Lecture 5: Univariate Volatility
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1 Lecture 5: Univariate Volatility Modellig, ARCH and GARCH Prof. Massimo Guidolin Financial Econometrics Spring 2015
2 Overview Stepwise Distribution Modeling Approach Three Key Facts to Remember Volatility Clustering in the Data Naïve Variance Forecast Models: Rolling Window Variance Estimation and the RiskMetrics System ARCH Models GARCH Models; Comparisons with RiskMetrics Leverage Effects and Component GARCH Estimation of GARCH Models Refinements over GARCH modeling strategies: NGARCH and other GARCH models Variance Model Evaluation 2
3 Overview and General Ideas In all finance applications, modeling and forecasting volatility or the covariance structure of asset returns, is crucial Financial economists are concerned with modeling volatility in (individual) asset and portfolio returns because volatility is considered a measure of risk, and investors want a risk premium As you known, banks and other financial institutions apply so-called value-at-risk (VaR) models to assess their risks Modeling and forecasting volatility or, in other words, the covariance structure of asset returns, is therefore important Three steps of a stepwise distribution modeling (SDM) approach: ❶ Establish a variance forecasting model for each of the assets individually and introduce methods for evaluating the performance of these forecasts ❷ Consider ways to model conditionally non-normal aspects of the returns on assets in our portfolio i.e., aspects that are not captured by conditional means, variances, and covariances 3
4 Overview and General Ideas Most time series of asset returns, bond yields and exchange rates exhibit time-varying means, variances, and covariances ❸ Link individual variance forecasts with a correlation model The variance and correlation models together will yield a timevarying covariance model, which can be used to calculate the variance of an aggregate portfolio of assets The idea that second moments vary over time has an even deeper importance While most classical finance is built on the assumption that both asset returns and their underlying fundamentals are IID Normal over time, casual inspection of GDP, financial aggregates, interest rates, exchange rates etc. reveals that these series display time-varying means, variances, and covariances Fundamentals = quantities that justify asset prices in a rational framework 4
5 Three Key Results to Bear in Mind When variances and covariances are time-varying we speak about conditional HETEROSKEDASTICITY 3 simple facts to remember and understand: 1 The fact that the conditional variance may change in heteroskedastic fashion, does not necessarily mean the series is non-stationary Even though the variance may go through high and low periods, the unconditional (long-run, steady-state, average) variance may exist and be actually constant 2 Conditional heteroskedasticity implies that the unconditional, long-run distribution of asset returns will be non-normal 3 Many models of conditional heteroskedasticity, but in the end we care for their forecasting performance E.g., consider the (dividend-corrected) realized returns on a valueweighted index (by CRSP) of NYSE, AMEX, and NASDAQ stocks 5
6 Volatility Clustering in the Data For most (all?) financial time series, volatility clusters : high (low) volatility tends to be followed by high (low) volatility Sample period is 1972: :12, monthly data Value-Weighted NYSE/AMEX/NASDAQ Returns Turbulence Turbulence Turbulence Turbulence Quiet period Quiet period Quiet period Our objective is to develop models that can fit the sequence of calm and turbulent periods and especially forecast them Note: value-weighted NYSE/AMEX/NASDAQ are ptf. returns! 6
7 Volatility Clustering & Serial Correlation in Squares A standard autocorrelogram can be used to test the hypothesis that asset returns are IID variables As seen in Lecture 1, there is weak serial correlation in returns This lack of correlation means that, given yesterday s return, today s return is equally likely to be positive or negative The autocorrelation estimates from a standard autocorrelogram can be used to test the hypothesis that the process generating observed returns is a series of independent and identically distributed (IID) variables The asymptotic standard error of an autocorrelation estimate is approximately 1/(T) 1/2, where T is the sample size The IID hypothesis can be tested using the Portmanteau Q-statistic of Box and Pierce (1970), calculated from the first k autocorrelations as: 7
