2. Multivariate ARMA

Transcription

1 2. Multivariate ARMA JEM 140: Quantitative Multivariate Finance IES, Charles University, Prague Summer 2018 JEM 140 () 2. Multivariate ARMA Summer / 19

2 Multivariate AR I Let r t = (r 1t,..., r kt ) 0 denote the log returns of k assets at time t I Assume that r t is stationary I De ne its mean as µ = E (r t ) where the expectation is over the joint distribution of r t. JEM 140 (IES) 2. Multivariate ARMA Summer / 19

3 Covariance Matrix I De ne the covariance matrix of r t as Γ 0 = E (r t µ) (r t µ) Γ 11 (0) Γ 12 (0) Γ 13 (0) Γ 1k (0) Γ 21 (0) Γ 22 (0) Γ 23 (0) Γ 2k (0) = Γ 31 (0) Γ 32 (0) Γ 33 (0) Γ 3k (0) Γ k1 (0) Γ k2 (0) Γ k3 (0) Γ kk (0) where h Γ ii (0) = E (r it µ i ) 2i (variances) h i Γ ij (0) = E (r it µ i ) r jt µ j for i 6= j (covariances) JEM 140 (IES) 2. Multivariate ARMA Summer / 19

4 Vector Autoregressive Models I A multivariate time series r t is a VAR(1) process if it follows the model r t = φ 0 + Φr t 1 + a t (1) where: I I I I φ 0 is a k-dimensional vector Φ is a k k matrix of coe cients fat g is a sequence of serially uncorrelated random vectors with mean zero and covariance matrix Σ It is often assumed that a t is multivariate Normal I Consider the case with k = 2 : r 1t = φ 10 + Φ 11 r 1,t 1 + Φ 12 r 2,t 1 + a 1t r 2t = φ 20 + Φ 21 r 1,t 1 + Φ 22 r 2,t 1 + a 2t JEM 140 (IES) 2. Multivariate ARMA Summer / 19

5 Moments of VAR(1) I Assume that the VAR(1) process r t is stationary I Taking expectations and using E (a t ) = 0 we obtain I Since r t is time-invariant, if the matrix (I I Moreover, E (r t ) = φ 0 + ΦE (r t 1 ) µ E (r t ) = (I Φ) 1 φ 0 Φ) 1 is nonsingular. φ 0 = (I Φ) µ and hence VAR(1) in (1) can also be written as r t µ = (I Φ) µ + Φr t 1 µ + a t = I µ Φµ + Φr t 1 µ + a t = Φ (r t 1 µ) + a t er t = Φer t 1 + a t (2) JEM 140 (IES) 2. Multivariate ARMA Summer / 19

6 Stationarity of VAR(1) I By repeated substitutions, we can rewrite (2) as er t = Φer t 1 + a t = Φ (Φer t 2 + a t 1 ) + a t = Φ 2 er t 2 + Φa t 1 + a t = Φ 2 (Φer t 3 + a t 2 ) + Φa t 1 + a t = Φ 3 er t 3 + Φ 2 a t 2 + Φa t 1 + a t =. = a t + Φa t 1 + Φ 2 a t 2 + Φ 3 a t I For the series to have a nite sum, we need Φ j! 0 as j! (all eigenvalues of Φ should be less than 1) JEM 140 (IES) 2. Multivariate ARMA Summer / 19

7 Vector Autoregressive Models of Order p I The time series r t follows a VAR(p) model if it satis es r t = φ 0 + Φ 1 r t 1 + Φ 2 r t Φ p r t p + a t for p > 0. I Using the backshift operator B j de ned by r t j = B j r t the AR(p) model can also be written as or more compactly as where is a matrix polynomial. (I Φ 1 B... Φ p B p ) r t = φ 0 + a t Φ(B)r t = φ 0 + a t Φ(B) = I Φ 1 B... Φ p B p JEM 140 (IES) 2. Multivariate ARMA Summer / 19

8 Moments of AR(p) I If r t is stationary then we have µ = E (r t ) = (I Φ 1... Φ p ) 1 φ 0 = [Φ(1)] 1 I Moreover, Cov(r t, a t ) = Σ Cov(r t j, a t ) = 0 for all j > 0 JEM 140 (IES) 2. Multivariate ARMA Summer / 19

9 Determining the order of VAR(p) I Consider the following consecutive VAR models: r t = φ 0 + Φ 1 r t 1 + a t r t = φ 0 + Φ 1 r t 1 + Φ 2 r t 2 + a t. =. r t = φ 0 + Φ 1 r t 1 + Φ 2 r t Φ i r t i + a t. =. I We can estimate the model by the maximum likelihood (MLE) and then use information criteria (e.g. AIC) to select the model order JEM 140 (IES) 2. Multivariate ARMA Summer / 19

