Econometria dei mercati finanziari c.a. A.A AR, MA and ARMA Time Series Models. Luca Fanelli. University of Bologna

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1 Econometria dei mercati finanziari c.a. A.A AR, MA and ARMA Time Series Models Luca Fanelli University of Bologna

2 For each class of models considered in these slides (AR, MA, ARMA) we focus on: - representation - estimation. We also consider forecasting issues for the AR model alone.

3 AR model: representation We start from the simplest AR model, the autoregressive model of order one, AR(1). We say that the process generating the log-return {r t } belongs to the class of covariance stationary AR(1) models if where r t = β 0 + β 1 r t 1 + u t, u t WN(0,σ 2 u) 1 <β 1 < 1 r 0 is fixed.

4 Two important remarks. 1. The AR(1) can be interpreted in the conventional way: F t := {r t,r t 1,...,r 1 } r t = E(r t F t 1 )+u t u t is a MDS with respect to F t E(r t F t 1 ):=β 0 + β 1 r t 1 hence it is aspecialcaseof the dynamic linear regression model discussed in Slides The fact that u t is a White Noise such that Var(u t ):=σ 2 u t, does not conflict with the possibility that Var(u t F t 1 ):=σ 2 t F t 1. Recall that σ 2 u:=e(u 2 t ):=E(E(u2 t F t 1))!

5 3. The stationarity restriction 1 <β 1 < 1canbe derived by following the definition (Slides 3) and imposing the following conditions: E(r t ):=μ const t Var(r t ):=σ 2 r const t Cov(r t,r t τ ):=depends on τ, not on t. Exercise: prove the statement above!

6 Define the LAG OPERATOR L: L j x t :=x t j, j integer x t :=(1 L)x t. The AR(1) can be re-written as (1 β 1 L)r t = β 0 + u t, u t WN(0,σ 2 u) where β(l):=(1 β 1 L) is known as theautoregressivepolynomial. With this definition, the AR(1) can conventionally be written as β(l)r t = β 0 + u t, u t WN(0,σ 2 u).

7 The covariance stationarity of the AR(1) model is equivalent to the condition (also known as asymptotic stationarity): every solution z C of the characteristic equation is such that z > 1. (1 β 1 z)=0

8 Recall Every real number in R can be uniquely associated with a point that lies on the real line. Every complex number in C can be uniquely associated with a point in the R R plane. Formally, if z C, then z (a, b) a + bi where (a, b) R R, i is such that i 2 =-1, a is its the real part and b is its immaginary part. If z C, z := a + bi :=(a 2 + b 2 ) 1/2 :=( (a, b) 0 ) 1/2 is the absolute value of the complex number.

9 Now consider any (scalar) polynomial of degree n: p n (x):=a 0 + a 1 x + a 2 x a n x n where x R and a i R, i =0, 1,...,n are real numbers. The equation p n (x) =0 may have solutions x that do not belong to R but to C. Consider as an example the quadratic case (n:=2) p 2 (x):=a 0 + a 1 x + a 2 x 2 ; here we know that if (a 1 ) 2 4a 2 a 0 < 0 the equation p 2 (x) =0has no real solution. A fundamental theory of algebra (due to Gauss) says that has always n solutions in C. p n (x) =0 Accordingly, we can represent any of these solutions in the R R plane. In the R R plane we can also represent the unit circle, which is the circle that crosses the points (1,0), (0,-1), (-1,0), (0,1).

10 Therefore, (covariance) stationarity requires that the solutions to the characteristic equation are such that z > 1. β(z)=(1 β 1 z)=0 If it happens that there exist solutions such that z :=1, we say that the AR(1) has a unit root and is nonstationary. (Note that all points (1,0), (0,-1), (-1,0), (0,1) are such that z :=1; it can be shown that the roots (0,- 1), (-1,0), (0,1) are related to the seasonal behavior of r t ). If it happens that there exist solutions such that z <1, we say that the AR(1) is non-stationary and explosive.

11 If the AR(1) is asymptotically stationary, i.e. if very solution z C of the characteristic equation (1 β 1 z)=0is such that z > 1, then it exists (1 β 1 L) 1 :=1 + β 1 L +(β 1 ) 2 L 2 +(β 1 ) 3 L where := j=0 (β 1 ) j L j := j=0 θ j L j :=θ(l) θ j := (β 1 ) j 0asj and θ(1):= j=0 θ j := j=0 (β 1 ) j (1 β 1 ) 1 <.