8 Volatility Clustering & Serial Correlation in Squares A standard autocorrelogram can be used to test the hypothesis that asset returns are IID variables The asymptotic distribution of the Q k statistic, under the null of an IID process, is Chi-square, with k degrees of freedom VW CRSP Stock Returns 10 Year U.S. Govt. Bond Returns Does this mean that stock & bond returns are (approximately) IID? Unfortunately not: it turns out that the squares and absolute values of stock and bond returns display high and significant autocorrelations 8
9 Volatility Clustering & Serial Correlation in Squares The high dependence in series of absolute returns proves that the returns process is not made up of IID random variables VW CRSP Squared Stock Returns 10 Year U.S. Govt. Squared Bond Returns Of course, similar evidence applies to REIT and 1M T-bills The high dependence in series of absolute returns proves that the returns process is not made up of IID random variables Large squared returns are more likely to be followed by large squared returns than small squared returns are 9
10 Volatility Clustering & Serial Correlation in Squares The high dependence in series of absolute/squared returns proves that the return process is not IID Here the conceptual point is that while the opposite does not hold: How to explain this phenomenon? If changes in price volatility create clusters of high and low volatility, this may reflect changes in the flow of relevant information to the market These stylized facts can be explained by assuming that volatility follows a stochastic process where today s volatility is positively correlated with the volatility on any future day This is what ARCH and GARCH models are for, but first we introduce a few alternatives that historically were developed before ARCH and GARCH were 10
11 Naïve Models: Rolling Window Forecasts In a rolling window variance model the forecast of time t+1 variance is a moving average of m recent squared returns Consider the simple model for one asset (or ptf.) return: Here R t+1 is the continuously compounded return and z t+1 is a pure shock to returns, z t+1 = R t+1 / t+1 The model assumes (as in Christoffersen s book) that the mean = 0 This is an acceptable approximation on daily data; absent this assumption, the model is R t+1 = + t+1 z t+1 The assumption of normality will be discussed/removed in lecture 6 The easiest way to capture volatility clustering is by letting tomorrow s variance be the simple average of the most recent m observations, as in Constant weighting 11
12 Naïve Models: Rolling Window Forecasts A high m will lead to an excessively smoothly evolving t+1, and a low m will lead to an excessively jagged pattern of t+1 However, the fact that the model puts equal weights (equal to 1/m) on the past m observations yields unwarranted results When plotted over time, variance exhibits box-shaped patterns An extreme return (positive or negative) today will bump up variance by 1/m times the return squared for exactly m periods after which variance immediately will drop back down The autocorrelation plot of squared returns sug gests that a more gradual decline is warranted in the effect of past returns on today s variance Also: how shall we pick m? 12
13 Naïve Models: RiskMetrics In JP Morgan s RiskMetrics model the forecast of variance is a weighted avg. of today s variance and today s squared return A more interesting model is JP Morgan s RiskMetrics system: The weights on past squared returns decline exponentially as we move backward in time: 1,, 2, Also called the exponential variance smoother Because for = 1 we have 0 = 1, it is possible to re-write it as: which is equivalent to: See lecture notes for why this is the case A weighted avg. of today s variance and today s squared return 13
14 Naïve Models: RiskMetrics The key advantages of the RiskMetrics model are: ❶ Recent returns matter more for tomorrow s variance than distant returns do as is less than 1 and therefore gets smaller when the lag,, gets bigger ❷ It only contains one unknown parameter, When estimating on a large number of assets, RiskMetrics found that the estimates were quite similar across assets, and therefore simply set = 0.94 for every asset for daily data In this case, no estimation is necessary ❸ Little data need to be stored in order to calculate tomorrow s variance; in fact, after including 100 lags of squared returns, the cumulated weight is already close to 100% Of course, once R 2 t, is calculated, past returns are not needed Given all these advantages of the RiskMetrics model, why not simply end the discussion on variance forecasting here? 14
15 Naïve Models: RiskMetrics 15
16 Naïve Models: RiskMetrics 16
