garchreferencestime seriesvariancevolatility-forecasting
Lets say I have a GARCH(1,1) model,
First, I model the conditional MEAN,
$$Y_t=\delta+\beta Y_{t-1}+\varepsilon_t$$
NextI gather the residuals $\varepsilon_t$ and model the conditional variance,
$$h_t=\omega + \alpha_i\varepsilon_{t-1}^2+\beta_ih_{t-1}$$
I need to get the standard residuals,
my attempt at this is,
$$U_t=\varepsilon_t/\sqrt{h_{t}}$$
Is this correct?
I am struggling to find this in any textbooks or papers so a reference would be appreciated if possible!
I found an answer in the "vignette" to the "rugarch" package in R. Here is a quote from pages 7-8 (emphasis is mine):
Because of the presence of the indicator function, the persistence of the model now crucially depends on the asymmetry of the conditional distribution used. The persistence of the model $\hat P$ is,
$$ \hat P = \sum_{j=1}^q \alpha_j + \sum_{j=1}^p \beta_j + \sum_{j=1}^q \gamma_j\kappa $$
where $\kappa$ is the expected value of the standardized residuals $z_t$ below zero (effectively the probability of being below zero),
$$ \kappa = \mathbb{E}(\mathbb{I}_{t-j} z_{t-j}^2) = \int_{-\infty}^0 f(z,1,0,\dotsc) dz $$
where $f$ is the standardized conditional density with any additional skew and shape parameters $(\dotsc)$. In the case of symmetric distributions the value of $\kappa$ is simply equal to 0.5.
This is best done through simulation. See my MATLAB code example and explanation below:
%% Get S&P 500 price series
d=fetch(yahoo,'^GSPC','Adj Close','1-jan-2014','30-dec-2014');
n = 1; % # of shares
p = d(end:-1:1,2); % share price, the dates are backwards
PV0 = n*p(end); % portfolio value today
%%
r=price2ret(p,[],'Continuous'); % get the continous compounding returns
Mdl = arima('ARLags',1,'Variance',garch(1,1),'Constant',0);
fit = estimate(Mdl,r) % fit the AR(1)-GARCH(1,1)
fit.Variance
[E0,V0] = infer(fit,r); % get the estimated errors and variances
%% Simulate periods ahead
[Y] = simulate(fit,2,'Y0',r,'E0',E0,'V0',V0, 'NumPaths',1e5);
ret = sum(Y); % compund the return
histfit(PV0*(exp(ret)-1),100,'normal')
title 'P&L distribution'
ret2d = prctile(ret,0.01); % get the 99% lowest return
VaR = PV0*(exp(ret2d)-1);
fprintf('Today portfolio value: %f\n2-days ahead 99%% VaR: %f\n',PV0,VaR);
Output:
ARIMA(1,0,0) Model:
--------------------
Conditional Probability Distribution: Gaussian
Standard t
Parameter Value Error Statistic
----------- ----------- ------------ -----------
Constant 0 Fixed Fixed
AR{1} -0.020272 0.0697731 -0.290541
GARCH(1,1) Conditional Variance Model:
----------------------------------------
Conditional Probability Distribution: Gaussian
Standard t
Parameter Value Error Statistic
----------- ----------- ------------ -----------
Constant 7.43521e-06 2.14546e-06 3.46556
GARCH{1} 0.653488 0.126818 5.15295
ARCH{1} 0.206016 0.0881823 2.33626
fit =
ARIMA(1,0,0) Model:
--------------------
Distribution: Name = 'Gaussian'
P: 1
D: 0
Q: 0
Constant: 0
AR: {-0.020272} at Lags [1]
SAR: {}
MA: {}
SMA: {}
Variance: [GARCH(1,1) Model]
ans =
GARCH(1,1) Conditional Variance Model:
--------------------------------------
Distribution: Name = 'Gaussian'
P: 1
Q: 1
Constant: 7.43521e-06
GARCH: {0.653488} at Lags [1]
ARCH: {0.206016} at Lags [1]
Today portfolio value: 2090.570000
2-days ahead 99% VaR: -79.986280
For a portfolio consisting of one share of S&P 500 index, I got the current value of \$2091, and 2-day VaR at 99% is \$80.
I also plotted the P&L distribution with Normal fit, so you can see that the normal distribution is not a very good fit. That's why you have to simulate when using GARCH.
You can also look at MATLAB's own GARCH example here.
The idea is that you fit AR(1)-GARCH(1,1) to the returns. Then you Monte-Carlo simulate the returns two days ahead. Then you compound two days of returns in each path, and find 0.01 percentile. This is how much or more you'll lose at 99% confidence in terms of returns, and in terms of dollars it's a simple arithmetic using the current portfolio value.
You can find worked out examples in this book: Carol Alexander, Market Risk Analysis, Volume IV, Value at Risk Models, February 2009
Best Answer
This is correct.
References: