Introduction to ARMA and GARCH processes

Transcription

1 Introduction to ARMA and GARCH processes Fulvio Corsi SNS Pisa 3 March 2010 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

2 Stationarity Strict stationarity: (X 1, X 2,..., X n) d = (X 1+k, X 2+k,..., X n+k ) for any integer n > 1, k Weak/second-order/covariance stationarity: E[x t] = µ E[X t µ] 2 = σ 2 < + (i.e. constant and indipendent of t) E[(X t µ)(x t+k µ)] = γ( k ) (i.e. indipendent of t for each k) Interpretation: mean and variance are constant mean reversion shocks are transient covariance between X t and X t k tends to 0 as k Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

3 Stationarity Strict stationarity: (X 1, X 2,..., X n) d = (X 1+k, X 2+k,..., X n+k ) for any integer n > 1, k Weak/second-order/covariance stationarity: E[x t] = µ E[X t µ] 2 = σ 2 < + (i.e. constant and indipendent of t) E[(X t µ)(x t+k µ)] = γ( k ) (i.e. indipendent of t for each k) Interpretation: mean and variance are constant mean reversion shocks are transient covariance between X t and X t k tends to 0 as k Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

4 Stationarity Strict stationarity: (X 1, X 2,..., X n) d = (X 1+k, X 2+k,..., X n+k ) for any integer n > 1, k Weak/second-order/covariance stationarity: E[x t] = µ E[X t µ] 2 = σ 2 < + (i.e. constant and indipendent of t) E[(X t µ)(x t+k µ)] = γ( k ) (i.e. indipendent of t for each k) Interpretation: mean and variance are constant mean reversion shocks are transient covariance between X t and X t k tends to 0 as k Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

5 Withe noise weak (uncorrelated) E(ǫ t) = 0 t V(ǫ t) = σ 2 t ρ(ǫ t, ǫ s) = 0 s t where ρ γ( t s ) γ(0) strong (independence) ǫ t I.I.D.(0, σ 2 ) Gaussian (weak=strong) ǫ t N.I.D.(0, σ 2 ) Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

6 Lag operator the Lag operator is defined as: LX t X t 1 is a linear operator: L(βX t) = β LX t = βx t 1 L(X t + Y t) = LX t + LY t = X t 1 + Y t 1 and admits power exponent, for instance: L 2 X t = L(LX t) = LX t 1 = X t 2 L k X t = X t k L 1 X t = X t+1 Some examples: X t = X t X t 1 = X t LX t = (1 L)X t y t = (θ 1 + θ 2 L)LX t = (θ 1 L + θ 2 L 2 )X t = θ 1 X t 1 + θ 2 X t 2 Expression like (θ 0 + θ 1 L + θ 2 L θ nl n ) with possibly n =, are called lag polynomial and are indicated as θ(l) Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

7 Lag operator the Lag operator is defined as: LX t X t 1 is a linear operator: L(βX t) = β LX t = βx t 1 L(X t + Y t) = LX t + LY t = X t 1 + Y t 1 and admits power exponent, for instance: L 2 X t = L(LX t) = LX t 1 = X t 2 L k X t = X t k L 1 X t = X t+1 Some examples: X t = X t X t 1 = X t LX t = (1 L)X t y t = (θ 1 + θ 2 L)LX t = (θ 1 L + θ 2 L 2 )X t = θ 1 X t 1 + θ 2 X t 2 Expression like (θ 0 + θ 1 L + θ 2 L θ nl n ) with possibly n =, are called lag polynomial and are indicated as θ(l) Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

8 Lag operator the Lag operator is defined as: LX t X t 1 is a linear operator: L(βX t) = β LX t = βx t 1 L(X t + Y t) = LX t + LY t = X t 1 + Y t 1 and admits power exponent, for instance: L 2 X t = L(LX t) = LX t 1 = X t 2 L k X t = X t k L 1 X t = X t+1 Some examples: X t = X t X t 1 = X t LX t = (1 L)X t y t = (θ 1 + θ 2 L)LX t = (θ 1 L + θ 2 L 2 )X t = θ 1 X t 1 + θ 2 X t 2 Expression like (θ 0 + θ 1 L + θ 2 L θ nl n ) with possibly n =, are called lag polynomial and are indicated as θ(l) Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

