Lecture 6: Non Normal Distributions and their Uses in GARCH Modelling Prof. Massimo Guidolin 20192 Financial Econometrics Spring 2015
Overview 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) 2
Overview and General Ideas In this lecture we learn how to model departures of the marginal conditional densities from normality 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 R PF,t and possibly assume a GARCH has been fitted Link individual variance forecasts with correlations Recall baseline model: 3
Why an Interest in the Conditional Density? (G)ARCH models fail to produce sufficient non-normalities 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 nonnormality This can be seen in the fact that the standardized residuals from most GARCH models fail to be normally distributed Matching Gaussian Kernel density 4
Tools to Test for Normality: Jarque-Bera For instance, in a Gaussian GARCH(1,1) model, R t+1 = t+1 z t+1, z t+1 N(0,1) 2 t+1 = + R 2 t + 2 t and z t+1 = R t+1 / t+1 N(0,1) is a testable implication This GARCH is called Gaussian because z t+1 N(0,1), where z t 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 5
Tools to Test for Normality: Jarque-Bera Skewness is the scaled third central moment and reveals whether the empirical distributions of standardized residuals is asymmetric around the mean 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 6
Tools to Test for Normality: Jarque-Bera 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 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: 7
Tools to Test for Normality: Kernel Estimators A kernel density estimator is a smoother of a standard empirical histogram 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 : 8
Tools to Test for Normality: Kernel Estimators (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: 9
Tools to Test for Normality: Kernel Estimators 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 10
Tools to Test for Normality: Q-Q Plots A Q-Q plot represents the quantiles of an empirical density vs. the quantile of some theoretical distribution 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 z t = R PF,t /σ PF,t in ascending order, and call the ith sorted value z i 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 11
Tools to Test for Normality: Q-Q Plots 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 Raw S&P 500 returns After GARCH(1,1) 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) nonnormalities occur 12
Non-Normality: What Can We do? Two key approaches to deal with non-normalities: to model conditional Gaussian moments; change the marginal density 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 13