Statistics for nance - 2018-2019 Part II Fabio Bacchini University Rome 3
. 1 Financial time series: prices and returns 2 Normality conditions 3 ARCH-GARCH Extension of GARCH: TGARCH EGARCH Bacchini (Univ. Rome 3) Statistica per la nanza 1 / 40
Introductional concepts Most nancial studies involve returns, instead of prices, of assets: an asset is a complete and scale-free summary of the investment opportunity return series are easier to handle than price series because the former have more attractive statistical properties. One-Period Simple Return: Holding the asset for one period from date t 1 to date t would result in a simple gross return 1 + R t = P t P t 1 or P t = P t 1 (1 + R t ) The corresponding one-period simple net return or simple return is R t = P t P t 1 1 = P t P t 1 P t 1 Bacchini (Univ. Rome 3) Statistica per la nanza 2 / 40
Multiperiod Simple Return 1+R t [k] = P t = P t X P t 1 X... P t k+1 = (1+R t )(1+R t 1 )... (1+R t k+1 ) P t k P t 1 P t 2 P t k = Π k 1 j=0 (1 + R t j) Thus, the k-period simple gross return is just the product of the k one-period simple gross returns involved. This is called a compound return. The k-period simple net return is R t [k] = P t P t k P t k Bacchini (Univ. Rome 3) Statistica per la nanza 3 / 40
Daily Prices of Apple Stock from December 10 to 14, 2018 the weekly simple gross return of the stock is R t [5] = 165.48 169.60 = 0.9757075 and the weekly simple return is 0.0242925 = 2.4% Bacchini (Univ. Rome 3) Statistica per la nanza 4 / 40
Continuously Compounded Return The natural logarithm of the simple gross return of an asset is called the continuously compounded return or log return: r t = ln(1 + R t ) = ln( P t P t 1 ) = p t p t 1 where p t = ln(p t ). Continuously compounded returns r t enjoy some advantages over the simple net returns R t r t = ln(1 + R t [k]) = ln(1 + R t ) + ln(1 + R t 1 ) +... + ln(1 + R t k+1 ) Bacchini (Univ. Rome 3) Statistica per la nanza 5 / 40
. prices and trading volume of Apple stock Bacchini (Univ. Rome 3) Statistica per la nanza 6 / 40
. prices and trading volume of Apple stock last 4 months Bacchini (Univ. Rome 3) Statistica per la nanza 7 / 40
. closing prices of Apple stock Bacchini (Univ. Rome 3) Statistica per la nanza 8 / 40
. daily returns of Apple stock (log) Bacchini (Univ. Rome 3) Statistica per la nanza 9 / 40
U.S. Nasdaq stock returns Bacchini (Univ. Rome 3) Statistica per la nanza 10 / 40
U.S. Nasdaq stock returns vs normal distribution Bacchini (Univ. Rome 3) Statistica per la nanza 11 / 40
U.S. Nasdaq stock returns, annual variance Bacchini (Univ. Rome 3) Statistica per la nanza 12 / 40
. Characteristics of nancial time series slightly dierent from the normal distribution the hp of stationarity for the variance does not seem to hold so we need for tests to check the normal distribution introduce models that are able to manage dierence in the variance along the time Bacchini (Univ. Rome 3) Statistica per la nanza 13 / 40
Detect normality In statistics, skewness and kurtosis, which are normalized third and fourth central moments of X, are often used to summarize the extent of asymmetry and tail thickness. Specically, the skewness and kurtosis of X are dened as: S(x) = E[ (X µ x) 3 σ 3 x ] K(x) = E[ (X µ x) 4 ] The quantity K(x) 3 is called the excess kurtosis because K(x) = 3 for a normal distribution. Thus, the excess kurtosis of a normal random variable is zero. A distribution with positive excess kurtosis is said to have heavy tails, implying that the distribution puts more mass on the tails of its support than a normal distribution does. Such a distribution is said to be leptokurtic. On the other hand, a distribution with negative excess kurtosis has short tails (e.g., a uniform distribution over a nite interval). Such a distribution is said to be platykurtic. In nance, the rst fourth moments of a random variable are used to describe the behavior of asset returns. σ 4 x Bacchini (Univ. Rome 3) Statistica per la nanza 14 / 40