9 Lag operator the Lag operator is defined as: LX t X t 1 is a linear operator: L(βX t) = β LX t = βx t 1 L(X t + Y t) = LX t + LY t = X t 1 + Y t 1 and admits power exponent, for instance: L 2 X t = L(LX t) = LX t 1 = X t 2 L k X t = X t k L 1 X t = X t+1 Some examples: X t = X t X t 1 = X t LX t = (1 L)X t y t = (θ 1 + θ 2 L)LX t = (θ 1 L + θ 2 L 2 )X t = θ 1 X t 1 + θ 2 X t 2 Expression like (θ 0 + θ 1 L + θ 2 L θ nl n ) with possibly n =, are called lag polynomial and are indicated as θ(l) Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

10 Moving Average (MA) process The simplest way to construct a stationary process is to use a lag polynomial θ(l) with j=0 θ2 j < to construct a sort of weighted moving average of withe noises ǫ t, i.e. MA(q) Y t = θ(l)ǫ t = ǫ t + θ 1 ǫ t 1 + θ 2 ǫ t θ qǫ t q Example, MA(1) Y t = ǫ t + θǫ t 1 = (1 + θl)ǫ t being E[Y t] = 0 γ(0) = E[Y ty t] = E[(ǫ t + θǫ t 1 )(ǫ t + θǫ t 1 )] = σ 2 (1 + θ 2 ); and, γ(1) = E[Y ty t 1 ] = E[(ǫ t + θǫ t 1 )(ǫ t 1 + θǫ t 2 )] = σ 2 θ; γ(k) = E[Y ty t k ] = E[(ǫ t + θǫ t 1 )(ǫ t k + θǫ t k 1 )] = 0 k > 1 ρ(1) = γ(1) γ(0) = θ 1 + θ 2 ρ(k) = γ(k) γ(0) = 0 k > 1 hence, while a withe noise is 0-correlated, MA(1) is 1-correlated (i.e. it has only the first correlation ρ(1) different from zero) Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

11 Moving Average (MA) process The simplest way to construct a stationary process is to use a lag polynomial θ(l) with j=0 θ2 j < to construct a sort of weighted moving average of withe noises ǫ t, i.e. MA(q) Y t = θ(l)ǫ t = ǫ t + θ 1 ǫ t 1 + θ 2 ǫ t θ qǫ t q Example, MA(1) Y t = ǫ t + θǫ t 1 = (1 + θl)ǫ t being E[Y t] = 0 γ(0) = E[Y ty t] = E[(ǫ t + θǫ t 1 )(ǫ t + θǫ t 1 )] = σ 2 (1 + θ 2 ); and, γ(1) = E[Y ty t 1 ] = E[(ǫ t + θǫ t 1 )(ǫ t 1 + θǫ t 2 )] = σ 2 θ; γ(k) = E[Y ty t k ] = E[(ǫ t + θǫ t 1 )(ǫ t k + θǫ t k 1 )] = 0 k > 1 ρ(1) = γ(1) γ(0) = θ 1 + θ 2 ρ(k) = γ(k) γ(0) = 0 k > 1 hence, while a withe noise is 0-correlated, MA(1) is 1-correlated (i.e. it has only the first correlation ρ(1) different from zero) Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

12 Properties MA(q) In general for a MA(q) process we have and γ(0) = σ 2 (1 + θ θ θ2 q ) γ(k) = q k σ 2 θ j θ j+k j=0 k q = 0 k > q q k ρ(k) = j=0 θ jθ j+k 1 + q j=1 θ2 j k q = 0 k > q Hence, an MA(q) is q-correlated and it can also be shown that any stationary q-correlated process can be represented as an MA(q). Wold Theorem: any mean zero covariance stationary process can be represented in the form, MA( ) + deterministic component (the two being uncorrelated). But, given a q-correlated process, is the MA(q) process unique? In general no, indeed it can be shown that for a q-correlated process there are 2 q possible MA(q) with same autocovariance structure. However, there is only one MA(q) which is invertible. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

13 Properties MA(q) In general for a MA(q) process we have and γ(0) = σ 2 (1 + θ θ θ2 q ) γ(k) = q k σ 2 θ j θ j+k j=0 k q = 0 k > q q k ρ(k) = j=0 θ jθ j+k 1 + q j=1 θ2 j k q = 0 k > q Hence, an MA(q) is q-correlated and it can also be shown that any stationary q-correlated process can be represented as an MA(q). Wold Theorem: any mean zero covariance stationary process can be represented in the form, MA( ) + deterministic component (the two being uncorrelated). But, given a q-correlated process, is the MA(q) process unique? In general no, indeed it can be shown that for a q-correlated process there are 2 q possible MA(q) with same autocovariance structure. However, there is only one MA(q) which is invertible. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

