The VAR and Econometric Models of the VAR Ragnar Nymoen

Stationarity of VAR processes Estimation of the VAR The VAR and structural models Econometric models of the VAR

E 4101/5101 Lecture 7: The VAR and econometric models of the VAR Ragnar Nymoen

Department of Economics, University of Oslo

10 March 2011

E 4101/5101 Lecture 7: The VAR and econometric models of the VAR Department of Economics, University of Oslo Stationarity of VAR processes Estimation of the VAR The VAR and structural models Econometric models of the VAR

IntroductionI

I Main references:

I Hamilton Ch 10 and 11. I Davidson and MacKinnon, Ch 13 (note their reference back to Ch 12.2 on Seemingly Unrelated Regressions, known from e.g. E-4160, and the unrestricted reduced form (URF) of the simultaneous equations model (also known from E-4160)

I Vector autoregressions, VARs has become widely adopted in macroeconomics due to the so called Sims’ critique, in Sims (1980).

I In line with this,Hamilton motivates the VAR by its “convenience for estimation and forecasting”: meaning that VARs are easy to estimate and are useful for forecasting.

IntroductionII

I The VAR has another very important role as well, as a statistical model that underlies identiﬁed structural econometric models.

I This role is important both for the stationary case and for the case with unit-roots and (potential) co-integration. Three text-books that develop this viewpoint are: Hendry (1955), Johansen (1995) and also Davidson and MacKinnon (2004).

The bivariate case—the ﬁnal equationI

The simplest example is a vector autoregressive process (VAR) with two variables and ﬁrst order dynamics as in:

y a a y e t = 11 12 t−1 + y,t , (1) xt a21 a22 xt−1 ex,t

where ey,t and ex,t are two white-noise variables (correlated or uncorrelated).

I Notation: Follow H and use yt for both stochastic variable and realization.

I To simplify we have dropped the intercept, (interpret yt and xt as deviations from their respective means)

The bivariate case—the ﬁnal equationII

I White noise disturbances can be replaced by the weaker assumption of innovations with conditional mean zero and constant conditional covariance matrix. The conditioning is on yt−1 and xt−1.

We now express yt as a function of yt−1 and ey,t and ex,t . Solve the ﬁrst equation for xt−1:

xt−1 = (1/a12)yt − (a11/a12)yt−1 − (1/a12)ey,t (2) Substitution in the second equation of the VAR gives

a22 a11 a22 xt = yt + (a21 − a22 )yt−1 − ey,t + ex,t , a12 6= 0 (3) a12 a12 a12

The bivariate case—the ﬁnal equationIII

Finally: Replace t by t + 1 in the ﬁrst equation in (1), and replace xt by the left hand side of (3):

a22 a11 yt+1 = a11yt + a12{ yt + (a21 − a22 )yt−1 a12 a12 a22 − ey,t + ex,t } + ey,t+1 (4) a12

(4) is called a ﬁnal equation. It shows that yt given by the VAR (1) follows the 2 order stochastic diﬀerence equation:

yt = (a11 + a22)yt−1 + (a12a21 − a22a11)yt−2 + εt . (5) | {z } | {z } φ1 φ2

The bivariate case—the ﬁnal equationIV The disturbance in the ﬁnal equation is:

εt = ey,t − a22ey,t−1 + a12ex,t−1

Write the ﬁnal equation more compactly

φ(L)yt = εt , where (6)

2 φ(L) = 1 − φ1L − φ2L ,

φ1 = (a11 + a22) and φ2 = (a12a21 − a22a11).

The associated characteristic polynomial:

2 p(λ) = λ − φ1λ − φ2. (7)

The bivariate case—the ﬁnal equationV From before we know that if the two roots of p(λ) = 0 are both diﬀerent from one in magnitude:

1. yt has a stationary solution and

2. yt is weakly stationary (since the stationary solution defines yt as a well defined filter of the stationary input series εt ). If the roots are inside the unit circle: The backward solution is globally asymptotically stable, and yt is a causal process. The same conclusion follows for xt , if you derive the final equation for the xt variable. 0 Have found: The stationarity of the vector time series [yt , xt ] can be investigated by deriving the final equations for one of the variables

The bivariate case—the ﬁnal equationVI

Dynamic multipliers and impulse response functions Note that the dynamic multipliers for yt

∂yt+s ∂ey,t

can be found by solving the final equation for yt (5). But from the same final equation we also find the cross-derivatives:

∂yt+s ∂ex,t The full set of multipliers is called impulse-response functions in the VAR terminology.

