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Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence

E 5101/9101 Time Series Econometrics

Lecture 1:Stochastic di¤erence equations

Ragnar Nymoen

Department of Economics, University of Oslo

20 January 2011

ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo

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References

I Hamilton Ch 1 and 2

I Davidson and MacKinnon, Ch 1, although not speci…c abouttime series, contains an instructive review of econometricmodelling concepts that we assume known, and which willused already in the …rst lecture.


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I Dynamic models consist of relationships among variables at

di¤erent points in time.I We will work with discrete time , so dynamic models include

variables from di¤erent time periods.

I The structure of dynamic models can be simple, for example

Y t D 0 C 1Y t 1 C "t (1)

where "t is “white-noise”and 1 a parameter, or

Y t D 0 C 1X t C "t C 1"t 1 (2)

with 1 and 1 as parameters.

I Or large and complicated as in an econometric forecastingmodel or a calibrated DSGE model.


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I Simple and complicated dynamic structures have common

features ) we can learn a lot about dynamic models by …rststudying relatively simple and “transparent” structures.

I Static models (where time plays no essential role) willcontinue to interest us:

I

As special cases or simpli…cation of dynamic models (forexample 1 D 0 in (2).I As the solved out relationship between steady-state values of a

dynamic model’s variables—we refer to this as steady-state ,long-run or equilibrium relationships, e.g.

Y D 0 C 1X

where Y and X are long-run, steady-state values of Y t andX t .


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I As either cointegration regressions or spurious regressions

between stochastic trends

Y t D 0 C 1X t C "t ,

and assume that both Y t and X t are random-walk variables.Then "t is a random walk in the case of a spurious regression

but "t is a stationary variable in the case of a cointegratingregression.

I Already these …rst bullet points show that concepts likestochastic trend , stationary variable, and steady-state will play

central roles in a course on dynamic econometrics.I We start this course by making clear the meaning of

stationarity and by working out its implications in theory andin practical examples.


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I We then (many lectures ahead) start working with thespecial-case of non-stationarity in the form of stochastictrends.

I The mathematical background for the stationary case isreviewed in this …rst lecture. It is also essential forunderstanding the extension to non-stationarity that comeslater.


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AR(1) model I

I Assume that we have t D 1, 2, : : : , T independent andidentically distributed random variables "t :

"t IID 0, 2

" , t D 1,

2, : : : ,

T

Then, from

Y t D 1Y t 1 C "t ,

1

< 1, "t IID

0, 2

"

, (3)

we know something precise about the conditional distributionof Y t given Y t 1, and more generally the history of Y up to period t 1.


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AR(1) model II

I But this is not enough to characterize the properties of theOLS estimator for 1, which may be surprising based on whatwe know from regression models with “classical disturbances”.

I We will refer to Y t as given by (3) as a 1st order autoregressive process , usually denoted AR(1). The OLS/MMestimator b1 is

b1 D

PT t D2 Y t Y t 1

PT t

D2 Y 2t

1

DT

Xt

D2

1Y 2t 1

PT t

D2 Y 2t

1

!C

T

Xt

D2

Y t 1"t

PT t

D2 Y 2t

1

!(4)H)

E b1 1

D E

PT t D2 Y t 1"t

PT t D2 Y 2t 1

!.


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AR(1) model III

I Even if we assume E .Y t 1"t / D 0, we cannot state that thedenominator and numerator are independent: For example will"2 “be in” the numerator and (because of Y 2 D 1 C "2) alsoin Y 2 Y 2 in the denominator.

I This means that Y t

1 cannot be regarded as exogenous in theeconometric sense, and therefore E b1 1 6D 0.

I What about asymptotic properties? With reference to theLaw of large numbers and Slutsky’s theorem we have

plim b1 1 D plim1T PT

t D2 Y t 1"t

plim 1T

PT t D2 Y 2t 1

D 0 2

"

121

D 0.

if E .Y t 1"t / D 0 and

1

< 1.


( )

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AR(1) model IVI The zero in the numerator seems trivial since it is just a sum

of terms with zero expectations, but closer inspection showsthat we need that the variance of Y t 1"t is …nite. Thespeci…cation of the AR(1) model above is su¢cient for thisresult.