14 Properties MA(q) In general for a MA(q) process we have and γ(0) = σ 2 (1 + θ θ θ2 q ) γ(k) = q k σ 2 θ j θ j+k j=0 k q = 0 k > q q k ρ(k) = j=0 θ jθ j+k 1 + q j=1 θ2 j k q = 0 k > q Hence, an MA(q) is q-correlated and it can also be shown that any stationary q-correlated process can be represented as an MA(q). Wold Theorem: any mean zero covariance stationary process can be represented in the form, MA( ) + deterministic component (the two being uncorrelated). But, given a q-correlated process, is the MA(q) process unique? In general no, indeed it can be shown that for a q-correlated process there are 2 q possible MA(q) with same autocovariance structure. However, there is only one MA(q) which is invertible. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

15 Invertibility conditions for MA first consider the MA(1) case: given the result Y t = (1 + θl)ǫ t (1 + θl) 1 = (1 θl + θ 2 L 2 θ 3 L 3 + θ 4 L ) = inverting the θ(l) lag polynomial, we can write (1 θl + θ 2 L 2 θ 3 L 3 + θ 4 L )Y t = ǫ t which can be considered an AR( ) process. ( θl) i If an MA process can be written as an AR( ) of this type, such MA representation is said to be invertible. For MA(1) process the invertibility condition is given by θ < 1. For a general MA(q) process Y t = (1 + θ 1 L + θ 2 L θ ql q )ǫ t the invertibility conditions are that the roots of the lag polynomial 1 + θ 1 z + θ 2 z θ qz q = 0 lie outside the unit circle. Then the MA(q) can be written as an AR( ) by inverting θ(l). i=0 Invertibility also has important practical consequence in application. In fact, given that the ǫ t are not observable they have to be reconstructed from the observed Y s through the AR( ) representation. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

16 Invertibility conditions for MA first consider the MA(1) case: given the result Y t = (1 + θl)ǫ t (1 + θl) 1 = (1 θl + θ 2 L 2 θ 3 L 3 + θ 4 L ) = inverting the θ(l) lag polynomial, we can write (1 θl + θ 2 L 2 θ 3 L 3 + θ 4 L )Y t = ǫ t which can be considered an AR( ) process. ( θl) i If an MA process can be written as an AR( ) of this type, such MA representation is said to be invertible. For MA(1) process the invertibility condition is given by θ < 1. For a general MA(q) process Y t = (1 + θ 1 L + θ 2 L θ ql q )ǫ t the invertibility conditions are that the roots of the lag polynomial 1 + θ 1 z + θ 2 z θ qz q = 0 lie outside the unit circle. Then the MA(q) can be written as an AR( ) by inverting θ(l). i=0 Invertibility also has important practical consequence in application. In fact, given that the ǫ t are not observable they have to be reconstructed from the observed Y s through the AR( ) representation. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

17 Auto-Regressive Process (AR) A general AR process is defined as φ(l)y t = ǫ t It is always invertible but not always stationary. Example: AR(1) (1 φl)y t = ǫ t or Y t = φy t 1 + ǫ t by inverting the lag polynomial (1 φl) the AR(1) can be written as Y t = (1 φl) 1 ǫ t = (φl) i ǫ t = φ i ǫ t i = MA( ) i=0 hence the stationarity condition is that φ < 1. From this representation we can apply the general formula of MA to compute γ( ) and ρ( ). In particular, ρ(k) = φ k k i.e. monotonic exponential decay for φ > 0 and exponentially damped oscillatory decay for φ < 0. In general an AR(p) process i=0 Y t = φ 1 Y t 1 + φ 2 Y t φ py t p + ǫ t is stationarity if all the roots of the characteristic equation of the lag polynomial are outside the unit circle. 1 φ 1 z φ 2 z 2... φ pz p = 0 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

18 Auto-Regressive Process (AR) A general AR process is defined as φ(l)y t = ǫ t It is always invertible but not always stationary. Example: AR(1) (1 φl)y t = ǫ t or Y t = φy t 1 + ǫ t by inverting the lag polynomial (1 φl) the AR(1) can be written as Y t = (1 φl) 1 ǫ t = (φl) i ǫ t = φ i ǫ t i = MA( ) i=0 hence the stationarity condition is that φ < 1. From this representation we can apply the general formula of MA to compute γ( ) and ρ( ). In particular, ρ(k) = φ k k i.e. monotonic exponential decay for φ > 0 and exponentially damped oscillatory decay for φ < 0. In general an AR(p) process i=0 Y t = φ 1 Y t 1 + φ 2 Y t φ py t p + ǫ t is stationarity if all the roots of the characteristic equation of the lag polynomial are outside the unit circle. 1 φ 1 z φ 2 z 2... φ pz p = 0 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

