2-day ahead prediction of value at risk with GARCH(1,1) in R

Question

Let's say I have a 10 year dataset of Tesla (example) and I am taking the percentage change of lag 2:

tsla <- quantmod::getSymbols("TSLA", from = base::as.Date("2011-01-01"), to = base::as.Date("2022-01-31"), auto.assign = F)
tsla = as_tibble(tsla)
head(tsla)
d = tsla%>%
  dplyr::select(TSLA.Adjusted)%>%
  dplyr::mutate(Close = TSLA.Adjusted)%>%
  dplyr::mutate(y = as.numeric((Close - dplyr::lag(Close, 2)) / Close))%>%
  dplyr::select(Close,y)%>%
  tidyr::drop_na();d

That look like this:

# A tibble: 2,786 × 2
   Close        y
   <dbl>    <dbl>
 1  5.37  0.00783
 2  5.58  0.0434 
 3  5.65  0.0499 
 4  5.69  0.0200 
 5  5.39 -0.0475 
 6  5.39 -0.0553 
 7  5.24 -0.0282 
 8  5.15 -0.0470 
 9  5.13 -0.0226 
10  4.81 -0.0716 
# … with 2,776 more rows

Now I want to fit the GARCH(1,1) model with normal innovations.

garnor1 = function(x){
  require(fGarch)
  t = length(x)
  fit = garchFit(~garch(1,1),data=x,trace=F,cond.dist="norm")
  m = fit@fitted
  cv = [email protected]
  var = m+cv*qnorm(0.01) # low tail 
  return(var[t])
}

What I have succeeded is that I can estimate the the lower value at risk for 2-day returns up to time $t$. This will give a number that is the VaR up until now (say today). Am I right until now?

If yes, I know that the VaR is being calculated from the predictive function for the $t+2$ quantile value. Doing so I have to predict the above function:

g11pre = function(x){
  require(fGarch)
  fit = garchFit(~garch(1,1),data=x,trace=F,cond.dist="norm")
  df=coef(fit)["shape"]
  p = predict(fit,2)
  m=p$meanForecast
  cv=p$standardDeviation
  var=m+cv*qnorm(0.01)
  return(var[2])
}

And this last predictive function I have to backtest or the previous one?

Edit

For the backtesting in the predictive function I tried something by my own.(in order to fully understand it):

db=  d%>%
dplyr::mutate(back_lower = zoo::rollapplyr(y,252,FUN = g11pre,by = 21,fill=NA))%>%
  tidyr::fill(back_lower)%>%
  tidyr::drop_na()

I know it looks strange.Let me explain.I am using the full 10 year dataset.The period of estimation are the first 252 days and then roll by one month (21) days.I am not interested on by 2 day assess-roll the model. Plotting the backtesting result:

p = ggplot() + 
  geom_line(data = db, aes(x =1:length(y) , y = y), color = "black") +
  geom_line(data = db, aes(x = 1:length(back_lower), y = back_lower), color = "red") +
  xlab('') +
  ylab('risk low')

That looks like a step graph (this what It must look like)

Answer 1

In general, forecasting Value at Risk (VaR) following a parametric GARCH framework follows standard practices of univariate (point) forecasting. Moreover, we are always interested in the accuracy of a VaR model based on the out-of-sample forecast performance.

Forecasting 2-day ahead Out-of-sample Value at Risk:

Assume that we are standing at time $t$ with information $\mathcal{F}_t$, then for a constant mean model $\mu$, we can define a univariate GARCH(1,1) framework for a return process $r_{t+1}$ as:

\begin{align*} r_{t+1} \vert \mathcal{F}_{t} &= \mu + \varepsilon_{t+1}\\ \varepsilon_{t+1} &= \sigma_{t+1} \cdot z_{t+1}\\ \sigma^2_{t+1} &= \omega + \alpha_1 \varepsilon_{t}^2 + \beta_1 \sigma_{t}^2, \end{align*} where $z_t \overset{iid}{\sim} D(0,1)$ (which in your case is the Gaussian distribution). It is the standardized distribution of the innovations, $z_t$, that determines the quantile in the forthcoming parametric VaR equation. Then the VaR at time $t+1$ given the filtration $\mathcal{F}_t$ can be defined as (equivalent to the 1-step ahead VaR):