12 By multiplying both sides of equation β(l)r t = β 0 + u t by θ(l):=β(l) 1,onegets 1 (1 β 1 L) (1 β 1L)r t = θ(l)β 0 + r t = θ(1)β 0 + θ(l)u t r t = β 0 1 β 1 + r t = β 0 1 β 1 + j=0 j=0 (β 1 ) j u t j θ j u t j. 1 (1 β 1 L) u t This is the infinite moving average MA( ) representation associated with the AR(1).

13 To sum up, the stationary AR(1) model r t = β 0 + β 1 r t 1 + u t, u t WN(0,σ 2 u) admits an equivalent MA( ) representation r t = β 0 1 β 1 + j=0 θ j u t j, θ j := (β 1 ) j. The MA representation is useful to derive the moments of the process: 1. it is immediate to see that μ r :=E(r t ):= β 0 1 β 1 :=θ(1)β 0 ; 2. it is immediate to see that σ 2 r:=var(r t ):=Var j=0 θ j u t j := j=0 ³ θj 2 Var(ut j ) := j=0 ³ β 2 1 j σ 2 u := σ2 u (1 β 2 1 );

14 3. it is immediate to see that e.g. for τ:=1 Cov(r t,r t 1 ):=Cov θ j u t j, j=0 h=0 θ h u t h 1 := j=1 θ j Var(u t j ):= j=1 θ j σ 2 u:= := β 1 1 β 2 σ 2 u. 1 β j 1 σ2 u j=1 More generally, hence σ2 u Cov(r t,r t τ ):= 1 β 2 1 (β 1 ) τ, τ=1,2,... ρ(τ):=corr(r t,r t τ ):= (β 1 ) τ, τ=1,2,...

15 Exercise: show that the AR(1) with β 1 :=1 has a uit root and is not non-stationary. Note that the AR(1) with β 1 :=1 is known as the Random Walk process. Thesequenceofcoefficients θ 1,θ 2,...θ j,... obtained from the MA( ) representation r t = β 0 + u t + θ 1 u t 1 + θ 2 u t β 1 is known as impulse response function. Ifoneinter- prets u t as a shock (an event that can not be predicted on the basis of existing information), then the parameter θ 1 measures the impact of the shock on the variable r t after one period, θ 2 measures the impact of the shock on the variable r t after two periods, and so on.

16 We can generalize the AR(1) to the AR(p), p 2. We say that the process generating the log-return {r t } belongs to the class of covariance stationary AR(p) models if r t = β 0 + β 1 r t β p r t p + u t, u t WN(0,σ 2 u) where r 0, r 1,..., r 1 p are fixed and the autoregressive polynomial β(l):=(1 β 1 L β 2 L 2... β p L p ) is such that every solution z C of the characteristic equation is such that z > 1. β(z) =0

17 Also in this case if the AR(p) is stationary it exists the inverse polynomial such that θ(l):=β(l) 1 := j=0 θ j L j θ j 0asj and θ(1):= j=0 θ j :=β(1) 1 :=(1 β 1 β 2... β p ) 1 <.

18 The AR(p) model β(l)r t = β 0 + u t, u t WN(0,σ 2 u) can also be represented as the MA( ): r t = β(1) 1 β 0 + θ(l)u t. E(r t ):=θ(1)β 0 := β 0 (1 β 1 β 2... β p ) ; ρ(τ):=corr(r t,r t τ ) slightly more involving.

19 AR model: estimation The estimation of stationary AR models does not need a separate treatment. Everything is the Slides 3. To see this, whatever the lag order p, thearmodel can be compacted in the expression r t = x 0 tβ + u t, t =1,...T where β:= β 0 β 1. β p, x t:= 1 r t 1. r t p. OLS and ML estimation of θ:=(β 0,σ 2 u) 0 is done. Testing hypotheses on β is done.