19 State Space Representation of AR(p) to gain more intuition on the AR stationarity conditions write an AR(p) in its state space form Y t φ 1 φ 2 φ 3... φ p 1 φ p Y t 1 ǫ t Y t Y. = t Y t p Y t p 0 X t = F X t 1 + v t Hence, the expected value of X t satisfy, E[X t] = F X t 1 and E[X t+j ] = F j+1 X t 1 is a linear map in R p whose dynamic properties are given by the eigenvalues of the matrix F. The eigenvalues of F are given by solving the characteristic equation λ p φ 1 λ p 1 φ 2 λ p 2... φ p 1 λ φ p = 0. Comparing this with the characteristic equation of the lag polynomial φ(l) 1 φ 1 z φ 2 z 2... φ p 1 z p 1 φ pz p = 0 we can see that the roots of the 2 equations are such that z 1 = λ 1 1, z 2 = λ 1 2,..., z p = λ 1 p Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

20 State Space Representation of AR(p) to gain more intuition on the AR stationarity conditions write an AR(p) in its state space form Y t φ 1 φ 2 φ 3... φ p 1 φ p Y t 1 ǫ t Y t Y. = t Y t p Y t p 0 X t = F X t 1 + v t Hence, the expected value of X t satisfy, E[X t] = F X t 1 and E[X t+j ] = F j+1 X t 1 is a linear map in R p whose dynamic properties are given by the eigenvalues of the matrix F. The eigenvalues of F are given by solving the characteristic equation λ p φ 1 λ p 1 φ 2 λ p 2... φ p 1 λ φ p = 0. Comparing this with the characteristic equation of the lag polynomial φ(l) 1 φ 1 z φ 2 z 2... φ p 1 z p 1 φ pz p = 0 we can see that the roots of the 2 equations are such that z 1 = λ 1 1, z 2 = λ 1 2,..., z p = λ 1 p Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

21 ARMA(p,q) An ARMA(p,q) process is defined as φ(l)y t = θ(l)ǫ t where φ( ) and θ( ) are p th and q th lag polynomials. the process is stationary if all the roots of φ(z) 1 φ 1 z φ 2 z 2... φ p 1 z p 1 φ pz p = 0 lie outside the unit circle and, hence, admits the MA( ) representation: Y t = φ(l) 1 θ(l)ǫ t the process is invertible if all the roots of θ(z) 1 + θ 1 z + θ 2 z θ qz q = 0 lie outside the unit circle and, hence, admits the AR( ) representation: ǫ t = θ(l) 1 φ(l)y t Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

22 Estimation For AR(p) Y t = φ 1 Y t 1 + φ 2 Y t φ py t p + ǫ t OLS are consistent and under gaussianity asymptotically equivalent to MLE asymptotically efficient For example, the full Likelihood of an AR(1) can be written as L(θ) f (y T, y T 1,..., y 1 ; θ) = f Y1 (y 1 ; θ) }{{} marginal first obs T f Yt Yt 1 (y t y t 1 ; θ) t=2 }{{} conditional likelihood under normality OLS=MLE For a general ARMA(p,q) Y t = φ 1 Y t φ py t p + ǫ t + θ 1 ǫ t θ qǫ t q Y t 1 is correlated with ǫ t 1,..., ǫ t q OLS not consistent. MLE with numerical optimization procedures. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

23 Estimation For AR(p) Y t = φ 1 Y t 1 + φ 2 Y t φ py t p + ǫ t OLS are consistent and under gaussianity asymptotically equivalent to MLE asymptotically efficient For example, the full Likelihood of an AR(1) can be written as L(θ) f (y T, y T 1,..., y 1 ; θ) = f Y1 (y 1 ; θ) }{{} marginal first obs T f Yt Yt 1 (y t y t 1 ; θ) t=2 }{{} conditional likelihood under normality OLS=MLE For a general ARMA(p,q) Y t = φ 1 Y t φ py t p + ǫ t + θ 1 ǫ t θ qǫ t q Y t 1 is correlated with ǫ t 1,..., ǫ t q OLS not consistent. MLE with numerical optimization procedures. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