The bivariate case—the companion formI

Let zt denote the vector time series made up of yt and xt . Deﬁne

0 0 zt = [yt , xt ] and et = [ey,t , ex,t ] ,

(1) can be expressed as

zt = Fzt−1 + et , (8)

where F is the matrix with VAR coeﬃcients, i.e.,

a a F = 11 12 . a21 a22

The bivariate case—the companion formII I Since equation (8) is a companion form, we have directly from Lecture 1 that the associated characteristic polynomial of F is

a11 − λ a12 |F−λI| = a21 a22 − λ where I is the identity matrix. I λ is an eigenvalue (root) of F if

|F−λI| = 0, (9)

If we write out (9) we have that if λ is a root in (9), it is also a root in the characteristic equation

2 λ − φ1λ − φ2 = 0

with φ1 = (a11 + a22) and φ2 = a12a21 − a11a22.

The bivariate case—the companion formIII

F is an example of a companion matrix. Since (9) is the same equation as the characteristic equation for 0 the final equation for yt , we have that [yt , xt ] is a stable process iff the eigenvalues of F have moduli different from 1.

The general case: Companion formI

The companion form is well suited for generalizations. Let yt be the n × 1 vector.

0 yt = [y1t , y2, ... , ynt ]

The VAR of or order p is:

yt = φ1yt−1 + φ2yt−2+...+φpyt−p + et (10)

where φi is a n × n matrix with coeﬃcients and et is a vector with white-noise disturbances that may be correlated

The general case: Companion formII

Write (10) in companion form:     φ1 φ2 ··· φp−1 φp     yt yt−1 et  I 0 0 ··· 0   yt−1     yt−2   0   .  =  0 I 0 ··· 0   .  +  . ,  .   . . . . .   .   .     . . . . .      y   y 0 t−p+1 0 0 0 I 0 t−p | {z } | {z } | {z } vt ξt | {z } ξt−1 n×p Fnp×np (11)

ξt = Fξt−1 + vt , (12) with the symbols in H equation (10.1.11) p 259.

The general case: Companion formIII

I This VAR (called covariance stationary vector process in Ch 10) is stationarity if all the eigenvalues of the companion matrix F have moduli from

|F−λI| = 0

that are diﬀerent from zero. See H page 259.

I The VAR is causal if the moduli of all roots are less than 1.

I Note that the number of roots is increasing in both p and n (the length and the size of the VAR).

Impulse-response functionsI

The vector variable ξt+s above can be written as

s−1 i s ξt+s = ∑ F vt+s−i + F ξt (13) i=0

I and for ξt : t−1 i t ξt = ∑ F vt−i + F ξ0. (14) i=0 In the maintained stationary case we also have

Ft → 0 t→∞

Impulse-response functionsII so ∞ i ξt = ∑ F vt−i (15) i=0 These expressions are multivariate generalization of the corresponding from Lecture 1.

I By writing out the ﬁrst n rows of ξt the solution of vector variable yt is seen to be an inﬁnite sum of et−j with declining weight, i.e., a convergent sum of the history of et .

I “The inﬁnite MA representation”

I See H p 260 for the details, which are matrix generalizations of the expressions we had in Lecture 1 for a single variable.

I The weights are determined by the companion matrix F.

Impulse-response functionsIII

I As in the bivariate case we can calculate not only the direct dynamic multipliers, but also the cross-derivatives.

I Together these multipliers are referred to as impulse-response functions: ∂y it+s , ∀ i and j ∂εjt

I Clearly, after estimation of the VAR, it is of interest to consider the empirical impulse-responses, which are directly obtainable in the software.

Estimation of the VARI

I Let the disturbance vector et be independent and identically distributed with mean zero and covariance Ω,In particular consider the Gaussian VAR et ∼ i.i.d N(0, Ω), as in Hamilton, p 291.

