lecture 1: fundamental concepts in time series analysis...

Lecture 1: Fundamental concepts in Time Series Analysis(part 1)

Florian Pelgrin

University of Lausanne, Ecole des HECDepartment of mathematics (IMEA-Nice)

Sept. 2011 - Jan. 2012

Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 1 / 61

Introduction

Motivation

Example 1 : Monthly log returns

Suppose that a financial analyst is interested to describe and tomodel the monthly (log-) return of different stocks markets.

Another alternative (among others !) could be to describe and tomodel the (log-) price of stock markets indexes (say, the S&P 500,CAC40, or Nikkei index), denoted by Pt .

Monthly returns are defined by :

rt =Pt − Pt−1

Pt−1or rt ' log

(Pt

Pt−1

).


Introduction

1980 1990 2000 2010-30

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0

10

20SP500

1980 1990 2000 2010-30

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0

10

20Nikkei

1980 1990 2000 2010-30

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0

10

20DAX

1980 1990 2000 2010-30

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20FTSE100


Introduction

Questions :

What are the main characteristics of the previous series ?

Descriptive statisticsDistributionPersistence, (partial) autocorrelationsEtc.

How can we model these series ?

Given the information available at time T , can we forecast the valueof a return at time T + h (h > 0) ?

· · · Univariate time series models seek to answer these questions · · ·


Introduction

Example 2 : How to model and forecast the monthly (effective) US fedfund rate ?


Introduction

Objectives

Discuss techniques for characterizing and modelling univariate timeseries.

These techniques can be employed for short-term (medium-term)prediction of asset prices or returns (or any macro variables), to testthe market efficiency hypothesis, etc.

In this block, the discussion is restricted to linear time series modelsand mainly focus on the class of autoregressive moving average(ARMA) models.

While financial (macro) time series exhibit in general more demandingstructures than ARMA models, this class of stochastic processes is afirst starting point and often serves as a benchmark.


Introduction

Class of ARMA(p,q) stochastic processes

Suppose that the value of a time series at time t, Xt , is a linearfunction of a constant term, the last p values of Xt , and thecontemporaneous and last q values of an (exogenous) stochasticterm, denoted by εt :

Xt = µ+ φ1Xt−1 + φ2Xt−2 + · · ·+ φpXt−p

+εt + θ1εt−1 + · · ·+ θqεt−q

= µ+

p∑k=1

φkXt−k +

q∑j=0

θjεt−j .

with θ0 = 1.

This model is called an ARMA(p,q) model in which p is the order ofthe autoregressive part (the past values of Xt) and q is the order ofthe moving average part (the contemporaneous and past value of theshock or the error term εt).


Introduction

Some references...

Box, G. et G. Jenkins, 1970, Time Series Analysis : Forecasting andControl, Holden-Day.

Brockwell, P. and A. Davis, 1987, Time Series : Theory and Methods,Springer Verlag.

Gourieroux, C. et A. Monfort, 1995, Time series and DynamicModels, Economica.

Hamilton, J., 1994, Times Series Analysis, Princeton University Press.

Harvey, A.C., 1993, Time Series Models, MIT Press.

Lutkepohl, H., 1991, Introduction to Multiple Time Series Analysis,Springer-Verlag.

Tsay, R., 2002, Analysis of Financial Time Series, Wiley Series.


Introduction

Road map

1 Definition of a time series

2 The backward and forward operators

3 Weak and strong stationarity

4 Linear time series and transformation

5 Innovation and Wold’s decomposition

6 Appendix


Definition of a time series

1. Definition of a time series

Suppose that :

One is interested in a variable (say, a price) or a transformation ofthat variable (say, a return or the log(price)) :

Asset priceAsset returnInterest ratesExchange ratesGDP, Unmployment rate, Consumption, Investment, etc.

This variable (the realizations) is observed at different (regular)periods (t = t1, · · · , tk) and the time elapsed between two realizationsis constant (say, daily, monthly, quarterly, yearly observations)



The notion of ”time” or ”duration” : The duration between two(consecutive) realizations (measures) may change for a given variable(the same variable can be published at different frequencies) andacross variables :

- Macro data are in general monthly, quarterly or yearly data.

- Financial data can be monthly, quarterly or yearly data but also daily orintra-day data (high frequency).

- For financial data, time is not necessarily the calendar time but it couldbe the transaction time (see application).

