Cointegration – Hypothesis testing and identification Testing restrictions
We have so far seen estimation and other issues associated with a cointegrated V AR model. But as in any empirical application, in a cointegrated V AR system also we can systematically test many hypotheses of interest, mainly dictated by theoretical considerations. Though such restrictions on both the cointegrating vectors as well as the loadings matrix do not help us in identifying these vectors, these are of interest by themselves. As an example, by testing for the exclusion of a particular variable in a cointegrating vector, potentially irrelevant variables may be tested out of the system, thus reducing the dimension of the analysis. Many other interesting propositions, like the concept of weak exogeneity, can be tested on the loadings matrix too. In this module, we shall see how to formulate such restrictions in two different ways – one, in terms of free or unrestricted parameters and another in terms of of restrictions explicitly, which is the more traditional way. Which way should one follow, is entirely a matter of taste; but as a practice, we shall demonstrate the use of both ways. ———————
Formulating hypotheses as restrictions on β :
Restrictions on the β vector can be imposed in terms of si free parameters or in terms of mi
restrictions. We first specify in terms of free parameters.
Let ϕi be the (si × 1) redefined coefficient vector, Hi is a(N 1 × si) design matrix of known
elements, N 1 is the dimension of Zt in the V AR model, where N 1 is N plus any deterministic
variable and constant included in the V AR model. And i = 1, . . . , r. These are the notations for formulating the hypotheses in terms of free parameters.
And in terms of restrictions, we specify Ri matrices of size (N 1 × mi), where mi = N 1 − si
restrictions on βi such that R01β1 = 0, . . . , R0rβr = 0.
Let us illustrate these with help of a vector of variables, Zt = (mrt, yrt, ∇pt, Rm,t, Rb,t, Ds)0. These
variables are typically used in macro/monetary relations and Dsis a dummy variable. Let us suppose
there are 3 cointegrating relations.
Note that, when we estimate, a cointegrating relation will contain all the variables in the Z vector. It may so happen, that the coefficient attached to a particular variable is very near zero, but we may like to check if it can be statistically considered to be so; and this is exactly what we check in any hypothesis testing. And hence, the first cointegrating relation, β01Zt should actually look like
β01Zt = β11mrt + β12ytr+ β13∇pt+ β14Rm,t+ β15Rb,t+ β16Ds, (Unrestricted).
But to illustrate hypotheses testing involving cointegration vectors, let us suppose for exposition sake, that the first cointegration looks like
β01Zt= [(mrt − y r
t) − b1(Rm,t− Rb,t) − b2DS] , (Restricted).
Note here that inflation rate has been omitted. Let us demonstrate how we arrived at the restricted cointegrating vector from the unrestricted one, first by using only the free parameters and then by using the restrictions only.
Treating (mrt − yr
t) and (Rm,t − Rb,t) as one variable, we may say that there are three f ree
parameters. Note also that in this expositional cointegration relation, the coefficients are such that β12 = −β11 and β15 = −β14. This simply means that the coefficients of mrt and ytr are equal in size
but opposite in sign. The same is true with the coefficients of Rm,t and Rb,t. Now, to differentiate
between the restricted and the unrestricted vectors, let us re-define the cointegrating vector β1 such
that, β11 = −β12 = ϕ11 and β14 = −β15 = ϕ12, ϕ13 = β16. With this, the re-defined cointegrating
β1 = H1ϕ1 = 1 0 0 −1 0 0 0 0 0 0 1 0 0 −1 0 0 0 1 ϕ11 ϕ12 ϕ13 = ϕ11 −ϕ11 0 ϕ12 −ϕ12 ϕ13
With this we can write the expositional cointegrating vector as ϕ11(mrt− ytr) + ϕ12(Rm,t− Rb,t) +
ϕ13Ds. Notice next that in the expositional cointegrating relation, we have normalized on the first
variable – that is, we have set the coefficient of the first variable to be 1, so that now the normalized cointegrating vector is (1, −1, 0, ϕ12/ϕ11, −ϕ12/ϕ11, ϕ13/ϕ11)0. We shall simplify further and
assume that b1 = −ϕ12/ϕ11 and b2 = −ϕ13/ϕ11 so that the normalized cointegrating vector is
(1, −1, 0, −b1, b1, ; −b2)0. Thus, we get the first expositional restricted cointegration vector as
(1, −1, 0, −b1, b1, −b2)Zt= [(mrt− ytr) − b1(Rm,t− Rb,t) − b2Ds] .
