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Time Series Analysis
Basic Concepts

Rachidi Kotchoni (rachidi.kotchoni@u-paris10.fr)
Université Paris Ouest Nanterre La Défense

September 30, 2016

R. Kotchoni ()

Time Series Analysis

September 30, 2016

1 / 47

Types of Time Series

A time series is a variable that describes a statistical entity over time.
The value taken by X at time t is denoted Xt , t = 0, 1, ..., ∞.

Two types of time series:
Stock: level of a variable recorded at a given point in time. Example:
the price of a stock; the wealth of an agent; the stock of capital in a
given economy.
Flow: variable of a stock. Example: GDP growth rate; return on a
…nancial asset; investments during a year in a given economy.

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Types of Time Series
Suppose we have monthly observations (Xt ) and want to obtain
quarterly data
If Xt is a ‡ow, we simply take its sum within each quarter
If Xt is a stock, we may pick the observations at the beginning, in the
middle or a the end of every quarter. Alternatively, we can take the
average value of Xt over the quarter.

Example: If Xt,j is the log-return on an asset at day j of month t:
Xt,j = log

Pt,j
,
Pt,j 1

the monthly returns are:
et,j =
X

mt

∑ (log Pt,j

log Pt,j

1)

= log Pt,m t

log Pt,1

j =1

where mt is the number of days in month t.
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Time Series Analysis

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Regularly Sampled Time Series

Time series that are observed at …xed lengh of intervals are said to be
regular.
Intra-daily series: e.g., observed every 5 minutes
daily series: one observation per day
weekly series: one observation per week
Likewise: monthly series; quarterly series; yearly series.

Examples:
quarterly GDP of Canada
monthly Consumer Price Index of Benin
daily close price of the S&P500 (US stock market index)

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Time Series Analysis

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Irregularly Sampled Time Series

Stock transaction prices are typically observed at irregular time
intervals
Example: Intradaily transaction prices of the IBM stock.
ask price: demanded for an immediate "sell"
bid price: o¤ered for an immediate "buy".

The "bid price" is always below the "ask price".
A transaction occurs when a seller chooses to sell at the buyer’s price
(bid), or when a buyer chooses to buy at the seller’s price (ask).
Pt,j = price of transaction #j during day t.
If several transactions occur at the same second, the recorded price will
be an average of these transactions’prices.

R. Kotchoni ()

Time Series Analysis

September 30, 2016

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Irregularly Sampled Time Series

Transactions occur at random time intervals
The time elapsed between Pt,j and Pt,j +1 is a random variable.

For certain stocks, hundreds or thousands of daily transactions
For some other stock, only a few transactions.
The number of transactions on an asset is a proxy for its level of
liquidity
The di¤erence between the "ask" and the "bid" is also a measure of
liquidity.

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Time Series Analysis

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Frequency of a Series

High frequency data: intra-daily, daily, weekly (or even, monthly for
macroeconomists).
Low frequency data: one observation per month, quarter or year.
The frequency of a regular series is:
freq=

1 unit of time
time between 2 consecutive obs.

If the unit of time is a day, month or year, then freq gives the number
of observations per day, month or year.

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Features of times series

Time series data are usually not independent and identically
distributed
Correlation over time: xt can often be used to predict the behavior of
xt + 1
Trend: xt may be increasing or decreasing over time so that
E ( xt ) 6 = E ( xt + 1 ) .
Volatility: The variance of xt may be time varying
Conditional Heteroscedasticity: The variance of xt conditional on
past information may be time varying

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Features of times series

Cycles: the trajectory of xt may display predictable swings over time.
Seasonality: usually refers to cycles within a year. For instance, the
sales of toy shops are quite in Q4 and low in Q1
Long cycles: usually refers to cycles of several years, like economic
expansions and recessions.

Erratic ‡uctuations: ‡uctuations that are hard to predict.

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Why Study Time Series?

1

Find the model that best describes an economic series

2

Forecast future realizations of economic time series

3

Test economics theories. Example of theories are: the Phillips curve;
The consumption CAPM; The e¢ cient Market Hypothesis; The
Expectation Hypothesis; The Purchasing Power Parity etc.