The information provided by the frequency of the data is notnecessarily the same (aggregation of processes is not a simplematter !).



The information provided by the two figures is not the same :

The top panel shows that the estimated average number of trades persecond (i) is roughly the same for every day and (ii) varies within aday (transactions are generally more important when the market isjust opened or near the closure time).

The bottom panel conveys a different information (trades per daywhen the market is opened) : fluctuations may reflect financial andeconomic news, etc.

Trades per day can be obtained after aggregating the average numberof trades per second : this aggregation leads to a different pattern ofthe times series and thus a different modelling.



Taking the frequency of the data...

The sequence of realizations is a time series, i.e. the time-orderedsequence of observations, say X1,X2, · · · ,XT , is a time series.

The data generating process underlying these realizations is called astochastic process.

The building block of time series theory is the concept of stochasticprocess.

...In the sequel, the two concepts—stochastic process and time series—arebriefly discussed...



Definition (Stochastic processes)

A real-valued stochastic process is a sequence of random variables index byt ∈ Z on the probability space (Ω,A,P) :

Z× Ω→ R(t, ω) 7→ X (t, ω) = Xt(ω).

where Ω is the sample space, ω ∈ Ω is a state of the nature such thatxt = Xt(ω), A is a σ-algebra, and P is a probability measure.

Remarks : In the sequel :

1 No difference is made between the random variables and theirrealizations for the sake of notation.

2 Only discrete stochastic processes are considered.



Definition

A realization of the stochastic process (Xt)t∈Z for a given ω ∈ Ω is themapping defined by :

Z→ Rt 7→ xt(ω).

Definition (Time series)

The realization of a stochastic process is said to be a time series or achronological series.


The backward and forward operators

2. The backward and forward operators

The main objective of this section is to introduce notation that arevery convenient to write linear time series models and to characterizetheir properties.

Consider a situation where the value of a time series at time t, Xt , isa linear function of a constant term, the last p values of Xt , and thecontemporaneous and last q values of an (exogenous) stochasticterm, denoted by εt :

Xt = µ+ φ1Xt−1 + φ2Xt−2 + · · ·+ φpXt−p

+εt + θ1εt−1 + · · ·+ θqεt−q

= µ+

p∑k=1

φkXt−k +

q∑j=0

θjεt−j .

with θ0 = 1.



This process rewrites :

Φ(L)Xt = µ+ Θ(L)εt

with

Φ(L) = 1− φ1L− φ2L2 − · · · − φpLp

Θ(L) = 1 + θ1L + · · ·+ θqLq

where L is the lag (backward) operator such that :

LXt = Xt−1 and LkXt = Xt−k

L−1Xt = FXt = Xt+1

and F is the forward (lead, delay) operator.

In the sequel, the backward and forward operators are defined andsome properties are briefly discussed.



The lag operator

Definition

The lag operator or backward (shift) operator, denoted by L, is anoperator that shifts the time index backward by one unit.

Examples :1 Applying the backward operator to a variable at time t, say Xt , yields

the value of that variable at time t − 1 :

LXt = Xt−1

2 Applying the backward operator twice (L2) amounts to lagging thevariable twice :

L2Xt = L(LXt) = LXt−1 = Xt−2.

3 More generally,

LkXt = Xt−k



More formally, the lag operator transforms one time series, say

(Yt)t∈Z = (Y−∞, · · · ,Y−1,Y0,Y1, · · · ,Y+∞) ,

into another time series, say (Xt)t∈Z where :

Xt = Yt−1

A constant can be viewed as a special series, namely :

(Yt)t∈Z

where Yt = c for all t. Therefore,

Lc = c .



The lead operator

Definition

The lead operator or forward (shift) operator, denoted by F , is an operatorthat shifts the time index forward by one unit.

Examples :1 Applying the forward operator to a variable at time t, say Xt , yields

the value of that variable at time t + 1 :

FXt = Xt+1

2 Applying the forward operator twice (F 2) amounts to leading thevariable twice :

F 2Xt = F (FXt) = FXt+1 = Xt+2.

3 More generally,

F kXt = Xt+k



Remarks :

Raising L to a negative (integer) power yields the forward operator atthat power :

L−kXt = F kXt = Xt+k

Applying the lag and lead operator to a variable at time t yields thevalue of that variable at time t :

FLXt = LFXt = Xt .