———————————
We shall now demonstrate for the same vector, how to arrive at the restricted cointegrating vector, using only the implied restrictions on the first cointegrating vector. First we shall fix the dimension of the R matrix, which is a (6 × 3) matrix. In terms of restrictions notice, that −β11 = β12 so that
β11+ β12 = 0; β13 = 0 and −β14 = β15 so that β14+ β15 = 0. With this we get the first restricted
cointegrating vector as R01β1 = 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 β11 β12 β13 β14 β15 β16 = 0 ———————————
Using the same logic on the second restricted cointegrating vector, β02Zt= ytr− b3(∇pt− Rb,t),
we can write using the free parameters. Notice that there are only two free parameters, so that si = 2
and H2 is now (6 × 2) matrix:
β2 = H2ϕ2 = 0 0 1 0 0 1 0 0 0 −1 0 0 ϕ21 ϕ22 = 0 ϕ21 ϕ22 0 −ϕ22 0 .
Similarly, in terms of restrictions, with the R matrix now being a (6 × 4) matrix, we have
R02β2 = 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 1 0 0 0 0 0 0 0 1 β21 β22 β23 β24 β25 β26 = 0 ———————————
And, for the third restricted cointegrating vector given by β03Zt= (Rm,t− Rb,t) + b4Ds
one can show that, using free parameters and with H3 being a (6 × 2) matrix,
β3 = H3ϕ3 = 0 0 0 0 0 0 1 0 −1 0 0 1 ϕ31 ϕ32 = 0 0 0 ϕ31 −ϕ31 ϕ32 .
And, in terms of restrictions, we have the R matrix as a (6 × 4) matrix, and arrange them as
R03β3 = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 1 0 β31 β32 β33 β34 β35 β36 = 0
Note the following:
• Ri = H⊥⊥⊥,i, that is R0iHi = 0.
• Since such testing is done normally after the rank has been determined, such restrictions are null hypothesis on the stationary linear combination of variables.
———————————
Same restrictions on all cointegrating vectors
For some reason, we may be interested in testing for the exclusion of a particular variable from all cointegrating vectors. This results in the ‘same’ exclusion restrictions on all the cointegrating vectors. Or, we may want to check if some well known economic relation is common to all relations. To be more specific, suppose we want to test if the relation (mr
t− ytr) is common to all cointegrating
relations. How do we test it?
We continue with the same vector of variables, Zt, as before and also assume that we have three
cointegrating relations. But we shall not refer back to the three expositional restricted cointegrating vectors used before. So for example, for this set up, our cointegrating relations are simply given as β01Zt, β02Zt, β03Zt respectively. Writing out the first cointegrating relation explicitly,we have
β01Zt = β11mrt + β12ytr+ β13∇pt+ β14Rm,t+ β15Rb,t+ β16Ds.
The restriction we want to test implies, that for this vector, β11(mrt − ytr). Since our aim is to check
if this is common to all cointegrating relations, we have for the other two cointegrating vectors, β21(mrt − ytr) and β31(mrt − ytr). If we re-define our restricted cointegrated vectors as ϕ1, ϕ2 and
ϕ3, the restrictions imply that ϕ21 = −ϕ11, ϕ22 = −ϕ12, ϕ23 = −ϕ13 With this, we can impose
these restrictions either using the free parameters or the restrictions. Using the free parameters, for example, we have the (6 × 5) H matrix, the restrictions can be expressed compactly as
Hϕ = 1 0 0 0 0 −1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 ϕ11 ϕ12 ϕ13 ϕ21 ϕ22 ϕ23 ϕ31 ϕ32 ϕ33 ϕ41 ϕ42 ϕ43 ϕ51 ϕ52 ϕ53
However, expressing this in terms of restrictions alone, is easier since R matrix is of dimension (N 1 × m) where m is the number of restrictions in each vector. For the present case, this simply means
R0β = 0, R0 = (1 1 0 0 0 0). And the transformed data vector for this set up becomes,
H0Zt= (mr− yr) t ∇pt Rm,t Rb,t Ds
Note that, if this restriction is accepted, an important consequence is that, instead of using two variables mr
t and ytr separately, we use the relation (mrt − ytr) as one variable, so that the dimension
of the variables, now is 5. Such restrictions are common in economic theory. The restriction (mr−yr) t
is generally understood as measuring money income velocity; (Rm,t−Rb,t) is defined as the rate spread;
and (R − ∇p) measures real interest rate.
