CS代写 GARCH01 in EViews). It is visually clear that the conditional variance foll

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University of Wales
School of Economics

Financial Econometrics
Tutorial 6

1. (ARCH model characteristics)

(a) The specification 𝜀”|Ω”%& ∼ 𝑁(0, 𝜎”-) implies that the conditional mean of 𝜀” is 0. The

conditional mean of 𝑦” is 𝑐 + 𝜙&𝑦”%&. The unconditional mean of 𝜀” is 0, by iterated expectations.

The unconditional mean of 𝑦” is also obtained by Rule 5: 𝐸(𝑦”) = 𝑐 + 𝜙&𝐸(𝑦”%&) and stationarity

𝐸(𝑦”) = 𝐸(𝑦”%&) = 𝑐/(1 − 𝜙&).

(b) The specification 𝜀”|Ω”%& ∼ 𝑁(0, 𝜎”-) implies that the conditional variance of 𝜀” is 𝜎”-. The

conditional variance of 𝑦” is the same as that of 𝜀”, 𝜎”-, because the conditional mean 𝑐 + 𝜙&𝑦”%& of

𝑦” is fixed for given information set Ω”%&. As the conditional mean of 𝜀” is zero, the unconditional

variance of 𝜀” is obtained by 𝐸(𝜀”-) = 𝐸{𝐸(𝜀”-|Ω”%&)} = 𝐸(𝜎”-) = 𝛼; + 𝛼&𝐸(𝜀”%&
– ) + 𝛼-𝐸(𝜀”%-

stantionarity 𝐸(𝜎”-) = 𝐸(𝜀”%&
– ) = 𝐸(𝜀”%-

– ) = 𝛼;/(1 − 𝛼& − 𝛼-).

For the unconditional variance of 𝑦”, we find Var(𝑦”) = 𝜙&
-Var(𝑦”%&) + Var(𝜀”) because 𝜀”

is uncorrelated with 𝑦”%&. Then, by stationarity, Var(𝑦”) = Var(𝑦”%&) = Var(𝜀”)/(1 − 𝜙&
-). Finally,

Var(𝜀”) = 𝐸(𝜎”-) = 𝛼; + 𝛼&𝐸(𝜀”%&
– ) + 𝛼-𝐸(𝜀”%-

– ) = 𝛼;/(1 − 𝛼& − 𝛼-) as stationarity implies

Var(𝜀”) = 𝐸(𝜀”%&
– ) = 𝐸(𝜀”%-

(a) and (b)) and autocovariances are zero for any 𝑗 > 0: (iterated expectations)

𝛾B = CovF𝜀”, 𝜀”%BG = 𝐸F𝜀”𝜀”%BG = 𝐸H𝐸F𝜀”𝜀”%BIΩ”%&GJ = 𝐸H𝐸(𝜀”|Ω”%&)𝜀”%BJ = 𝐸{0} = 0.

However, 𝜀” is NOT an independent WN process because the conditional variance of

𝜀” is a function of 𝜀”%& and 𝜀”%-, by definition.

(d) The ARCH model differ from the standard homoscedastic model in that the conditional
variance of the shock (or error term) is a function of lagged shocks whereas the conditional variance

of the shock in any standard ARMA model is a constant. The variance equation in the ARCH model is

designed to capture “clustering” or the dependence structure in squared shocks.

(e) It is easily seen from the variance equation: 𝜕𝜎”-/𝜕𝜀”%&

(f) Now the variance equation is simplified to 𝜎”- = 𝛼; + 𝛼&𝜀”%&
– . First, we find 𝐸(𝜀”L) =

𝐸{𝐸(𝜀”L|Ω”%&)} = 𝐸{3𝜎”L} = 3𝐸{𝛼;
– + 2𝛼;𝛼&𝜀”%&

L }. Because 𝐸(𝜀”%&
– ) = 𝛼;/(1 − 𝛼&) by

part (b) and 𝐸(𝜀”L) = 𝐸(𝜀”%&
L ) by stationarity, it then follows that

𝐸(𝜀”L) = 3{𝛼;
– + 2𝛼;𝛼&[𝛼;/(1 − 𝛼&)]}/(1 − 3𝛼&

-(1 + 𝛼&)/[(1 − 𝛼&)(1 − 3𝛼&

Finally, we find the unconditional kurtosis,

Kurtosis =
QF[RS%Q(RS)]TG

and conclude that the unconditional distribution of 𝜀” is non-normal with heavy tails (kurtosis > 3),

noting that its conditional distribution is normal.

