Nonlinear Econometrics for Finance Lecture 7
Nonlinear Econometrics for Finance Lecture 7 1 / 22
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Review: ARCH and GARCH
Review: ARCH(1) and ARCH(p)
εt = htut, with Et−1(ut) = 0 and ht = μ∗+φ∗ε2t−1.
Et−1(u2t ) = 1
htut, with Et−1(ut) = 0 and μ∗ + φ∗1ε2t−1 + … + φ∗pε2t−p.
Et−1(u2t ) = 1
Nonlinear Econometrics for Finance Lecture 7
Review: ARCH and GARCH
Review: GARCH(1,1) and GARCH(p,q)
GARCH(1,1)
εt = htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1 ht = μ∗ + δ∗ht−1 + φ∗ε2t−1.
GARCH(p,q)
htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1 μ∗ + δ1∗ht−1 + … + δp∗ht−p + φ∗1ε2t−1 + … + φ∗qε2t−q.
Nonlinear Econometrics for Finance Lecture 7
Review: ARCH and GARCH
Review: TGARCH(1,1) and GARCH(1,1)-M
TGARCH(1,1):
εt = htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1 ht = μ∗ + δ∗ht−1 + φ∗ε2t−1 + η∗ε2t−11(εt−1<0).
GARCH(1,1)-in-mean:
rM,t−rf = βht+εt,
htut, with Et−1(ut) = 0 and Et−1(u2t ) = 1 μ∗ + δ∗ht−1 + φ∗ε2t−1.
Nonlinear Econometrics for Finance Lecture 7
Applications
Applications to risk management Value at Risk (VaR)
Nonlinear Econometrics for Finance Lecture 7 5 / 22
Applications
Why is time-varying conditional volatility important?
It is a key input in fundamental finance problems:
1 Option pricing
2 Portfolio allocation
3 Risk management
Let us see a Value-at-Risk (VaR) calculation. More on this for those
taking Financial Risk Management ...
Nonlinear Econometrics for Finance Lecture 7 6 / 22
Applications
Value-at-Risk I
The real risk of a portfolio is the risk that its value will decline, not that its value will increase. Hence, downside risk is natural.
The 1% VaR is the value such that there is a probability 99% of making more than that value (or, in other words, the value such that there is a probability 1% of losing more than that value).
Suppose you invest $1, 000, 000. The 1% VaR is: $1, 000, 000 × |x|, where p(r < x) = 1%.
Nonlinear Econometrics for Finance Lecture 7 7 / 22
Applications
Value-at-Risk II
Simple idea, but how do we calculate it?
One possible answer: we look at the empirical distribution of returns and identify
the first percentile. Suppose it is -0.0154. Hence, the VaR is
$1, 000, 000 × | − 0.0154| = $15, 400.
Common practice: Do not look at the empirical distribution of returns. Assume a
Normal distribution, for example. Given the model rt = the 1% VaR (for a $1 investment) is
htut, where ut ∼ N(0,1),
What to choose for
out-of-sample (at time T + 1) using a GARCH-type model. For example, with a GARCH(1,1) model, your forecast for variance would be
∗ˆ∗ˆ∗2 hT+1=μˆ +δhT+φεT.
Using the estimation results for the baseline model in the code (with S&P 500 data), we would have
2 hT +1 = 0.00000101 + 0.9238hT + 0.0686εT .
Suppose now that given hT and ε , hT+1 turns out to be 0.0056.
Then, the 1% VaR is $1, 000, 000 × 2.33 × 0.0056 = $13, 072.
ht? Simply use past data (up to T) to forecast variance
Nonlinear Econometrics for Finance Lecture 7 8 / 22
Non-normal errors
GARCH with non-normal errors
Nonlinear Econometrics for Finance Lecture 7 9 / 22
Non-normal errors
GARCH models: what if the uts are not normal? A typical example: GARCH(p,q) model with tν-distributed errors
Recall: The t-student distribution with ν degrees of freedom has fatter tails than the normal distribution. However, when ν → ∞, the t-student distribution converges to the (standard) normal.
GARCH(p,q) with tν-distributed errors
ν − 2 2 ν
εt = ν htut, with Et−1(ut)=0 and Et−1(ut)= ν−2 ht = μ∗ + δ1∗ht−1 + ... + δp∗ht−p + φ∗1ε2t−1 + ... + φ∗qε2t−q.