However, the following important points need to be noted.
• Note that the number of restrictions m that we can impose on the endogenous variables is constrained by the fact that N − m ≥ r. This means that we can impose only one more restriction. For example we can check if the interest rate spread (Rm− Rb) is common to all
cointegrating relations. And if this is also accepted, then the transformed data vector becomes H0Zt=(mr− yr)t, ∇pt, (Rm− Rb)t, Ds
0
. This model will produce exactly three eigen values and is testable.
• Remember that the restricted model is not going to re-estimate the number of cointegrating relations. What we are interested in, is the log likelihood value of the restricted model, given that we have already estimated three cointegrating vectors against the ‘unrestricted’ model which had identified these cointegrating relations with the full set of variables. So, what we are basically looking at is, restrictions within the estimated cointegrating vectors; so the number of such vectors is not going to change in the restricted version but the number of variables within each vector may change. For instance, if the above restriction that both the relation (mr− yr)
t
and the interest rate spread relation (Rm− Rb)t are common to all three cointegrating vectors
has been accepted, then the number of variables in each cointegrating relations will now be four instead of the original six variables. The number of cointegrating rank, however, will remain the same.
————————-Estimation in restricted cointegrated systems
We have not yet mentioned anything about how we are going to estimate restricted models. Note that in all cases of hypotheses testing involving the cointegrated V AR model, the statistical test that is generally used is the likelihood ratio (LR) test. Thus, following the general practice, we shall estimate both the restricted and the unrestricted versions and use the respective log likelihood values in the test.
————————-We shall outline below the steps involved in the estimation of the restricted model subject to the condition that the same restriction is applied to all the cointegrating vectors. Let us take, for example, that the relation (mr− yr)
t is common to all the vectors.
• Estimate the unrestricted model following the steps given on pages 22 through 25 and calculate the log -likelihood value using the first three eigen values.
• Before estimating the restricted model, note that the transformed V ECM looks like, ∇Zt = αϕ0H0Zt−1+ Γ1∇Zt−1+ Γ2∇Zt−2+ · · · + Γp−1∇Zt−p+1+ et
implying that the unrestricted cointegrating vector, β has now been replaced by the restricted cointegrating vector βc = Hϕ.
• We again go through the same steps outlined on pages 25 through 27, only with the difference that in I.2 we regress HZt−1 on the short run dynamics and go through the rest of the steps.
• Now we calculate the LR statistic as follows
2(L∗A−L∗0) = Tnln(1 − λc1) − ln(1 − ˆλ1) + ln(1 − λc2) − ln(1 − ˆλ2) + · · · ln(1 − λcr) − ln(1 − ˆλr)
o . • Recall that the entire restricted modeling is done on stationary cointegrating relations and
hence all statistical testing can be done using Gaussian properties. Hence, this statistic is distributed as χ2(j) where j = rm and m is the number of restrictions in each cointegrating
vector. So, there are j degrees of freedom. In the present case, there was only one restriction per cointegrating vector; hence j = 3.
• Suffice it to say that the above mentioned steps are generally those that are used to estimate any restricted model. So we shall not outline the steps of all restricted models.
————————-There are many other interesting restrictions that can be tested on the β vectors. For illustration we can conduct a joint test that the interest rate spread is common to all the relation and the dummy variable can be excluded from all the relations. One can also check if a particular relation is stationary. For example, one can check if ∇pt is stationary. In this case the cointegrating vector
is (1 − 1) and this known can be tested for its correctness. The basic framework is the same as before. Hence, we shall not pursue them here. Interested readers can refer to the book by K.Juselius on this subject.
We shall next examine the implications of testing restrictions on the α vector.