The difference operator, denoted by ∆, is used to express thedifference between two consecutive realizations of a time series :

∆Xt = Xt − Xt−1

More generally, the differentiation of order k is defined by :

∆k = (1− L)k



Properties...

Definition

Suppose that∞∑k=0

|ak | < +∞,∞∑k=0

|bk | < +∞, and :

A(L) =∞∑k=0

akLk ,B(L) =∞∑k=0

bkLk .

Then :

(i) ∀α ∈ R, αA(L) = (αA)(L) = α

( ∞∑k=0

akLk

)=∞∑k=0

(αak)Lk ;

(ii) A(L) + B(L) = (A + B)(L) =∞∑k=0

akLk +∞∑k=0

bkLk =∞∑k=0

(ak + bk)Lk ;

(iii) A(L) B(L) = (AB)(L) = B(L) A(L) with

(AB)(L) = (BA)(L) = C (L) =∞∑k=0

ckLk with ck =+∞∑j=0

ajbk−j .



Definition

Let P(z) define a polynomial of order p, P(z) =p∑

k=0

akzk with ak ∈ R. A

lagged polynomial P(L) can be defined from P(z) as follows :

P(L) =

p∑k=0

akLk

with :

P(L)Xt =

(p∑

k=0

akLk

)Xt

=

p∑k=0

akXt−k .



In particular, assuming that P(z) has distinct roots (zi ∈ C), it canbe factored as :

P(z) =

p∏i=1

(z − zi ) =

p∏i=1

(−zi )

p∏i=1

(1− z

zi

)

= α

p∏i=1

(1− λiz)

with λi = 1/zi .

Consequently,

P(L) = α

p∏i=1

(1− λiL).



Armed with the previous definitions, some characteristics of a time-seriesare presented :

1 Weak and strong Stationarity

2 Stationarity and invertibility

3 Autocovariance function, autocorrelation function (ACF), partialautocorrelation function (PACF)


Weak and strong stationarity

3. Weak and strong stationarity

Stationarity describes a statistical property of a stochastic process(time series model).

Briefly speaking, it means that the process achieves a certain state ofstatistical equilibrium so that the distribution of the process does notchange much.

In particular, it guarantees that the essential properties of a timeseries remain constant over time (as for instance, the mean, thevariance, and the autocovariances).

Depending on whether all characteristics (as for instance, allmoments) or only some particular ones (as for instance, the first andsecond moments, and the covariances) are of interest, different typesof stationarity are distinguished (e.g., strong stationarity and weakstationarity).



Definition (Strong stationarity)

A stochastic process (or a time series) (Xt)t∈Z is said to be strongly orstrictly stationary if the joint distribution of (Xt1 , · · · ,Xtk ) is identical tothat of (Xt1+t , · · · ,Xtk+t) for all t, where k is an arbitrary positive integerand (t1, t2, · · · , tk) is a collection of k positive integers.



An equivalent definition is given as follows.

Definition (Strong stationarity)

A stochastic process (or a time series) (Xt)t∈Z is said to be strongly orstrictly stationary if the distribution of (Xt)t∈Z is identical to that of(Xt)t∈Z with Yt = Xt+h.

Remarks :

1 Strong stationarity is equivalent to say that the distribution isinvariant over time.

2 Strong stationarity is often too restrictive since it requires that thetime series is completely invariant over time, i.e. all moments areconstant over time.



Definition (Weak stationarity)

A stochastic process (Xt) is weakly stationary, covariance or second-orderstationary if it satisfies the following properties :

1 E(Xt) = m is independent of t ;

2 V(Xt) is time-invariant.

3 Cov(Xt ,Xt+h) = E [(Xt −m)(Xt+h −m)] = γX (h) is time-invariant.

Remarks :

1 More formally, the second condition writes ”V(Xt) exists (i.e.E(X 2

t ) <∞)”.

2 γX (h) is the j-th lag autocovariance and γX (0) = V(Xt).

3 γX (h) = γX (−h) for all h, i.e.

Cov(Xt+h,Xt) = Cov(Xt ,Xt−h).



Remarks :1 Weak stationarity exploits the ”stability” of the first two moments

whereas strong stationarity implies the stability of all the moments(among others).