24 Prediction write the model in its AR( ) representation: η(l)(y t µ) = ǫ t then the optimal prediction of Y t+s is given by [ η(l) 1 ] E[Y t+s Y t, Y t 1,...] = µ + L s η(l)(y t µ) + which is known as Wiener-Kolmogorov prediction formula. with [ L k] + = 0 for k < 0 In the case of an AR(p) process the prediction formula can also be written as E[Y t+s Y t, Y t 1,...] = µ + f (s) 11 (Yt µ) + f(s) 12 (Y t 1 µ) f (s) 1p (Y t p+1 µ) where f (j) 11 is the element (1, 1) of the matrix F j. The easiest way to compute prediction from AR(p) model is, however, through recursive methods. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

25 Prediction write the model in its AR( ) representation: η(l)(y t µ) = ǫ t then the optimal prediction of Y t+s is given by [ η(l) 1 ] E[Y t+s Y t, Y t 1,...] = µ + L s η(l)(y t µ) + which is known as Wiener-Kolmogorov prediction formula. with [ L k] + = 0 for k < 0 In the case of an AR(p) process the prediction formula can also be written as E[Y t+s Y t, Y t 1,...] = µ + f (s) 11 (Yt µ) + f(s) 12 (Y t 1 µ) f (s) 1p (Y t p+1 µ) where f (j) 11 is the element (1, 1) of the matrix F j. The easiest way to compute prediction from AR(p) model is, however, through recursive methods. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

26 Box-Jenkins Approach check for stationarity: if not try different transformation (ex differentiation ARIMA models) Identification: check the autocorrelation (ACF) function: a q-correlated process is an MA(q) model check the partial autocorrelation (PACF) function: for an AR(p) process, while the k lag ACF can be interpreted as simple regression Y t = ρ(k)y t k + error, the k lag PACF can be seen as a multiple regression Y t = b 1 Y t 1 + b 2 Y t b k Y t k + error it can be computed by solving the Yule-Walker system: b 1 b 2. b k = 1 γ(0) γ(1)... γ(k 1) γ(1) γ(1) γ(0)... γ(k 2) γ(2) γ(k 1) γ(k 2)... γ(0) γ(k) Importantly, AR(p) processes are p partially correlated identification of AR order Validation: check the appropriateness of the model by some measure of fit. AIC/Akaike = T log ˆσ 2 e + 2 m BIC/Schwarz = T log ˆσ 2 e + m log T with σ 2 e estimation error variance, m = p + q + 1 n of parameters, and T n of obs Diagnostic checking of the residuals. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

27 Box-Jenkins Approach check for stationarity: if not try different transformation (ex differentiation ARIMA models) Identification: check the autocorrelation (ACF) function: a q-correlated process is an MA(q) model check the partial autocorrelation (PACF) function: for an AR(p) process, while the k lag ACF can be interpreted as simple regression Y t = ρ(k)y t k + error, the k lag PACF can be seen as a multiple regression Y t = b 1 Y t 1 + b 2 Y t b k Y t k + error it can be computed by solving the Yule-Walker system: b 1 b 2. b k = 1 γ(0) γ(1)... γ(k 1) γ(1) γ(1) γ(0)... γ(k 2) γ(2) γ(k 1) γ(k 2)... γ(0) γ(k) Importantly, AR(p) processes are p partially correlated identification of AR order Validation: check the appropriateness of the model by some measure of fit. AIC/Akaike = T log ˆσ 2 e + 2 m BIC/Schwarz = T log ˆσ 2 e + m log T with σ 2 e estimation error variance, m = p + q + 1 n of parameters, and T n of obs Diagnostic checking of the residuals. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

28 Box-Jenkins Approach check for stationarity: if not try different transformation (ex differentiation ARIMA models) Identification: check the autocorrelation (ACF) function: a q-correlated process is an MA(q) model check the partial autocorrelation (PACF) function: for an AR(p) process, while the k lag ACF can be interpreted as simple regression Y t = ρ(k)y t k + error, the k lag PACF can be seen as a multiple regression Y t = b 1 Y t 1 + b 2 Y t b k Y t k + error it can be computed by solving the Yule-Walker system: b 1 b 2. b k = 1 γ(0) γ(1)... γ(k 1) γ(1) γ(1) γ(0)... γ(k 2) γ(2) γ(k 1) γ(k 2)... γ(0) γ(k) Importantly, AR(p) processes are p partially correlated identification of AR order Validation: check the appropriateness of the model by some measure of fit. AIC/Akaike = T log ˆσ 2 e + 2 m BIC/Schwarz = T log ˆσ 2 e + m log T with σ 2 e estimation error variance, m = p + q + 1 n of parameters, and T n of obs Diagnostic checking of the residuals. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