4

etc.

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Trend and Seasonality
Let yt be a quarterly series in which we suspect the presence of a
trend, time dependence and seasonality.
We may specify the following model for yt :
yt
µt

= µt + ρ yt 1 µt 1 + εt avec
= α0 + α1 t + δ1 Qt,1 + δ2 Qt,2 + δ3 Qt,3 , 8 t

µt is the mean of yt
α0 : a constant
α1 t: a linear tend
ρ: autocorrelation or time dependence,
δ1 Qt,1 , δ2 Qt,2 and δ3 Qt,3 : Seasonality
εt : error term.

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Trend and Seasonality

In general, the trend can be any deterministic function of time g (t )
linéaire tend: g (t ) = α1 t
logarithmic tend: g (t ) = α1 log t
quadratic tend: g (t ) = α1 t 2

Qt,k , k = 1, 2, 3, 4 are seasonal dummy variables:
Qt,k =

1 if t is the k th quarter of the current year
0, sinon

Qt,4 is not included to avoid multicolinearity.
δ1 , δ2 and δ3 are called seasonal oe¢ cients.

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De-trending and de-Seasonalization

Substitute µt and µt
yt

1

in the expression of yt :

= (α0 + α1 t + δ1 Qt,1 + δ2 Qt,2 + δ3 Qt,3 ) +
ρ (yt 1 α0 α1 (t 1) + δ1 Qt 1,1 + δ2 Qt
+ εt .

1,2

+ δ3 Qt

1,3 )

This yields an equation that can be estimated by OLS:
yt

R. Kotchoni ()

= b
β0 + b
β1 t + b
ρyt 1 + b
δ1 Qt,1 + b
δ2 Qt,2 + b
δ3 Qt,3
+b
β2 Qt 1,1 + b
β3 Qt 1,2 + b
β4 Qt 1,3 + bεt

Time Series Analysis

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De-trending and de-Seasonalization

α0 and α1 are estimated as:
b
β1 = b
α1 (1

b
ρ) and b
β0 = (1

b
ρ) b
α0 + b
ρb
α1

After estimating the coe¢ cients, one computes the de-trended and
de-seasonalized series as

where

yet = yt

bt
µ

bt = b
µ
α0 + b
α1 t + b
δ1 Qt,1 + b
δ2 Qt,2 + b
δ3 Qt,3

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Time Series Analysis

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De-trending and de-Seasonalization

This procedure can be adapted for the case of an autoregresive
process of order p.
p

yt

µt

=

∑ ρi

yt

i

µt

i

+ εt avec

i =1

µt

= α0 + α1 t + δ1 Qt,1 + δ2 Qt,2 + δ3 Qt,3 ,

Simply substitue µt , ..., µt p in the expression of yt and estimated
the resulting equation by OLS
Note that it is possible to …lter µt nonparametrically as well.

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Strong Stationarity

yt is said to be strongly stationary if the distribution of
(yt , yt +1 ..., yt +p ) is the same for all t, 8 p 2 Z.

This de…nition does not require that the moments of yt exist
It is hard to use empirically.
It can be used to check if a speci…ed theoretical model is stationary

Moments of a random variable X :
Any linear combination of quantities of type E X k , k =2 N.

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Famous moments
First moment: µ = E (X )
Second moments:
h
m2 = E X 2 (non centered); µ2 = E (X

i
µ)2 (centered)

Third moments:
h
m3 = E X 3 (non centered); µ3 = E (X
Skewness: S =

µ3
σ3/2

i
µ)3 (centered)

Fourth moments:
h
m4 = E X 4 (non centered); µ4 = E (X
Kurtosis: K =

R. Kotchoni ()

Time Series Analysis

µ4
σ4

i
µ)4 (centered)
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Covariance Stationarity

yt is said to be second order stationary or covariance stationary if :
E (yt ) = µ (constant 8 t)

Var (yt ) = σ2 (constant 8 t)

Cov (yt , yt

h)

= γh (Only depend on h)

If yt is strongly stationary and has …nite second moments, then it is
also covariance stationary.
A series that has a trend or a seasonality is not stationary.
A series that is heteroscedastic is not stationary
"Stationary" with no other precision means "covariance stationary".

