CS考试辅导 Nonlinear Econometrics for Finance Lecture 7

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+φε􏰀T. Using the estimation results for the baseline model in the code (with S&P 500 data), we would have 2 􏰀hT +1 = 0.00000101 + 0.9238􏰀hT + 0.0686ε􏰀T . 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 =ν􏰁1+. 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, θ􏰀T = 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) 􏰓  N0,−E    ∂θ∂θ⊤    􏰐 􏰏􏰎 􏰑 N(0,Ω−1) 0   􏰒 ∂ log p(xt|xt−1, ..., θ0) ∂ log p(xt|xt−1, ..., θ0) 􏰓  N0,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⊤ 1􏰋T ∂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 程序代写 CS代考 加微信: powcoder QQ: 1823890830 Email: powcoder@163.com