Financial Econometrics Non-linearity and Autoregressive Conditional Heteroskedasticity (ARCH) Models

[Audio] Financial Econometrics Non-linearity and Autoregressive Conditional Heteroskedasticity ( ARCH) Models.

[Audio] Non-linearity Motivation: the linear structural (and time series) models cannot explain a number of important features common to much financial data: Leptokurtosis: Tendency for financial asset returns to have distributions that exhibit fat tails and excess peakedness at the mean. Volatility clustering or volatility pooling: The tendency for volatility in financial markets to appear in bunches: The information arrivals occur in bunches rather than being evenly spaced over time. "...large changes tend to be followed by large changes, of either sign and small changes tend to be followed by small changes..." . Leverage effects: The tendency for stock prices to be negatively correlated with changes in stock volatility. The tendency for volatility to rise more following a large price fall than following a price rise of the same magnitude..

[Audio] Daily S&P 500 Returns for January 1990 – December 1999.

Non-linearity. .

[Audio] Non-linearity The linear paradigm is a useful one. Many apparently non-linear relationships can be made linear by a suitable transformation. On the other hand, it is likely that many relationships in finance are intrinsically non-linear. There are many types of non-linear models, e.g.: ARCH / GARCH models used for modelling and forecasting volatility. Switching models: allowing the behaviour of a series to follow different processes at different points in time. Bilinear models..

[Audio] Non-linearity The traditional tools of time series analysis (e.g. ACF) may find no evidence that we could use a linear model, but the data may still not be independent. General tests are designed to detect many departures from randomness in data and will detect a variety of non-linear structures but not the type of non-linearity. One particular non-linear model that has proved very useful in finance is the Autoregressive Conditional Heteroskedasticity ( ARCH) model..

[Audio] Volatility Volatility is one of the most important concepts in the whole of finance. Econometricians called upon to determine how much one variable will change in response to a change in some other variable. Increasingly however, they are being asked to forecast and analyse the size of the errors of the model. Most investors dislike risk taking and require a premium for holding assets with risky payoffs. The variance of an asset has been used to measure risk and split the risk into a company specific component ( diversifiable), and a market component which cannot be diversified. This measure of the unconditional volatility does not recognize that there may be predictable patterns in stock market volatility..

[Audio] Volatility Although volatility is not directly observable, it has some characteristics that are commonly seen in asset returns. Consider: Volatility clusters: volatility evolves over time in a continuous manner, volatility does not diverge to infinity and leverage effect. We will analyse models of conditional (on information at time t- 1) volatility. These type of models have the implication for finance that investors can predict the risk. These type of models successfully characterise the fact that stock prices seem to go through long periods of high and long periods of low volatility..

Heteroskedasticity. .

Autoregressive Conditional Heteroskedasticity (ARCH) Models.

ARCH Models. .

ARCH Models. .

Non-Negativity Constraints. .

Testing for ARCH Effects. .

Testing for ARCH Effects. .

Problems with ARCH(q) Models. .

Generalised Autoregressive Conditional Heteroskedasticity (GARCH) Model.

[Audio] GARCH Models But in general, a GARCH( 1,1) model will be sufficient to capture the volatility clustering in the data. Why is GARCH Better than ARCH? more parsimonious - avoids over fitting less likely to breach non-negativity constraints.

The Unconditional Variance under the GARCH Specification.

[Audio] Estimation of ( G) ARCH Models Since the model is no longer of the usual linear form, we cannot use OLS. We use another technique known as maximum likelihood. The method works by finding the most likely values of the parameters given the actual data. It involves choosing values for the parameters that maximize the chance (or likelihood) of the data occurring. More specifically, we form a log-likelihood function and maximise it..

Estimation of (G)ARCH Models. .

[Audio] Extensions to the Basic GARCH Model Since the GARCH model was developed, a huge number of extensions and variants have been proposed. Three of the most important examples are EGARCH, GJR, and GARCH-M models. Problems with GARCH(p,q) Models: Non-negativity constraints may still be violated GARCH models cannot account for leverage effects Leverage effects – The tendency for volatility to rise more following a large price fall than following a price rise of the same magnitude. Possible solutions: the exponential GARCH (EGARCH) model or the GJR model, which are asymmetric GARCH models..

The EGARCH Model. . abstract.

The GJR Model. .

GARCH-in Mean. .

What Use Are GARCH-type Models?. .

[Audio] Forecasting Variances using GARCH Models Consider an example using data on daily foreign exchange returns. Using a GARCH( 1,1) model we can quite easily calculate one, two, three, four and five step-ahead variance forecasts. i.e., a forecast can be constructed for each day of the next trading week. Important note: Remember the forecasted variance for the whole week is the sum of the five daily variance forecasts. What Use Are Volatility Forecasts? Option pricing Conditional betas Dynamic hedge ratios The Hedge Ratio - the size of the futures position to the size of the underlying exposure, i.e., the number of futures contracts to buy or sell per unit of the spot good..

What is the optimal value of the hedge ratio?. .

[Audio] ( G) ARCH Modelling in STATA ARCH model of y with first- and second-order ARCH components and regressor x using tsset data: arch y x, arch(1, 2) Add a second-order GARCH component: arch y x, arch(1, 2) garch(2) Add an autoregressive component of order 2 and a moving-average component of order 3: arch y x, arch(1, 2) garch( 2) ar(2) ma(3) As above, but with the conditional variance included in the mean equation: arch y x, arch(1, 2) garch(2) ar(2) ma(3) archm EGARCH model of order 2 for y with an autoregressive component of order 1: arch y, earch(2) egarch(2) ar(1) Lecture.

Application in STATA. .

[Audio] Application in STATA The model confirms the presence of ARCH effects at the 5% significance level..

[Audio] Application in STATA. Application in STATA.

Empirical Application: ARCH model. .

[Audio] Empirical Application: ARCH model The graph of the differenced series clearly shows periods of high volatility and other periods of relative tranquillity. This makes the series a good candidate for ARCH modelling. Indeed, price indices have been a common target of ARCH models. Engle ( 1982) presented the original ARCH formulation in an analysis of U.K. inflation rates..

[Audio] Empirical Application: ARCH model We fit a constant-only model by OLS and test ARCH effects by using Engle's Lagrange multiplier test estat archlm.

[Audio] Empirical Application: ARCH model Because the LM test shows a p-value of 0.0038, which is well below 0.05, we reject the null hypothesis of no ARCH(1) effects. Thus, we can further estimate the ARCH(1) parameter by specifying arch(1). The first-order generalized ARCH model ( GARCH, Bollerslev 1986) is the most commonly used specification for the conditional variance in empirical work and is typically written GARCH( 1, 1). We can now estimate a GARCH(1, 1) process for the log-differenced series..

[Audio] Empirical Application: GARCH model. Empirical Application: GARCH model.

[Audio] Empirical Application: GARCH model. Empirical Application: GARCH model.

Empirical Application: ARCH model. .

[Audio] Application in STATA. Application in STATA.

[Audio] Application in STATA. Application in STATA.

[Audio] Bloomberg Volatility index Compare volatility in Bloomberg with estimated volatility using Bloomberg data in STATA..