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Time Series Analysis

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Conditional Heteroscedasticity

A series can be stationary but conditionally heteroscedastic:
constant mean: E (yt ) = µ for all t.
constant unconditional variance: Var (yt ) = σ2 for all t.
stable autocovariance structure: Cov (yt , yt h ) = γh for all t.
but, time varying conditional variance:
Var (yt jyt 1 , yt 2 ...y0 ) = σ2t

In this case, we have:
Var (yt )

E (Var (yt jyt

= E

R. Kotchoni ()

σ2t

+ Var

1 , yt 2 ...y0 )) + Var
(µ) = E σ2t .

Time Series Analysis

(E (yt jyt

1 , yt 2 ...y0 ))

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Autocovariance

Let yt be a covariance stationary series.
The autocovariance of order h of yt is:
γ(h ) = Cov (yt , yt

h)

γh .

Note that:

= Cov (yt , yt h )
= Cov (yt , yt +h ) γ h and
γ(0) = Cov (yt , yt ) = Var (yt ) = σ2
γh

R. Kotchoni ()

Time Series Analysis

γ0 .

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Autocorrelation

The autocorrelation of order h is:
ρ (h ) =

γh
γ0

ρh

By de…nition, we have: ρ0 = 1.

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Partial Autocorrelation

The partial autocorrelation of order h of yt , denoted e
ρh , is the
correlation between yt and yt h that does not transit by observations
yt 1 , ..., yt h +1 .
It is the "response" of yt to yt
(yt 1 , ..., yt h +1 ).

h

when we control for the e¤ects of

It can be obtained via the following OLS regression:
yt = β0 + β1 yt

1

+ ... + βt

h + 1 yt h + 1

By de…nition, e
ρ0 = ρ0 = 1 and e
ρ1 = ρ1 .

+e
ρh yt

h

+ εt

To …nd e
ρ2 ,...,e
ρh one need to estimate separate regressions at di¤erent
lags and pick the last coe¢ cient.
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Time Series Analysis

September 30, 2016

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Autoregressive Model of Order 1

Consider the autoregressive model of order 1, denoted AR(1):
yt = µ + ρyt

1

+ εt

where εt is a "white noise" with mean 0 and variance σ2ε .
Weak white noise: εt is uncorrelated with is past and future
realizations:
E (εt εt h ) = 0 for all h 2 Z
Strong white noise: εt is IID on top of being a weak white noise.

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Time Series Analysis

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Autoregressive Model of Order 1

The AR(1) model implies that:
yt
Substituting for yt
yt

1

1

= µ + ρyt

2

+ εt

1

into yt yields:

= µ + ρ (µ + ρyt 2 + εt 1 ) + εt
= µ (1 + ρ) + ρ2 yt 2 + εt + ρεt

1

But we also have:
yt

R. Kotchoni ()

2

= µ + ρyt

Time Series Analysis

3

+ εt

2

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Autoregressive Model of Order 1

Substitute again yt
yt

2

in this expression

= µ (1 + ρ) + ρ2 (µ + ρyt 3 + εt 2 ) + εt + ρεt
= µ 1 + ρ + ρ2 + ρ3 yt 3 + εt + ρεt 1 + ρ2 εt

∑ ρi + ρ3 yt

3

+

t 1

yt = µ



∑ ρi εt

3,

yt

4,

etc., we obtain:

ρi + ρt y0 +

i =0

R. Kotchoni ()

i

i =0

i =0

By substituting recursively yt

2

3 1

3 1

= µ

1

Time Series Analysis

t 1

∑ ρi εt

i

i =0

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Autoregressive Model of Order 1

Let us consider the case ρ 6= 1. We have:
E (yt ) = µ

1
1

ρt
+ ρt E (y0 )
ρ

Var (yt ) = ρ2t Var (y0 ) +

t 1

∑ ρ2i σ2ε

i =0

= ρ2t Var (y0 ) + σ2ε

1
1

ρ2t
.
ρ2

Note that ρ 6= 1 is necessary for E (yt ) and Var (yt ) to exist.