I For the Gaussian VAR, the OLS estimators of φi , i = 1, 2, ... , p, are the Maximum-Likelihood estimators.

I This is a direct extension of the ML theory for the AR-model of a single time series I The ML estimators are obtained by OLS on each of the n equations in the VAR—also when Ω contains non-zero elements oﬀ the main diagonal. Why?

Estimation of the VARII

I For the weaker assumption et ∼ i.i.d with mean zero and covariance Ω,the OLS estimators have the same asymptotic distribution as the ML estimators of the Gaussian VAR.

I Note that if the disturbances in one equation are for example autocorrelated, the theory does not apply. We then have a situation represented by the models in EBs lecture note 4, 5 and 6:

I Then need IV estimators, including GMM

I Hence et ∼ i.i.d N(0, Ω) with mean zero and covariance Ω is a deﬁning characteristic of the Gaussian VAR, as well as a limitation.

How do we test the VAR assumptions?I In practice we need to 1. Determine the lag length p of the VAR (because it is seldom known a priori) 2. Check the assumption about the disturbances.

Seldom the case that these two issues can regarded in isolation: Under-speciﬁcation of p might result in residuals that are autocorrelated. However, we mention the main tools for each problem in turn:

How do we test the VAR assumptions?II Determination of p (speciﬁcation testing)

I Information criteria. PcGive provides the Schwarz, Hannan-Quin and Akaike information criteria (SC, HQ and AIC in the output)

I Start with a large p and test successively that the coeﬃcients of the largest lag in the VAR is zero: i.e., a sequence of F-tests.

I Information criteria and sequence of tests can of course be combined—in PcGive 13 this combined procedure can be done with the use of Autometrics (example later).

How do we test the VAR assumptions?III Testing assumptions about εit (mis-speciﬁation testing)

I Since each equation is estimated by OLS, we can use (E-4160) test-battery: autoregresive autocorrelation, ARCH disturbance, White tests of heteroskedasticity, non-normality tests.

I PcGive also has vector versions of these mis-speciﬁcation tests.

I Note the degrees of freedom tend to be very large for these tests, so even if the size of the test is OK, mis-speciﬁcation may be hidden (due to low power of test).

How do we test the VAR assumptions?IV Signiﬁcant departures from the hypothesis of Gaussian disturbances can often be resolved by

I Larger p

I Increase the dimension (n) of the VAR: more variables in the yt vector. I Introduce exogenous stochastic explanatory variables: VAR-X model, conditional or partial model.

I Introduce deterministic variables in the VAR. NOTE:

I The economic relevance of the statistically well speciﬁed VAR is a matter in itself.

How do we test the VAR assumptions?V

I Little help if p is set so large that there are no degrees of freedom left, I or if VAR-X introduce variables that are diﬃcult to rationalize or interpret theoretically or historically.

I May then want to estimate simpler model with GMM for each equation instead.

I However, in this part of the lecture we follow the VAR approach and assume that a congruent (not mis-speciﬁed) and relevant VAR can be established.

Building ﬂexibility into the VARI One natural way to obtain a congruent VAR is (in addition to increase p and n which are subject to the above caveats) is to include deterministic terms. Speciﬁcally, the Maximum-Likelihood interpretation of OLS estimators is also valid for the augmented VAR

yt = cdt + φ1yt−1 + φ2yt−2+...+φpyt−p + ﬁxt + et (16)

where dt is a vector with deterministic variables. For example:

dt = (1, Trendt , dum1t , ... , dumKt )

Building ﬂexibility into the VARII

dumkt is a dummy variable representing

I Seasonal dummy

I Dummy for structural break (change in mean of yt ), or outlier

xt is a vector of conditioning economic variables, and ﬁ is a matrix of coeﬃcients for these variables.

