11 1 ARCH GARCH Models STAT 510

In particular, we would want indication to believe the series in not independent, which would visually be seen as lags for which the ACF and PACF values are outside the blue area. This would imply a significant difference in the ACF or PACF from 0 and would allow us to model the ARCH effects. We have mentioned previously that although we are limited in calculating the predicted price of a financial asset in the stock market due to the efficient market hypothesis, we can predict volatility. Albeit a very simplified example, we can see that with the added volatility of company B there comes an added notion of risk with the asset. However, this should not be confused with an implication that there is no way to make a profit.

1. The seasonality highly depends on the particular market where the trading happens, and possibly on the specific asset.
2. It should be noted that the COVID 19 period should be seen as a stress-test for these models, and given the very similar performance of GARCHNet we would like to emphasise that it performs well under all conditions.
3. In other words, it has conditional heteroskedasticity, and the reason for the heteroskedasticity is that the error term is following an autoregressive moving average pattern.
4. We use arch_model() from the arch package and specify that the data is of mean zero and modeled with a GARCH process.

https://1investing.in/ processes are widely used in finance due to their effectiveness in modeling asset returns and inflation. GARCH aims to minimize errors in forecasting by accounting for errors in prior forecasting and enhancing the accuracy of ongoing predictions. The result is that the conclusions and predictive value drawn from the model will not be reliable. GARCH is a statistical model that can be used to analyze a number of different types of financial data, for instance, macroeconomic data. Financial institutions typically use this model to estimate the volatility of returns for stocks, bonds, and market indices. Generally, when testing for heteroskedasticity in econometric models, the best test is the White test.

Forecasting Volatility: Deep Dive into ARCH & GARCH Models

However, when dealing with time series data, this means to test for ARCH and GARCH errors. None are perfect and which to use probably depends on what you want to achieve. If you have less than about 1000 daily observations, then the estimation is unlikely to give you much real information about the parameters. That is going to be one with about the right persistence (see below), with the alpha1 parameter somewhere between 0 and 0.1, and the beta1 parameter between 0.9 and 1. Using the GARCH model from above, we are able to forecast volatilities 𝜎ₜ² at time t, which are non-trivial predictions.

GARCH

GARCHNet would be more often chosen by regulators than by company management itself due to its relatively higher opportunity cost. We also note a rather large advantage of this model—by obtaining much more data (see p values greater than 10) the model can generate predictions of the same or better quality than GARCH models that consider smaller data samples. Value-at-Risk (VaR) defines the worst possible loss with a given probability \(\alpha \), assuming normal market conditions for a specific time period t (Philippe, 2006). In other words, VaR is a quantile of the distribution of the observed financial time series. In our case, these are log returns of the price quotations of the respective stock index.

GARCHNet with a skewed t-distribution is worse by a small margin, which is not consistent with its GARCH counterpart—GARCH with a skewed t-distribution is the best model compared to other members of its family. This may indicate that the distribution parameter estimation approach is inefficient for Adam’s optimizer, or that the approach we used should be reconsidered. For example, the parameter estimation should be changed to an estimation for the entire training sample, rather than based on a fairly small sample (of length p) of time series in the prediction phase. GARCHNet with a normal distribution tends to be worse than its standard counterpart, but in the last two periods this relationship is much smaller.

Another advantage of their approach is that it can use explained artificial intelligence (XAI) methods. However, instead of using estimation, they assumed the values of additional (in addition to the first and second moments) parameters of the distribution. Another research by Nguyen et al. (2019) proposes a fairly similar approach, but to a stochastic volatility (SV) model, which is related to GARCH. In their research, they propose an SV-LSTM model that uses LSTM NN instead of using the AR(1) process to model volatility.

2 Bollerslev’s GARCH Model

In the Tables 1, 2 and 3 we have presented the results of the statistical tests and the number of exceptions for each index tested. The results are mostly the same for all indexes, but the biggest difference is seen for the WIG 20. There is no GARCH model that outperforms its GARCHNet counterpart across all periods tested and for all p sequence lengths tested in terms of number of exceptions. However, GARCHNet with a t-distribution appears to have the largest excess. In the case of the WIG 20, for only two cases was the number of exceptions higher than for GARCH with a t distribution, for the S&P 500 it was five cases and for the FTSE 100 nine cases.