I The denominator is due to the assumption 1 < 1, whichentails that the variance of Y t in (3) is …nite and equal to 2

"/.1 21/. We will return to derivation and proofs later.

I The OLS/MM estimator

b1 is consistent, and it can be shown

to be asymptotically normal:

p T b1 1

d ! N 0,

1 2

1

(5)

which entails that t-tests can be compared with critical valuesfrom the normal distribution.


I d i B i i AR(1) d l Di¤ i C i f L f F d d

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Monte-Carlo analysis of AR(1)

In (3) the …nite sample bias can be shown to be approximately

E b1 1

t

21

T ,

this is called the Hurwitz-bias after Leo Hurwitz (1958). We can

make this more concrete with a Monte-Carlo analysis.In the experiment, the DGP is

Y t D 0.5Y t 1 C "Yt , "Yt NIID .0, 1/ ,

and T D

10, 11,. . . , 99, 100. We use 1000 replications for each T and estimate the bias:

OE b1.T / 1

D 1

1000

1000Xi D1

b1.T /i 1

.


I t d ti B i i AR(1) d l Di¤ ti C i f L t f F t d d

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Bias in the AR(1) model

10 20 30 40 50 60 70 80 90 100

.08

.07

.06

.05

.04

.03

.02

.01 OE

b1.10/ 0.5

D

0.058 >

... t 20.

510 D 0.1

OE b1.100/ 0.5

D0.008 >

...t

2

0.5

100 D 0.

01.


Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag operator form Future dependence

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Monte Carlo analysis of ARX(1)

Y t D 0 C 1Y t 1 C 0X t C "t , j1j < 1, "t IID

0, 2"

.(6)

which we will also refer to as an Autoregressive Distributed Lagmodel, ADL.We will investigate OLS properties formally later, but at this pointwe decide to trust the Monte Carlo:

Y t D

0.5Y t 1 C

1

X t C

"Yt

, "Yt

NIID .0, 1/ ,

X t D 0.5X t 1 C "Xt , "Xt NIID .0, 2/ ,

There are now two biases, OE

O1.T / 0.5

and OE

O 0.T / 1



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Biases in the ADL model

10 20 30 40 50 60 70 80 90 100

.04

.03

.02

.01

0.00

10 20 30 40 50 60 70 80 90 100

0.00

0.02

0.04

OE O1.T / 0.

5OE

O 0.T / 1



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Conclusions from the Monte Carlos

I The biases are small, and the speeds of convergence to zeroare high

I If the Monte Carlo can be trusted ,we see that the use of conventional estimation methods for dynamic models is“unproblematic”, given that the model is correctly speci…ed .

I However: Estimation can be re…ned, correctness of speci…cation is obviously a key point, and there are limits to

the conventional statistical analysis. All these issues will beaddressed in the course.



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Central concept: Solution of the AR(1) model I

By repeated substitution (backward)

Y t D 0 C 1Y t 1 C "t , (7)

we obtain

Y t D 0

t 1Xi D0

i 1 C t

1Y 0 Ct 1Xi D0

i 1"t i (8)

as the solution.

Proof: Insert solutions for Y t and Y t 1 in (7) and show that youget an identity.The solution is a function of t , the whole sequence "t ,"t 1, . . . "1

and the initial condition Y 0.



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g g ( ) q p g p p

Central concept: Solution of the AR(1) model II

The mathematical solution does not require "t IID 0, 2

"

or any

other speci…c distributional assumption for "t . But the statisticalproperties of the statistical variables Y t given by the solution willdepend on distributional assumptions. ARMA models will beconsidered early in our course.

Note also that the solution (8) does not depend on the assumption

1

< 1. But 1is nevertheless essential both for the nature of the

solution, as stable, unstable or explosive, and for the statistical

properties of the solution (cf ARMA models). Here be brie‡y lookat the importance of 1 for the stability of Y t from the AR(1).