20 Forecasting issues: the AR(p) model Given the AR(p): r t = β 0 + β 1 r t β p r t p + u t u t WN(0,σ 2 u) t =1, 2,...,T we consider the problem of computing forecasts of actual returns. We start from the one step-ahead forecast: ˆr T +1 :=E(r T +1 F T ). From the AR(p) it follows that the quantity is given by ˆr T +1 :=E(r T +1 F T ) :=β 0 + β 1 r T + β 2 r T β p r T p.

21 In practice, the actual one step-ahed forecast is ˆr a T +1 :=x0 T +1ˆβ := ˆβ 0 + ˆβ 1 r T + ˆβ 2 r T ˆβ p r T p that means that we need to estimate the parameters from the data in order to obtain the forecast! We also need forecast intervals. Define the forecast error: e T +1 :=r T +1 ˆr T +1 :=u T +1 If u T +1 F T N(0,σ 2 u), then e T +1 F T N(0,σ 2 u) which implies r T +1 ˆr T +1 F T N(0,σ 2 u).

22 From r T +1 ˆr T +1 F T N(0,σ 2 u) we have Pr q 1 α r T +1 ˆr T +1 q 1 α :=1 α σ u where q 1 α is a quantile of the N(0, 1) distribution, also known as (1 α)100 percentile. Hence Pr ˆr T +1 σ u q 1 α r T +1 ˆr T +1 + σ u q 1 α :=1 α is the probability that the unknown future return r T +1 falls within the interval ˆr T +1 ± σ u q 1 α. In practice, in large samples (T ), we have that Pr hˆr a T +1 ˆσ uq 1 α r T +1 ˆr a T +1 +ˆσ uq 1 α i :=1 α

23 Two step-ahead forecast: ˆr T +2 :=E(r T +2 F T ) :=E(β 0 + β 1 r T +1 + β 2 r T β p r T p+1 F T ) :=β 0 + β 1 E(r T +1 F T )+β 2 r T β p r T p+1 :=β 0 + β 1ˆr T +1 + β 2 r T β p r T p+1. The actual forecast will be: ˆr T a +2 :=ˆβ 0 + ˆβ 1ˆr T a +1 + ˆβ 2 r T ˆβ p r T p+1.

24 Forecast error: e T +2 :=r T +2 ˆr T +2 :=β 1 (r T +1 ˆr T +1 )+u T +2 therefore Note that :=β 1 e T +1 + u T +2 e T +2 F T N(0, (1 + β 2 1 )σ2 u). Var(e T +2 ) Var(e T +1 ):=σ 2 u because as the forecast horizon increases, the uncertainty of the forecast increases as well. The (1 α)100 forecast interval is therefore given by ˆr a T +2 ± (1 + ˆβ 2 1) 1/2ˆσ u q 1 α. In general, the step-ahed forecast will be function of the 1 step-ahed forecast and the variance of the corresponding forecast error will tend to increase.

25 It is possible to show that ˆr T + :=E(r T + F T ) E(r t ) and this is due to the mean-reversion property of the stationary AR(p). Exercise: model. prove the property above for the AR(1)

26 MA model: representation We start from the simplest MA model, the moving average model of order one, MA(1). We say that the process generating the log-return {r t } belongs to the class of MA(1) models if r t = β 0 + u t + θ 1 u t 1, u t WN(0,σ 2 u) where θ 1 is any real number 6= 1andu 0 is fixed. Important remark: the MA(1) is covariance stationary for any value of θ 1 (including θ 1 :=1). Exercise: prove this statement! Every MA(1) (MA(q))processisastationaryprocess.

27 We can re-write the MA(1) model as r t = β 0 + θ(l)u t where θ(l):=1 + θ 1 L is the MA polynomial. As we know, if every solution z C of the characteristic equation (1 + θ 1 z)=0 is such that z > 1, then it exists such that θ(l) 1 :=β(l):=1+ j=1 β j L j β j := (θ 1 ) j 0asj and β(1):=1+ j=1 β j <.