29 ARIMA Integrated ARMA model: ARIMA(p,1,q) denote a nonstationary process Y t for which the first difference Y t Y t 1 = (1 L)Y t is a stationary ARMA(p,q) process. Y t is said to be integrated of order 1 or I(1). If 2 differentiations of Y t are necessary to get a stationary process i.e. (1 L) 2 Y t then the process Y t is said to be integrated of order 2 or I(2). I(0) indicate a stationary process. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

30 ARIMA Integrated ARMA model: ARIMA(p,1,q) denote a nonstationary process Y t for which the first difference Y t Y t 1 = (1 L)Y t is a stationary ARMA(p,q) process. Y t is said to be integrated of order 1 or I(1). If 2 differentiations of Y t are necessary to get a stationary process i.e. (1 L) 2 Y t then the process Y t is said to be integrated of order 2 or I(2). I(0) indicate a stationary process. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

31 ARFIMA The k-difference operator (1 L) n with integer n can be generalized to a fractional difference operator (1 L) d with 0 < d < 1 defined by the binomial expansion (1 L) d = 1 dl + d(d 1)L 2 /2! d(d 1)(d 2)L 3 /3! +... obtaining a fractionally integrated process of order d i.e. I(d). If d < 0.5 the process is cov stationary and admits an AR( ) representation. The usefulness of a fractional filter (1 L) d is that it produces hyperbolic decaying autocorrelations i.e. the so called long memory. In fact, for ARFIMA(p,d,q) processes φ(l)(1 L) d Y t = θ(l)ǫ t the autocorrelation functions is proportional to ρ(k) ck 2d 1 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

32 ARFIMA The k-difference operator (1 L) n with integer n can be generalized to a fractional difference operator (1 L) d with 0 < d < 1 defined by the binomial expansion (1 L) d = 1 dl + d(d 1)L 2 /2! d(d 1)(d 2)L 3 /3! +... obtaining a fractionally integrated process of order d i.e. I(d). If d < 0.5 the process is cov stationary and admits an AR( ) representation. The usefulness of a fractional filter (1 L) d is that it produces hyperbolic decaying autocorrelations i.e. the so called long memory. In fact, for ARFIMA(p,d,q) processes φ(l)(1 L) d Y t = θ(l)ǫ t the autocorrelation functions is proportional to ρ(k) ck 2d 1 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

33 ARCH and GARCH models Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

34 Basic Structure and Properties of ARMA model standard time series models have: Y t = E[Y t Ω t 1 ] + ǫ t E[Y t Ω t 1 ] = f (Ω t 1 ; θ) ] Var [Y t Ω t 1 ] = E [ǫ 2 t Ω t 1 = σ 2 hence, Conditional mean: varies with Ω t 1 Conditional variance: constant (unfortunately) k-step-ahead mean forecasts: generally depends on Ω t 1 k-step-ahead variance forecasts : depends only on k, not on Ω t 1 (again unfortunately) Unconditional mean: constant Unconditional variance: constant Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

35 AutoRegressive Conditional Heteroskedasticity (ARCH) model Engle (1982, Econometrica) intruduced the ARCH models: Y t = E[Y t Ω t 1 ] + ǫ t E[Y t Ω t 1 ] = f (Ω t 1 ; θ) [ ] Var [Y t Ω t 1 ] = E ǫ 2 t Ω t 1 = σ (Ω t 1 ; θ) σt 2 hence, Conditional mean: varies with Ω t 1 Conditional variance: varies with Ω t 1 k-step-ahead mean forecasts: generally depends on Ω t 1 k-step-ahead variance forecasts : generally depends on Ω t 1 Unconditional mean: constant Unconditional variance: constant Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

36 ARCH(q) How to parameterize E [ ǫ 2 t Ω t 1] = σ((ωt 1 ; θ) σ 2 t? ARCH(q) postulated that the conditional variance is a linear function of the past q squared innovations q σt 2 = ω + α i ǫ 2 t i = ω + α(l)ǫ2 t 1 i=1 Defining v t = ǫ 2 t σ 2 t, the ARCH(q) model can be written as ǫ 2 t = ω + α(l)ǫ 2 t 1 + vt Since E t 1 (v t) = 0, the model corresponds directly to an AR(q) model for the squared innovations, ǫ 2 t. The process is covariance stationary if and only if the sum of the positive AR parameters is less than 1. Then, the unconditional variance of ǫ t is Var(ǫ t) = σ 2 = w/(1 α 1 α 2... α q). Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