————————-Formulating hypotheses as restrictions on α :
Restrictions on the α vector is closely associated with the concept of weak exogeneity. A test of zero row vector is equivalent to testing for a particular variable to be weakly exogenous to the long run parameters. When the null is accepted, then the particular variable is singled out as the common driving trend. One can check also for a known vector in α.
—————-Testing for long run weak exogeneity
The hypothesis that a particular variable influences, but at the same time, is not influenced by other variables in the system, is called the ‘no levels feedback’ hypothesis or the concept of weak exogeneity. The way to test is as before, both with the free parameters and restrictions. We express the restricted vector as
Hαααc : α = Hαc,
where α is (N × r) matrix; H is a (N × s) matrix, where s is the number of free parameters; αc
is (s × r) matrix of non-zero α coefficients. The equivalent form with the restrictions in the α vector is
Hαααc : R0α = 0, where R = H⊥⊥⊥.
Note the following important point.
• When the null of a zero row vector has been accepted, it means that particular variable does not adjust to the deviations in the long run relations, meaning that the variable can be considered a common stochastic trend. And since, there cannot be greater than N − r common trends in a system with r cointegrating relations, the number of such zero row restrictions can be at most, (N − r).
Empirical illustration
Let us work again with the same Ztvector where now the vector is Zt = (mrt, ytr, ∇pt, Rm,t, Rb,t)0.
We want to test if the bond rate, Rb,t is long run weakly exogenous for the long run parameters in
the data – that is, we want to test if α51= α52 = α53= 0. So here s = 4 and m = 1. So our V ECM
∇mr t ∇yr t ∇2p t ∇Rm,t ∇Rb,t | {z } ↓ ∇Zt = · · · + α11 α12 α13 · · · · α51 α52 α53 β11Z1,t−1+ β12Z2,t−1+ β13Z3,t−1+ β14Z4,t−1+ β15Z5,t−1 β21Z1,t−1+ β22Z2,t−1+ β23Z3,t−1+ β24Z4,t−1+ β25Z5,t−1 β31Z1,t−1+ β32Z2,t−1+ β33Z3,t−1+ β34Z4,t−1+ β35Z5,t−1 | {z } ↓ β ββ0Zt−1
Since our interest is the equation for Rb,t, we shall write out the equation explicitly as
∇Rb,t = α51(β11Z1,t−1+ β12Z2,t−1+ β13Z3,t−1+ β14Z4,t−1+ β15Z5,t−1) +
α52(β21Z1,t−1+ β22Z2,t−1+ β23Z3,t−1+ β24Z4,t−1+ β25Z5,t−1) +
α53(β31Z1,t−1+ β32Z2,t−1+ β33Z3,t−1+ β34Z4,t−1+ β35Z5,t−1)
If the null of α51= α52 = α53 = 0 is accepted in the equation for Rb,t, then we consider
that the bond market is weakly exogenous. Hence, we restrict it the following way:
= · · · + 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 αc 11 αc12 αc13 · · · · αc 41 αc42 αc43 β01Zt−1 β02Zt−1 β03Zt−1 = · · · + αc 11 αc12 αc13 · · · · α41c αc42 αc43 0 0 0 β01Zt−1 β02Zt−1 β03Zt−1 ———————————–
The same specification in terms of R matrix is easily seen to be R0 = [0, 0, 0, 0, 1] .
Some issues about the estimation of V ECM under this restriction is worth noting.
• Assuming that the bond rate is weakly exogenous implies that valid statistical inference on β can be obtained from the four dimensional system consisting of all variables but the bond rate. Such an analysis is called the partial system analysis and the model that uses those four variables is called the partial model and the lone equation explaining the bond rate is called the marginal model. Thus evidence of weak exogeneity actually gives us a condition when a partial model can be used to efficiently estimate β without loss of information. This argument is based on a partitioning of the density function into conditional and marginal densities. We shall explain this below.
• The implication of the above property is that, when there are N − r = m zero rows in the α matrix, – note that m ≤ N − r – we can partition the N equations into N − m equations that exhibit levels feedback and m equations that do not. Since m equations do not contain information about long-run relations, one can estimate a system of N − m variables conditional on the m marginal models of the weakly exogenous variables.