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g g ( ) q p g p p

Solution without initial condition I

A …rst hint about the importance of 1 emerges if we considerfurther substitutions “back to in…nity”:

Y t D 0

1

Xi D0

i 1 C

1

Xi D0

i 1"t 1

If we assume 1 < 1, we can write

Y t D0

1 1

C1

Xi D0

i 1"t i (9)

(9) is a solution of (7). However

Y t D C t 1 C 0

1 1

C1

Xi D0

i 1"t i , (10)



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Solution without initial condition II

for an arbitrary C is also a solution.To see that, insert (10) into (7):

C t 1 C 01 1

C 1Xi D0

i 1"t i

D 0 C 1

C t 1

1 C 0

1 1

C1

Xi D0

i 1"t i 1

!C "t ,

since the two sides are identical, (10) is a solution of (7).



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Reconciling the solutions with/without initial condition I

We will show that if we call (9) a “particular solution”, Y

part

t , andC t 1 a “homogenous solution”. Y ht , then (10) is the same solution

as our …rst solution in (8) subject only to one condition: C in mustbe unique.We …rst solve the issue about the arbitrariness of C .

Since (10) must hold for all periods, we have, for t D 0:

Y 0 D C C 0

1 1

C1X

i D0

i 1"i (11)

In a causal solution “history is in principle known” includingP1i D0 i

1"i . Hence, we can regard C as determined from (11):

C D Y 0 0

1 1

1

Xi D0

i 1"i ,

1

< 1 (12)



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Reconciling the solutions with/without initial condition II

Since we can regard Y 0 as known, the arbitrariness of C isremoved. As we shall see, we can make this notion of uniquenessprecise by conditioning on the history of the process .The next slides show that, subject only to (12), the solutions (8)and (10) are indeed the same:

C t 1 |{z }Y ht C

0

1 1 C

1

Xi D0

i 1"t i | {z }

Y part t

0

t 1

Xi D0

i 1

Ct

1Y 0

C

t 1

Xi D0

i 1"t i | {z }

(8)



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Reconciling the solutions with/without initial condition IIISubstitution the solution for C in (12) back into (10) gives:

Y t D(

Y 0 0

1 1

1X

i D0

i 1"i

)t

1 C 0

1 1

C1X

i D0

i 1"t i

D0

1 1 C Y 0 0

1 1t 1

t 1

1Xi D0

i 1"i C

1Xi D0

i 1"t i

1

Xi D0

i 1"i

C

1

Xi D0

i 1"t i

D t

1 01"0

C1

1"1

C...

C01"t C 1"t 1 C ... C t

1"0 C t C11 "1 C ...

D 01"t C 1"t 1 C ...t 1

1 "1 Dt 1

Xi D0

i 1"t i .



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Reconciling the solutions with/without initial condition IVHence, we have

Y t D0

1 1

C

Y 0 0

1 1

t

1 Ct 1Xi D0

i 1"t i . (13)

If we can show that

0

1 1

C

Y 0 0

1 1

t

1 0

t 1Xi D0

i 1 C t

1Y 0 (14)

we have that the solution in (13) is the same as our …rst solution(8). We use the result that (assuming 1 < 1/

0

t 1

Xi D0

i 1 D 0

1 t 1

1 1

,



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Reconciling the solutions with/without initial condition Vhence

0

t 1Xi D0

i 1 C t

1Y 0 D 0

1 t 1

1 1

C t 1Y 0 D 0

1 1

0

t 1

1 1

C t 1Y 0

D0

1 1 C Y 0

0

1 1t

1

showing the identity in (14).



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Reconciling the solutions with/without initial condition VI

We summarize:

I If we know C , the solutions in (8) and (10) are the same.

I (8) is a general solution and can be interpreted as the sum of

a homogenous solution and a particular solution.I The particular solution corresponds to the solution without

initial condition, (9)

I The expression (13) is another way of writing the generalsolution (8) (for the case of 1 < 1/.

I As we will see, (13) is particularly useful for understanding theproperties of econometric forecasts.



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Causal and non-causal solutions

I Hamilton Ch 2.5, discusses at some length the solution whenwe cannot condition on Y 0.

I It may be unknown,I or irrelevant because

1

< 1 is not an economicallymeaningful assumption to make.

I Intuitively we understand that this will a¤ect the solution, forexample how the parameter C in the general solution isdetermined.