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Autoregressive Model of Order 1
Let us …nd the limit of E (yt ) and Var (yt ) as t ! ∞.
When jρj < 1, we have:
lim E (yt ) =

t !∞

µ
1

ρ

and lim Var (yt ) =
t !∞

σ2ε
.
1 ρ2

Moreover,
Cov (yt , yt

h)

= ρh Var (yt ) .

Hence, the AR(1) is second order stationary if and only if jρj < 1. In
this case, we have:
E (yt ) =

R. Kotchoni ()

µ
1

ρ

and Var (yt ) =

Time Series Analysis

σ2ε
.
1 ρ2

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Explosive Process

When jρj > 1, we have
lim E (yt ) =

t !∞

∞ and

lim Var (yt ) = ∞.

t !∞

In this case, the AR(1) is explosive and therefore non stationary.

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Time Series Analysis

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Stochastic Trend
Unit Root

Let us now consider the case ρ = 1 (unit root).
We have:

t 1

yt = µt + y0 +

∑ εt

i

i =0

so that:
E (yt ) = µt and Var (yt ) = tσ2ε
The series behaves as though it has a linear trend but its variance is
in…nite.
Hence the term "stochastic trend".

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Time Series Analysis

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Stochastic Trend
Unit Root

Likewise, when ρ =
yt = µ

1 we have:
t 1

t 1

i =0

i =0

∑ ( 1)i + ( 1)t y0 +

∑(

1)i εt

i

Hence:
E (yt jy0 ) =

µ + y0 and Var (yt ) = tσ2ε if t even
y0 and Var (yt ) = tσ2ε if t is odd

E (yt jy0 ) = µ

so that yt is non stationary
Economic time series will mostly have ρ = 1.

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Time Series Analysis

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Stochastic Trend
Testing for Unit Root

Several tests exist in the literature to detect the presence of unit root
e.g.: Dickey-Fuller test.

Suppose ρ = 1 so that
yt = µ + yt
The …rst di¤erence ∆yt = yt

yt

1

1

+ εt

is therefore stationary:

∆yt

= µ + εt
E (∆yt ) = µ; Var (∆yt ) = σ2ε
Cov (∆yt , ∆yt h ) = 0 for all h

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Time Series Analysis

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Stochastic Trend
Testing for Unit Root

The Dickey-Fuller test exploits the idea that if ρ = 1, the coe¢ cient
α should not be signi…cant in the following regression:
∆yt = µ + αyt

1

+ εt

Versions of the test exist where a trend and/or longer lags are added
in the RHS.
∆yt = α0 + µ1 t + α1 yt

1

+ α2 yt

2

+ ... + αp yt

p

+ εt

In all cases, the test is based on the slope coe¢ cient of the …rst lag
yt 1
The distribution of the test statistic depends on whether there is a
trend or not or whether lags of yt are added or not
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Time Series Analysis

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Integration of Order d
If the data generating process is more complicated than an AR(1), it
can be necessary to di¤erenciate yt more than once before obtaining a
stationary series.
∆d yt is the series yt di¤erenciated d times:
∆yt
∆ yt
2

∆3 yt

=
=
=
=
=

yt

yt

1

∆ (∆yt ) = ∆ (yt

yt

(yt yt 1 ) (yt 1
yt 2yt 1 + yt 2
∆ ∆2 yt = ...etc.

1)

yt

2)

If ∆d yt is stationary while ∆n yt is not for n < d, then we say that yt
is integrated at order d
i.e., yt have a unit root with order of multiplicity d
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Time Series Analysis

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Autoregressive Model of Order p: AR(p)
The AR(p) model is of the form:
p

yt = c + ∑ ρi yt

i

+ εt

i =1

where εt is a white noise.
Let us de…ne the "lag" operator as:
Lyt = yt

1.

Then we have:
L2 yt
p

L yt

R. Kotchoni ()

= LLyt = Lyt
= yt p .

