lecture 6: non normal distributions and their uses...

Lecture 6: Non Normal Distributions and their Uses in GARCH Modelling

Prof. Massimo Guidolin

20192– Financial Econometrics Spring 2015

Overview

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Non-normalities in (standardized) residuals from asset return models

Tools to detect non-normalities: Jarque-Bera tests, kernel density estimators, Q-Q plots

Conditional and unconditional t-Student densities; MLE vs. method-of-moment estimation

Cornish-Fisher density approximations and their applications in risk managements

Hints to Extreme Value Theory (EVT)

Lecture 6: Non-normal distributions – Prof. Guidolin

Overview and General Ideas

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Let’s recap where we are at in the course. This is what we said… We will proceed in three steps following a stepwise distribution

modeling (SDM) approach: Establish a variance forecasting model for each of the assets

individually and introduce methods for evaluating the performance of these forecasts DONE!

Consider ways to model conditionally non-normal aspects of the returns on the assets in our portfolio—i.e., aspects that are not captured by conditional means, variances, and covariances NEXT • We still study RPF,t and possibly assume a GARCH has been fitted

Link individual variance forecasts with correlations Recall baseline model:

In this lecture we learn how to model departures of the marginal conditional densities from normality


Why an Interest in the Conditional Density?

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In lecture 6, we have studied dynamic univariate models of conditional heteroskedasticity • It has been stressed that these induce unconditional return

distributions which are non-normal However ARCH models do not seem to induce sufficient non-

normality • This can be seen in the fact that the standardized residuals from most

GARCH models fail to be normally distributed

(G)ARCH models fail to produce sufficient non-normalities


Matching Gaussian Kernel density

Tools to Test for Normality: Jarque-Bera

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For instance, in a Gaussian GARCH(1,1) model, Rt+1 = t+1zt+1, zt+1 N(0,1) 2

t+1 = + R2t + 2

t and zt+1 = Rt+1/t+1 N(0,1) is a testable implication

• This GARCH is called “Gaussian” because zt+1 N(0,1), where zt is the standardized residual series

Therefore non-normalities keep plaguing standardized residuals from many types of Gaussian GARCH models

Two issues: (A) How can we detect non-normalities in an empirical density (for either returns or standardized residuals)? (B) What can we do about it?

Jarque-Bera test based on sample skewness & kurtosis If X is a r. v. with mean μ and standard deviation , the skewness

measures the asymmetry of the density function:


In our case, standardized residuals – but this can be applied generally


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Skewness is computed as an odd power scaled central moment Its sign depends on the relative weight of the observations below

the mean respect to those above the mean: • Skew = 0, symmetric distribution (e.g., Normal) • Skew > 0, asymmetric to the right (e.g., Log-normal) • Skew < 0, asymmetric to the left (e.g., many empirical densities for

realized asset returns)

Kurtosis is instead defined as: • This measure gives large weights to the observations far from the

mean, i.e. the observations that falls in the tails of the distribution • The normal distribution has kurtosis of 3, so that its excess of kurtosis

(kurt-3) is 0; a kurtosis larger than 3 means tails fatter than in the normal case

Skewness is the scaled third central moment and reveals whether the empirical distributions of standardized residuals is asymmetric around the mean



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Jarque and Bera (1980) proposed a test that measures departure from normality in terms of the skewness and kurtosis • Under the null of normally distributed errors, the asymptotic

distribution of sample estimators of skewness and kurtosis are:

• Asymptotic means that the normal approximation becomes increasingly good as the sample size grows

• Because they are asymptotically independent, the squares of their standardized forms can be added to obtain the Jarque-Bera statistic:

Kurtosis is the scaled fourth central moment and reveals whether the empirical distributions of standardized residuals has tails thicker than a Gaussian distribution

Jarque-Bera test summarizes any non-zero skewness and any non-zero excess kurtosis in a formal test of hypothesis


Tools to Test for Normality: Kernel Estimators

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• Large values of this statistic indicate departures from normality • Example on S&P 500 daily returns, 1926-2010:

A kernel density estimator is an empirical density “smoother” based on the choice of two objects, the kernel function K(x) and the bandwidth parameter h:

• It generalizes the “histogram estimator”:

A kernel density estimator is a “smoother” of a standard empirical histogram



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• (x) is the delta (Dirac) function, with (x) always zero but at x=0, when (0) = 1

• Let’s give a few examples. The most common type of kernel function used in applied finance is the Gaussian kernel:

• A K(x) with optimal (in a Mean-Squared Error sense) properties is Epanechnikov’s:

• Other popular kernels are the triangular and box kernels:



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• The bandwidth parameter h is usually chosen according to the rule (T here is sample size):

• The choice of the bandwidth in this way depends on the fact that it minimizes the integrated MSE:

• Do different choices of K(x) make a big differences? • It seems not, financial returns are typically leptokurtic, i.e., they have

fat tails and highly peaked densities around mean


Moment-matched Gaussian

Tools to Test for Normality: Q-Q Plots

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• A less formal and yet powerful method to visualize non-normalities consists of quantile-quantile (Q-Q) plots

The idea is to plot the quantiles of the returns against the quantiles of the normal (or otherwise selected) theoretical distribution

• If the returns are truly normal, then the graph should look like a straight line on a 45-degree angle • Systematic deviations from the 45-degree line signal that the returns

are not well described by the normal distribution • The recipe is: sort all standardized returns zt = RPF,t/σPF,t in ascending

order, and call the ith sorted value zi • Then calculate the empirical probability of getting a value below the

actual as (i−0.5)/T , where T is number of obs. • The subtraction of .5 is an adjustment allowing for a continuous

distribution

A Q-Q plot represents the quantiles of an empirical density vs. the quantile of some theoretical distribution


Tools to Test for Normality: Q-Q Plots

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• Calculate the standard normal quantiles as where denotes the inverse of the standard normal density

• We can scatter plot the standardized and sorted returns on the Y-axis against the standard normal quantiles on the X-axis

Why do risk managers care? Because differently from JB test and kernel density estimators, Q-Q plots provide information on where (in the support of the empirical return distribution) non-normalities occur


Raw S&P 500 returns After GARCH(1,1)

Non-Normality: What Can We do?

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An obvious question is then: if all (most) financial returns have non-normal distributions, what can we do about it?

Probably, to stop pretending asset returns are “more or less” Gaussian in many applications and conceptualizations

Given that, there are two possibilities. First, to keep assuming that asset returns are IID, but with marginal, unconditional distributions different from the Normal • Such marginal distributions will have to capture the fat tails and

possibly also the presence of asymmetries Second, stop assuming that asset returns are IID and model instead

the presence of dynamics/time-variation in conditional densities • You have done this already: GARCH models!

It turns out that both approaches are needed by high frequency (e.g., daily) return data

Two key approaches to deal with non-normalities: to model conditional Gaussian moments; change the marginal density


lecture 6: non normal distributions and their uses...

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