slides sales-forecasting-session2-web

Arthur CHARPENTIER - Sales forecasting.

Sales forecasting # 2

Arthur Charpentier

[email protected]

1


Agenda

Qualitative and quantitative methods, a very general introduction

• Series decomposition

• Short versus long term forecasting

• Regression techniques

Regression and econometric methods

• Box & Jenkins ARIMA time series method

• Forecasting with ARIMA series

Practical issues : forecasting with MSExcel

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Time series decomposition

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Time series decomposition, modeling the random part

Histogram of residuals (v2)

Den

sity

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Normal QQ plot of residuals (v2)

Theoretical Quantiles

Sam

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Time series decomposition, forecasting

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Time series decomposition, modeling the seasonal

componant

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componant

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componant

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Modeling the random component

The unpredictible random component is the key element when forecasting. Most

of the uncertainty comes from this random component εt.

The lower the variance, the smaller the uncertainty on forecasts.

The general theoritical framework related to randomness of time series is related

to weakly stationary.

25


De�ning stationarity

Time series (Xt) is weakly stationary if

� for all t, E(X2t

)< +∞,

� for all t, E (Xt) = µ, constant independent of t,

� for all t and for all h, cov (Xt, Xt+h) = E ([Xt − µ] [Xt+h − µ]) = γ (h),independent of t.

Function γ (·) is called autocovariance function.

Given a stationary series (Xt) , de�ne the autocovariance function, as

h 7→ γX (h) = cov (Xt, Xt−h) = E (XtXt−h)− E (Xt) .E (Xt−h) .

and de�ne the autocorrelation function, as

h 7→ ρX (h) = corr (Xt, Xt−h) =cov (Xt, Xt−h)√V (Xt)

√V (Xt−h)

=γX (h)γX (0)

.

26


De�ning stationarity

A process (Xt) is said to be strongly stationary if for all t1, ..., tn and h we have

the following law equality

L (Xt1 , ..., Xtn) = L (Xt1+h, ..., Xtn+h) .

A time series (εt) is a white noise if all autocovariances are null, i.e. γ (h) = 0 for

all h 6= 0. Thus, a process (εt) is a white noise if it is stationary, centred and

noncorrelated, i.e.

E (εt) = 0, V (εt) = σ2 and ρε (h) = 0 for any h 6= 0.

27


Statistical issues

Consider a set of observations {X1, ..., XT }.

The empirical mean is de�ned as

XT =1T

T∑t=1

Xt.

The empirical autocovariance function is de�ned as

γ̂T (h) =1

T − h

T−h∑t=1

(Xt −XT

) (Xt−h −XT

),

while the empirical autocorrelation function is de�ned as

ρ̂T (h) =γ̂T (h)γ̂T (0)

.

Remark those estimators can be biased, but asymptotically unbiased. More

precisely γ̂T (h)→ γ (h) and ρ̂T (h)→ ρ (h) as T →∞.

28


Backward and forward operators

De�ne the lag operator L (or B for backward) the linear operator de�ned as

L : Xt 7−→ L (Xt) = LXt = Xt−1,

and the forward operator F ,

F : Xt 7−→ F (Xt) = FXt = Xt+1,

Note that L ◦ F = F ◦ L = I (identity operator) and further F = L−1 and

L = F−1.

� it is possible to compose those operators : L2 = L ◦ L, and more generally

Lp = L ◦ L ◦ ... ◦ L︸︷︷︸ where p ∈ N

with convention L0 = I. Note that Lp (Xt) = Xt−p.

� Let A denote a polynom,A (z) = a0 + a1z + a2z2 + ...+ apz

p. Then A (L) is the

29


operator

A (L) = a0I + a1L+ a2L2 + ...+ apL

p =p∑k=0

akLk.

Let (Xt) denote a time series. Series (Yt) de�ned by Yt = A (L)Xt satis�es

Yt = A (L)Xt =p∑k=0

akXt−k.

or, more generally, assuming that we can formally the limit,

A (z) =∞∑k=0

akzk et A (L) =

∞∑k=0

akLk.

30


Backward and forward operators

Note that for all moving average A and B, thenA (L) +B (L) = (A+B) (L)

α ∈ R, αA (L) = (αA) (L)

A (L) ◦B (L) = (AB) (L) = B (L) ◦A (L) .

Moving average C = AB = BA satis�es( ∞∑k=0

akLk

)◦

( ∞∑k=0

bkLk

)=

( ∞∑i=0

ciLi

)où ci =

i∑k=0

akbi−k.