The VAR interpreted as a unrestricted reduced formI Consider again the bivariate case of, Xt and Yt and dynamics of the ﬁrst order: The simultaneous equations representation of this dynamic system is 1 b y b b y ε 12,0 t = 11,1 12,1 t−1 + y,t b21,0 1 xt b21,1 b22,1 xt−1 εy,t (17) where ey,t are ey,t uncorrelated Gaussian disturbances. If we start from (17), the VAR in (1) is seen to be a reduced form of the simultaneous equations model:

−1 a a 1 b b b 11 12 = 12,0 11,1 12,1 a21 a22 b21,0 1 b21,1 b22,1 | {z } in (1)

The VAR interpreted as a unrestricted reduced formII

and −1 e 1 b ε y,t = 12,0 y,t . ex,t b21,0 1 εx,t | {z } in (1)

I Since the simultaneous equation model (17) is “unrestricted”, in fact it is not identiﬁed, we call the reduced form of that model the unrestricted reduced form, URF, see Davidson and MacKinnon page 596.

I The VAR disturbances ey,t and ex,t are correlated even if (as here) εy,t and εx,t are uncorrelated ( b12,0 6= 0, or b21,0 6= 0).

The VAR interpreted as a unrestricted reduced formIII I This removes the interpretability of the impulse responses since for example ∂yt+s ∂ey,t

is in general not interpretable as the eﬀect on yt+s of a shock to y in period t.

I This shows that the unrestricted VAR is not a structural model.

I It would not help to estimate (17) in this case, since neither equation is identiﬁed on the order condition.

I In fact, the under-identiﬁcation of the simultaneous equation model (17) and the non-structural VAR is one and the same thing: It is all about lack of identiﬁcation.

The VAR interpreted as a unrestricted reduced formIV

I The VAR has unidentiﬁed impulse-responses I The simultaneous equations model (17) is not identiﬁed on the rank and order conditions for a simultaneous equation model.

I “Fork in the road”:

I “Sim’s followers to the left”. I “Those who want to keep the systems of equations model, SEM, to the right”

I We will take the SEM route in this course.

I Our programme will be to regard the VAR as the statistical model of the system and try to specify structural econometric models that do not “throw away information”, but instead encompass the VAR.

SVARs and identiﬁed SEMsI

I One popular way to identify the impulse-response, thus obtaining a structural VAR, a SVAR, is to replace ey,t and ex,t by a pair of uncorrelated disturbances (orthogonalized innovations in Hamilton’s terminology).

I This is called the Cholesky decomposition/factorization (the theorem is in H Ch 4.4 p 91-91), while the application to our case is on page 320 (orthogonalization) and 327-330 (identiﬁcation of impulse-responses) I The Cholesky factorization is equivalent to choosing a recursive system of equations model. I “xt causally prior to yt ” ⇐⇒ b21,0 = 0

SVARs and identified SEMsII I So could estimate the recursive SEM in the first place. (There is nothing saying that a structural economic model cannot be recursive!) I Other ways to identify the VAR impulse-responses involves more complicated operations on the VAR covariance matrix. See the last sections of Hamilton’s Ch 11. I The main, and general, message are that these restrictions are equivalent to restrictions on the contemporaneous and/or lagged coefficients of the SEM representation of the system. I There is no way around solving identification with reference to a theoretical framework.

I The diﬀerence between the SVAR route and the SEM route may not be as large as one may think I A choice of “how to” express one’s theory?

Conﬁdence intervals for impulse-responesI

I Hypotheses that can be formulated as linear restrictions on the VAR are easy to test: t-test and F-tests.

I Inference about the signiﬁcance of the impulse-responses is another matter

I Apart from the simplest case (AR(1)) the standard errors needed to construct conﬁdence intervals are complicated, as Hamilton Ch 11.7 shows.

I In addition, difficult to interpret the relevance of a significant but unidentified dynamic multiplier.

I For (identiﬁed) SVARs, an given the increased computation power, the current practice is to use Monte-Carlo simulation to construct conﬁdence intervals.

Conﬁdence intervals for impulse-responesII

I Regard the estimated VAR as the true DGP (may want to relax assumption about exact Gaussian distribution perhaps). I Generate many thousand data sets. I Estimate and calculate the impulse-response functions for each replication. I The conﬁdence interval can be constructed as the intervall that excludes the lowest 2.5 percent and the highest 2.5 percent for example

I The simulation based method can also be used for (large scale) SEM, which is one way of obtaining identiﬁed VARs.