Time Series Analysis

1

= yt

2

and

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Stationarity of an AR(p)
Let us de…ne P (L) = 1

p
∑i =1 ρi Li . Then the AR(p) becomes:

P ( L ) yt = c + ε t
If there exist a polynomial P 1 (L) = ∑i∞=0 θ i Li such that
2
∑i∞=0 θ i < ∞ and P 1 (L)P (L) = 1, then P (L) is invertible.
For an AR(1), P (L) = 1 ρL.
If jρj < 1, we have:

P 1 (L) =



∑ ρ i Li

i =0

If jρj

1, then P 1 (L) does not exist.

P (L) is invertible if all the solutions of P (x ) = 0 are outside the unit
circle.
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Time Series Analysis

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Stationarity of an AR(p)
If P (L) is invertible, we can write:
P (L)yt
yt

= c + εt ,
= P

1



(c + εt ) =

∑ θ i Li ( c + ε t ) = c

i =0





i =0

i =0

∑ θ i + ∑ θ i εt

1.

with the concention that: Li c = c.
If εt

1

is a white noise with mean 0 and variance σ2ε , then:


E (yt ) = c

∑ θi

and Var (yt ) = σ2ε

i =0



∑ θ 2i

i =0

The AR(p) model P (L)yt = c + εt is stationary if and only if the
polynomial P (L) is invertible.
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Time Series Analysis

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Mean of a stationary AR(p)
Consider a stationary AR(p) given by:
p

yt = c + ∑ ρi yt

i

+ εt

i =1

Take the expectation on both sides of the equality:
p

E (yt ) = c + ∑ ρi E (yt i ) + E (εt ) ,
i =1

where E (εt ) = 0 for all t.
Moreover, E (yt ) = E (yt i ) = µ because of stationarity. Hence:
p

µ = c + ∑ ρi µ ) µ =
i =1

R. Kotchoni ()

Time Series Analysis

1

c
p
∑ i =1 ρ i
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Mean of a stationary AR(p)

Stationarity requires that ∑pi=1 ρi 6= 1.
p
∑i =1 ρi = 1 implies the existence of a unit root.
With no loss of generality, we can write the AR(p) as:
p

yt

µ=

∑ ρi (yt

i

µ ) + εt ,

i =1

using the fact that c = µ

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p

∑i =1 ρi µ.

Time Series Analysis

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Autocovariances of a stationary AR(p)
Multiply both sides of equality by (yt
h
E (yt

i
µ )2 =

µ) and take the expectation:

p

∑ ρi E [(yt

i

µ) (yt

µ)] + E [εt (yt

µ)]

i =1

But note that:

E [(yt

i

h
E (yt

µ )2

µ) (yt

i

= Var (yt ) = γ0

µ)] = Cov (yt i , yt ) = γi

Likewise:
p

E [εt (yt

µ)] =

∑ ρi E [εt (yt

i

µ)] + E ε2t = σ2ε .

i =1

since εt is uncorrelated with lags of yt .
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Time Series Analysis

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Autocovariances of a stationary AR(p)

Finally we have:

p

γ0 =

∑ ρi γi + σ2ε ,

i =1

This de…nes a …nite di¤erence equation of order p for the sequence γi .
Classical methods exist to solve this kind of equation, but not relevant
for this case.

Instead, we will try to …nd a system of linear equation of which
γ0 , γ1 , γ2 , ..., γp is solution
The equation above contains p + 1 unknowns. Hence, we need p other
equations.

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Time Series Analysis

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Autocovariances of a stationary AR(p)
Multiply again the AR(p) by (yt
E [(yt

µ) (yt

h

µ) and take the expectation:

h

µ)]

p

=

∑ ρi E [(yt

µ) (yt

i

h

µ)] + E [εt (yt

h

µ)]

i =1

for h = 1, ..., p.
This leads to:
p

γh

=

∑ ρ i γ ji

i =1

= ρ1 γh

1

hj

+ ρ2 γh

2

+ ... + ρh γ0 + ρh +1 γ1 + ... + ρp γp

h

Together with the previous equation, we obtain, a system of p + 1
equations with p + 1 unknowns.
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Time Series Analysis

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Example for an AR(2)

