Università di Pavia. Forecasting. Eduardo Rossi

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1 Università di Pavia Forecasting Eduardo Rossi

2 Mean Squared Error Forecast of Y t+1 based on a set of variables observed at date t, X t : Yt+1 t. The loss function MSE(Y t+1 t ) = E[Y t+1 Y t+1 t ]2 The forecast with the smallest MSE is Y t+1 t = E[Y t+1 X t ] Suppose Y t+1 t is a linear function of X t: Ŷ t+1 t = α X t if E[(Y t+1 α X t )X t] = 0 then α X t is the linear projection of Y t+1 on X t. Eduardo Rossi c - Time Series Econometrics 11 2

3 Linear Projection The LP projection produces the smallest MSE among the class of linear forecasting rule using P(Y t+1 X t ) = α X t MSE[ P(Y t+1 X t )] MSE[E(Y t+1 X t )] E[(Y t+1 α X t )X t] = 0 E[Y t+1 X t] = α E[X t X t] α = E[Y t+1 X t]e[x t X t] 1 Eduardo Rossi c - Time Series Econometrics 11 3

4 Properties of Linear Projection The MSE associated with a LP is given by E[(Y t+1 α X t ) 2 ] = E[(Y t+1 ) 2 ] 2E(α X t Y t+1 )+E(α X t X t α) Replacing α E[(Y t+1 α X t ) 2 ] = E[(Y t+1 ) 2 ] 2E(Y t+1 X t)[e(x t X t)] 1 E(X t Y t+1 ) +E(Y t+1 X t)[e(x t X t)] 1 [E(X t X t)][e(x t X t)] 1 E(X t Y t+1 ) E[(Y t+1 α X t ) 2 ] = E[(Y t+1 ) 2 ] E(Y t+1 X t)[e(x t X t)] 1 E(X t Y t+1 ) If X t includes a constant term, then P[(aY t+1 +b) X t ] = a P(Y t+1 X t )+b The forecast error is [ay t+1 +b] [a P(Y t+1 X t )+b] = a[y t+1 P(Y t+1 X t )] is uncorrelated with X t as required of a linear projection. Eduardo Rossi c - Time Series Econometrics 11 4

5 LP and OLS LP is closely related to OLS regression y t+1 = β X t +u t [ ] 1 [ 1 T β = X t X 1 t T T t=1 ] T X t Y t+1 t=1 β is constructed from the sample moments, while α is constructed from population moments. If {X t,y t+1 } is covariance stationary and ergodic for second moments, then the sample moments will converge to the population moments as the sample size T goes to infinity 1 T 1 T T X t X t t=1 p E[X t X t] T p X t y t+1 E[Xt Y t+1 ] t=1 Eduardo Rossi c - Time Series Econometrics 11 5

6 LP and OLS implying β p α β is consistent for the LP coefficient. Eduardo Rossi c - Time Series Econometrics 11 6

7 Forecast based on an infinite number of observations Forecasting based on lagged ǫ s. Infinite MA: (Y t µ) = ψ(l)ǫ t ǫ t WN(0,σ 2 ) ψ 0 = 1, j=0 ψ j <. An infinite number of obs on ǫ through date t: {ǫ t,ǫ t 1,...}. We know the values of µ and {ψ 1,ψ 2,...} Y t+s = µ+ǫ t+s +ψ 1 ǫ t+s 1 +ψ 2 ǫ t+s ψ s ǫ t +ψ s+1 ǫ t Eduardo Rossi c - Time Series Econometrics 11 7

8 Forecast based on an infinite number of observations The optimal linear forecast is: Ê[Y t+s ǫ t,ǫ t 1,...] = µ+ψ s ǫ t +ψ s+1 ǫ t where Ê[Y t+s X t ] P(Y t+s 1,X t ). The unknown future ǫ s are set to their expected value of zero. The forecast error is Y t+s Ê[Y t+s ǫ t,ǫ t 1,...] = ǫ t+s +ψ 1 ǫ t+s 1 +ψ 2 ǫ t+s ψ s 1 ǫ t+1 Eduardo Rossi c - Time Series Econometrics 11 8