Why are conﬁdence intervals so wide?I

I As noted by Hamilton, p 339, the practical experience with conﬁdence intervals from VARs are that they are “disappointingly wide”. H gives two interpretations and cures:

I Impose more restrictions on the estimated VAR before calculating impulse-responses (their insigniﬁcance is due to ineﬃcient estimation) I Impose more restriction on the system dynamics from the outset (do Bayesian economtrics instead, See Ch 12)

Why are conﬁdence intervals so wide?II

I There is third interpretation as well: The information set (the variables in the VAR and whether exogenous explanatory variables are included or not) may have little relevance for the task of estimating the dynamic multipliers with any precision.

I Dynamic multipliers are like other partial derivatives: They are extremely responsive to omitted variables bias. I This suggest that more modeling, rather than less, may the solution in some cases.

Open and closed systemsI

I The extended VAR in (16) contained a vector of endogenous variables yt and a vector of unmodeled variables xt . Therefore this kind of system is called an open VAR.

I Open systems impose untested exogeneity assumptions. I This does not mean that they are “suspect” or “inferior” to other systems.

I Not obvious that it is interesting test the exogeneity of world GDP for a small open economy for example. But more about that later.

I We consider closed systems at this stage because they make it easier to show how certain popular econometric model can be developed from the VAR, as the statistical model of the system.

Open and closed systemsII

I We will use the bivariate VAR in (17) since the low dimensionally of the VAR and the short lag length ease exposition without (much) loss of generality.

I For simplicity we also assume Gaussian errors.

Simultaneous equation and recursive models

I The ﬁrst model class, which is the simultaneous equations model representation of the VAR has already been mentioned.

I Relevance of this model hinges on identiﬁcation. I Estimation of identiﬁed model structures can be done by familiar system methods: 2SLSL, 3SLS and FIML. We will see examples

I Recursive system

I As we have seen, a special form of identiﬁcation I Estimation by OLS, equation by equation

I Combinations are possible: We then speak of block-recursive systems. Operational macro models are often of this combined type. Will also combine several estimation methods

Conditional and marginal modelI

I A VAR can always be re-parameterized in terms of a conditional model and a marginal model.

I The relevance of the conditional model stems from the fact that it can often, but not always, contain parameters of interest (i.e., in terms of economic theory or for the purpose of the analysis). Write the VAR (1), the URF relative to the SEM (17), as

yt = µy,t−1 + ey,t (18)

xt = µx,t−1 + ex,t (19)

Conditional and marginal modelII where

2 ex,t σx ωxy ∼ N 0, 2 | xt−1, yt−1 . (20) ey,t ωxy σy

The conditional distribution of ex,t and ey,t is bi-variate normal (with expectation zero and covariance

2 σx ωxy 2 . ωxy σy

The conditioning is on xt−1 and yt−1. The correlation coeﬃcient between ex,t and ey,t is

ωxy ρxy = . σx σy

Conditional and marginal modelIII

µx,t−1 and µx,t−1 are the expectations of Xt and Yt conditional on the pre-history, xt−1 and yt−1:

µy,t−1 = E[yt | xt−1, yt−1] = a11yt−1 + a12xt−1 (21)

µx,t−1 = E[xt | xt−1, yt−1] = a21yt−1 + a22xt−1 (22)

yt and xt in (18) and (19) are also bi-variate normally distributed.

Conditional and marginal modelIV The distribution of yt conditional on xt , xt−1, yt−1 is also normal, with expectation

σy σy E[Yt | xt , xt−1, yt−1] = µy,t−1 − ρxy µx,t−1 + ρxy xt σx σx ωx y ωx y = 2 xt + (a12 − 2 a22)xt−1 σx σx ωx y + (a11 − 2 a21)yt−1. σx If we deﬁne εt = yt − E[yt | xt , xt−1, yt−1] (23) we obtain the ﬁrst order autoregressive distributed lag model, ARDL: yt = φ1yt−1 + β0xt + β1xt−1 + εt . (24)

Conditional and marginal modelV

The disturbance εt can alternatively be written εt

ωx y εt = ey,t − 2 ex,t (25) σx because ωx y E[ey,t | ex,t , xt−1, yt−1] = 2 ex,t σx from the properties of the normal distribution. Since ex,t = [xt − E[xt | xt−1, yt−1], we can write the conditioning on xt , hence. ωx y E[ey,t | xt , xt−1, yt−1] = 2 ex,t . σx

Conditional and marginal modelVI The disturbance εt in the ARDL model for yt is conditionally normal with mean 0, and variance σ2

2 εt ∼ N(0, σ | xt , xt−1, xt−1).