31


Geometry and probability

Recall that it is possible to de�ne an inner product in L2 (space of squared

integrable variables, i.e. �nite variance),

< X,Y >= E ([X − E(X)] · [Y − E(Y )]) = cov([X − E(X)], [Y − E(Y )])

Then the associated norm is ||X||2 = E([X − E(X)]2

)= V (X).

Two random variables are then orthogonal if < X,Y >= 0, i.e.cov([X − E(X)], [Y − E(Y )]) = 0.

Hence conditional expectation is simply a projection in the L2, E(X|Y ) is the theprojection is the space generated by Y of random variable X, i.e.

E(X|Y ) = φ(Y ), such that

� X − φ(Y ) ⊥ X, i.e. < X − φ(Y ), X >= 0,� φ(Y ) = Z∗ = argmin{Z = h(Y ), ||X − Z||2}� E(φ(Y )) <∞.

32


Linear projection

The conditional expectation E(X|Y ) is a projection if the set of all functions

{h(Y )}.

In linear regression, the projection if made in the subset of linear functions h(·).

We call this linear function conditional linear expectation, or linear projection,

denoted EL(X|Y ).

In purely endogeneous models, the best forecast for XT+1 given past

informations {XT , XT−1, XT−2, · · · , XT−h, ...} is

X̂T+1 = E(XT+1|{XT , XT−1, XT−2, · · · , XT−h, · · · }) = φ(XT , XT−1, XT−2, · · · , XT−h, · · · ).

Since estimating a nonlinear function is di�cult (especially in high dimension),

we focus on linear functions, i.e. autoregressive models,

X̂T+1 = EL(XT+1|{XT , XT−1, XT−2, · · · , XT−h, · · · }) = α0XT+α1XT−1+α2XT−2+· · ·+αhXT−h+· · · .

33


De�ning partial autocorrelations

Given a stationary series (Xt), de�ne the partial autocorrelation function

h 7→ ψX (h) as

ψX (h) = corr(X̂t, X̂t−h

),

where X̂t−h = Xt−h − EL (Xt−h|Xt−1, ..., Xt−h+1)

X̂t = Xt − EL (Xt|Xt−1, ..., Xt−h+1) .

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Autocorrelations of residuals (v2)

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Partial autocorrelations of residuals (v2)

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Time series decomposition, modeling the detrended series

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Autocorrelations of detrended series

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Partial autocorrelations of detrended series

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Time series decomposition, modeling Yt = Xt −Xt−12

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Partial autocorrelations of lagged detrended series

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A13 Highway: forecasting detrended series (ARMA)

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A13 Highway: forecasting detrended series (ARMA)

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Estimating autocorrelations with MSExcel

48


A white noise

A white noise is de�ned as a centred process (E(εt) = 0), stationary(V (εt) = σ2), such that cov (εt, εt−h) = 0 for all h 6= 0.

The so-called Box-Pierce test can be used to test H0 : ρ (1) = ρ (2) = ... = ρ (h) = 0

Ha : there exists i such that ρ (i) 6= 0.

The idea is to use

Qh = Th∑k=1

ρ̂2k,

where h is the lag number and T the total number of observations.

Under H0, Qh has a χ2 distribution, with h degrees of freedom.

49


A white noise

Another statistics with better properties is a modi�ed version of Q,

Q′h = T (T + 2)h∑k=1

ρ̂2k

T − k,

Most of the softwares return Qh for h = 1, 2, · · · , and the associated p-value. If p

exceeds 5% (the standard signi�cance level) we feel con�dent in accepting H0,

while if p is less than 5% , we should reject H0.

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A white noise

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Simulated white noise

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White noise autocorrelations

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00.

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White noise partial autocorrelations

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Time series decomposition, testing for white noise

Box−Pierce statistic, testing for white noise on lagged detrended series

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ox−P

ierc

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p−va

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Time series decomposition, testing for white noise

Box−Pierce statistic, testing for white noise on residuals (v2)

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Autoregressive process AR(p)

We call autoregressive process of order p, denoted AR (p), a stationnary process

(Xt) satisfying equation

Xt −p∑i=1

φiXt−i = εt for all t ∈ Z, (1)

where the φi's are real-valued coe�cients and where (εt) is a white noise process

with variance σ2. (1) is equivalent to

Φ (L)Xt = εt where Φ (L) = I− φ1L− · · · − φpLp

54


Autoregressive process AR(1), order 1

The general expression for AR (1) process is

Xt − φXt−1 = εt for all t ∈ Z,

where (εt) is a white noise with variance σ2.