I We return to some of Hamiltons points when we distinguish

between causal solutions and non-causal solutions todi¤erence equations at the end of this lecture.I causal solutions and models correspond to

1

< 1I non-causal (future determined) solutions and models

correspond to

1

> 1



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A main tool in this course will be higher order stochastic di¤erence

equations with constant coe¢cients.A p th order equation for the stochastic variable Y t is

Y t D 0 C 1Y t 1 C 2Y t 2 C ... C p Y t p C "t (15)

where "t is a “white noise” variable.From a mathematical point of view, (15) is exactly the sameproperties as an inhomogenous deterministic di¤erence equation. If we for example write such an equation as

a0x t C a1x t 1 C ... C ap x t p D b t (16)

for x t I t D 0, 1, 2, .., we see the parallel by de…ning: x t D Y t ,b t D 0 C "t , a0 D 1 and ai D 1 .i D 1, 2, : : : , p ).



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From mathematics we know that, for known initial conditions x 0,x 1, . . . ,x t p 1 the general solution of (16) is:

x t D x ht C x part t (17)

where x ht is the solution of the homogenous di¤erence equation

a0x t C a1x t 1 C ... C ap x t p D 0 (18)

for known initial conditions and x part t is a particular solution of

(16).We can de…ne the characteristic polynomial associated with the

homogenous part of (15) as

p .y / D y p 1y p 1 ... p , (19)

for a number y .



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Let be at root in the characteristic equation given by p ./ D 0,namely

p 1p 1 ... p D 0 (20)

as in Proposition 1.1 in Hamilton, who refers to the roots aseigenvalues.Then Y ht

DC t , for an arbitrary constant C , is a solution of the

homogenous part of the stochastic di¤erence equation (15). Thisis seen directly from

Y ht

1Y ht

1

...

p Y ht

p

DC .t

1t 1

...

p

t p /

D C t p .p 1p 1 ... p / D 0

Since the order of the characteristic equation is p , there are p roots.



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The general homogenous solution is

Y ht Dk X

l D1

ml X j D1

C lj t j 1 t l (21)

where ml is the number of repetition of each distinct root, hencem1 C m2 C : : : C mk D p .

Set k D p and ml D 1, 8 l

Y ht Dp X

l D1

1X j D1

C lj t j 1 t l D

p Xl D1

1X j D1

C lj t j 1 t l D

p Xl D1

C l 1 t l

This is the case of p distinct roots, and by simplifying the notationwe get



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Y ht D C 1t 1 C C 2t

2 C ... C C p t p (22)

p initial values are assumed known, they determine C 1, C 2,...,

C p .In the following we will drop the explicit notation for homogenous(Y ht ) and special solution, and let it be understood from thecontext which solution we have in mind.



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Stability I

The …rst, and principal stability, concept we need is globalasymptotic stability.Equation (16) is globally asymptotically stable if the generalsolution of the associated homogenous equation tend to 0 ast

! 1of all values of the constants C lj . Then the e¤ect of the

initial conditions “dies out” as t ! 1.

TheoremA necessary and su¢cient condition for global asymptotical stability of a pth order deterministic di¤erence equation with

constant coe¢cients is that all roots of the associated characteristic equation (e.g. ( 20 )) have moduli strictly less than 1.

I For the case of real roots, moduli means “absolute values”,



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Stability III

more generally it refers to the magnitudes of complex roots(see lecture note on that).

I In general the condition states that all roots must be locatedinside the complex unit circle . Roots on the circle give rise tounstable solutions. Roots outside the unit circle gives rise to

explosive solutions.

I Often, in the time series literature, the stability conditions isstated as “all root outside the unit circle”. This is confusingbut is has a simple explanation.

I Consider multiplying both sides of our charateristic equationby p . This gives:

1 11 ... p p D 0



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Stability III

I Next de…ne z D 1 and write

1 1z ... p z p D 0 (23)

which de…nes p roots that are the reciprocals of the

eigenvalues.I In terms of the “z -roots”, the stability condition becomes:

“no roots are larger than 1” (outside the unit circle).

I Most of the time we follow Hamilton an stick to the

eigenvalues from the charateristic equation (20). But since wealso can write the model in terms of lag-operators , the form in(23) is sometimes practical.



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The importance of conditioning I

Above, we have mentioned the importance of known (…xed) initialvalues for the uniqueness of the solution of the deterministicdi¤erence equations.

When Y t is stochastic and the di¤erence equation is (15) it maystill seem “…shy” to base the solution on known initial values.After all the Y t s are stochastic.