Conditional and marginal modelVII In sum the VAR is modelled by the ARDL equation

yt = φ1yt−1 + β0xt + β1xt−1 + εt . (26)

E[εt | xt , xt−1, yt−1] = 0 2 2 2 Var[εt | xt , xt−1, yt−1] ≡ σ = σy (1 − ρxy ), (show) ωx y ωx y φ1 = a11 − 2 a21, β0 = 2 , σx σx ωx y β1 = a12 − 2 a22. σx

and the marginal equation for xt :

xt = a21yt−1 + a22xt−1 + ex,t (27)

Conditional and marginal modelVIII

(26) can be estimated by OLS which is conditional FIML for the case of Gaussian disturbances.

I If our purpose is to test hypotheses about the parameters in the conditional model , we do not need to estimate the marginal model (27).

I If the purpose is forecasting, or policy analysis (what happens if there is a change in the marginal model?), the marginal model must also be estimated.

Sequential conditioning—the general caseI

I The derivation of the ARDL(1, 1) was by sequential conditioning: First we conditioned on the history (yt−1 and xt−1), and then we conditioned on current xt . I This approach is general and can be used to derive, from a large joint distribution (many variables and long lags), a conditional model.

I For example, the ARDL(p, p)

yt − φ1yt−1 − ... − φpyt−p = β0xt + β1xt−1 + ... + βpxt−p + εt , (28) can be derived from a pth order Gaussian in yt and xt . I By the same token: Can also have n − 1 diﬀerent x variables in the conditional model, if the initial VAR is of dimension n.

Equilibrium correction modelI The conditional ARDL can be re-parameterized as an equilibrium correction model, ECM. For the ﬁrst order model (with intercept φ0 added):

∆yt = φ0 + β0∆xt + (φ1 − 1)yt−1 + (β0 + β1)xt−1 + εt

∆yt = β0∆xt        φ0 (β0 + β1)  + (φ1 − 1) yt−1 − − xt−1 + εt  (1 − φ1) (1 − φ1)     | {z∗ }  yt−1 

Equilibrium correction modelII Deﬁne ∗ φ0 (β0 + β1) yt = − xt−1 (1 − φ1) (1 − φ1)

as the conditional equilibrium path for yt , then equilibrium-correction is seen to be the obvious name for this parameterization of the model. Recall from the ﬁrst lecture that equilibrium correction is inherent for stationary variables. For β0 = β1 = 0, the solution can be written:       t−1 φ0  φ0  t i yt = + y0 − φ1 + ∑ φ1εt−i . 1 − φ1  1 − φ1  i=0 | {z }  | {z } y ∗  y ∗ 

Equilibrium correction modelIII

For p = 4 (with intercept φ0 added)

4 4 y − y = + x + , (29) t ∑ φi t−i φ0 ∑ βi t−i εt i=1 i=0 one possibility is to put the levels term at the fourth lag

3 3 y = + † y + † x (30) ∆ t φ0 ∑ φi ∆ t−i ∑ βi ∆ t−i i=1 i=0

+ (φ(1) − 1)yt−4 + β(1)xt−4 + εt

Equilibrium correction modelIV where φ(1) is φ(L) with L = 1, β(1) is the same for β(L).

i † φi = ∑ φj − 1, i = 1, 2, 3, (31) j=1 i † βi = ∑ βj , i = 1, 2, 3. j=0

Alternatively,put the level-terms at the ﬁrst-lag

3 3 y = + ‡ y + ‡ x (32) ∆ t φ0 ∑ φi ∆ t−i ∑ βi ∆ t−i i=1 i=0

+ (φ(1) − 1)yt−1 + β(1)xt−1 + εt

Equilibrium correction modelV 4 ‡ = − , i = 1, 2, 3, φi ∑ φj j=i+1 ‡ β0 = β0 (33) 4 ‡ βi = − ∑ βj , i = 1, 2, 3. j=i+1

In both cases the long-run multiplier with respect to xt is

4 β(1) ∑j=0 βj K1 = = 4 (34) 1 − φ1(1) (1 − ∑j=1 φj )

† ‡ β0 = β0 = β0, (35)

Equilibrium correction modelVI but the other parameters are not the same in the two versions.

