The experiment consists of estimating Value-at-Risk forecasts one day ahead in a window of 250 test days (approximately one trading year) and comparing them with equivalent garch models. The experiment was written and conducted in Python and pytorch (Paszke et al., 2019). This book provides a comprehensive and systematic approach to understanding GARCH time series models and their applications whilst presenting the most advanced results concerning the theory and practical aspects of GARCH. The probability structure of standard GARCH models is studied in detail as well as statistical inference such as identification, estimation, and tests. The book also provides new coverage of several extensions such as multivariate models, looks at financial applications, and explores the very validation of the models used.

In the most extreme approach we assumed that it is only trained fully once (on the first 1000 observations) and then we have increased the frequency of training up to 500 updates (update very other training sample). Such approach has been proven faulty in results comparison, due to large jumps in volatility estimates. GARCH processes, because they are autoregressive, depend on past squared observations and past variances to model for current variance.

- Since model diagnostics are often an overlooked step, we will spend some time assessing whether our fitted model is valid and provide direction for next steps.
- It can be seen that the GARCHNet predictions do not deviate from the rate of return, even more—for some intervals GARCHNet confirms the presence of volatility shocks much faster.
- It generates spikes in predicted volatility that line up with periods of actual higher variance of daily returns.
- Our results show that GARCHNet is an outstanding model that can explain conditional variance at least at the same level as traditional approaches.

For inference (and maximum likelihood estimation) we would also assume that the \(\epsilon_t\) are normally distributed. GARCH models are used when the variance of the error term is not constant. Heteroskedasticity describes the irregular pattern of variation of an error term, or variable, in a statistical model. Simulation is dependent on the estimated parameters, but not as seriously as with prediction. Model errors compound as we simulate farther into the future, but they compound with a vengeance when we predict far into the future. Using the empirical distribution — the standardized residuals from the fitted model — is often the best choice for the innovations.

We will more formally test these effects using the McLeod-Li test for ARCH and GARCH effects. The nuances and mathematical formulations of the test statistics are left out below but the concept and hypothesis are summarized. Recall, in various cases such as linear regression and even the ARMA(p,q) model, we want a homoskedastic model (constant variance across observations).

## CF4103 Financial Time Series Analysis Questions and Answers

Another interesting dimension of model comparison are loss functions (LF). There are two parties who are usually interested in these values—the regulator and the companies themselves. From the regulator’s point of view, the most important aspect is the value lost on the occurrence of a VaR exception, while from the company’s point of view, the opportunity cost of holding excess reserves. 2, we present the theoretical background of https://1investing.in/Net and the necessary background for VAR backtesting. 3, we describe the empirical experiment with data and model descriptions. 5 we include concluding remarks and paths for extending our framework in future research.

This section illustrates

how to forecast volatility using the GARCH(1,1) model. I think Bayes estimation of garch models is a very natural thing to do. We have fairly specific knowledge about what the parameter values should look like.

## Returns

A mean model is quite simply a model that predicts the average outcome value based on one or more predictor variables. The use of these mean models are plentiful across several domains; however, there also exist series of models to predict the conditional variance, which have various uses in financial data. Our results show that GARCHNet is an outstanding model that can explain conditional variance at least at the same level as traditional approaches. Value-at-risk forecasts are rather conservative, but fewer exceptions are observed for this reason.

The model makes these forecasts by accessing data from log returns Xₜ from which 𝜎ₜ₊₁² can be calculated (n-step-ahead forecast). We then use these forecasts to extend the forecast of volatility to times t+1, t+2, …, t+n. We do not continually update Xₜ as time t increases, but rather limit our forecasts for all future dates to be made solely on the current date. We can use all of the above to intuitively see that if Xₜ swings rapidly (i.e., the log return increases or decreases swiftly), Xₜ² is high and thus the volatility 𝜎ₜ² at time t is high. This is how ARCH models capture price swings and their impact on the volatility of an underlying asset.

## Gaussian process-driven GARCH

In chapter 5, it was shown that daily asset

returns have some features in common with monthly asset returns and

some not. In particular, daily returns have empirical distributions

with much fatter tails than the normal distribution and daily returns

are not independent over time. Absolute and squared returns are positively

autocorrelated and the correlation dies out very slowly. Daily volatility

appears to be autocorrelated and, hence, predictable. In this chapter

we present a model of daily asset returns, Robert Engle’s (ARCH) model, that can capture these

stylized facts that are specific to daily returns.

To ensure that this is an independent series, or rather to test that it is not, we remember that an independent series is one in which transformations of the series are themselves independent. Thus, we can repeat the process above using the squared returns or the absolute returns. We utilize the absolute returns, as this is generally done in financial data to diminish the effects of outliers. The GARCHNet models were compared with the original GARCH models in an empirical study. VAR estimates were created using a rolling window method—we trained the model using 1000 observations and estimated a forecast, then moved one time step forward and estimated another forecast. Logarithmic returns of the WIG20 index (Warsaw Stock Exchange, Poland), S&P 500 (New York Stock Exchange, the USA) and FTSE 100 (London Stock Exchange, the UK) were used as data.

On a plot of returns, for example, stock returns may look relatively uniform for the years leading up to a financial crisis such as that of 2007. The generalized autoregressive conditional heteroskedasticity (GARCH) process is an econometric term developed in 1982 by Robert F. Engle, an economist and 2003 winner of the Nobel Memorial Prize for Economics. GARCH describes an approach to estimate volatility in financial markets. That the plotting function is pp.timeplot is an indication that the names of the input returns are available on the output — unlike the output in the other packages up to here. Figure 4 compares this estimate with a garch(1,1) estimate (from rugarch but they all look very similar). The estimation procedure “tries” to make the residuals conform to the hypothesized distribution.

## Autoregressive conditional heteroskedasticity

All the GARCH model variations seek to incorporate the direction, positive or negative, of returns in addition to the magnitude (addressed in the original model). Essentially, wherever there is heteroskedasticity, observations do not conform to a linear pattern. Therefore, if statistical models that assume constant variance are used on this data, then the conclusions and predictive value one can draw from the model will not be reliable. The only model this package does is the garch(1,1) with t distributed errors. This has the idea of a specification for a model that is a separate object. Then there are functions for fitting (that is, estimating parameters), predicting and simulation.

The persistence of a garch model has to do with how fast large volatilities decay after a shock. For the garch(1,1) model the key statistic is the sum of the two main parameters (alpha1 and beta1, in the notation we are using here). We know that returns do not have a normal distribution, that they have long tails. It is perfectly reasonable to hypothesize that the long tails are due entirely to garch effects, in which case using a normal distribution in the garch model would be the right thing to do. However, using the likelihood of a longer tailed distribution turns out to give a better fit (almost always).