Volatility Modeling - Science topic

Thank You it helped a lot

Long memory volatility data are best modelled as FIGARCH models. There are perhaps no known ways of transformation from long memory to short memory data.

The estimated model AR(1)-GARCH(1,1) is different from the true model AR(3)-GARCH(1,1). The estimates must therefore be biased.

The Ljung-Box test is aimed at testing the independance of errors using residuals of an ARMA model estimated on the same data. But it makes use of autocorrelations so it is not powerful when the errors are uncorrelated but not independent. When applied to squared residuals, it can reveal ARCH and GARCH effects. Note that the errors of a ARCH-GARCH model are uncorrelated but not independent. Have a look at the excellent book by Francq and Zakoian entitled "GARCH Models: Structure, Statistical Inference and Financial Applications" published by Wiley in 2010.

Have you looked at the Job Openings for Economists website? Many of the deadlines may have passed, but they may still accept for post-doc.

Hi,

Faisal Nawaz Please refer to the article at:

https://www.efmaefm.org/0EFMAMEETINGS/EFMA%20ANNUAL%20MEETINGS/2012-Barcelona/papers/EFMA2012_0173_fullpaper.pdf

There is no single, universal, multi-faceted model built of many indicators selected from fundamental and technical analysis, including market, economic and financial analyzes, indicator analyzes, due diligence and others that could be used to analyze the situation, economic situation, volatility, etc. for every capital market, including the currency market, stock market, market of raw materials and other production factors, and for every situation, i.e. bear market, bull market, balance, strong changes in trends, different levels of investment risk, various phases in the economic cycle of the economy, and irrespective of the national specifics of the operation of financial markets. Specific analytical models should be built for a specific economic situation, for a specific market. Then it is possible to achieve a high level of decision-making accuracy on the basis of forecasts formulated from this type of multi-faceted, complex indicator models for analyzing the current and prospective situation on a specific stock exchange market or other specific capital market operating in a specific macroeconomic environment, in a specific financial system, in a specific financial system country.

Greetings.

Dariusz Prokopowicz

It is a broad issue and there is no a single answer. It depends on what is the aim of your research. One important parameter is what are the major developments and events during the time of research. Are there breakpoints? What is the frequency of data? There is always a minimum of observations. I think you should test using different duration and after comparing results, you will be able to choose the optimal solution. Experience in modeling is a major factor of success.

It is a general project area not a particular paper.

This Matlab toolbox might help:

The exponential smoothing method gives greater weight to the more new observations, while moving average give equal weight to all observations

Definitely, lambda can be estimated within maximum-likelihood framework. Refer to the page 507 in Tsay (2010) for the log-likelihood structure.

Tsay (2010). Analysis of financial time series. John Wiley.

Yes, that is right. This what one does in practice. EViews output returns three info crit: AIC, SIC, and Hannan Quinn.

An option trader needs the implied volatility, at least of liquid options, not to price, the price is given by the market, but to hedge the options, by means of the underlying assets and a proxy of a bankaccount. I advise you, to pay a visit to option traders, in order to get a picture of what is going on in the trading pit!

Volatility modelling may be by generalized autoregressive conditional heteroscedasticity (Garch) modelling in which case the conditional variance is modelled. The autocorrelation function of the squares of the data may be used as a diagnostic tool in this regard.

I have to clarify myself a bit here. In any analysis of time series data, we always ensure the series is stationary before anything else. Now in regime-switching models, is that necessary? I hope that makes the question clearer.

The GARCH-MIDAS applications are gaining ground in the recent literature.

See, for example, among others:

Wei et al. (2018). Hot money and China’s stock market volatility: Further evidence using the GARCH–MIDAS model. Physica A: Statistical Mechanics and its Applications.

Pan et al. (2017). Oil price volatility and macroeconomic fundamentals: A regime switching GARCH-MIDAS model. Journal of Empirical Finance.

Hi: the random walk equation for sigma_t is

sigma_t = sigma_t-1 + epsilon_t.

so the coefficient of the previous sigma value would be +1.0 if RW for volatility is true.

the "naive model" versus igarch ) is probably that volatility follows a random walk. so, besides what the other people said about forecasting comparisons, another way is to check if volatility follows a random walk by estimating the model above and seeing if the coefficient on sigma_t-1 is +1.

actually, both methods should probably be done. but, the forecasting procedure should be done on out of sample observations that are not part of the respective estimation procedures. so, maybe split the data into two parts and estimate using half and then forecast using the other half. I hope this helps.

"I am also required to carry out the forecasting procedure mathematically for a few steps ahead."

You simply need to take conditional expectation from both side of variance equation and derive the formula for h-step ahead variance forecasts.

We provided the formula in our recent paper on volatility forecasting. See the Appendix B Forecasting formulae on page 323-324.

A.S. Hasanov et al. (2018). Forecasting volatility in the biofuel feedstock markets in the presence of structural breaks: A comparison of alternative distribution functions. Energy Economics 70, 307–333.