28 In this case we say that the MA(1) process is invertible, i.e. it can be represented as AR( ): θ(l) 1 r t = θ(1) 1 β 0 + u t β(l)r t = β(1)β 0 + u t The invertibility condition allow us to express r t as linear function of infinite past returns plus the White Noise term: this means that one can forecast r t by using past values r t 1,r t 2,... If the MA(1) is not invertible, it can be shown that the poynomial θ(l) 1 also involves powers of the type L j (not L j ): we have the paradox that in order to forecast r t one needs also future values of r t!! MA model: estimation, see ARMA

29 ARMA model: representation We start from the simplest ARMA process, which is the ARMA(1,1) process. We say that the process generating the log-return {r t } belongs to the class of covariance stationary and invertible ARMA(1, 1) models if r t = β 0 + β 1 r t 1 + u t + θ 1 u t 1, u t WN(0,σ 2 u) where r 0 and r 1,arefixed, u 0 := 0=:u 1,where 1 <β 1 < 1, 1 <θ 1 < 1. In this model, the condition on 1 <β 1 < 1 ensures the (covariance) stationarity of the process because β(z):=(1 β 1 z)willhavesolutions z > 1, while the condition 1 <θ 1 < 1 ensures the invertibility of the MA part, beacuse θ(z):=(1 + θ 1 z) will have solutions z > 1.

30 Moments of stationary ARMA(1,1) process. Given r t = β 0 + β 1 r t 1 + u t + θ 1 u t, u t WN(0,σ 2 u) 1 <β 1 < 1 we have μ r :=E(r t ):= β 0 1 β 1 ; σ 2 r:=var(r t ):= (1 + θ2 1 +2β 1θ 1 )σ 2 u 1 β 2 1 Exercise: prove the formulas above. ρ(τ):=corr(r t,r t τ ):=β 1 + β 1σ 2 u σ 2. r

31 Generalization to the ARMA(p, q) case,p 2, q 2. We say that the process generating the log-return {r t } belongs to the class of covariance stationary and invertible ARMA(p, q) models if β(l)r t = β 0 + θ(l)u t, u t WN(0,σ 2 u) where r 0, r 1,..., r 1 p are fixed, u 0 := 0=:u 1,...,=:u 1 q are fixed, β(l):=(1 β 1 L β 2 L 2... β p L p ) θ(l):=(1 + θ 1 L + θ 2 L θ q L q ) and the polynomials β(l) and θ(l) are such that the solutions, z 1,z 2 C of the characteristic equations β(z 1 )=0, θ(z 2 )=0 are such that z 1 > 1and z 2 > 1.

32 ARMA model: estimation We consider the estimation problem of the ARMA(p,q) model: (1 β 1 L... β p L p )r t = β 0 +(1+θ 1 L+...+θ q L q )u t u t WN(0,σ 2 u) r 0,r 1,...,r 1 p are fixed u 0 :=u 1 :=...:=u 1 q :=0 ϑ:=(β 0,β 1,..., β p,θ 1,...,θ q ) unknown parameters. Hp: u t F t 1 is Gaussian.

33 With this additional hypothesis, we are able to write the log-likelihood function of the ARMA(p,q). First, given the time series r 1,...,r T where r 0,r 1,..., r 1 p are treated as given (nonstochastic), the likelihood function is L(ϑ):=f(r 1,...,r T ; r 0,r 1,..., r 1 p,ϑ). We know that for dependent observations the likelihood function admits the conditional factorization: L(ϑ):=Π T t=1 f(r t r t 1,...,r 1 ; r 0,r 1,..., r 1 p,ϑ).

34 Under HP: where f(u t F t 1 ):=(2πσ 2 u) 1/2 exp =(2πσ 2 u) 1/2 exp ( ( u2 t 2σ 2 u ) [r t E(r t F t 1 )] 2 2σ 2 u u t :=r t E(r t F t 1 ) ) :=r t β 0 β 1 r t 1... β p r t p θ 1 u t 1... θ q u t q :=r t μ t so that if u t F t 1 N(0,σ 2 u), t =1, 2,...,T then r t F t 1 N(μ t,σ 2 u), t =1, 2,...,T.

35 This means that L(ϑ):=Π T t=1 f(r t r t 1,..., r 1 ; r 0,r 1,...,r 1 p,ϑ) and = Π T t=1 (2πσ2 u) 1/2 exp ( u2 t 2σ 2 u ) log L(ϑ):=- T 2 log(2π) T 2 log(σ2 u) TX t=1 u 2 t 2σ 2 u. There is not analytic solution to ˆϑ:= max ϑ log L(ϑ) hence log L(ϑ) must be maximized numerically. ˆϑ will be the ML estimator of the parameters of the ARMA(p, q) model.