9 Forecast based on an infinite number of observations E[(Y t+s Ê[Y t+s ǫ t,ǫ t 1,...]) 2 ] = (1+ψ ψ 2 s 1)σ 2 when s the MSE converges to the unconditional variance σ 2 j=0 ψ2 j. MA(q): ψ(l) = 1+θ 1 L+...+θ q L q Y t+s = µ+ǫ t+s +θ 1 ǫ t+s θ t+s q ǫ t+s q The optimal linear forecast is µ+θ s ǫ t +θ s+1 ǫ t θ q ǫ t q+s s = 1,...,q Ê[Y t+s ǫ t,ǫ t 1,...] = µ s = q +1,... Eduardo Rossi c - Time Series Econometrics 11 9

10 Forecast based on an infinite number of observations MSE: σ 2 s = 1 (1+θ θ 2 s 1)σ 2 s = 2,3,...,q (1+θ θ 2 q)σ 2 s = q +1,q +2,... The MSE increases with the forecast horizon up until s = q. For s > q the forecast is the unconditional mean and the MSE is the unconditional variance of the series. Eduardo Rossi c - Time Series Econometrics 11 10

11 Forecast based on an infinite number of observations Compact lag operator ψ(l) L s = L s +ψ 1 L 1 s +ψ 2 L 2 s +...+ψ s 1 L 1 +ψ s L 0 +ψ s+1 L 1 +ψ s+2 L the annihilation operator replaces negative powers of L by zero [ ] ψ(l) = ψ s L 0 +ψ s+1 L 1 +ψ s+2 L L s + Ê[Y t+s ǫ t,ǫ t 1,...] = µ+ [ ] ψ(l) L s ǫ t + Eduardo Rossi c - Time Series Econometrics 11 11

12 Forecast based on an infinite number of observations Forecasting based on lagged Y s. In the usual forecasting situation we have obs on lagged Y s. Suppose the infinite MA process has an Infinite AR representation η(l)(y t µ) = ǫ t η(l) = j=0 η jl j, η 0 = 1 and j=0 η j < η(l) = [ψ(l)] 1. A c.s. AR(p) satisfies (1 φ 1 L φ 2 L φ p L p )(Y t µ) = ǫ t φ(l)(y t µ) = ǫ t η(l) = φ(l) ψ(l) = [φ(l)] 1 Eduardo Rossi c - Time Series Econometrics 11 12

13 Forecast based on an infinite number of observations For an MA(q): Y t µ = (1+θ 1 L+...+θ q L q )ǫ t Y t µ = θ(l)ǫ t ψ(l) = θ(l) η(l) = [θ(l)] 1 provided that is based on an invertible representation. Eduardo Rossi c - Time Series Econometrics 11 13

14 Forecast based on an infinite number of observations ARMA(p,q) can be represented as an AR( ) with ψ(l) = θ(l) φ(l) provided that the roots of φ(z) and θ(z) lie outside the unit circle. When the restrictions are satisfied obs on {Y t,y t 1,Y t 2,... } will be sufficient to construct {ǫ t,ǫ t 1,...}. Eduardo Rossi c - Time Series Econometrics 11 14

15 Forecast based on an infinite number of observations For example for an AR(1): (1 φl)(y t µ) = ǫ t given φ and µ and Y t,y t 1, the value of ǫ t can be constructed from ǫ t = (Y t µ) φ(y t 1 µ) For an invertible MA(1): (1+θL) 1 (Y t µ) = ǫ t given an infinite number of obs on Y, we can compute: ǫ t = (Y t µ) θ(y t 1 µ)+θ 2 (Y t 2 µ) θ 3 (Y t 3 µ)+... Eduardo Rossi c - Time Series Econometrics 11 15