Volatility persistence in GJR model is given by

alpha + beta + gamma / 2 < 1

-0.014168 + 0.880880 + 0.149107/2 = 0.941266

In EViews output, RESID(-1) < 0 is an indicator (i.e., dummy) variable

(I_(t-1) = 1 if RESID(-1) < 0).

For a leverage effect, we look at gamma > 0 (0.149107 > 0).

Condition for non-negativity of variance equation coefficients:

c > 0

alpha > 0 (i.e., coefficient of RESID(-1)^2)

beta >= 0 (i.e., coefficient of GARCH(-1))

alpha + gamma >= 0.

That is, variance model is still valid even if alpha < 0 or gamma < 0 provided that alpha + gamma >= 0

Following

Thanks Peter for the links !!

The DCC model has become a standard model in many Econometric software: RATS, G@RCH under OxMetric, Microfit, MATLAB, rmgarch package in R.

Thanks..

Volatility analysis is done before modelling. Arch test, for instance, is part of the analysis which, if significant, provides ground for modelling.

Dear Abdul Rishad,

ARCH and GARCH models have become important tools in the analysis of time series data, particularly in financial applications. These models are especially useful when the goal of the study is to analyze and forecast volatility. Attached article may help you to understand more about the economic significance of GARCH model.

Best Wishes,

JJ test  only captures when there is a linear association between  sets data, however to assess linear and nonlinear association between sets of data, I advise using nonlinear cointegration test approach , like rank test  of Breitung.

best luck

Hi Carlos,

Thanks for you anser.

1) The data is stationary, so I would not say I have really extreme ouliers.

2)Yes, I am using Gaussian GARCH, I have also tried using GED distribution, but it appears to also be non-stationary. 

3) Yes, of course I have specified the mean equation first.

GARCH models are applied to returns, not prices. Returns are stationary, and therefore you do not need to worry about this issue.

 You might find my recent paper Pricing derivatives in a regime switching market with time inhomogeneous volatility relevant. link: http://arxiv.org/abs/1611.02026

In this model, the volatility is modeled as a particular type of pure jump process. The class of pure jump processes includes Markov chains. Although Markovian volatility is well studied, the analysis in a non-Markovian setup is not so common. In our paper, we have taken volatility as an independent semi-Markov process and analyzed the model.

A good reference for a comparison of linear and log returns is the working paper of Attilio Meucci : "Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management". 

If the securities  used for computing returns are not too volatile and the time step of the return is short, the distribution of linear and compounded returns is similar but as the time step grows, so does the difference between the two distributions.

You decide whether to consider linear or compounded (log) returns based on your particular objective. In fact linear returns aggregate across assets thus risk and portfolio managers use linear returns for risk analysis, performance attribution, and portfolio optimization.  Compounded return aggregates across time and  thus log returns  are used for projections since it is easy to estimate their one period distribution and to project this distribution to a generic horizon into the future.

If your aim is to model the volatility of a series of futures contracts then you'll need to pay attention to the Samuelson effect and eventual seasonality in the volatility. You may take a look at these papers we wrote with Lorenz Schneider:

If your aim is to model a time series of a rolling proxy for a commodity's spot price (e.g. CL1 time series) then using a GARCH based econometrics model seems suitable. Depending on the chosen commodity you'll need a seasonal component (e.g. Natural Gas, Wheat)

Can you tell me how to check BEKK Framework in Eviews? 

Dear Malik,

I am including a PPT to explain how to model any GARCH type model in Eviews. I hope it will be beneficiary for you. Take care.

Mustafa Ozer

ISSUE: How to calculate the Skorohod integral?

STOCHASTIC PROCESS: Assume that there are random variables W(h) in Hilbert space H. The Hilbert space is a generalized Euclidean space, i.e. two dimension XY-coordinate system. The Hilbert H is comprised of elements h₁, h₂, … H. The random variable set W(h) in normally distributed in the Gaussian fashion (bell shaped with mean = 0) and there the mapping of h to W(h) is linear with mean:

(1)   E[W(h)] = 0

and variance:

(2)   E[W(g)W(h)]]

Assume now that there is a function u, and that u is given by:

(3)   U = sum(F_jh_j)

… where F_j = smoothing function and h_j element of H in Hilbert space.

The Skorohod integral (sigma) is given by:

… where DF = Malliavin derivative.

If you are working with financial data, you probably have the random variable set W(h) already, and the smoothing function F_j that accompanies it. I assume that the remaining issue is to find the Malliavin derivative for the second part of the equation: [DF_j, h_j]_H. Since the first part sum(F_jW(h_j)) is a sum, you must have series of observations. Let's call this smoothed observations (F_jW(h_j)) as f₁, f₂, ... These are series of points once connected is a smoothed curve. in that same topology, there is a corresponding series of changes of the curve whose rate of change may be given by the derivative of the function that i associated with the curve; this part is the second portion of the equation: [DF_j, h_j]_H. At the end, the Skohorod integral is simply the sum of all points defined by the function that provides the characteristics of the change minus the rate of change of that smoothed curve. Do not be confused by H and h from Hilbert space notation---it is line in space, i.e. just like XY space. H and h is used because we are dealing with special kind of stochastic process and the integration for that special stochastic process.