• Note that, ironically, if we want to estimate β from a partial model, we have to estimate the full system first and test for weak exogeneity of a variable! If the null has been accepted, then it may be profitable to re-estimate the partial model conditioned on the weakly exogenous
variable. Why re-estimate? Because, re-estimating a partial system, after accepting the null of weak exogeneity of a variable, results in a more balanced model some times. This may be true especially if there nonlinearities in the system or non-constant parameters.
• However, in many cases, it may be of interest to estimate the partial model from the outset. In our system we typically include variables which we know apriori to be weakly exogenous. For example, we know that US interest rates affect Indian rates but we also know for sure that Indian rates do not influence US rates at all! A variable like the oil prices affects all other macro variables in a system, without being affected by them. If we have such variables in our model, it is profitable to go for the partial model right from the outset. This is normally the strategy adopted by researchers, especially if they have a large number of variables.
• However, if we go for the partial model on such apriori considerations from the outset, we have to refer to a different set of asymptotic critical values. Assuming that our initial classification of weak exogeneity is correct, we can refer to the tables calculated by Johansen et.al in Journal of Business and Economic Statistics, 1998, pp.388-399.
• Now, just what is this partial model? How is this related to the concept of weak exogeneity? To understand the link, we digress a bit and recall some basic statistical results. Details follow. —————————–
Weak exogeneity and partial models. We recall some basic statistical results. Marginal distributions:
Let X ≡ (X1, X2) be a bivariate random vector with a joint distribution function F (x1, x2).
The question that naturally arises is if we could separate X1 and X2 and consider them as individual
random variables. The answer to this question leads to the concept of marginal distribution. Given that the probability model has been defined in terms of the joint density functions, it is necessary to define these in terms of the marginal density functions. Hence, the marginal density functions of X1
and X2 is f1(x1) = Z ∞ −∞ f (x1, x2)dx2 f2(x2) = Z ∞ −∞ f (x1, x2)dx1
Literally this means the marginal density of Xi(i = 1, 2) is obtained by integrating or throwing
out Xj(j 6= i) from the joint density. The algebra behind this assertion should be available in any
elementary book on statistics. Conditional distributions
Another useful idea is to consider the question of deriving the density of a subset of random vectors by conditioning with respect to some other subset of random vectors given the joint density, which leads us to the concept of conditional distributions. This is of great value in the context of the probability model, because it offers a way to decompose the joint density function. Formally, if we need the conditional density of X2 given X1,
f (x1, x2) = (f1(x1)) · (f (x2|x1)).
Needless to say that if X1 and X2 are independent
Given the importance of these concepts, we shall define these in the context of bivariate normal density function, which takes the form f (x1, x2; µ, Σ) and we write X ∼ (µ, Σ) where we can deduce
the following. µ = µ1 µ2 ; Σ = σ2 1 σ12 σ21 σ22 = σ2 1 ρσ1σ2 ρσ2σ1 σ22 , where ρ = σ12 σ1σ2 =⇒ (det Σ) = σ12σ22(1 − ρ2) > 0 − 1 < ρ < 1. The marginal and conditional distributions in this case are denoted by,
X1 ∼ N (µ1, σ12), X2 ∼ N (µ2, σ22) (1) (X1|X2) ∼ N µ1+ ρ σ1 σ2 (x2− µ2), σ21(1 − ρ2 (2) (X2|X1) ∼ N µ2+ ρ σ2 σ1 (x1− µ1), σ22(1 − ρ2 (3) How does one retrieve the model implied by these distributions?
From (2) we can write the model for X1 given X2 as
X1 = a + bX2+ υ1 where X2 = µ2+ υ2, υ2 ∼ N (0, σ22) Here a = µ1− bµ2; b = ρ σ1 σ2 , υ1 ∼ N (0, σ2) σ2 = σ12(1 − ρ 2)
Similarly, from (3) we can write the model for X2 given X1 as
X2 = a∗+ b∗X1+ υ2∗ where X1 = µ2 + υ∗1, υ ∗ 1 ∼ N (0, σ 2 1) Here a∗ = µ2 − b∗µ1; b∗ = ρ σ2 σ1 , υ∗2 ∼ N (0, ˜σ2) ˜σ2 = σ22(1 − ρ2) ————————–
Now we can generalize these points to the N vector to get the multivariate density and conditional density functions.