2 Strong stationarity implies weak stationarity as long as the first twomoments exist.

3 The converse is not true in general.

4 If (Xt) is a Gaussian stochastic process, weak stationarity isequivalent to strong stationarity (why ?).

5 An independent and identically distributed stochastic processsatisfies :

γX (h) = 0 for all h 6= 0.



Examples :

1. Independent White Noise (IWN(0,σ2)) :

Xt = εt with εt ∼ i.i.d.(0, σ2)

where E(Xt) = 0,V(Xt) = σ2, and γX (h) = 0 for all h 6= 0.

2. Gaussian White Noise (GWN(0,σ2)) :

Xt = εt with εt ∼ i.i.d.N (0, σ2)

where E(Xt) = 0,V(Xt) = σ2, and γX (h) = 0 for all h 6= 0.

3. White Noise (WN(0,σ2)) :

Xt = εt with εt ∼ (0, σ2)

where E(Xt) = 0,V(Xt) = σ2, and Cov(εt , εt−h) = 0 for all h 6= 0.



More formally...

Definition (Weak White Noise)

A discrete stochastic process (εt) is said to be a weak white noise if :

E(εt) = 0 for all t

V(εt) = E(ε2t ) = σ2

ε <∞ for all t

Cov(εt , εt−h) = E(εtεt−h) = 0 for all t and h 6= 0.

Remark : The assumption of independence is not required—only theabsence of correlation is needed.



Definition (Strong White Noise)

A discrete stochastic process (εt) is said to be a strong white noise if :

1 E(εt) = 0 and V(εt) = E(ε2t ) = σ2

ε <∞ for all t ;

2 The εt random variables are identically distributed ;

3 εt and εt−h are independent for all t and h 6= 0.

Remarks :

1 Independence is required.

2 If εt ∼ N (0, σ2ε), (εt) is a Gaussian white noise process.

3 White noise processes are the building block used in most time series models.



0 50 100 150 200 250 300-3

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3Simulation of a white noise with mean 0 and variance 1



Examples (cont’d) :

4. Moving average process of order 1 (Xt ∼ MA(1)) :

Xt = εt + θεt−1,

where θ ∈ R∗ and εt ∼WN(0, σ2ε ) and :

E(Xt) = 0

V(Xt) = σ2ε (1 + θ2)

Cov(Xt ,Xt−h) =

0 if |h| > 1

θσ2ε if |h| = 1.



Examples (cont’d) :

5. Autoregressive process of order 1 (X ∼ AR(1))

Xt = ρXt−1 + εt ,

where |ρ| < 1, εt ∼WN(0, σ2ε ), and :

E(Xt) = 0

V(Xt) =σ2ε

1− ρ2

Cov(Xt ,Xt−h) = ρhσ2ε

1− ρ2.



Stationarity versus non-stationarity : A first discussion...

Do macro and financial times series (weakly) stationary ?

Well, this is debatable ! ! !

The assumption of stationarity is generally appropriate for financialreturns, but not for the (log) asset prices, interest rates, or exchangesrates....

The assumption of stationarity has been hotly debated inmacroeconomics times series (GDP, consumption, investment,unemployment) and for good reasons !



1980 1985 1990 1995 2000 2005 20100

2000

4000Value of the S&P 500 index

1980 1985 1990 1995 2000 2005 20104

6

8

10Value of the log of S&P 500 index

1980 1985 1990 1995 2000 2005 2010-40

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20Monthly log-return of the S&P 500 index

Note the difference between the (log-) price and the return of theS&P500...the latter is mean-reverting around 0, the former is not !



Note that :

1 (Xt)t∈Z is stationary but (Yt)t∈Z and (Zt)t∈Z are not !

2 The non-stationarity of (Zt)t∈Z is mostly due to the failure of the mean revertingproperty (or the time-dependence of the mean)

3 The non-stationarity of (Yt)t∈Z is mostly due to the time-dependence of thevariance.



Examples of nonstationary processes

1. Deterministically trending process :

Xt = a + bt + εt

where εt ∼WN(0, σ2ε ) and b 6= 0. (Xt)t∈Z is nonstationary since :

E(Xt) = a + bt depends on t.

Remark : A simple detrending transformation (a and b beingknown...) yields a stationary process :

Xt ≡ Xt − a− bt = εt



More formally...