You should contact Tom Doan at Estima Software.

I'm sure he could help.

Hi,

The answer lies in the relationship between the VIX index and the 30-day variance swap rate. Essentially, the squared VIX index approximates the conditional risk-neutral expectation of the annualized return variance, over the following 30 days, which is the 30-day variance swap rate.

The realized return variance can be split in different components showing that only OTM calls and puts give a contribution in its replication.

Please have a look at the attached paper (pp. 15-16), where an extensive explanation is provided.

Best regards

Why not taking the squared returns (or absolute returns) over the five minutes, take averages over days (for each five minute interval) and test the equality of means?

Implied Volatility as it is only the market's prediction of it from a Black- Scholes model, with quoted option prices , need not necessarily follow the statistical properties of a variance- co variance matrix .'Volatility smile' is an example for it

Maria, Refk and the great LUMENGO: Thanks for your suggestions

Looks like the free Ox version and G@rch package is available for version 6. Download link is at the end of chapter 1 http://www.timberlake.co.uk/slaurent/G@RCH/default.htm

EXCHANGE RATE & UNDERLYING REGIME

Exchange rate under a given regime would display very stable pattern. If the patterns change and that change is significant, it may be due to either (i) regime change, or (ii) economic shock. Regime change will cause the exchange rate to change as the market would adjust and readjust to the new regime. This new adjustment would become stabilized (see y(t) below). However, if it is a shock, the change would readjust itself to the old pattern because there is no institutional or fundamental change that would affect the exchange rate.

EXISTING REGIME

In order to detect regime change, one must have adequate data series over a period of time to allow the data to show its characteristic. The time series data would produce a characteristic pattern that confirms to the linear structure of:

(1) Y’ = a1 + bX + e

… where a1 = Y-intercept, b = independent factor, and e = forecast error. This model is a normal run of the existing regime.

In subsequent periods, b may vary, but the pattern of Y’ would still obey the same characteristic pattern. Assume then that there is a change:

(2) Y’’ = a2 + bX + e

And that a1 > a2 or a1 < a2. the Y-intercept of the two periods are not equal. This signifies that there has been a change that could be explained by the law of probability. It also follows the change of a1 to a2 was not also predictable. For this reason, it is necessary to construct a new model to accommodate a2.

REGIME SWITCHING

Generally, the abrupt change of the Y-intercept described above is not deterministic, i.e. could not have been reasonably predicted. Regime switching occurs in such a manner. While Y’ obeys a predictable patter with a fixed Y-intercept, the subsequent change produces a Y-intercept that is due to institutional change. Thus, Y’’ may be rewritten as:

(3) Y(regime) = C + bX + e

… where the Y-intercept C results from a two-state Markov chain: a period of a prior regime and a period under a new regime. The difference between (a1 – C) > 0 or (a1 – C) < 0; that is the difference is real and significant. The challenge is to locate the period when the switching occur.

DETECTION INDICATOR

The indicator used for regime switching is the Y-intercept in the equation Y = a + bX + c. Assume then that several samples had been taken, say several time periods, i.e. t1, t2, … tn, and that regression analysis is run for each period; the result shows that a1, a2, …, a^n are not equal. Does it mean that there is a regime switching? No. In order to constitute regime switching we need to use only two periods: one prior to regime switching and one after the regime switching. The challenge is to find this pair of Y-intercept at various time period.

One way to find the pair is to undertake series of data set over several time periods and list all their Y-intercepts for each period. A period that shows a peak or a trough from the mean is a suspected pair. The second step is to select the pair and verify whether there is a regime change. The verification may be accomplished by the Anderson-Darling test.

The Anderson-Darling test requires n > 5 or minimum sample size of 6. Here, the sample is the individual a1, a2, …, a^n or Y-intercepts for each period, i.e. 3 before and 3 after the suspected change. The critical value for the Anderson-Darling test A^2 at 0.95 confidence interval is 0.787; the decision rule follows: if A^2(observed) > 0.787, there has been a regime change, otherwise accept the null hypothesis (no regime change).

In order to prove that no Type I or Type II error had been committed, several segments of the data length of 6 to 7 elements (periods) may be retested. Prior periods where there is no regime change, the A^2 value would be less than 0.787. There will be a data segment that would break this pattern showing A^2 > 0.787.

NUMBER OF REGIMES

For purposes of constructing a predicting model, this issue may be moot. A system does not switch regime so often so as to necessitate the accounting for number of regimes used---unless one runs a data series that covers a span of many years or decades. For practical purposes, i.e. a specified segment of time series data, to proof the regime switching the number of regime is not relevant to the extent that it does not play a role in determining the current exchange rate. However, if the time series data prove that there had been several regimes switching in the period, then the switching frequency may be observed. However, this data (regime switching frequency) cannot be used to forecast potential switching in the new future---it only shows a past count.

GOODNESS-OF-FIT

After the evidence of regime has been verified then a model may be constructed. Use any conventional methods to test the goodness-of-fit and readjust the model accordingly. At this point, basic statistical tests combined with econometrics would be helpful.