If X ∼ N (µ, Σ) then the marginal distribution of any (K × 1) subset of X, where X = X1 X2 ; µ µ1 µ2 ; Σ = Σ11 Σ12 Σ21 Σ22
Marginal distributions of X1 and X2 are easily seen to be,
X1 ∼ N (µ1, Σ11), and X2 ∼ N (µ2, Σ22)
For the same partition, the conditional distributions are given by,
(X1|X2) ∼ N µ1+ Σ12Σ−122(X2− µ2), Σ11− Σ12Σ−122Σ21 and (X2|X1) ∼ N µ2+ Σ21Σ−111(X1− µ1), Σ22− Σ21Σ−111Σ12 ——————————–
Now we shall use these concepts and establish how to derive the partial model in the cointegrated V AR framework. Let Zt= (Z1t, Z2t)0. Let us partition
α = α1 α2 ; Γi = Γ1i Γ2i ; et = e1t e2t
With these partitions given let us partition the V ECM as follows:
∇Z1t = α1β0Zt−1+ p−1 X i=1 Γ1i∇Zt−i+ e1t ∇Z2t = α2β0Zt−1+ p−1 X i=1 Γ2i∇Zt−i+ e2t
For this partition scheme, we have
Ω = Ω11 Ω12 Ω21 Ω22 . And, µ1 = E(∇Z1t) = α1β0Zt−1+ p−1 X i=1 Γ1i∇Zt−i µ2 = E(∇Z2t) = α2β0Zt−1+ p−1 X i=1 Γ2i∇Zt−i
Mapping with the definition of conditional density defined before, we have X1 = ∇Z1t, X2 = ∇Z2t
and Σ = Ω so that, from the formula for the conditional density of (X1|X2), we have the conditional
model for ∇Z1t, given ∇Z2t and given the past,
∇Z1t = ω∇Z2t+ (α1− ωα2) β0Zt−1+ p−1 X i=1 ˜ Γ1i∇Zt−i+ ˜e1t where ω = Ω12Ω−122; Γ˜1i = (Γ1i− ωΓ2i) ; e˜1t = (e1t− ωe2t)
and this partial model has variance
Ω11.2 = Ω11− Ω12Ω−122Ω21
Since β enters both the equations for ∇Z1t and ∇Z2t, we cannot analyse the conditional model
for ∇Z1t alone, unless α2 = 0. If we can show this, then
∇Z1t = ω∇Z2t+ α1β0Zt−1+ p−1 X i=1 ˜ Γ1i∇Zt−i+ ˜e1t ∇Z2t = p−1 X i=1 ˜ Γ2i∇Zt−i+ e2t
With this, a fully efficient estimate of β can be obtained from the partial model explained by the equation for Z1t. We estimate it by the usual method of concentrating out the short run dynamics
as well as ∇Z2t. Such an estimation delivers a total of (N − m) eigenvalues from which we use r
nonzero eigen values to decide the number of cointegrating vectors. More details are available in the book by Johansen.
Identification in cointegrated systems
With nonstationary data, cointegration is a real possibility. We had in the previous discussion seen the issues connected with a cointegrated data set up. Recall that, one could get r < N such cointegrating relations given a vector of N variables. But in a cointegrated model, we have both a long-run structure (given by the cointegrating relations) and the short-run structure given by the equations in differences. The classical concept of identification is related to prior economic structure. But here Johansen approaches the identification as a purely statistical process and lists out three different meanings:
• generic identification, which is related to a statistical model,
• empirical identification, which is related to the estimated parameter values, so that we do not accept basically any over identification restriction on parameters, and
• economic identification, which is tested to the economic interpretability of the estimated coef-ficients of an empirically identified structure.
Ideally all three must be fulfilled for an empirical model to be considered satisfactory.