The solution to this problem is both simple and valid: We replace

the statement “known initial values” with the statement“conditional on the pre-history Y t 1 , Y t 1 , : : : , Y t p ”.



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Again: Subject to conditioning on the history of the series, the

solution in (21) is uniqueHowever there is another way to obtain the unique solution thatgives additional insight

That approach is based in the Companion form, which we also will

use several times later

To simplify notation, and without loss of generality, we now dropthe intercept from the equation, 0 D 0.As in Hamilton Ch 1.2 we can write

Y t D 1Y t 1 C 2Y t 2 C ... C p Y t p C "t

as a 1. order equation for a p dimensional vector.



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Yt D26664

Y t

Y t 1.

.

.

Y t p C1

37775

D2666664

1 2 p 1 p 1 0 0 00 1 0 0

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

0 0 0 1 0

3777775

26664

Y t 1

Y t 2.

.

.

Y t p

37775

C26664

"t

0.

.

.

0

37775(24)

or, more compactly:

Yt D FYt 1 C et (25)

with vector and matrix de…nitions given by (24). Multiplying out

we see that the …rst row in (25) gives us back (15) while the otherrows gives identities:

Y t j Y t j j D 1, 2.., p 1.



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(25) is called the companion form.Assume that Yt 1 is known (conditioning!).Repeated substitution backwards gives the solution for Y

t

Yt D Ft Y0 C F

t 1e1 C F

t 2e2 C C Fet 1 C et (26)

and for the single variable

Y t D

f .t /

11Y

0 Cf .t /

12Y

1 C f .t /

1p Y

.p 1/Cf .t 1/11 "1 C f .t 2/

11 "2 C : : : C f 11 "t 1 C "t (27)

where f .t /11 represent element .1, 1/ in Ft , f .t /

12 is element .1, 2/ inFt and so on.

(Note: Hamilton conditions on Y 1,Y 2, Y p , so solves for oneperiod further back).As expected (27) describes Y t as a function of the p initial values that we condition on, and the history of the "t series from timeperiod 1 to t .



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Dynamic multipliers I

Use companion form to solve for Y t C j assuming thate

t C j ( j D 0, 1, 2, : : : /, are known

Y t C j D f . j C1/

11 Y t C f . j C1/

12 Y t 1 C f . j C1/

1p Y t p Cf

. j /

11 "t C f . j

1/

11 "t C1 C : : : C f .1/

11 "t C j 1 C "t C j ,

(28)

The e¤ect on Y t C j of a unit increase in "t is obtained from (28)directly. The dynamic multiplier is:

j D@ Y

t C j @"t

D f . j /11 . (29)

@ Y t C1

@"t D f .0/

11 D 1



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Dynamic multipliers II

Higher order multipliers f . j

1/

11 depend on the eigenvalues of thematrix F.An eigenvalue for F is a scaler that satis…es the characteristicequation:

jF

I

j D0 (30)

which can be written:

p 1p 1 2p 2 p 1 p D 0, (31)

which the same characteristic equation that we had for thedi¤erence equation (15).The proof that (30) above is equivalent with (31) is in Hamilton p21 (appendix to Ch 1)



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Dynamic multipliers IIIAssume that F has p distinct eigenvalues. In this case, from matrixalgebra, we have that F can be diagonalized as

F D G3G1 (32)

where 3 is the p

p diagonal matrix with the p eigenvalues alongthe main diagonal. G is the associated p p matrix with linearlyindependent eigenvectors (as columns).It follows that

F j D G3 j

G1 (33)

An element along the diagonal of 3 j is j i (i D 1, 2, , p ).

Let g ij denote the element in row i , column j in G, and let g ij

denote the corresponding element in the inverse matrix witheigenvectors G1.



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Dynamic multipliers IV

Expanding (33), we obtain f . j /

11 as:

f . j /

11 D g 11 g 11

j 1 C g 12 g 21

j 2 C C g 1p g p 1

j

p (34)

or

f . j /11 D c 1 j 1 C c 2 j 2 C C c p j p (35)

where

c i D

g 1i g i 1

.