If φ = ±1, process (Xt) is not stationary. E.g. if φ = 1, Xt = Xt−1 + εt (called

random walk) can be written

Xt −Xt−h = εt + εt−1 + ...+ εt−h+1,

and thus E (Xt −Xt−h)2 = hσ2.

But it is possible to prove that for any stationary process

E (Xt −Xt−h)2 ≤ 4V (Xt). Since it is impossible to have for any h,

hσ2 ≤ 4V (Xt), it means that the process cannot be stationary.

55



If |φ| < 1 it is possible to invert the polynomial lag operator

Xt = (1− φL)−1εt =

∞∑i=0

φiεt−i (as a function of the past) (εt) ). (2)

For a stationary process,the aucorelation function is given by ρ (h) = φh.

Further, ψ(1) = φ and ψ(h) = 0 for h ≥ 2.

56


A AR(1) process, Xt = 0.7Xt−1 + εt

Simulated AR(1)

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−2

02

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F

AR(1) autocorrelations

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CF

AR(1) partial autocorrelations

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Simulated AR(1)

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CF


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A AR(1) process, Xt = −0.5Xt−1 + εt

Simulated AR(1)

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02

4

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50.

00.

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F


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6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF


59



Simulated AR(1)

0 100 200 300 400 500

−10

−5

05

10

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F


0 10 20 30 40

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF


60



Those processes are also called Yule process, and they satisfy(1− φ1L− φ2L

2)Xt = εt,

where the roots of Φ (z) = 1− φ1z − φ2z2 are assumed to lie outside the unit

circle, i.e. 1− φ1 + φ2 > 0

1 + φ1 − φ2 > 0

φ21 + 4φ2 > 0,

61



Autocorrelation function satis�es equation

ρ (h) = φ1ρ (h− 1) + φ2ρ (h− 2) for any h ≥ 2,

and the partial autocorrelation function satis�es

ψ (h) =

ρ (1) for h = 1[ρ (2)− ρ (1)2

]/[1− ρ (1)2

]for h = 2

0 for h ≥ 3.

62


A AR(2) process, Xt = 0.6Xt−1 − 0.35Xt−2 + εt

Simulated AR(2)

0 100 200 300 400 500

−4

−2

02

0 10 20 30 40

−0.

20.

00.

20.

40.

60.

81.

0

Lag

AC

F


0 10 20 30 40

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF


63


A AR(2) process, Xt = −0.4Xt−1 − 0.5Xt−2 + εt

Simulated AR(2)

0 100 200 300 400 500

−4

−2

02

0 10 20 30 40

−0.

4−

0.2

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F


0 10 20 30 40

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF


64


Moving average process MA(q)

We call moving average process of order q, denoted MA (q), a stationnary

process (Xt) satisfying equation

Xt = εt +q∑i=1

θiεt−i for all t ∈ Z, (3)

where the θi's are real-valued coe�cients, and process (εt) is a white noise

process with variance σ2. (3) processes can be written equivalently

Xt = Θ (L) εt whereΘ (L) = I + θ1L+ ...+ θqLq.

The autocovariance function satis�es

γ (h) = E (XtXt−h)

= E ([εt + θ1εt−1 + ...+ θqεt−q] [εt−h + θ1εt−h−1 + ...+ θqεt−h−q])

=

[θh + θh+1θ1 + ...+ θqθq−h]σ2 if 1 ≤ h ≤ q0 if h > q,

65


Moving average process MA(q)

If h = 0, then γ (0) =[1 + θ21 + θ22 + ...+ θ2q

]σ2. This equation can be written

γ (k) = σ2

q∑j=0

θjθj+k with convention θ0 = 1.

Autocovariance function satis�es

ρ (h) =θh + θh+1θ1 + ...+ θqθq−h

1 + θ21 + θ22 + ...+ θ2qif 1 ≤ h ≤ q,

and ρ (h) = 0 if h > q.

66


Moving average process MA(1), order 1

The general expression of MA (1) is

Xt = εt + θεt−1, for all t ∈ Z,

where (εt) is a white noise with variance σ2. Autocorrelations are given by

ρ (1) =θ

1 + θ2, and ρ (h) = 0, for h ≥ 2.

Note that −1/2 ≤ ρ (1) ≤ 1/2 : MA (1) processes only have small

autocorrelations.