$$ VaR_{t+1\vert t}^\alpha = \mu + \sigma_{t+1\vert t}\cdot\Phi^{-1}_\alpha\left(0,1\right) $$ Note that in some cases we negate the above calculation, in order to get positive VaR estimates. This makes sense when taking about losses in notional amount. Nevertheless the 2-step ahead VaR forecast can be obtained via a recursive forecasting scheme:

$$ VaR_{t:t+2\vert t}^\alpha = \mu + \sigma_{t:t+2\vert t}\cdot\Phi^{-1}_\alpha\left(0,1\right), \tag{1} $$ where we observe that the only forecastable object in the Gaussian VaR framework is the conditional volatility process. Note that the 2-step ahead conditional volatility forecast, $\sigma_{t:t+2\vert t}$, is recovered by utilizing the prediction at time $t+1$. Using the fact that the multistep recursive forecast for the GARCH(1,1) is given by (see this link p. 449):

$$ \mathbb{E}\left[\sigma_{t+h}^2 \vert \mathcal{F}_t\right] = \sum_{i=0}^{h-1} (\alpha_1 + \beta_1)^i \cdot \omega + (\alpha_1+\beta_1)^{h-1}(\alpha_1 \varepsilon_t^2+\beta_1 \sigma^2_t) $$

we get for $t+2$:

\begin{align*} \sigma^2_{t:t+2\vert t}&:=\mathbb{E}\left[\sigma_{t+2}^2 \vert \mathcal{F}_t\right] \\ &= \omega + (\alpha_1 + \beta_1) \mathbb{E}\left[\sigma_{t+1}^2\vert \mathcal{F}_t\right]\\ &= \omega + (\alpha_1+\beta_1)\omega + (\alpha_1+\beta_1) \alpha_1 \varepsilon_t^2 + (\alpha_1+\beta_1) \beta_1 \sigma_t^2. \end{align*} Inserting this in $(1)$ yield the 2-step ahead VaR forecast/prediction in the GARCH(1,1) framework. Now you can do the following recursive procedure to acquire your forecasts:

1. Fit the GARCH(1,1) model via a rolling or expanding window,
2. get estimates $(\hat{\omega},\hat{\alpha}_1, \hat{\beta}_1, \hat{\mu})$ at each time point $t$,
3. calculate the 2-step ahead VaR forecast, $VaR_{t:t+2\vert t}^\alpha $.

Alternative approach:

Instead of the above methodology, you could fit the GARCH(1,1) model on sparse sampled returns which are sampled every second day, ie. $\{r_t,r_{t+2},r_{t+4},\ldots\}$.

This will yield conditional volatilities every second day. Then the 1-step ahead VaR forecast,$VaR_{t+1\vert t}^\alpha$, defined in $(1)$ gives you a VaR estimate for the next 2 days.

This methodology is easy to implement as you can follow most literature on how they define the 1-step ahead forecast without the need of additional derivations. For this matter, you can get some inspiration from the R code provided here (however, they only estimate VaR in-sample, not out-of-sample).

---

If you want to specify a time-varying mean model $\mu_t$, then you also need the 2-step ahead conditional forecast of $\mu_{t+2\vert t}$ in order to recover your VaR forecasts. These can very often get recovered using a similar recursive scheme. I hope this provide some insight.

Answer 2

Your question seems to be whether to evaluate/backtest in-sample VaRs or out-of-sample VaRs. I presume you want to get a sense of your model's anticipated performance on unseen future data. If your model was exactly correct, both ways would be fine. However, in reality models are not exactly correct. Then out-of-sample VaRs are the relevant ones, as they (unlike the in-sample VaRs) mimic how you will be using your model.

Just note that if your 2-day periods overlap, your effective sample size is half of what the nominal sample size is. (I cannot tell from your code whether you only look at every second day or at every day. It is in the latter case that the neighboring 2-day returns will overlap.)