivations are very intuitive. There is a (possibly) less intuitive, but more commonplace, way to write the same thing.
εt = htut,
where ut is white noise with standard deviation equal to 1.
• Suppose that
h t = μ ∗ + φ ∗ ε 2t − 1 .
Chapter 2 Nonlinear econometrics for finance • What about V art−1(rt) = Et−1[rt − Et−1(rt)]2? This is simply
Et−1[ε2t] = = = =
as before.
• Again, we can generalize and write
Et−1[htu2t ]
Et−1[(μ∗ + φ∗ε2t−1)u2t ]. μ ∗ + φ ∗ ε 2t − 1
ht = μ∗ + φ∗1ε2t−1 + ... + φ∗pε2t−p.
• This is, once more, an ARCH model of order p: ARCH(p).
• This model is useful to capture persistence (and variance is persistent). However, it is not very parsimonious. There are too many parameters to estimate. The GARCH model provides a solution to this issue.
• In sum: ARCH(1):
εt = htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1
ht = μ∗+φ∗ε2t−1.
Chapter 2 Nonlinear econometrics for finance
εt = htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1
ht = μ∗ + φ∗1ε2t−1 + ... + φ∗pε2t−p.
Generalized Autoregressive Conditional Heteroskedas- ticity (GARCH) - Bollerslev (1986)
εt = htut,
Et−1[ε2t ]
δ∗ht−1 + φ∗ε2t−1.
Extra piece
Et−1[htu2t ]
μ∗ + δ∗ht−1 + φ∗ε2t−1
• Intuition: conditional variance is a weighted average of two components, i.e., past conditional variance (ht−1 depends on past information) and current innovations (i.e., current squared residuals). It is as if the previous expectation (ht−1) were updated using the current squared residuals (ε2t−1). Empirically, the weights δ∗ and φ∗ are generally positive. Empirically, their sum is usually close to one.
Chapter 2 Nonlinear econometrics for finance
• Why does the GARCH model capture persistence without requiring too many pa- rameters? Write:
ht = μ∗+δ∗ht−1+φ∗ε2t−1 ⇒
ht = μ∗ + δ∗(μ∗ + δ∗ht−2 + φ∗ε2t−2) + φ∗ε2t−1
= μ∗ + δ∗μ∗ + (δ∗)2ht−2 + δ∗φ∗ε2t−2 + φ∗ε2t−1.
We could, of course, plug in ht−2, then ht−3, and keep on going. What we would obtain is an expression for ht as an ARCH(∞) model with restrictions on the param- eters. In other words, the parameters associated with the terms ε2t−1,ε2t−2,...,ε2t−.. would just be functions of δ∗ and φ∗. Namely, they would be φ∗, δ∗φ∗, (δ∗)2φ∗, and so on.
• The model we presented is called GARCH(1,1). GARCH(1,1):
εt = htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1
ht = μ∗ + δ∗ht−1 + φ∗ε2t−1.
• As always, this model can be generalized. We could write:
ht = μ∗ + δ1∗ht−1 + ... + δp∗ht−p + φ∗1ε2t−1 + ... + φ∗qε2t−q.
• This is a GARCH(p,q) model.
Chapter 2 Nonlinear econometrics for finance
GARCH(p,q):
εt = htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1
ht = μ∗ + δ1∗ht−1 + ... + δp∗ht−p + φ∗1ε2t−1 + ... + φ∗qε2t−q.
• For most practical purposes in finance, a simple GARCH(1,1) model does extremely
well! Let’s see using the data “S&P500daily-level.xls.”
Chapter 2 Nonlinear econometrics for finance
GARCH(1,1) for S&P 500 market returns
Dependent Variable: DAILYRET
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps) Date: Time:
Sample (adjusted): 1/03/1980 12/29/2006
Included observations: 6814 after adjustments
Convergence achieved after 28 iterations
Coefficient covariance computed using outer product of gradients Presample variance: backcast (parameter = 0.7)
GARCH = C(1) + C(2)*RESID(-1)ˆ2 + C(3)*GARCH(-1)
Variable Coefficient Std. Error z-Statistic Variance Equation
0.0000 0.0000 0.0000
0.010344 −6.573424 −6.570419 −6.572387
C 1.01E − 06 RESID(-1)ˆ2 0.068566 GARCH(-1) 0.923857
R-squared −0.001357
1.28E − 07 7.912140 0.001473 46.55053 0.002842 325.0986
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter.
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
−0.001210 0.010351 0.730036 22398.66 1.957469
We notice that the weights δ∗ and φ∗ are
one but smaller than one (as required for a stationary conditional variance process). The coefficient associated with past conditional variance (i.e., δ∗) is particularly large, thereby demonstrating that conditional variance is highly persistent. Statistical significance is strong.