2. (GARCH model characteristics)

(a-b) From 𝜀”|Ω”%& ∼ 𝑁(0, 𝜎”-), it is clear that 𝐸(𝜀”|Ω”%&) = 0 and Var(𝜀”|Ω”%&) = 𝜎”-. It

follows that 𝐸(𝑦”|Ω”%&) = 𝑐 + 𝜙&𝑦”%& and Var(𝑦”|Ω”%&) = 𝜎”-. The unconditional means are

obtained by iterated expectations:

𝐸(𝜀”) = 𝐸{𝐸(𝜀”|Ω”%&)} = 𝐸{0} = 0,

𝐸(𝑦”) = 𝐸{𝐸(𝑦”|Ω”%&)} = 𝑐 + 𝜙&𝐸(𝑦”%&) and

𝐸(𝑦”) = 𝐸(𝑦”%&) = 𝑐/(1 − 𝜙&).

Because the (conditional) mean of 𝜀” is zero, we find

Var(𝜀”) = 𝐸{Var(𝜀”|Ω”%&)} = 𝐸{𝜎”-} = 𝛼; + 𝛼&𝐸(𝜀”%&- ) + 𝛽&𝐸(𝜎”%&- )

and, by stationarity,

Var(𝜀”) = 𝐸{𝜎”-} = 𝐸(𝜀”%&- ) = 𝐸(𝜎”%&- ) = 𝛼;/(1 − 𝛼& − 𝛽&).

Further Var(𝑦”) = 𝜙&-Var(𝑦”%&) + Var(𝜀”) because 𝜀” is uncorrelated with 𝑦”%&. Again, by

stationarity, we find Var(𝑦”) = Var(𝑦”%&) = Var(𝜀”)/(1 − 𝜙&-).

(a) and (b)) and autocovariances are zero for any 𝑗 > 0:

𝛾B = CovF𝜀”, 𝜀”%BG = 𝐸F𝜀”𝜀”%BG = 𝐸H𝐸F𝜀”𝜀”%BIΩ”%&GJ = 𝐸H𝐸(𝜀”|Ω”%&)𝜀”%BJ = 𝐸{0} = 0.

However, 𝜀” is NOT an independent WN process because the conditional variance of 𝜀” is a

function of 𝜀”%&, by definition.

(d) It is easily seen from the variance equation: 𝜕𝜎”-/𝜕𝜀”%&- = 𝛼&. Note that 𝜎”%&- in the

variance equation is a function of Ω”%-.

(e-i) We show that 𝑤” = 𝜀”- − 𝜎”- has a zero mean and zero autocorrelations. First, because

𝐸(𝜀”|Ω”%&) = 0, 𝐸(𝑤”|Ω”%&) = 𝐸(𝜀”-|Ω”%&) − 𝜎”- = 𝜎”- − 𝜎”- = 0, implying 𝐸(𝑤”) = 0.

Second, CovF𝑤”, 𝑤”%BG = 𝐸F𝑤”𝑤”%BG = 𝐸H𝐸(𝑤”|Ω”%&)𝑤”%BJ = 𝐸{0} = 0, for all 𝑗 ≥ 1.

(e-ii) It only involves some substitutions:

𝜀”- = 𝜎”- + 𝑤” = 𝛼; + 𝛼&𝜀”%&- + 𝛽&𝜎”%&- + 𝑤”

= 𝛼; + 𝛼&𝜀”%&- + 𝛽&(𝜀”%&- − 𝑤”%&) + 𝑤”

= 𝛼; + (𝛼& + 𝛽&)𝜀”%&- + 𝑤” − 𝛽&𝑤”%&,

i.e., an ARMA(1,1) for 𝜀”-.

(f) When 𝛼& + 𝛽& = 1, the variance equation

𝜎”- = 𝜔(1 − 𝛼& − 𝛽&) + 𝛼&𝜀”%&- + 𝛽&𝜎”%&- = (1 − 𝛽&)𝜀”%&- + 𝛽&𝜎”%&-

becomes an EWMA of 𝜀”-.

COMPUTING EXERCISES

3. (Estimation of ARCH)

(a) The time series plot, histogram and correlogram of the return series are given below. The

correlogram shows little autocorrelation in the return (large p-values for Ljung-Box Q-statistics). The

data distribution in the histogram is roughly “bell shaped” but has a negative skewness (-0.147) and

large kurtosis (5.016). The normality is rejected (JB = 171.74 with a tiny p-value).

100 200 300 400 500 600 700 800 900

-6 -4 -2 0 2 4

Sample 1 994
Observations 993

Mean 0.015735
Median 0.007796
Maximum 4.964596
Minimum -7.043759
Std. Dev. 1.301875
Skewness -0.146823
Kurtosis 5.016094

Jarque-Bera 171.7420
Probability 0.000000

(b) For the squared return series, the clustering is prominent in the time series plot and the

correlogram reveals strong autocorrelations (tiny p-values for the Q-statistics).

We note that the auxiliary equation in the test have large coefficients at lag 1,2 and 5. We conclude

that the ARCH effect should be accounted for in the model for the return series.