Definition (Deterministic trending process)

(Xt)t∈Z is a (nonstationary) deterministically trending process (i.e., trendstationary) if it can be written as :

Xt = f (t) + Zt

where f is a deterministic function of t and (Zt)t∈Z is a covariancestationary process.



Examples of nonstationary processes (cont’d)2. Random walk process :

Xt = Xt−1 + εt

where εt ∼WN(0, σ2ε) and X0 is fixed. This can be written as :

Xt = Xt−2 + εt−1 + εt = · · ·

= X0 + ε1 + · · ·+ εt = X0 +t−1∑j=0

εt−j .

A random walk is nonstationary since :

V(Xt) = V

(X0 +

t−1∑j=0

εt−j

)= V

(t−1∑j=0

εt−j

)=

t−1∑j=0

V(εt−j)

= σ2ε

t−1∑j=0

1 = tσ2ε .

Remark : A simple transformation (first-difference transformation) yields astationary process :

∆Xt ≡ Xt − Xt−1 = εt .Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 43 / 61


0 50 100 150 200 250 300 350 400 450 500-30

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40Simulation of a random walk

no driftwith drift mu=0.1



More formally...

Definition

(Xt)t∈Z is a (nonstationary) stochastically trending process (differencestationary) or is integrated of order d if the d−differenced process is(asymptotically) stationary :

Zt ≡ (1− L)dXt = ∆dXt

where L is the lag operator, d is a positive integer, and ∆ is thedifferentiation operator with ∆ = 1− L.

Remarks :

1 A trend stationary process and a difference stationary process havedifferent statistical properties (see Block 3).

2 In practise, they can be difficult to disentangle (see Block 3).



Remarks :

1 The lag operator transforms Xt into Xt−1 = LXt .

2 The difference operator ∆ expresses the difference between twoconsecutive random variables (or realizations).

3 ∆d is defined by :

∆d = (1− L)d =d∑

k=0

n!

k!(n − k)!(−1)kLk .

4 A random walk (with or without drift) is an I (1) process, i.e. isintegrated of order one (d = 1).



Does it matter ?

YES ! ! !

In which sense(s) ?

Statistical properties are fundamentally different !

Modeling choice : ARMA(p,q) versus ARIMA(p,d,q)

Nature of shocks : permanent versus temporary shocks

Inference and asymptotic distributions

Forecasting

Etc.

⇒ The determination of stationarity/nonstationarity is a prerequisitein time series analysis...(see further—block 3.1.)


Linear time series and transformation

4. Linear time series, Wold’s decomposition andstationarity

Definition (Linear time series)

A time series (Xt) is said to be linear if it can be written as :

Xt = µ+∞∑k=0

θkεt−k

where µ is the mean of Xt , θ0 = 1, and (εt) is a white noise process withE(εt) = 0, V(εt) = σ2

ε < +∞, and Cov(εt , εt−k) = 0, for all k 6= 0.

Remark : More generally, a linear stochastic process has the following form :

Xt = µ++∞∑

k=−∞

θkεt−k



Properties of a linear process

Definition

Let (Xt) denote a weakly stationary stochastic process that has thefollowing linear representation :

Xt = µ+∞∑k=0

θkεt−k

Then :

E [Xt ] = µ

V [Xt ] = σ2ε

∞∑k=0

θ2k

Cov [Xt ,Xt−h] = σ2ε

∞∑k=0

θkθk+h.

Proof : See Appendix 1.Florian Pelgrin (HEC) Univariate time series Sept. 2011 - Jan. 2012 49 / 61


Transformation of a linear process

Proposition

Let (Xt)t∈Z be a weakly stationary process and (ai )i∈Za sequence of realnumbers such that : ∑

i∈Z|ai | <∞.

Then Yt =∑i∈Z

aiXt−i defines a new weakly stationary process.



Transformation of a linear process (cont’d)

1 Yt is defined (almost surely) for all t, i.e. Yt ∈ L1(Ω,A,P), ∀t ∈ Z.

2 Yt ∈ L2(Ω,A,P), ∀t ∈ Z.

3 (Yt)t∈Z is a weakly stationary process such that :

EYt = mY = mX

∑i∈Z

ai

γY (h) =∑i∈Z

∑j∈Z

aiajγ(h + j − i)

∀h ∈ Z.