We shall start with a V AR and the associated V ECM. Being reduced form models, how does one retrieve the so called structure behind these reduced form models? Let us demonstrate this with the simplest of the V ECM models:
∇Zt= αβ0Zt−1+ Γ1∇Zt−1+ et et ∼ N (0, Ω)
A structural model is defined by the economic formulation of the problem and can be, for instance, given by
B0∇Zt = Bβ0Zt−1+ B1∇Zt−1+ vt, vt ∼ N (0, Σ) with
Γ1 = B−10 B1; α = B−10 B; et = B−10 vt; Ω = B−10 ΣB0
In a VECM, for a unique identification of the short run structural parameters, given by the set {B0, B1, B, Ω} , we have to normally impose N (N − 1) restrictions on the N equations. Note
however that the set of long-run parameters is the same in both forms, implying that identification of the long-run structure can be done in either form. In order to identify the long-run relations, we formulate restrictions on the individual cointegrating relations. The problem of identifying the long run structure is similar to the one encountered in econometrics in connection with identifying a simultaneous system equations model. The classical result in identification of the system is given by a rank condition. (See Goldberger,1964, Econometric Theory) and this has been extended to the VECM context by Johansen(1995,Journal of Econometrics,69,111-132). Just as in the classical case, where we impose restrictions on the parameters such that the parameter matrix satisfies a rank con-dition, in the VECM context also we have to impose (r − 1) restrictions on each cointegrating vector, so that in general we need to impose r(r − 1) just-identifying restrictions on β. Since Goldberger’s scheme of identification is based on parameters, which are generally unknown, Johansen defines the rank conditions based on the observable matrices, H and R. The idea here is to choose these matrices in such a way that the linear restrictions implied by them satisfy a rank condition.
Let us demonstrate this with the help of both free parameters and restrictions in a cointegrating relation. Accordingly, let Hi = Ri,⊥ be a (N 1 × si) matrix of full rank; and let Ri be a (N 1 × mi)
matrix of full rank, with (si + mi = N 1) so that R0iHi = 0. Thus, there are mi restrictions and si
parameters to be estimated in the ith relation. Thus, the cointegrating relations are thus assumed to satisfy the restrictions R0iβi = 0 or equivalently, βi = Hiϕi for some si−vector ϕi; that is,
where the matrices H1, . . . , Hr express some linear economic hypotheses to be tested against the
data. Herein we specify the condition for identification: The first cointegrating relation is identified, if and only if,
rank(R01β1, R01β2, . . . , R01βr) = rank(R01H1ϕ1, . . . , R 0
1Hrϕr) = r − 1.
The intuitive meaning behind this is very simple. When applying the restrictions of one coin-tegrating on other coincoin-tegrating vectors, we get a matrix of rank r − 1. Hence it is not possible to obtain linear combination of β2, . . . , βr to construct a vector in the same way as β1 which could be confused with β1. Hence β1 can be recognized among all linear combinations of β1, . . . , βr as the only one that satisfy the restrictions R1. But how does one check this if the parameter values are
unknown? And we can estimate the parameters only if the restrictions are identifying. So, to make the condition operational, Johansen provides us with a condition to check which of the cointegrating vectors are identified based only on the known coefficient matrices, Ri and Hi. The condition is: For
all i and k = 1, . . . , r − 1 and any set of indices 1 ≤ i1 ≤ . . . ≤ ik≤ r, not containing i, it holds that,
rank (R0iHi1, . . . , R0iHik) ≥ k.
If the condition is satisfied for any particular i, then the restrictions are satisfying that partic-ular cointegrating vector. If all βi vectors similarly satisfy this rank condition, then the model is generically identified. Basically, this is the first criterion that must be satisfied, if one is interested in identifying a particular cointegrating relation. As an example consider r = 2 where the condition that must be satisfied is
ri.j = rank (R0iHj) ≥ 1, i 6= j.
For r = 3, we have the conditions,
ri.j = rank (R0iHj) ≥ 1, i 6= j
ri.jm = rank (R0i(Hj, Hm)) ≥ 2, i, j, m different
So, if one is interested in identifying structures in a cointegrated model, then the above rank condition should be verified before one proceeds with model estimation. Note that this condition tackles only equation by equation restrictions. More specifically, only exclusion or zero restrictions are allowed. Cross equation restrictions or restrictions on covariance matrix are not allowed.
————————-We shall fix this with an example.