17 ARCH Models In a ARCH(q) model the forecasts of variance depends on q lags of squared returns, 2 t+1 = + 1 R 2 t + 2 R 2 t q R 2 t-q+1 The RiskMetrics model has a number of shortcomings, but these can be understood only after introducing ARCH models Historically, ARCH models were the first-line alternative developed to compete with exponential smoothers In the zero-mean return case, their structure is very simple: This is a ARCH(1) However, historically it was soon obvious that just using one lag of past squared returns would not be sufficient One needs to use a large number q > 1 of lags on the RHS This means that squared returns are best modeled using an AR(q) instead of a simple AR(1) Yet ARCH(1) already implies one complication: they require nonlinear parameter estimation 17
18 Generalized ARCH (GARCH) Models In a GARCH(1,1) the forecast of variance depends on one lag of squared returns and on past variance, 2 t+1 = + R 2 t + 2 t But from the first part of this course, you know already where to look for: ARMA models The simplest generalized autoregressive conditional heteroskedasticity (GARCH(1,1)) model is:, α > 0, β > 0 The implied, unconditional, or long-run average, variance, 2, is This derives from Furthermore, if one solves for from the long-run variance expression and substitutes it into the GARCH equation: 18
19 GARCH Models: Forecasting Long-run forecasts in a GARCH(1,1) are a function of the persistence ( + ) and of initial deviations from steady-state Therefore a GARCH(1,1) implies that tomorrow s variance is a weighted average of the long-run variance, today s squared return, and today s variance Or, tomorrow s variance is predicted to be the long-run average variance with: something added (subtracted) if today s squared return is above (below) its long-run average, and something added (subtracted) if today s variance is above (below) its long-run average How do you forecast variance in a GARCH model? The one-day forecast of variance, 2 t+1 t, is given directly by the model as 2 t+1; as for multi-periods forecasts, one can show that: 19
20 GARCH Models: Forecasting This implies that as the forecast horizon H grows, because for ( + ) < 1 implies that ( + ) H-1 0, then E t [ 2 t+h] 2 For shorter horizons instead, E t [ 2 t+h] > 2 when 2 t+1 > 2 and viceversa when 2 t+1 < 2 The conditional expectation, E t [ ], refers to taking the expectation using all the information available at the end of day t, which includes the squared return on day t itself ( + ) plays a crucial role and it is commonly called the persistence level/index of the model A high persistence, ( + ) close to 1, implies that shocks which push variance away from its long-run average will persist for a long time Of course, eventually the long-horizon forecast will be the long-run average variance, 2 In asset allocation problems, we sometimes care for the variance of long-horizon returns, 20
21 GARCH Models: Forecasting The variance of cumulative, H-period returns is not simply H times the long run variance of each return As we assume that returns have zero autocorrelation (from their sample stats), the variance of the cumulative H-day returns is: Solving in the GARCH(1,1) case, we have: You will see in a moment why establishing a difference w.r.t. H 2 t+1 is so important Let s now compare GARCH(1,1) and RiskMetrics: are they so different? In a way they are not: comparing with you can see that RiskMetrics is just a a special case of GARCH(1,1) 21
22 GARCH Models: Comparison with RiskMetrics A JPM RiskMetrics(1,1) model is just a special, restricted, nonstationary case of GARCH(1,1) A RiskMetrics model is a special case in which = 0 and = = 1 so that, equivalently, + = 1; this has a number of implications: Implication 1: because = 0 and + = 1, under RiskMetrics the long-run variance does not exist as gives an indeterminate ratio 0/0 While RiskMetrics ignores that long-run average variance tends to be relatively stable over time, a GARCH model with ( + ) < 1 does not Equivalently, while a GARCH with ( + ) < 1 is a stationary process, a RiskMetrics model is not Implication 2: because, under RiskMetrics + = 1, so that H which means that any shock to current variance is destined to persist forever 22
23 GARCH Models: Comparison with RiskMetrics In a JPM RiskMetrics(1,1) model long-run variance does not exist (it explodes), any shock to current variance persists forever, and the variance of long-horizon returns is H 2 t+1 If today is a high-variance day, then the RiskMetrics model predicts that all future days will be high-variance A GARCH more realistically assumes that eventually in the future variance will revert to the average value Implication 3: Under RiskMetrics, the variance of long-horizon returns is: t+h /H What is the density, the distribution of long-horizon returns implied by these models? Impossible to show in closed form, see the posted notes Lecture 5: Univariate volatility α = 0.05, β = 0.90, σ 2 = GARCH(1,1) RiskMetrics 23