16 Forecast based on an infinite number of observations Under the conditions Ê[Y t+s Y t,y t 1,...] = µ+ [ ] ψ(l) L s the forecast of Y t+s as a function of lagged Y s. Using η(l) = [ψ(l)] 1 Ê[Y t+s Y t,y t 1,...] = µ+ [ ] ψ(l) L s Wiener-Kolmogorov prediction formula. + + η(l)(y t µ) [ψ(l)] 1 (Y t µ) Eduardo Rossi c - Time Series Econometrics 11 16

17 Wiener-Kolmogorov prediction formula - AR(1) For example for an AR(1): (1 φl)(y t µ) = ǫ t 1 ψ(l) = 1 φl = 1+φL+φ2 L φ s L s +... the annihilation operator is: [ ] ψ(l) L s = φ s +φ s+1 L 1 +φ s+2 L = φs 1 φl + [ ] ψ(l) Ê[Y t+s Y t,y t 1,...] = µ+ where ǫ t = (1 φl)(y t µ). L s Ê[Y t+s Y t,y t 1,...] = µ+φ s (Y t µ) + η(l)(y t µ) = µ+ φs 1 φl (1 φl)(y t µ) the forecast decays geometrically from (Y t µ) toward µ as s increases. Eduardo Rossi c - Time Series Econometrics 11 17

18 Wiener-Kolmogorov prediction formula - AR(1) Given that ψ j = φ j, from the MSE of a MA( ), we have that the MSE s-period-ahead forecast error is: as s [1+φ φ 2(s 1) ]σ 2 MSE = σ2 1 φ 2 Eduardo Rossi c - Time Series Econometrics 11 18

19 Wiener-Kolmogorov prediction formula - AR(p) Stationary AR(p) process Y t+s µ = f (s) 11 (Y t µ)+f (s) 12 (Y t 1 µ)+...+f (s) 1p (Y t p+1 µ)+ ψ j = f (j) 11 ǫ t+s +ψ 1 ǫ t+s 1 + +ψ s 1 ǫ t+1 the optimal s-period-ahead forecast is Ŷ t+s t = µ+f (s) 11 (Y t µ)+...+f (s) 1p (Y t p+1 µ) forecast error Y t+s t Ŷt+s t = ǫ t+s +ψ 1 ǫ t+s 1 + +ψ s 1 ǫ t+1 Eduardo Rossi c - Time Series Econometrics 11 19

20 Wiener-Kolmogorov prediction formula - AR(p) To calculate the optimal forecast we use a recursion. Start with the forecast Ŷt+1 t Ŷ t+2 t+1 : Ŷ t+1 t µ = φ 1 (Y t µ)+...+φ p (Y t p+1 µ) Ŷ t+2 t+1 µ = φ 1 (Y t+1 µ)+...+φ p (Y t p+2 µ) Law of Iterated Projections: Forecast Ŷt+2 t+1 projected on date t information set then we obtain Ŷ t+2 t : Ŷ t+2 t µ = φ 1 (Ŷ t+1 t µ)+...+φ p (Y t p+2 µ) substituting Ŷ t+1 t Ŷ t+2 t µ = φ 1 [φ 1 (Y t µ)+...+φ p (Y t p+1 µ)]+ φ 2 (Y t µ)+...+φ p (Y t p+2 µ) Eduardo Rossi c - Time Series Econometrics 11 20

21 Wiener-Kolmogorov prediction formula - AR(p) Ŷ t+2 t µ = (φ 2 1 +φ 2 )(Y t µ)+(φ 1 φ 2 +φ 3 )(Y t 1 µ)+...+ (φ 1 φ p 1 +φ p )(Y t p+2 µ)+φ 1 φ p (Y t p+1 µ) The s-period-ahead forecast of an AR(p) process can be obtained by iterating on Ŷ t+j t µ = φ 1 (Ŷ t+j 1 t µ)+...+φ p (Ŷ t+j p t µ) Eduardo Rossi c - Time Series Econometrics 11 21