If X is a normal random variable, then S(x) and K(x) 3 are distributed asymptotically as normal with zero mean and variances 6/T and 24/T, respectively; see Snedecor and Cochran (1980, p. 78). These asymptotic properties can be used to test the normality of asset returns. These asymptotic properties can be used to test the normality of asset returns. Given an asset return series (r 1,..., r T ) to test the skewness of the returns, we consider the null hypothesis H o : S(r) = 0 versus the alternative hypothesis H a : S(r) 0. The t ratio statistic of the sample skewness is t = Ŝ(r) 6/T Similarly, one can test the excess kurtosis of the return series using the hypothesis H o : K(r) 3 = 0: t = ˆK(r) 3 24/T Bacchini (Univ. Rome 3) Statistica per la nanza 15 / 40
Jarque and Bera (1987) combine the two prior tests and use the test statistic JB = Ŝ 2 (r) 6/T + ( ˆK(r) 3) 2 24/T which is asymptotically distributed as a chi-squared random variable with 2 degrees of freedom, to test for the normality of r t. One rejects H o of normality if the p value of the JB statistic is less than the signicance level. Bacchini (Univ. Rome 3) Statistica per la nanza 16 / 40
Examples: white noise vs Nasdaq White noise x=rnorm(100,1) wn <- ts(x, start=2000,frequency=12) (Skewness) t=-0.8497506 (Kurtosis) t=0.03926036 JB X-squared: 0.7712 P VALUE: Asymptotic p Value: 0.68 Nasdaq (Skewness) t=-4.924424 (Kurtosis) t=5.763947 JB X-squared: 59.1128 P VALUE: Asymptotic p Value: 1.458e-13 Bacchini (Univ. Rome 3) Statistica per la nanza 17 / 40
Use of Ljung-box statistic the Ljung Box statistic of the residuals can be used to check the adequacy of a tted model If the model is correctly specied, then Q(m) follows asymptotically a chi-squared distribution with m g degrees of freedom, where g denotes the number of AR or MA coecients tted in the model Ljung Box statistic could be used to check the characteristics of the returns. For example for Nasdaq X-squared = 12.064, df = 12, p-value = 0.4405 Bacchini (Univ. Rome 3) Statistica per la nanza 18 / 40
volatility An important measure in nance is the risk associated with an asset and asset volatility is perhaps the most commonly used risk measure Volatility is a key factor in options pricing and asset allocation one measure of volatility could be represented by the absolute returns. In this case the LB statistic has a p value close to zero X-squared = 222.88, df = 12, p-value < 2.2e-16 Bacchini (Univ. Rome 3) Statistica per la nanza 19 / 40
volatility similar situation appears if we look at square of residuals e 2 t X-squared = 200.59, df = 12, p-value < 2.2e-16 Bacchini (Univ. Rome 3) Statistica per la nanza 20 / 40
(x µ) 2 where x is generated by a normal ditribution Bacchini (Univ. Rome 3) Statistica per la nanza 21 / 40
(x µ) 2 where x is the U.S. Nasdaq stock returns, Bacchini (Univ. Rome 3) Statistica per la nanza 22 / 40