24 GARCH Models with Leverage According to the leverage (asymmetric variance) effect, a negative return increases variance by more than a positive return of the same magnitude A number of papers have emphasized that a negative return increases variance by more than a positive return of the same magnitude This may be because, in the case of stocks, as a negative return on a stock implies a drop in the equity value, which implies that the company becomes more highly levered and thus riskier (assuming the level of debt stays constant) We can modify GARCH models so that the weight given to the return depends on whether the return is positive or negative This is described by the (sample) news impact curve (NIC) The NIC measures how new information is incorporated into volatility, i.e. it shows the relationship between the current return R t and conditional variance one period ahead 2 t+1, holding constant all other past and current information 24
25 GARCH Models with Leverage: EGARCH The news impact curve describes the relationship between the current return R t and conditional variance 2 t+1 In a GARCH(1,1) model we have: NIC(R t 2 t+1 = 2 ) = + R 2 t + 2 = A + R 2 t which is a quadratic function of R 2 t and therefore symmetric around 0 (with intercept A + 2 ) Problem: for most return series, the empirical NIC fails to be symmetric EGARCH is probably the most prominent asymmetric GARCH Asymmetric NIC As in ARCH models, in GARCH models the negativity of parameters may create difficulties in estimation Nelson (1991) has proposed a new form of GARCH, the Exponential GARCH (EGARCH), in which positivity of the conditional variance is ensured by the fact that ln( 2 t+1 ) is directly modeled GARCH 25
26 GARCH Models with Leverage: EGARCH An EGARCH model features both asymmetries and nonlinearities and it avoids constrained ML estimation Two types of EGARCH(1,1) found in the applied literature; the first type is the one originally proposed by Nelson Letting z t [R t / t ], the logconditional variance is: R The sequence g(z t ) is a zero-mean, i.i.d. random sequence: If z t > 0, g(z t ) is linear in z t with slope + 1 If z t < 0, g(z t ) is linear in z t with slope - 1 Thus, g(z t ) is function of both the magnitude and the sign of z t and it allows the conditional variance process to respond asymmetrically to rises and falls in stock prices Can be rewritten as t t 26
27 GARCH Models with Leverage The term represents a magnitude effect: If 1 > 0 and = 0, the innovations in the conditional variance are positive (negative) when the magnitude of z t is larger (smaller) than its expected value If 1 = 0 and < 0, the innovation in conditional variance is positive (negative) when returns innovations are negative (positive), in accordance with empirical evidence for stock returns Another way of capturing the leverage effect is to define an indicator variable, I t, to take on the value 1 if day t return is negative and zero otherwise The variance dynamics can now be specified as Equivalent to have 2 t+1 = + (1 + )R 2 t + 2 t after negative returns and 2 t+1 = + R 2 t + 2 t after positive ones 27
28 GARCH Models with Leverage: Threshold GARCH In a TARCH model the reaction to shocks is captured by a stepwise linear fnct. that differs for positive vs. negative shocks A positive θ will again capture the leverage effect This model is sometimes referred to as the GJR-GARCH model or threshold GARCH (TARCH) model GJR = Glosten, Jagannathan, and Runkle In this model, because when 50% of the shocks will be negative and the other 50% positive, long run variance is [ /(1 - ( θ) - )]; the persistence index is [ ( θ) + ] There is also a smaller literature that has connected time-varying volatility not to time series features, but to observable economic phenomena, especially at daily frequencies For instance, days where no trading takes place days that follow a weekend or a holiday have higher variance: where IT=1 in correspondence to a day that follows a weekend 28
29 GARCH Models with Predetermined Variables Other predetermined variables could be yesterday s trading volume or prescheduled news announcement dates such as company earnings and FOMC meetings dates Option implied volatilities have quite a high predictive value in forecasting next-day variance, e.g., the CBOE VIX (squared) In general, such models that use explanatory variables to capture time-variation in variance are represented as: where X t are predetermined variables Important to ensure that the GARCH model always generates a positive variance forecast You need to ensure that,,, and g(x t ) are all positive How do you estimate a GARCH model? This means, how do you estimated the fixed but unknown parameters,, and? 29
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