37 ARCH(q) How to parameterize E [ ǫ 2 t Ω t 1] = σ((ωt 1 ; θ) σ 2 t? ARCH(q) postulated that the conditional variance is a linear function of the past q squared innovations q σt 2 = ω + α i ǫ 2 t i = ω + α(l)ǫ2 t 1 i=1 Defining v t = ǫ 2 t σ 2 t, the ARCH(q) model can be written as ǫ 2 t = ω + α(l)ǫ 2 t 1 + vt Since E t 1 (v t) = 0, the model corresponds directly to an AR(q) model for the squared innovations, ǫ 2 t. The process is covariance stationary if and only if the sum of the positive AR parameters is less than 1. Then, the unconditional variance of ǫ t is Var(ǫ t) = σ 2 = w/(1 α 1 α 2... α q). Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

38 AR(1)-ARCH(1) Example: the AR(1)-ARCH(1) model Y t = φy t 1 + ǫ t σ 2 t = ω + αǫ 2 t 1 ǫ t N(0, σ 2 t ) - Conditional mean: E(Y t Ω t 1 ) = φy t 1 - Conditional variance: E([Y t E(Y t Ω t 1 )] 2 Ω t 1 = ω + αǫ 2 t 1 - Unconditional mean: E(Y t) = 0 - Unconditional variance: E(Y t E(Y t)) 2 = 1 ω (1 φ 2 ) (1 α) Note that the unconditional distribution of ǫ t has Fat Tail. In fact, the unconditional kurtosis E(ǫ 4 t )/E(ǫ 2 t ) 2 is [ ] [ ] [ ] E ǫ 4 t = E E(ǫ 4 t Ω t 1) = 3E σt 4 = 3[Var(σ t 2 ) + E(σ2 t )2 ] = 3[Var(σ t 2 }{{} ) +E(ǫ 2 t )2 ] > 3E(ǫ 2 t )2. >0 Hence, Kurtosis(ǫ t) = E(ǫ 4 t )/E(ǫ 2 t ) 2 > 3 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

39 AR(1)-ARCH(1) Example: the AR(1)-ARCH(1) model Y t = φy t 1 + ǫ t σ 2 t = ω + αǫ 2 t 1 ǫ t N(0, σ 2 t ) - Conditional mean: E(Y t Ω t 1 ) = φy t 1 - Conditional variance: E([Y t E(Y t Ω t 1 )] 2 Ω t 1 = ω + αǫ 2 t 1 - Unconditional mean: E(Y t) = 0 - Unconditional variance: E(Y t E(Y t)) 2 = 1 ω (1 φ 2 ) (1 α) Note that the unconditional distribution of ǫ t has Fat Tail. In fact, the unconditional kurtosis E(ǫ 4 t )/E(ǫ 2 t ) 2 is [ ] [ ] [ ] E ǫ 4 t = E E(ǫ 4 t Ω t 1) = 3E σt 4 = 3[Var(σ t 2 ) + E(σ2 t )2 ] = 3[Var(σ t 2 }{{} ) +E(ǫ 2 t )2 ] > 3E(ǫ 2 t )2. >0 Hence, Kurtosis(ǫ t) = E(ǫ 4 t )/E(ǫ 2 t ) 2 > 3 Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

40 GARCH(p,q) Problem: empirical volatility very persistent Large q i.e. too many α s Bollerslev (1986, J. of Econometrics) proposed the Generalized ARCH model. The GARCH(p,q) is defined as q p σt 2 = ω + α i ǫ 2 t i + β j σt j 2 = ω + α(l)ǫ2 t 1 + β(l)σ2 t 1 i=1 j=1 As before also the GARCH(p,q) can be rewritten as ǫ 2 t = ω + [α(l) + β(l)] ǫ 2 t 1 β(l)v t 1 + v t which defines an ARMA[max(p, q),p] model for et z. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

41 GARCH(p,q) Problem: empirical volatility very persistent Large q i.e. too many α s Bollerslev (1986, J. of Econometrics) proposed the Generalized ARCH model. The GARCH(p,q) is defined as q p σt 2 = ω + α i ǫ 2 t i + β j σt j 2 = ω + α(l)ǫ2 t 1 + β(l)σ2 t 1 i=1 j=1 As before also the GARCH(p,q) can be rewritten as ǫ 2 t = ω + [α(l) + β(l)] ǫ 2 t 1 β(l)v t 1 + v t which defines an ARMA[max(p, q),p] model for et z. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