See Hamiltion page 12. Note that

p Xi D1

c i D 1



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Dynamic multipliers V

since g 11g 11C g 12 g

21C C g 1p g p 1 is element .1

,

1/ inGG

1.With p D 1,this gives c 1 D 1 and f

. j /11 D

j 1 Since 1 D 1, the

multiplier for the 1st order case is

j @ Y t C j @"t

D j 1

With p D 2:

j

@ Y t C j

@"t Dc 1

j 1

C.1

c 1/

j 2

where

c 1 D 1

1 2



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Dynamic multipliers VI

see Hamilton Prop 1.2, as well as below for a direct derivation.

For the case of repeated roots, the algebra only requires a di¤erentdiagonalization than (32)—Jordan decomposition, see H p 18-19.

To summarize:I The dynamic multiplier ' j is a linear combination of all the

roots i .

I Y t C j in (28) is a linear combination of all multipliers and p

initial values with weights f . j

C1/

1i for (i D 1, 2, , p ). Alsothese weights are linear combination of all the p roots i .



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A forecasting corollary I

Set E "t C j j Y t 1 D 0. We then have:

E

Y t C j

D f . j C1/

11 Y t 1 C f . j C1/

12 Y t 2 C f . j C1/

1p Y t p , j D 0, 1, : : :

(36)For the case of real roots we see directly how the stability Theoremabove applies to this forecasting solution:

E[Y t C j ] ! j !1

0, i¤ ji j < 1 8 i (37)

CorollaryForecasts from stable dynamic equations always equilibrium correct to the long-run solution.



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A forecasting corollary II

If we include the constant 0 in the di¤erence equation, (37) isgeneralized to

E

Y t C j

!

j !10

1

Pp i

D1 i

Y , i¤ ji j < 1 8 i (38)

where Y denotes the long-run mean of Y .We shall see that this results extends to:

I Equations with exogenous variable, X t with non-zero

E[X t C j j Y t 1] 6D 0I Systems of equations with intercepts and non-zero mean

forcing variables.



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Roots on and outside the unit-circle I

One (single) root equal to 1 means that the history (in the form of initial values) is projected inde…nitely into the future, this is typicalof forecasts from unstable models such as a random-walk.

These insights also extend to the case of complex (imaginary)roots.

All moduli less than one: Dampened cycles, in thesolution/forecasts and in the multipliers.

One modus equal to one: Repeated cycles.



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Lag-polynomial

Consider the 2nd order autoregressive process (AR(2)):

.1 1L 2L2/

| {z }.L/ D

Y t D "t (39)

where L is the lag-operator , see Hamilton p 26-27, and

.L/ D .1 1L 2L2/

is a polynomial in the lag operator L, and is therefore called a

lag-polynomial .Note that just at L operates on Y t to produce Y t 1, thepolynomial .L/ operates on Y t to produce the new time series "t .



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Factorizing the lag-polynomial I

It will prove essential to factorize the polynomial .L/. A …rst stepis write .L/ as:

.L/ D L2.L2 1L1 2/. (40)

For a moment, let us regard L as a number. With the use of thescalar x D L1,the right hand side of (40) can be written asx 2p .x / where

p .x / D .x 2 1x 2/.

With reference to the Fundamental Theorem of Algebra we havethat the polynomial p .x / can be factorized as:

p .x / D .x 1/.x 2/,



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Factorizing the lag-polynomial II

where 1 and 2 are two (for simplicity distinct) roots to thecharacteristic equation:

2 1 2 D 0.

Then x

2p .x / becomes

x 2p .x / D x 2.x 1/.x 2/

D x 1.1 1x 1/.x 2/

D.1

1x 1/.1

2x 1/.

Re-introducing x D L1 we have

L2.L2 1L1 2/ D .1 1L/.1 2L/



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Factorizing the lag-polynomial III

which gives the desired factorization of .L/:

.L/ D .1 1L/.1 2L/. (41)

Hamilton p 30-32 contains a derivation of the factorization without

the use of the Fundamental Theorem of Algebra.For the general AR(p) process

Y t D 1Y t 1 C 2Y t 2 C ... C p Y t p C "t ,

the polynomial .L/ D 1 Pp j D1 j L j is factorized as

.L/ D .1 1L/.1 2L/....1 p L/, (42)



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Factorizing the lag-polynomial IV

where we have assumed p distinct roots in p ./ D 0, i.e.

p 1p 1... p D 0.