for nancial time series the Hp of homoskedasticity is inappropriate moreover an asset holder could be interested more in forecasts of the rate or return and its variance over a short period (the holding period). one approach to forecasting the variance is to explicitly introduce an independent variable that helps to capture the cluster volatility ɛ t = u t ht where u t is a WN process with mean 0 and variance 1 and h t and independent variable that could be also a stochstic process but independent from u t. As u t and h t are independent V (ɛ t ) = E(h t u 2 t ) = E(h t )E(u 2 t ) = E(h t ). If h t is costant the process is homoskedastic otherwise we have heteroskedasticity Bacchini (Univ. Rome 3) Statistica per la nanza 23 / 40
the introduction of h t allows for processes with a kurtosis greater than the one associated to a normal distribution k ɛ = E(ɛ4 t ) E(ɛ 2 t ) 2 = E(h2 t u 4 t ) E(h t u 2 t ) 2 but using the hp of independency for u t and h t k ɛ = E(h2 t )E(u 4 t ) E(h t ) 2 E(u 2 t ) 2 = E(h2 t ) E(h t ) 2 k u but E(h 2 t ) > E(h t ) 2 implying k ɛ > k u. If u t is normal ɛ t will be leptokurtic Bacchini (Univ. Rome 3) Statistica per la nanza 24 / 40
Specication of h t, ARCH As seen before, h t could be dened as in Engle (1982) V (ɛ t I t 1 = E(h t I t 1 ) = h t or, more general h t = α 0 + α 1 ɛ 2 t 1 h t = α o + p α i ɛ 2 t i the previous equation is called an autoregressive conditional heteroskedastic ARCH model. The coecients α i must satisfy some regularity conditions to ensure that the unconditional variance is positive and of nite order. In the rst order case α 0 > 0 and 0 < α 1 < 1 i=1 Bacchini (Univ. Rome 3) Statistica per la nanza 25 / 40
Specication of h t, GARCH ARCH model provide cluster of variance as well a kurtosis behaviour in line with empirical ndings associated to the nancial time series however ARCH model could be not parsimonious and rather restrictive the order of an ARCH model could be investigated looking at the PACF of the square of the residuals To match these problems we introduce the GARCH (p,q) model h t = α o + where G stays for Generalised. p α i ɛ 2 t i + i=1 q β i h t i i=1 Bacchini (Univ. Rome 3) Statistica per la nanza 26 / 40
For p = q = 1 h t = α o + α 1 ɛ 2 t 1 + β 1 h t 1 Only lower order GARCH models are used in most applications, say GARCH(1,1), GARCH(2,1), and GARCH(1,2) models. In many situations, a GARCH(1,1) model appears to be adequate Bacchini (Univ. Rome 3) Statistica per la nanza 27 / 40
Building a volatility model for an asset return series consists of four steps (Tsay, pag. 181) Specify a mean equation by testing for serial dependence in the data and, if necessary, building an econometric model (e.g., an ARMA model) for the return series to remove any linear dependence. Use the residuals of the mean equation to test for ARCH eects. Specify a volatility model (ARCH/GARCH) if ARCH eects are statistically signicant, and perform a joint estimation of the mean and volatility equations. Check the tted model carefully and rene it if necessary. Bacchini (Univ. Rome 3) Statistica per la nanza 28 / 40
Checking for ARCH eects The square series of the residual ɛ 2 t is used to check for conditional heteroskedasticity. The tests are available use the Ljung Box statistics on the ɛ 2 t.the null hypothesis of the test statistic is that the rst m lags of ACF of the ɛ 2 t are zero use the Lagrange multiplier introduced by Engle (1982). This test is equivalent to the usual F statistic for testing α i = 0 ɛ 2 t = α 0 + α 1 ɛ 2 t 1 +... + α m ɛ 2 t m + e t t = m + 1... T where e t denotes the error term, m is a prespecied positive integer and T is the sample size. H 0 : α 1 =... = α m and the alternative hp α i 0 for some i between 1 and m. Let SSR 0 = T t=m+1 (ɛ2 t ω) 2 where ω is the sample mean of ɛ 2 t and SSR 1 = T t=m+1 ê2 where ê is the square of the residuals of the previous linear regression. F = (SSR o SSR 1 )m SSR 1 (T 2m 1 When T is suciently large, one can use mf as the test statistic, which is asymptotically a chi-squared distribution with m degrees of freedom under the null hypothesis. The decision rule is to reject the null hypothesis if mf > χ m (α) Bacchini (Univ. Rome 3) Statistica per la nanza 29 / 40