42 GARCH(1,1) By far the most commonly used is the GARCH(1,1): σt 2 = ω + α ǫ 2 t 1 + β j σt 1 2 with ω > 0, α > 0, β > 0. By recursive substitution, the GARCH(1,1) may be written as the following ARCH( ): σt 2 = ω(1 β) + α β i 1 ǫ 2 t i i=1 which reduces to an exponentially weighted moving average filter for ω = 0 and α + β = 1 (sometimes referred to as Integrated GARCH or IGARCH(1,1)). Moreover, GARCH(1,1) implies an ARMA(1,1) representation in the ǫ 2 t ǫ 2 t = ω + [α + β]ǫ 2 t 1 βv t 1 + v t Forecasting. Denoting the unconditional variance σ 2 ω(1 α β) 1 we have: ˆσ 2 t+h t = σ2 + (α + β) h 1 (σ 2 t+1 σ2 ) showing that the forecasts of the conditional variance revert to the long-run unconditional variance at an exponential rate dictated by α + β Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

43 GARCH(1,1) By far the most commonly used is the GARCH(1,1): σt 2 = ω + α ǫ 2 t 1 + β j σt 1 2 with ω > 0, α > 0, β > 0. By recursive substitution, the GARCH(1,1) may be written as the following ARCH( ): σt 2 = ω(1 β) + α β i 1 ǫ 2 t i i=1 which reduces to an exponentially weighted moving average filter for ω = 0 and α + β = 1 (sometimes referred to as Integrated GARCH or IGARCH(1,1)). Moreover, GARCH(1,1) implies an ARMA(1,1) representation in the ǫ 2 t ǫ 2 t = ω + [α + β]ǫ 2 t 1 βv t 1 + v t Forecasting. Denoting the unconditional variance σ 2 ω(1 α β) 1 we have: ˆσ 2 t+h t = σ2 + (α + β) h 1 (σ 2 t+1 σ2 ) showing that the forecasts of the conditional variance revert to the long-run unconditional variance at an exponential rate dictated by α + β Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

44 Asymmetric GARCH In standard GARCH model: σ 2 t = ω + αr 2 t 1 + βσ2 t 1 σ 2 t responds symmetrically to past returns. The so called news impact curve is a parabola conditional variance News Impact Curve symmetric GARCH asymmetric GARCH standardized lagged shocks Empirically negative r t 1 impact more than positive ones asymmetric news impact curve GJR or T-GARCH σ 2 t = ω + αr 2 t 1 + γr2 t 1 D t 1 + βσ 2 t 1 with D t = - Positive returns (good news): α - Negative returns (bad news): α + γ - Empirically γ > 0 Leverage effect Exponential GARCH (EGARCH) ln(σt 2 ) = ω + α r t 1 σ + γ r t 1 + βln(σt 1 2 t 1 σ ) t 1 { 1 if rt < 0 0 otherwise Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

45 Asymmetric GARCH In standard GARCH model: σ 2 t = ω + αr 2 t 1 + βσ2 t 1 σ 2 t responds symmetrically to past returns. The so called news impact curve is a parabola conditional variance News Impact Curve symmetric GARCH asymmetric GARCH standardized lagged shocks Empirically negative r t 1 impact more than positive ones asymmetric news impact curve GJR or T-GARCH σ 2 t = ω + αr 2 t 1 + γr2 t 1 D t 1 + βσ 2 t 1 with D t = - Positive returns (good news): α - Negative returns (bad news): α + γ - Empirically γ > 0 Leverage effect Exponential GARCH (EGARCH) ln(σt 2 ) = ω + α r t 1 σ + γ r t 1 + βln(σt 1 2 t 1 σ ) t 1 { 1 if rt < 0 0 otherwise Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24

46 Estimation A GARCH process with gaussian innovation: has conditional densities: r t Ω t 1 N(µ t(θ), σ 2 t (θ)) f(r t Ω t 1 ; θ) = 1 ( σt 1 (θ) exp 2π 1 2 ) (r t µ(θ)) σt 2 (θ) using the prediction error decomposition of the likelihood L(r T, r T 1,..., r 1 ; θ) = f(r T Ω T 1 ; θ) f(r T 1 Ω T 2 ; θ)... f(r 1 Ω 0 ; θ) the log-likelihood becomes: log L(r T, r T 1,..., r 1 ; θ) = T T 2 log(2π) log σ t(θ) 1 T 2 t=1 t=1 (r t µ(θ)) σ 2 t (θ) Non linear function in θ Numerical optimization techiques. Fulvio Corsi Introduction to ARMA () and GARCH processes SNS Pisa 3 March / 24