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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Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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
!.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag operator form Future dependence
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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).
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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)
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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 ).
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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”,
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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 ”.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
(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 .
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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)
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence
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.
ECON 5101/9101: Lecture 1 Department of Economics, University of Oslo
Introduction Beginnings: AR(1) model Di¤erence equations Companion form Lag-operator form Future dependence