Why are we multiplying by ν−2? Because Vt−1[ut] = ν and we want ν ν−2
the time-varying conditional variance of εt to be ht, like in other models. In fact:
ν−2 ν−2 ν−2 ν
Vt−1[εt] = Vt−1 htut = htVt−1[ut] = ht = ht.
Nonlinear Econometrics for Finance Lecture 7 10 / 22
Non-normal errors
GARCH models: what if the uts are not normal? A typical example: GARCH(p,q) model with tν-distributed errors
The conditional probability density function of εt is:
Γ(2) ε2t 2
p(εt|εt−1)= ν 1+ Γ(2) π(ν−2)ht
where Γ(.) is the “gamma” function. The likelihood function is:
L({ε}, θ) = p(εT |εT −1, ..)p(εT −1|εT −2, ...)p(ε1)
T ν+1 −ν+1
Γ(2) ε2t 2 =ν1+.
t=1 Γ(2) The log-likelihood function is:
ν+1 T Γ(2)
ν+1 ε2t
l({ε},θ)=Tlog ν − log π(ν−2)ht−
Γ( 2 ) t=1 t=1 2
Thus, θT = maxθ T1 l({ε}, θ), where θ includes ν.
Nonlinear Econometrics for Finance Lecture 7
Doing MLE with wrong model Introduction to Quasi-MLE
Nonlinear Econometrics for Finance Lecture 7 12 / 22
Quasi-Maximum Likelihood Estimation (QMLE)
What happens if we use a Gaussian likelihood even though the conditional probabilities are not Gaussian (for example, what if they are tν -distributed, as we assumed earlier)?
The model: (Note: E∗ denotes expectation with respect to the true - in general, non-Gaussian - probabilities.)
rt = mt(θ0,m) + ht(θ0,h)ut with E∗t−1(ut) = 0 and E∗t−1(u2t ) = 1 The conditional (on time t − 1 information) mean and variance are:
E∗t−1 [rt ] mt (θ0,m ) V∗t−1 [rt ] = ht (θ0,h )
The log-likelihood (divided by T, as always) becomes:
1 1 T log l({r}, θ) =
1 −(rt−mt(θ0,m))2
e 2ht(θ0,h)
1 1 1 (rt −mt(θ0,m))2
−2 log(2π) − T t=1 2 log(ht(θ0,h)) − T t=1 2ht(θ0,h) .
2πht(θ0,h)
Nonlinear Econometrics for Finance Lecture 7 13 / 22
Why does it work? Because E∗t−1[scoret] = 0.
scoret = =
∂ log p(rt |rt−1 , ..., θ0 ) ∂θ
∂ log p(rt |rt−1 ,...,θ0 ) ∂θm
∂ log p(rt|rt−1,...,θ0) ∂θh
+ (rt−mt(θ0,m))2 ∂ht(θ0,h)
2h2t (θ0,h) ∂θh
(rt−mt(θ0,m) ∂mt(θ0,m) ht (θ0,h ) ∂ θm
− 1 2ht(θ0,h)
Now, because
1 E∗t−1[rt − mt(θ0,m)] = 0,
2 E∗t−1[(rt − mt(θ0,m))2] = ht,
we have E∗t−1[scoret] = 0.
a MLE is, effectively, GMM with the score as a moment condition. It is
standard to say: “MLE is GMM on the score.”
b Because E∗t−1[scoret] = 0 even in QMLE, QMLE delivers consistent and
asymptotically normal estimates (even with the wrong likelihood).
Nonlinear Econometrics for Finance Lecture 7 14 / 22
Inference in MLE: a summary
MLE: recap
We have, since B0 = −Ω0,
TθT−θ0 → =
Ω0 Note: −B0 is often called “Fisher’s information” matrix.
N(0,B Ω0B ) 00
N(0,−B−1) 0
−1 ∂2 log p(xt|xt−1, ..., θ0)
N0,−E ∂θ∂θ⊤
N(0,Ω−1) 0
∂ log p(xt|xt−1, ..., θ0) ∂ log p(xt|xt−1, ..., θ0) N0,E .
∂θ ∂θ⊤
Nonlinear Econometrics for Finance Lecture 7 15 / 22
Inference in MLE: a summary
MLE: Recap
We need to estimate either Ω0 or B0:
V(θ ) = 1 Ω −1
1 1 T ∂ log p(xt|xt−1, ..., θT ) ∂ log p(xt|xt−1, ..., θT )−1 T Tt=1 ∂θ ∂θ⊤
V(θ ) = 1 −B −1
1 1 ∂2 log p(x |x , ..., θ ) =− tt−1T.