10 Determining the order of VAR(p) I The MLE estimate of is eσ i = 1 T T h ba (i) t ba (i) t t=i+1 where ba (i) t is the MLE estimate of a t for VAR(i) and T is the legth of the time series. I The AIC of a VAR(i) model under the normality assumption is de ned as AIC (i) = ln eσ i + 2k 2 i/t I For a given vector time series, one selects the VAR order p such that i 0 AIC (p) = min 0ip0 AIC (i) where p 0 is a prespeci ed positive integer. JEM 140 (IES) 2. Multivariate ARMA Summer / 19

11 Multivariate MA I A vector moving-average model of order q, or VMA(q), is in the form r t = θ 0 + a t 1 Θ 1 a t 1 Θ q a t q or r t = θ 0 + Θ(B)a t where θ 0 is a k-dimensional vector, Θ i are k k matrices, and Θ(B) = I Θ 1 B Θ q B q is the MA matrix polynomial in the back-shift operator B. I The mean of VMA(q) is µ = E (r t ) = θ 0 JEM 140 (IES) 2. Multivariate ARMA Summer / 19

12 Multivariate ARMA I A VARMA(p, q) model can be written as Φ(B)r t = φ 0 + Θ(B)a t where Φ(B) and Θ(B) are two matrix polynomials. I For v > 0, the (i, j)th elements of the coe cient matrices Φ v and Θ v measure the linear dependence of r 1t on r j,t v and a j,t v, respectively. I The necessary and su cient condition of weak stationarity for rt is the same as that for the VAR(p) model with matrix polynomial Φ(B). JEM 140 (IES) 2. Multivariate ARMA Summer / 19

13 Example I Consider the monthly log returns of IBM stock and the S&P 500 index from January 1926 to December 1999 with 888 observations. JEM 140 (IES) 2. Multivariate ARMA Summer / 19

14 Order Determination I The following table shows the AIC output for the data: I The AIC results indicate that a VAR(3) model would be adequate. I The M(i) statistic indicates that only lags 1 and 3 are signi cant. I We can now estimate the model using MLE. r t = φ 0 + Φ 1 r t 1 + Φ 2 r t 3 + a t JEM 140 (IES) 2. Multivariate ARMA Summer / 19

15 MLE Estimates Note: In the simpli ed model, coe cients that are not signi cant at 5% were set to zero. JEM 140 (IES) 2. Multivariate ARMA Summer / 19

16 Outcomes I The model shows that IBM t = SP5 t SP5 t 3 + a 1t SP5 t = SP5 t SP5 t 3 + a 2t I The two log return series have positive and signi cant means, implying that the log prices of the two series had an upward trend over the data span. I At the 5% signi cance level, there is a unidirectional dynamic relationship from the monthly S&P 500 index return to the IBM return. I If the S&P 500 index represents the U.S. stock market, then IBM return is a ected by the past movements of the market. I However, past movements of IBM stock returns do not signi cantly a ect the U.S. market, even though the two returns have substantial concurrent correlation. JEM 140 (IES) 2. Multivariate ARMA Summer / 19

17 Outcomes I In matrix notation, the tted model can be written as IBMt = + SP5 SP5 t t 1 SP t 3 + indicating that SP5t is the driving factor of the bivariate series. a1t a 2t JEM 140 (IES) 2. Multivariate ARMA Summer / 19

18 Forecasting I For a VAR(p) model, the 1-step ahead forecast at the time origin h is r h (1) = φ 0 + and the associated forecast error is p i=1 e h (1) = a h+1 Φ i r h+1 I For 2-step ahead forecasts, we substitute r h+1 by its forecast to obtain r h (2) = φ 0 + Φ 1 r h (1) + and the associated forecast error is p i=2 i Φ i r h+2 e h (2) = a h+2 + Φ 1 [r t r h (1)] = a h+2 + Φ 1 a h+1 I If r t is stationary, then the l-step ahead forecast r h (l) converges to its mean vector µ as the forecast horizon increases. JEM 140 (IES) 2. Multivariate ARMA Summer / 19 i

19 Forecasting I The following table provides 1-step to 6-step ahead forecasts of the monthly log returns, in percentages, of IBM stock and the S&P 500 index at the forecast origin h = 888. I These forecasts are obtained by the re ned VAR(3) model. JEM 140 (IES) 2. Multivariate ARMA Summer / 19

20 Reference I Reference: Tsay (2005), chapter 8 JEM 140 (IES) 2. Multivariate ARMA Summer / 19