22 Wiener-Kolmogorov prediction formula - MA(1) Invertible MA(1) (Y t µ) = (1+θL)ǫ t with θ < 1. Wiener-Kolmogorov formula [ ] ψ(l) Ŷ t+s t = µ+ (1+θL) 1 (Y t µ) Forecast s = 1 [ (1+θL) L 1 ] + L s = θ + Ŷ t+1 t = µ+ θ 1+θL (Y t µ) = µ+θ(y t µ) θ 2 (Y t 1 µ)+θ 3 (Y t 2 µ)+... Eduardo Rossi c - Time Series Econometrics 11 22

23 Wiener-Kolmogorov prediction formula - MA(1) Alternatively ǫ t = (1+θL) 1 (Y t µ) in practice ǫ t = (Y t µ) θ ǫ t 1. For s = 2,3,... [ (1+θL) L s ] + = 0 Ŷ t+s t = µ Eduardo Rossi c - Time Series Econometrics 11 23

24 Wiener-Kolmogorov prediction formula - MA(q) (Y t µ) = θ(l)ǫ t θ(l) = (1+θ 1 L+θ 2 L θ q L q ) [ 1+θ1 L+...+θ q L q ] Ŷ t+s t = µ+ [ 1+θ1 L+...+θ q L q ] For L s + = L s Ŷ t+s t = µ+(θ s +θ s+1 L+...+θ q L q s ) ǫ t + 1 θ(l) (Y t µ) 1+θ s L+θ s+1 L θ q L q s s = 1,...,q 0 s = q +1,... ǫ t = (Y t µ) θ 1 ǫ t 1... θ q ǫ t q Eduardo Rossi c - Time Series Econometrics 11 24

25 Wiener-Kolmogorov prediction formula - ARMA(1,1) (1 φl)(y t µ) = (1+θL)ǫ t Stationarity: φ < 1. Invertibility: θ < 1. [ ] 1+θL 1 φl Ŷ t+s t = µ+ (1 φl)l s 1+θL (Y t µ) [ 1 (1 φl) = 1+φL+φ2 L ] [ ] 1 = (1 φl)l + θl s (1 φl)l s 1+θL (1 φl)l s + = + [ (1+φL+φ 2 L ) L s + θl(1+φl+φ2 L ) L s = (φ s +φ s+1 L+φ s+2 L )+ θ(φ s 1 +φ s L+φ s+1 L ) + ] + Eduardo Rossi c - Time Series Econometrics 11 25

26 Wiener-Kolmogorov prediction formula - ARMA(1,1) [ ] 1+θL (1 φl)l s + = φ s (1+φL+φ 2 L )+ θφ s 1 (1+φL+φ 2 L ) = (φ s +θφ s 1 )(1+φL+φ 2 L ) = φs +θφ s 1 1 φl Eduardo Rossi c - Time Series Econometrics 11 26

27 Wiener-Kolmogorov prediction formula - ARMA(1,1) Ŷ t+s t = µ+ [ ] 1+θL (1 φl)l s + = µ+ φs +θφ s 1 1 φl 1 φl 1+θL (Y t µ) 1 φl 1+θL (Y t µ) = µ+ φs +θφ s 1 1+θL (Y t µ) For s = 2,3,... the forecast Ŷ t+s t µ = φ(ŷ t+s 1 t µ) the forecast decays geometrically at the rate φ toward the unconditional mean µ. The one-period-ahead forecast (s=1) is given by Ŷ t+1 t = µ+ φ+θ 1+θL (Y t µ) Eduardo Rossi c - Time Series Econometrics 11 27

28 Wiener-Kolmogorov prediction formula - ARMA(1,1) Ŷ t+1 t = µ+ φ(1+θl)+θ(1 φl) (Y t µ) 1+θL = µ+φ(y t µ)+ 1 φl 1+θL (Y t µ) ǫ t = 1 φl 1+θL (Y t µ) = (Y t µ) φ(y t 1 µ) θ ǫ t 1 ǫ t = Y t Ŷ t t 1 Eduardo Rossi c - Time Series Econometrics 11 28

29 Wiener-Kolmogorov prediction formula - ARMA(1,1) s = 2, Ŷ t+2 t = µ+ φ2 +θφ 1+θL (Y t µ) = µ+φ φ+θ 1+θL (Y t µ) = µ+φ(φ+θ)(1 θl+θ 2 L 2 θ 3 L )(Y t µ) = µ+φ(φ+θ)(y t µ) φ(φ+θ)θ(y t 1 µ)+... Eduardo Rossi c - Time Series Econometrics 11 29