100 200 300 400 500 600 700 800 900

(d) From the ARCH(5) estimation results, 𝛼b; (1.029) is significantly positive. The point estimates

of (𝛼&, … , 𝛼d) are all non-negative. The point estimate of 𝛼& +⋯+ 𝛼d is positive and less than one.

Hence the restrictions on the ARCH parameters are satisfied. We plot the return over the plot of 𝜎”-

(called GARCH01 in EViews). It is visually clear that the conditional variance follows the variations

in the return and the clustering closely. The LM test for ARCH effect cannot reject the null of no

ARCH effect in the standardised residual series (large p-value).

100 200 300 400 500 600 700 800 900

(e) The histograms indicate that the distributions of both the residuals (E) and the standardised

residuals (V) are non-normal (large JB statistics and tiny p-values). The correlograms show that the

squared residuals (E2) have strong autocorrelations but the squared standardised residuals (V2) have

little. This is consistent with the LM test results in parts (c) and (d). The ARCH(5) model does a good

job in accounting for the autocorrelations in the squared residuals.

(f) We may use AIC or SIC to choose the number of lags in ARCH models. For ARCH(q), the

AIC and SIC for q = 1,2,…,9 are displayed in the bar chart below, where AICs are in group 1 on the

left and SICs are in group 2 on the right. Clearly, AIC selects q = 8 but SIC selects q = 5.

-5.00 -3.75 -2.50 -1.25 0.00 1.25 2.50 3.75

Sample 1 994
Observations 993

Mean -0.036835
Median -0.035182
Maximum 3.546326
Minimum -4.940958
Std. Dev. 0.999824
Skewness -0.253526
Kurtosis 4.338229

Jarque-Bera 84.73425
Probability 0.000000

-6 -4 -2 0 2 4

Sample 1 994
Observations 993

Mean -0.033842
Median -0.041781
Maximum 4.915019
Minimum -7.093336
Std. Dev. 1.301875
Skewness -0.146823
Kurtosis 5.016094

Jarque-Bera 171.7420
Probability 0.000000

Based on SIC and the principle of parsimony, the ARCH(5) model stays for the variance equation.

For the mean equation, as the returns have little autocorrelation, a more sophisticated ARMA

specification is unlikely to improve the model’s fit.

4. (Estimation of GARCH)

(a) The estimation results at Part (e) below indicate that all restriction on the parameters

are satisfied: 𝛼; > 0, 𝛼& ≥ 0, 𝛽& ≥ 0 and 𝛼& + 𝛽& < 1. We also note that the 𝛽& estimate is about 0.9, the 𝛼& estimate is about 0.1 and the 𝛼& + 𝛽& estimate is very close to 1. (b) The plot below contain the absolute return series (RA) and 𝜎" series from ARCH(5) and GARCH(1,1) respectively. Both 𝜎" series match the variations in the return. However the two 𝜎" series differ markedly in a number of places. The GARCH 𝜎" appears smoother than the ARCH 𝜎". (c) The LM test for ARCH effect (see below) cannot reject that the null hypothesis that there is no ARCH effect in the standardised residuals, as the p-value is quite large (0.17). The correlogram of the squared standardised residuals also indicate that there are no 100 200 300 400 500 600 700 800 900 RA ARCH5SGM GARCH11SGM 100 200 300 400 500 600 700 800 900 ARCH5SGM-GARCH11SGM © Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this autocorrelations. Hence the GARCH variance equation has adequately captured the clustering in the error term 𝜀". (d) The histogram of the standardised residual show that the null hypothesis of normality is strongly rejected (virtually zero p-value). Nonetheless, the kurtosis is much smaller than the kurtosis in the return series, which confirms that the variance equation can partially explain the excess kurtosis in the return series. (e) The estimation results with or without the quasi ML robust standard errors are given below. They are different and may lead to different conclusions. Generally, the robust standard errors in the variance equation are larger. -3.75 -2.50 -1.25 0.00 1.25 2.50 Series: Standardized Residuals Sample 2 994 Observations 993 Mean -0.037499 Median -0.029158 Maximum 3.045820 Minimum -4.590708 Std. Dev. 1.000489 Skewness -0.367698 Kurtosis 4.339797 Jarque-Bera 96.64633 Probability 0.000000 © Copyright University of Wales 2020. All rights reserved. This copyright notice must not be removed from this (f) The GARCH models can be easily estimated to obtain AIC and SIC in EViews. The AIC and SIC for GARCH(1,1), GARCH(2,1), GARCH(1,2) are GARCH(2,2) are presented in the bar chart below (AIC on the left and SIC on the right). Clearly, both AIC and SIC select GARCH(1,1). GARCH(1,1) GARCH(2,1) GARCH(1,2) GARCH(2,2) 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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