Adding lags is now trivial. Let us consider a GARCH(1,2) model, for instance.
indeed positive and their sum is very close to
Chapter 2 Nonlinear econometrics for finance
Dependent Variable: DAILYRET
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps)
Date: Time:
Sample (adjusted): 1/03/1980 12/29/2006
Included observations: 6814 after adjustments
Convergence achieved after 28 iterations
Coefficient covariance computed using outer product of gradients
Presample variance: backcast (parameter = 0.7)
GARCH = C(1) + C(2)*RESID(-1)ˆ2 + C(3)*RESID(-2)ˆ2 + C(4)*GARCH(-1)
Variable Coefficient Variance
C 7.54E − 07 RESID(-1)ˆ2 0.098520 RESID(-2)ˆ2 −0.044298
GARCH(-1) 0.939825 R-squared −0.001357
Std. Error Equation
1.11E − 07 0.003562 0.006186 0.003942
z-Statistic
6.802118 27.65606 −7.160494 238.3886
0.0000 0.0000 0.0000 0.0000
0.010344 −6.574587 −6.570580 −6.573204
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
−0.001210 0.010351 0.730036 22403.62 1.957469
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter.
The improvement in terms of fit is marginal. The estimated coefficient δ∗ is effectively unaltered and the estimated coefficients associated with the first and the second lagged squared residuals cancel each other out to deliver an overall estimate very similar to φ∗ from the previous regression.
7.1 Long-run variance
Consider a GARCH(1,1). Write
εt = htut, 21
Chapter 2 Nonlinear econometrics for finance where
ht = μ∗ + δ∗ht−1 + φ∗ε2t−1.
Hence, the unconditional variance of the returns can be written as:
E(ε2t ) = E(ht) = μ∗ + δ∗E(ht−1) + φ∗E(ε2t−1) ⇒ E(ht) = μ∗ + (δ∗ + φ∗)E(ht)
E(ht) = 1−(δ∗ +φ∗) = σ2
1. Conditional variance (from a GARCH(1,1) model) is time-varying around 2. We could write
μ∗ . 1−(δ∗+φ∗)
ht =σ2(1−δ∗ −φ∗)+δ∗ht−1 +φ∗ε2t−1
Thus, conditional variance is really an average of three objects: (1) long-run variance,
(2) squared return innovations, and (3) past conditional variance.
GARCH(1,1): a more explicit representation.
3. Note that
is equivalent to
htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1 σ2(1 − δ∗ − φ∗) + δ∗ht−1 + φ∗ε2t−1.
ht = μ∗ + δ∗ht−1 + φ∗ε2t−1
Chapter 2 Nonlinear econometrics for finance
ε2t + ht = μ∗ + δ∗ht−1 + φ∗ε2t−1 + ε2t + δ∗ε2t−1 − δ∗ε2t−1
ε2t =μ∗ +(φ∗ +δ∗)ε2t−1 +(ε2t −ht)−δ∗(ε2t−1 −ht−1)
which is an ARMA(1,1) structure for the squared returns. The squared returns mean-
revert to μ∗ (like h ) and have an autoregressive parameter equal to φ∗ + δ∗ < 1. 1−(δ∗+φ∗) t
Alternative specifications
The case δ∗ +φ∗ = 1 is a subcase of the GARCH(1,1) model. It is called IGARCH (or integrated GARCH). In this case, conditional volatility is highly persistent and evolves (almost) like a random walk.
εt = htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1
ht = (1 − φ∗)ht−1 + φ∗ε2t−1.
Back to an IGARCH model for the S&P 500 index
Let us impose the restriction δ∗ + φ∗ = 1 (and set μ∗ equal to zero).
Chapter 2 Nonlinear econometrics for finance
Dependent Variable: DAILYRET
Method: ML ARCH - Normal distribution (BFGS / Marquardt steps) Date: Time:
Sample (adjusted): 1/03/1980 12/29/2006
Included observations: 6814 after adjustments
Convergence achieved after 23 iterations
Coefficient covariance computed using outer product of gradients Presample variance: backcast (parameter = 0.7)
GARCH = C(1)*RESID(-1)ˆ2 + (1 - C(1))*GARCH(-1)
RESID(-1)ˆ2 GARCH(-1)
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood Durbin-Watson stat
Coefficient Std. Error Variance Equation
z-Statistic
60.81907 1159.114
0.0000 0.0000
0.010344 −6.561442 −6.560440 −6.561097
0.049854 0.950146
−0.001357 −0.001210 0.010351 0.730036 22355.83 1.957469
0.000820 0.000820
Mean dependent var S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter.
• Threshold GARCH or TGARCH. Consider
ht = μ∗ + δ∗ht−1 + φ∗ε2t−1 + ηε2t−11(εt−1<0).
This specification is useful to test whether negative innovations to stock returns (i.e., εt−
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