30 Wiener-Kolmogorov prediction formula - ARMA(p,q) ARMA(p,q): φ(l)(y t µ) = θ(l)ǫ t Ŷ t+1 t µ = φ 1 (Y t µ)+...+φ p (Y t p+1 µ)+θ 1 ǫ t +...+θ q ǫ t q+1 ǫ t = Y t Y t t 1 Ŷ τ t = Y τ τ t φ 1 (Ŷ t+s 1 t µ)+...+φ p (Y t+s p t µ)+θ 1 ǫ t +...+θ q ǫ t+s q for s = 1,...,q Ŷ t+s t µ = φ 1 (Ŷt+s 1 t µ)+...+φ p (Y t+s p t µ) for s = q +1,... Eduardo Rossi c - Time Series Econometrics 11 30

31 Forecasts based on a Finite number of observations {Y t,y t 1,...,Y t m+1 } observations. Presample ǫ s all equal to 0. Approximation Ê[Y t+s Y t,y t 1,...] = Ê[Y t+s Y t,...,y t m+1,ǫ t m = 0,ǫ t m 1 = 0,...] MA(q): ǫ t m = ǫ t m 1 =... = ǫ t m q+1 = 0 ǫ t m+1 = Y t m+1 µ ǫ t m+2 = Y t m+2 µ θ 1 ǫ t m+1 ǫ t m+3 = Y t m+3 µ θ 1 ǫ t m+2 θ 2 ǫ t m+1 The values are to be replaced in Ŷ t+s t = µ+(θ s +θ s+1 L+θ s+2 L θ q L q s ) ǫ t Eduardo Rossi c - Time Series Econometrics 11 31

32 Forecasts based on a Finite number of observations For s = q = 1: Ŷ t+s t = µ+θ(y t µ) θ 2 (Y t 1 µ)+...+( 1) m 1 θ m (Y t m+1 µ) truncated infinite AR. For m and θ small we have a good approximation. For θ = 1 the approximation may be poorer. Eduardo Rossi c - Time Series Econometrics 11 32

33 Exact Finite-sample Properties Exact projection of Y t+1 on its most recent values 1 Y t X t =. Linear Forecast If Y t is c.s. Y t m+1 α (m) X t = α m 0 +α m 1 Y t +...+α m my t m+1 E[Y t Y t j ] = γ j +µ 2 X t = [1,Y t,...,y t m+1 ] Eduardo Rossi c - Time Series Econometrics 11 33

34 Exact Finite-sample Properties implies α (m) = [ µ (γ 1 +µ 2 )... (γ m +µ 2 ) ] 1 µ... µ µ (γ 0 +µ 2 )... (γ m 1 +µ 2 )... µ (γ m 1 +µ 2 )... (γ 0 +µ 2 ) 1 when a constant term is included in X t it is more convenient to express variables in deviations from the mean. Eduardo Rossi c - Time Series Econometrics 11 34

35 Exact Finite-sample Properties Calculate the projection of (Y t+1 µ) on (Y t µ),(y t 1 µ),...,(y t m+1 µ) 1 α (m) = γ 0 γ 1... γ m 1... γ 1. γ m 1 γ m 2... γ 0 γ m s-period-ahead forecast Ŷ t+s t = µ+α (m,s) 1 (Y t µ)+...+α m (m,s) (Y t m+s µ) 1 α (m,s) 1 γ 0 γ 1... γ m 1 γ s. =.... α m (m,s) γ m 1 γ m 2... γ 0 γ s+m 1 Eduardo Rossi c - Time Series Econometrics 11 35

36 Exact Finite-sample Properties Inversion of an (m m) matrix. Two algorithms: 1. Kalman Filter to compute finite-sample forecast. 2. Triangular Factorization. Eduardo Rossi c - Time Series Econometrics 11 36