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 logreturn {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 rewritten 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 nonstationary 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 nonstationary. 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 logreturn {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 stepahead 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 stepahed 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 stepahead 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 stepahed forecast will be function of the 1 stepahed 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 meanreversion 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 logreturn {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 rewrite 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 logreturn {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 logreturn {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 loglikelihood 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.
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