Examples Case 1: WN ARCH LM-test; Null hypothesis: no ARCH effects data: x1 Chi-squared = 6.469, df = 12, p-value = 0.8906 Case 2: Nasdaq ARCH LM-test; Null hypothesis: no ARCH effects data: y Chi-squared = 59.234, df = 12, p-value = 3.113e-08 Bacchini (Univ. Rome 3) Statistica per la nanza 30 / 40
looking at the PACF order of ARCH 3 or 5 in this case Bacchini (Univ. Rome 3) Statistica per la nanza 31 / 40
Example Nasdaq Estimate Std. Error t value Pr(> t ) mu -1.732e-16 3.403e-01 0.000 1.00000 omega 1.357e+01 3.060e+00 4.436 9.18e-06 *** alpha1 3.020e-01 9.189e-02 3.287 0.00101 ** alpha2 2.237e-01 9.450e-02 2.368 0.01790 * alpha3 2.228e-01 9.812e-02 2.271 0.02317 * Statistic p-value Jarque-Bera Test R Chi^2 19.92496 4.713567e-05 Shapiro-Wilk Test R W 0.9815998 0.001436784 Ljung-Box Test R Q(10) 9.855101 0.4532965 Ljung-Box Test R Q(15) 22.12856 0.104477 Ljung-Box Test R Q(20) 25.57427 0.1803373 Ljung-Box Test R^2 Q(10) 12.82532 0.2336029 Ljung-Box Test R^2 Q(15) 20.38557 0.1576354 Ljung-Box Test R^2 Q(20) 29.16655 0.08453137 LM Arch Test R TR^2 10.72072 0.5529922 Bacchini (Univ. Rome 3) Statistica per la nanza 32 / 40
Example Nasdaq Estimate Std. Error t value Pr(> t ) mu -1.732e-16 3.521e-01 0.000 1.000000 omega 1.107e+01 3.270e+00 3.384 0.000714 *** alpha1 2.080e-01 8.468e-02 2.456 0.014040 * alpha2 1.801e-01 1.025e-01 1.757 0.078901. alpha3 2.071e-01 1.156e-01 1.791 0.073337. alpha4 6.020e-02 7.936e-02 0.759 0.448111 alpha5 1.298e-01 8.142e-02 1.593 0.111048 Bacchini (Univ. Rome 3) Statistica per la nanza 33 / 40
Example Nasdaq Estimate Std. Error t value Pr(> t ) mu -1.732e-16 3.157e-01 0.000 1.00000 omega 2.090e+00 1.204e+00 1.736 0.08258. alpha1 1.948e-01 6.515e-02 2.990 0.00279 ** beta1 7.652e-01 7.419e-02 10.314 < 2e-16 *** Statistic p-value Jarque-Bera Test R Chi^2 37.2409 8.189184e-09 Shapiro-Wilk Test R W 0.9716604 3.306684e-05 Ljung-Box Test R Q(10) 10.76728 0.3759345 Ljung-Box Test R Q(15) 21.89668 0.1105418 Ljung-Box Test R Q(20) 27.15074 0.131076 Ljung-Box Test R^2 Q(10) 3.200491 0.9763042 Ljung-Box Test R^2 Q(15) 7.957825 0.9254624 Ljung-Box Test R^2 Q(20) 12.8379 0.8842302 LM Arch Test R TR^2 2.977933 0.9956978 Bacchini (Univ. Rome 3) Statistica per la nanza 34 / 40
model checking standardized residuals ˆɛ t / ĥt Bacchini (Univ. Rome 3) Statistica per la nanza 35 / 40
model checking Volatility ĥt Bacchini (Univ. Rome 3) Statistica per la nanza 36 / 40
use of the garchfit instruction @residuals a numeric vector with the (raw, unstandardized) residual values. @fitted a numeric vector with the fitted values. @h.t a numeric vector with the conditional variances @sigma.t a numeric vector with the conditional standard deviation. example with the estimation of a GARCH(1,1) with no mean (only last osservation) µ = 0 and residual ( ɛˆ T = 5.960375 so tted values nasdaq ˆ T = 5.960375 m3@sigma.t = h T = 6.644798 and standardized residuals ɛˆ T / h ˆT = 5.960375/6.644798 = 0.8969987 Bacchini (Univ. Rome 3) Statistica per la nanza 37 / 40
estimated values Bacchini (Univ. Rome 3) Statistica per la nanza 38 / 40
Condence interval using µ ± 2 h t Bacchini (Univ. Rome 3) Statistica per la nanza 39 / 40
Bacchini (Univ. Rome 3) Statistica per la nanza 40 / 40