We used a rolling-window estimation approach (Zanin & Marra, 2012). For each forecast sample, we prepared a new model with new randomly initialized weights and trained it using the last 1000 observations. We have also tested a hypothesis that frequent updates of the model might not necessarily improve its quality, while only increasing the time overhead in training. We tested a framework where the model was fully reset (random weights fully initialized) less often than with each timestep forecast. The model might be refitted with fresh data, between resets to include new information.

That nastiness is just another aspect of us trying to ask a lot of the data. Assuming that you have enough data that it matters, even the best implementations of garch bear watching in terms of the optimization of the likelihood. We also propose an empirical experiment to verify the usefulness of the GARCHNet model.

Instead, an alternative estimation method called maximum likelihood
(ML) is typically used to estimate the ARCH-GARCH parameters. This
section reviews the ML estimation method and shows how it can be applied
to estimate the ARCH-GARCH model parameters. The farther ahead you predict, the closer to perfect your model has to be. If you are predicting with a time horizon of a month or more, then I’d be shocked if you got much value from a garch model versus a more mundane model. If you are predicting a few days ahead, then garch should be quite useful.

The ARCH model
is one of the most important models in the field of financial econometrics,
and its creator Robert Engle won the Nobel Prize in Economics in part
for his work on the ARCH model and its variants. Volatility clustering — the phenomenon of there being periods of relative calm and periods of high volatility — is a seemingly universal attribute of market data. GARCH (Generalized AutoRegressive Conditional Heteroskedasticity) models volatility clustering. If you have been around statistical models, you’ve likely worked with linear regression, logistic regression and several other mean modeling approaches.

Xₜ can be the price, returns or log returns of an asset, modeled by the sum of its squared historical values X²ₜ₋ⱼ. In these two cases, we see a clearly higher number of rejections of the null hypotheses—both due to underestimation and overestimation of risk. On average, garchNet models have very similar number of exceptions. Both model families were not able to respond correctly to the COVID-19 financial market crashes, hence the high number of exceptions in the last analyzed period. It should be noted that the COVID 19 period should be seen as a stress-test for these models, and given the very similar performance of GARCHNet we would like to emphasise that it performs well under all conditions. GARCH models describe financial markets in which volatility can change, becoming more volatile during periods of financial crises or world events and less volatile during periods of relative calm and steady economic growth.

Returns

But the parameters matter a lot when we are predicting out of sample. It generates spikes in predicted volatility that line up with periods of actual higher variance of daily returns. We also notice its ability to predict volatility clustering — certain periods have lower variance followed by periods of large variance. Additionally, it appears to do well in forecasting these volatility spikes in advance of the swings in the true daily returns. Where 𝜎ₜ² is the conditional variance at time t, X²ₜ₋₁ is the squared return at time t-1, and 𝜎²ₜ₋₁ is the conditional variance at time t-1.

It built on economist Robert Engle’s breakthrough 1982 work in introducing the Autoregressive Conditional Heteroskedasticity (ARCH) model. His model assumed the variation of financial returns was not constant over time but are autocorrelated, or conditional to/dependent on each other. For instance, one can see this in stock returns where periods of volatility in returns tend to be clustered together. The variance of the error term in GARCH models is assumed to vary systematically, conditional on the average size of the error terms in previous periods.

GARCH is the generalized auto-regressive conditional heteroskedastic model of order (P,Q) and is an extension of the ARCH(P) model. This addition to the model statement makes GARCH models more flexible and able to capture the persistence of volatility. We are able to capture volatility clustering — periods of high volatility or low volatility — via modeling with ARCH.

Two other widely used approaches to estimating and predicting financial volatility are the classic historical volatility (VolSD) method and the exponentially weighted moving average volatility (VolEWMA) method. The fGarch summary provides the Jarque Bera Test for the null hypothesis that the residuals are normally distributed and the familiar Ljung-Box Tests. Let’s use the fGarch package to fit a GARCH(1,1) model to x where we center the series to work with a mean of 0 as discussed above. The PACF (following) of the squared values has a single spike at lag 1 suggesting an AR(1) model for the squared series. Various studies have been conducted on the reliability of various GARCH models during different market conditions, including during the periods leading up to and after the Great Recession.