T T t=1 ∂θ∂θ⊤
Nonlinear Econometrics for Finance Lecture 7
Inference in QMLE
QMLE: the asymptotic variance does not simplify
Remember that
∂ ∂ log p(xt|xt−1, ..., θ0) p∗(xt|xt−1, ..., θ0)dxt ∂θ⊤ ∂θ
∗ ∂ log p(xt|xt−1, ..., θ0) ∗
Et−1 [scoret ] =
Taking derivatives of both sides with respect to θ:
p (xt |xt−1 , ..., θ0 )dxt = 0.
∂2 log p(xt|xt−1, ..., θ0) ∗
⊤ p (xt |xt−1 , ..., θ0 )dxt ∂ log p(xt|xt−1, ..., θ0) ∂p∗(xt|xt−1, ..., θ0)
∂θ ∂θ⊤ dxt
∂2 log p(xt|xt−1, ..., θ0) ∗
p (xt |xt−1 , ..., θ0 )dxt
∂ log p(xt|xt−1, ..., θ0)
∂ log p∗(xt|xt−1, ..., θ0) ∗ ∂θ⊤ p
(xt|xt−1, ..., θ0) dxt.
But, because p(xt|...) is Gaussian and p∗(xt|...) is the true probability, this is not implying ∂2logp(x |x ,...,θ ) ∂logp(x |x ,...,θ ) ∂logp(x |x ,...,θ )
=(a), by the properties of the logarithm
−B∗=−E∗ t t−1 0 =E∗ t t−1 0 t t−1 0 =Ω∗. 0 ⊤ ∂θ ⊤ 0
Nonlinear Econometrics for Finance Lecture 7
Inference in QMLE
QMLE inference
We have, since B0∗ ̸= −Ω∗0,
√d ∗−1∗∗−1
T θT−θ0 →N(0,(B0) Ω0(B0) ). Thus, we need to estimate both Ω∗0 and B0∗. As always:
Therefore,
∗ 1 T ∂ log p(xt|xt−1, ..., θT ) ∂ log p(xt|xt−1, ..., θT ) Ω = ,
0Tt=1 ∂θ ∂θ⊤
1T ∂2logp(x|x ,...,θ) B∗=− tt−1 T.
0 T t=1 ∂θ∂θ⊤
1 ∗−1 ∗ ∗−1 T
V(θT ) = B0
When doing QMLE, the full “sandwich” form has to be estimated.
Nonlinear Econometrics for Finance Lecture 7 18 / 22
Robustness vs. efficiency
Robustness vs. efficiency
Nonlinear Econometrics for Finance Lecture 7 19 / 22
Robustness vs. efficiency
QMLE vs. MLE
An important result: Cramer-Rao lower bound. Consider a probability density p(x,θ0) (satisfying certain regularity conditions) for which θ0 is a
single parameter. Let θT be any unbiased estimator of θ0 based on IID data
x1, ..., xT
distributed as p(x, θ0). Then,
1 ∂2 logp(x,θ ) −E 0 .
T ∂2θ
1 This result has been extended to dependent data, multiple parameters and so on.
2 The MLE estimates are consistent (i.e., asymptotically unbiased) and achieve the Cramer-Rao lower bound. In other words, there is no consistent estimator with a lower variance than the MLE estimator.
3 QMLE is more robust than MLE: it does not require correct specification of the likelihood but only correct specification of the first and second conditional moments.
4 QMLE is less efficient than MLE: larger asymptotic variance.
Nonlinear Econometrics for Finance Lecture 7 20 / 22
Robustness vs. efficiency
GMM vs. MLE
Similar comments apply to the comparison between GMM and MLE. (After all, the justification for QMLE is GMM.)
1 GMM is more robust than MLE: it does not require specification of the likelihood, only correct specification of the conditional moments.
2 GMM is less efficient than MLE: larger asymptotic variance.
Nonlinear Econometrics for Finance Lecture 7 21 / 22
Robustness vs. efficiency
Robustness vs. Efficiency
There is an important trade-off in econometrics between robustness and efficiency.
Think about the relation between QMLE and MLE, as an example.
1 QMLE is more robust: specifying two conditional moments imposes less structure on the problem than specifying the entire conditional distribution.
2 Because it is more robust, QMLE is less efficient: the resulting estimates have larger standard errors.
Conclude: “There are no free lunches:” The price of robustness is some loss in efficiency.
The same logic applies to the comparison between GMM and MLE.
Nonlinear Econometrics for Finance Lecture 7 22 / 22
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