Let us suppose that we have the following set of variables: Zt = (p1, p2, e12, i1, i2)0 where the
first two variables are prices in two different countries A and B, e12 is the exchange rate between
the two countries and the last two are the interest rates prevailing in the two countries. The vector (p1 − p2, ∇p1, e12, i1, i2) is found to be I(1). Let us suppose we have found three cointegrating
relations. We want to check if these are identified with some restrictions that satisfy some stylized facts, like the long run PPP relation, (p1− p2− e1t), and the uncovered interest rate parity (UIP)
relation, (i1t− i2t). It was found that, while PPP stationarity was accepted, UIP stationarity was
rejected. Next our intention was to check, if a combination or modification of these two hypotheses would give us, stationary relations. Accordingly, let us impose the following restrictions:
β = (H1ϕ1, H2ϕ2, H3ϕ3) , where H1 = 1 0 0 0 −1 0 0 1 0 −1 , H2 = 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 , H3 = 0 0 1 0 0 0 0 1 0 0 .
The first describes the relation between real exchange rates and the interest differential, the second is a modified PPP relation and the third describes a relation between price inflation and nominal interest rate. The rank condition would tell us if these restrictions identify the parameters of the long run relations. We don’t know the parameter values and hence we go for the generic identification. We calculate the following matrices:
R01H2 = 1 1 0 0 0 1 0 0 0 , R 0 1H3 = 0 0 0 1 1 0 , R 0 1(H2 : H3) = 1 1 0 0 0 0 0 1 0 1 0 0 0 1 0 , · · · etc. and find that,
rank(R01H2) = 2, rank(R01H3) = 2, rank(R01(H2 : H3)) = 3
rank(R02H1) = 1, rank(R02H3) = 2, rank(R02(H1 : H3)) = 2
rank(R03H1) = 2, rank(R03H2) = 3, rank(R03(H1 : H2)) = 3
Thus we see that for the proposed set of restrictions, all the cointegrating relations are satisfying the generic condition and almost all are identified. Next one may proceed to the empirical and economic identification of these relations.
———————-Just identification and normalisation
Johansen (1994,Journal of Econometrics,63,7-36) suggests a normalisation procedure for cointe-grating vectors that will be just identifying as well. In this case, the generic rank condition for identification of cointegrating vectors will be automatically satisfied. However, one has to still justify such restrictions imposed by the normalisation scheme as economically meaningful. If not, one can impose restrictions that satisfy some economic theory; but in such cases, one may have to test if the restrictions satisfy the generic rank condition.
The necessity to impose restrictions to identify cointegrating relations arises because the coin-tegrating linear combinations are not unique. For example, we can always translate Π = αβ0 as Π = αQQ−1β0 = ˜α ˜β0 where ˜α = αQ and ˜β = βQ0−1. We have to choose Q in such a way that it imposes (r − 1) restrictions on each cointegrating vector so that the rank condition for a generic identification is automatically satisfied. Johansen suggests that Q = β10 where β10 is a (r × r) nonsingular matrix, defined by β0 = [β01 β02] . In this case, αβ0 = (αβ01)β−11 0β0 = ˜α ˜β0 where
˜ α = (αβ01) , ˜β0 =hIr: β−1 0 1 β 0 2 i
. For example, assume that β is a (5 × 3) matrix and let us partition it the following way.
β11 β12 β13 β21 β22 β23 β31 β32 β33 · · · · β41 β42 β43 β51 β52 β53 = β1 · · · β2 so that ˜β implies β1 · · · β2 β−11 = 1 0 0 0 1 0 0 0 1 · · · · ˜ β41 β˜42 β˜43 ˜ β51 β˜52 β˜53
Notice that our choice of Q = β10 in our example has in fact imposed two zero restrictions and one normalisation on each cointegrating relation.
We shall consider a just identified structure describing long run relationship involving endogenous variables real money, inflation and short-term interest rate and exogenous variables, real income and bond rate, corresponding to the following restrictions on β.
β = (H1ϕ1, H2ϕ2, H3ϕ3) where H1 = 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 , H2 = 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 , H3 = 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1
H1 picks up real money, H2 explains inflation rate and H3 explains the short rate and the two
weakly exogenous variables enter all three relations. Note that this structure describes the long run ‘reduced form’ model for the endogenous variables in terms of the weakly exogenous variables. Note that, no testing for the generic rank condition is involved in this case, because as the r − 1 = 2 restrictions have been achieved by linear combinations of the unrestricted cointegrating relations, that is, by rotating the cointegrating space.