For an AR(2), we have to solve:
8
< γ0 = ρ1 γ1 + ρ2 γ2 + σ2ε
γ1 = ρ1 γ0 + ρ2 γ1
:
γ2 = ρ1 γ1 + ρ2 γ0

Solve this system for γ0 , γ1 and γ2 , assuming ρ1 , ρ2 and σ2ε are
known
Empirically, we often have to deal with the inverse problem:
b0 , γ
b 1 and γ
b2 .
try to estimate ρ1 , ρ2 and σ2ε for γ

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Time Series Analysis

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Estimation of AR(p) Model
Consider the last p equations of the previous system:
γh = ρ1 γh

1

+ ρ2 γh

2

+ ... + ρh γ0 + ρh +1 γ1 + ... + ρp γp

h,

for h = 1, ..., p.
In matrix notation, we have:
0
1 0
γ0
γ1
B γ C B
γ1
B 2 C B
B .. C = B
.
@ . A B
@ ..
γp
γp 1

γ1
γ0
..
.

..

.

..

.

γ1

γp
..
.
γ1
γ0

1

10
CB
CB
CB
C@
A

ρ1
ρ2
..
.
ρp

This justi…es the estimation of the coe¢ cients ρ1 , ..., ρp

R. Kotchoni ()

Time Series Analysis

1
C
C
C
A
by OLS.

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Estimation of AR(p) Models
OLS estimation of the slope coe¢ cients:
0
B
B
B
@
where

b
ρ1
b
ρ2
..
.

b
ρp

bh =
γ

1

C B
C B
C=B
A B
@
1

T

0

b0
γ

b1
γ
..
.

bp
γ

b1
γ

1

b0
γ
..
.

.

..

.

b1
γ

1

b1
γ
b0
γ

1
C
C
C
C
A

1

0
B
B
B
@

T

h t =∑
h +1

(yt

b ) (yt
µ

b
The constant is deduced as b
c=µ
R. Kotchoni ()

..

bp
γ
..
.

h
p

b ) and µ
b=
µ

1

b1
γ
b2
γ
..
.

C
C
C
A

bp
γ
1
T

T

∑ yt .

t =1

b.
ρi µ
∑ i =1 b

Time Series Analysis

September 30, 2016

44 / 47

Moving Average Model of Order q: MA(q)
The MA(q) model is of the form:
q

yt = c + εt

∑ θ i εt

j

j =1

where εt is a white noise.
Let us de…ne Q (L) = 1

q
∑j =1 θ i Li . Then the MA(q) becomes:

yt = c + Q ( L ) ε t
The MA(q) model is always stationary for …nite q:
q

E (yt ) = c and Var (yt ) = σ2ε

1+

∑ θ 2i

j =1

R. Kotchoni ()

Time Series Analysis

!

.

September 30, 2016

45 / 47

Invertibility of an MA(q)
The MA(q) model is invertible if the polynomial Q (L) is invertible.
Q (L) is invertible if all the solutions of Q (x ) = 0 are outside the unit
circle.
In this case, there exist Q 1 (L) = ∑i∞=1 ρi Li such that
Q (L)Q 1 (L) = 1.

An invertible MA(q) always admits an AR(∞) representation:


∑ ρi (yt

1

c ) = εt

i =1

Likewise, the inverse of a stationary AR(p) is an MA(∞).
The MA(q) model cannot be estimated by OLS.
Instead, one may use the maximum likelihood of the method of
moments.
R. Kotchoni ()

Time Series Analysis

September 30, 2016

46 / 47

ARMA Models
Stationarity and Invertibility

The ARMA(p,q) is of the following form:
q

p

yt = c + ∑ ρi yt

i

+ εt

i =1

∑ θ i εt

j

j =1

Using our previous notation, we have:
P ( L ) yt = c + Q ( L ) ε t
where

p

P (L) = 1

∑ ρi Li and Q (L) = 1

i =1

q

∑ θ i Li

j =1

The ARMA(p,q) model us stationary if P (L) is invertible.
The ARMA(p,q) model is invertible if Q (L) is invertible.
ARMA models may be estimated by maximum likelihood.
R. Kotchoni ()

Time Series Analysis

September 30, 2016

47 / 47


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