Proof : See Appendix


Innovation and Wold’s decomposition

5. Innovation and Wold’s decomposition

Definition (Strong innovation)

Let (Xt)t∈Z be a second order (stationary) stochastic process. Let Xt−1

denote all the past of Xt , Xt−1 = (Xt−1,Xt−2, · · · ). The stochasticprocess (εt)t∈Z defined by :

εt = Xt − E[Xt | Xt−1

]is said to be the strong innovation process of (Xt)t∈Z.



Definition (Linear past and Linear expectation)

Let (Xt)t∈Z be a second order (stationary) stochastic process.

1 The linear past of Xt , denoted Ht−1, is defined to be the vectorialspace made of all variables corresponding to linear combinations(finite or infinite) of 1,Xt−1,Xt−2, · · · .

2 The linear expectation of Xt conditionnally to its past is theprojection of Xt on Ht−1 :

X ∗t = EL [Xt | Ht−1]

Remark : X ∗t is the best approximation of Xt as linear combination of itspast.



Definition (Linear or weak innovation)

Let (Xt)t∈Z be a second order (stationary) stochastic process. The linearinnovation of (Xt) is defined to be :

εt = Xt − X ∗t

where

X ∗t = EL [Xt | Ht−1] .



Proposition

The stochastic process (εt) is a (weak) white noise.

Remarks :

1 (εt) corresponds to a sequence of homoscedastic (same variance),uncorrelated (autocovariances are zero), centered (mean zero)variables.

2 The innovation stochastic process contains all the new informationthat appears at time t and that is not predictable at time t − 1.

3 One has :

E(εt) = 0

E(εtXt−k) = Cov(εt ,Xt−k) = 0 ∀k > 0.



Definition (Wold’s decomposition)

Any covariance (weak) stationary time series (Xt) can be represented inthe form :

Xt = µ+∞∑k=0

θkεt−k

where µ is the mean of Xt , θ0 = 1, and (εt) is a white noise process withE(εt) = 0, V(εt) = σ2

ε < +∞, and Cov(εt , εt−k) = 0, for all k 6= 0.

Remark : This representation only exploits the covariance stationaryproperty.


Appendix

7. Appendix

1. Moments using the linear representation of time series.

2. Transformation of a linear time series.


Appendix

1. Moments using the linear representation of time series

Using the linear representation definition, the properties of Xt aregoverned by the θk coefficients.

In particular,

E(Xt) = µ

V(Xt) = E[(Xt − µ)2

]= E

( ∞∑k=0

θkεt−k

)2

=∞∑k=0

θ2kE[ε2t−k]

= σ2ε

∞∑k=0

θ2k


Appendix

Finally,

Cov(Xt ,Xt−h) = γX (h) = E [(Xt − µ)(Xt−h − µ)]

= E[(εt + θ1εt−1 + · · ·+ θhεt−h + · · · )×(εt−h + θ1εt−h−1 + · · ·+ θhεt−h−j + · · · )]

= σ2ε (θh + θh+1θ1 + · · · ) = σ2

ε

∞∑k=0

θkθk+h.


Appendix

2. Transformation of a linear time series

Proof :

1. The first property is a direct consequence of property (2) sinceL1(Ω,A,P) ⊂ L2(Ω,A,P).

2. The series with aiXt−i is convergent (in the standard way) in L2. Onehas : ∑

i∈Z‖aiXt−i‖2 =

∑i∈Z|ai |‖Xt−i‖2

=(γX (0) + m2

X

)1/2∑i∈Z|ai |

< +∞⇒ Yt ∈ L2(Ω,A,P).

3. The expression∑i∈Z

aiXt−i is then defined in L2. Therefore, the

moments of the stochastic process (Yt)t∈Z are defined. In particular,one can interchange the integral- and sum-operator (Fubini’stheorem).


Appendix

Proof (end) :

3. (end) One has :

EYt = E

(∑i∈Z

aiXt−i

)=∑i∈Z

aiE(Xt−i )

= mX

∑i∈Z

ai = mY independent of t

and

Cov(Yt ,Yt−h) = Cov

∑i∈Z

aiXt−i ,∑j∈Z

aiXt−h−j

=

∑i∈Z

∑j∈Z

aiajCov(Xt−i ,Xt−h−j)

=∑i∈Z

∑j∈Z

aiajγX (h + j − i).


lecture 1: fundamental concepts in time series analysis...

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