Note also that

p ./ D jF Ij D 1 2

1

where F is the coe¢cient matrix in the companion form.

We can therefore obtain the factorization (41) by …nding 1 and1 from the characteristic polynomial associated with F.



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Solution and multipliers IAssume

j1

j< 1 and

j2

j< 1, so that

1X j D0

. j i L

j / D .1 i L/1,

and.1 1L 2L/1 D .1 1L/1.1 2L/1 (43)

A (particular) solution of (39) is therefore:

Y t

D.1

1L

2L/1 "t

with the aid of the factorization we can write:

Y t D1

.1 1L/

1

.1 2L/"t (44)



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Solution and multipliers II

A very useful result, which is due to Sargent’s bookMacroeconomic Theory from 1987 (p 184) is

.1 2/1

1

1 1L 2

1 2L

D 1

.1 1L/

1

.1 2L/. (45)

By using Sargent’s result we have that the solution for Y t can bewritten as:

Y t D .1 2/1

1

1

1

L 2

1

2

L "t

D .1 2/1(

1

1X j D0

. j 1L j / 2

1X j D0

. j 2L j /

)"t .



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Solution and multipliers IIIor in extended form:

Y t D .c 1 C c 2/"t C .c 11 C c 22/"t 1

D C.c 121 C c 22

2/"t 2 C .c 131 C c 23

2/"t 3 C : : :(46)

where c 1 and c 2 are given by:

c 1 D 1

1 2, c 2 D 2

1 2. (47)

The closed form solution for the dynamic multipliers are nowreadily available:

j D@ Y t C j

@"t D c 1

j 1 C c 2

j 2 . (48)



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Solution and multipliers IV

Note:

I c 1 C c 2 D 1

I If complex root, then 2 D N1(the conjugate complex number

of 1)I With complex roots, then also c 1 and c 2 are complex and the

products c 1 j 1 and c 2

j 2 gives real numbers, see e.g., reference

note on complex numbers.

I

The multipliers, and therefore also Y t itself becomesdominated by the largest root.



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Solution and multipliers VI The long-run multiplier is

1X j D0

' j D c 11

1 1C c 2

1

1 2

D

1

.1 1/.1 2/with the use of the expressions for c 1 and c 2 in equation (47)

The generalization of (47) to AR(p) is straightforward. All rootsnow play a role for the multipliers, and have weights

c i D

p 1i Qp

j D1 j 6Di

.i j /,Xp

i D1c i D 1.



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Causal and non-causal processes I

I The statistical time series literature de…nes a variable (orstochastic process) Y t as causal if the stable solution can beexpressed as a function of initial conditions (conditioning) andthe sequence of exogenous variables t , t 1 , : : : and so onbackwards in time.

I For a causal process, the associated characteristic equationp ./ D 0 has all its roots inside the unit circle.

I Y t is a non-causal or future dependent process if the stablesolution is (linear) function of terminal conditions and the

sequence of exogenous variables t ,

t C 1,

: : : and so onforward in time.

I For a non-causal process, the associated characteristicequation p ./ D 0 has all its roots outs id e the unit circle.



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Causal and non-causal processes III An example:

Y t D 1Y t 1 C "t , 1 > 1, (49)

has one root larger than unity. A stable non-causal solution is:

Y t D .11 /N Y t CN CXN 1

i D1.1

1 /i "t Ci

where Y t CN is a terminal condition.The solution is stable since, if we look at the homogenouspart Y h

t !N !10 if

1> 1 as we have assumed.

I As we shall see later in the course, all causal models gives riseto stationary stochastic variables, but not all stationaryvariables have a causal solution.



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Causal and non-causal processes III

I Non-causal (future dependent) processes play a large role ineconomics,

I Although we …rst concentrate on causal models (bothstationary and non-stationary case), we will consider

non-causal processes in due course.

I Several names for the same thing

I “forward looking models” or “models with rationalexpectations”, “model with lead-terms”

I We will analyse these as non-causal time series models.



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References I

Hurwitz, L. (1950) Least squares bias in time series, in Statistical

Inference in Dynamic Economic Models, John Wiley & SonSarget, T. J. (1987) Macroeconomic Theory (Second Edition),Accademic Press


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