† 6= ‡, i = 1, 2, 3, (36) φi φi † ‡ βi 6= βi , i = 1, 2, 3. (37)

I ECMs are more ﬂexible than this.

I The AR lag length and the DL lag length need not be the same. I The levels of y and x can be on diﬀerent lags.

I The long run multiplier K1 is invariant to the diﬀerent ways of writing the ECM.

I But the interim multipliers are aﬀected (as illustrated)

Matrix notation for ECMI Matrix-notation for the ECM with one xt variable (m lags): 0 0 ∆yt = z1t b1 + z2t b2 + εt (38)

0 z1t = (1, ∆yt−1, ∆yt−2, ... , ∆yt−p+1, ∆xt , ∆xt−1, ... , ∆xt−m+1), 0 z2t = (yt−p,xt−m),   φ0  φ†   1   .   .   †  b1 =  φ   p−1   β†   0   .   .  † βm−1

Matrix notation for ECMII

(φ(1) − 1) φ∗ b = = . 2 β(1) β∗ (38) is a 1-1 linear transform of the ARDL(p, m) and therefore all results for the OLS estimators also holds for bˆ1and bˆ2. The estimate of the long-run multiplier K1 is non-linear

βd(1) βˆ∗ Kˆ = ≡ , 1 ∗ −(φ(\1) − 1) −φˆ

Matrix notation for ECMIII q However, Var[Kˆ 1]can be obtained by the so called Delta-method, see B˚ardsen(1989) :

2 2 ˆ∗ ! 1 ∗ β ∗ Var\[Kˆ 1] ≈ Var\(βˆ ) + Var\(φˆ ) (39) −φˆ∗ (−φˆ∗)2 ! 1 βˆ∗ + 2 Cov\(βˆ∗, φˆ∗). −φˆ∗ (−φˆ∗)2

Matrix notation for ECMIV

This also applies to an ECM with k explanatory variables xjt (with lag order mj ), i.e. the approximate variance of

∗ β[(1) βˆ Kˆ = j ≡ j , j ∗ −(φ(\1) − 1) −φˆ

is given by (39), with an obvious change in notation. q I Var[Kˆ j ] is available directly in PcGive.

A potential misundertanding—and looking aheadI

Although we have derived the ECM as a single equation model, estimateable by OLS, this does not mean that ECMs are synonymous with single equations and OLS estimation

I We derived our ECM from a bi-variate VAR.

I Already with three variables in the VAR , yt , xt , zt ,new possibilities for model formulation by sequential conditioning emerges:

I Can for example condition on xt (and the history of xt ). I This replaces the 3-variable VAR by a bi-variate VARX which is conditional on xt . I With Gaussian disturbances both the VAR and the VARX are Gaussian statistiscal models.

A potential misundertanding—and looking aheadII

I The VARX can be modelled in its turn, for example:

I Simultaneous equation model I Recursive system and conditional system

I If the order of dynamics of the VARX is 2 or higher, all these models can be put in “ECM form”.

I In the simultaneous equation model, the ECM structural equations are estimated by 2SLS or FIML

I We will return to these point after the co-integration theory

References

B˚ardsen,G.: “Estimation of Long-Run Coeﬃcients in Error-Correction Models” Oxford Bulletin of Economics and Statistics, 51, 345-350. Hendry, D. F. (1995) Dynamic Econometrics, Oxford University Press Johansen, S. (1995) Likelihood-based Inference in Cointegrated Vector Autoregressive Models, Oxford University Press. Sims, C. A. (1980) Macroeconomics and Reality, Econometrica 48,1—48

E 4101/5101 Lecture 7: The VAR and econometric models of the VAR Department of Economics, University of Oslo