Partial autocorrelation of order h is given by

ψ (h) =(−1)h θh

(θ2 − 1

)1− θ2(h+1)

.

67


A MA(1) process, Xt = εt + 0.7εt−1

Simulated MA(1)

0 100 200 300 400 500

−3

−2

−1

01

23

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

MA(1) autocorrelations

0 10 20 30 40

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF

MA(1) partial autocorrelations

68


A MA(1) process, Xt = εt − 0.6εt−1

Simulated MA(1)

0 100 200 300 400 500

−3

−2

−1

01

23

0 10 20 30 40

−0.

50.

00.

51.

0

Lag

AC

F

MA(1) autocorrelations

0 10 20 30 40

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF

MA(1) partial autocorrelations

69


Autoregressive moving average process ARMA(p, q)

We call autoregressive moving average process of orders p and q, denoted

ARMA (p, q), a stationnary process (Xt) satisfying equation

Xt =p∑j=1

φjXt−j + εt +q∑i=1

θiεt−i for all t ∈ Z, (4)

where the φj 's and θi's are real-valued coe�cients, and process (εt) is a white

noise process with variance σ2. (4) processes can be written equivalently

Φ (L)Xt = Θ (L) εt,

where Φ (L) = I− φ1L− ...− φqLq and Θ (L) = I + θ1L+ ...+ θqLq .

70


Autoregressive moving average process ARMA(p, q)

Note that under some technical assumptions, one can write

Xt = Φ−1 (L) ◦Θ (L) εt,

i.e. the ARMA(p, q) process is also an MA(∞) process, and

Φ (L) ◦Θ−1 (L)Xt = εt,

i.e. the ARMA(p, q) process is also an AR(∞) process.

Wald's theorem claims that any stationary process (satisfying further technical

conditions) can be written as a MA process.

More generally, in practice, a stationary series can be modeled either by an

AR(p) process, a MA(q), or an ARMA(p′, q′) whith p < p′ and q < q′.

71


A ARMA(1, 1) process, Xt = 0.7Xt−1εt − 0.6εt−1

Simulated ARMA(1,1)

0 100 200 300 400 500

−2

−1

01

23

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

ARMA(1,1) autocorrelations

0 10 20 30 40

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF

ARMA(1,1) partial autocorrelations

72


A ARMA(2, 1) process, Xt = 0.7Xt−1 − 0.2Xt−2εt − 0.6εt−1

Simulated ARMA(2,1)

0 100 200 300 400 500

−2

02

4

0 10 20 30 40

0.0

0.2

0.4

0.6

0.8

1.0

Lag

AC

F

ARMA(2,1) autocorrelations

0 10 20 30 40

−0.

6−

0.4

−0.

20.

00.

20.

40.

6

Lag

Par

tial A

CF

ARMA(2,1) partial autocorrelations

73


Fitting ARMA processes with MSExcel

74


Forecasting with AR(1) processes

Consider an AR (1) process, Xt = µ+ φXt−1 + εt then

• TX∗T+1 = µ+ φXT ,

• TX∗T+2 = µ+ φ.TX

∗T+1 = µ+ φ [µ+ φXT ] = µ [1 + φ] + φ2XT ,

• TX∗T+3 = µ+ φ.TX

∗T+2 = µ+ φ [µ+ φ [µ+ φXT ]] = µ

[1 + φ+ φ2

]+ φ3XT ,

and recursively TX∗T+h can be written

TX∗T+h = µ+ φ.TX

∗T+h−1 = µ

[1 + φ+ φ2 + ...+ φh−1

]+ φhXT .

or equivalently

TX∗T+h =

µ

φ+ φh

[XT −

µ

φ

]= µ

1− φh

1− φ︸︷︷︸1+φ+φ2+...+φh−1

+ φhXT .

75


Forecasting with AR(1) processes

The forecasting error made at time T for horizon h is

T∆h = TX∗T+h −XT+h =T X

∗T+h − [φXT+h−1 + µ+ εT+h]

= ...

= TX∗T+h −

[φh1XT +

(φh−1 + ...+ φ+ 1

)µ

+εT+h + φεT+h−1 + ...+ φh−1εT+1,

(6)

thus, T∆h = εT+h + φεT+h−1 + ...+ φh−1εT+1, with variance having variance

V̂ =[1 + φ2 + φ4 + ...+ φ2h−2

]σ2, where V (εt) = σ2.

thus, variance of the forecast error increasing with horizon.

76

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