A General Approach

sayβ1Y_t+β2X_twhich is integrated of orderd−b. The vector{β1,β2} is called the cointegrating vector.

A straightforward generalization of the above definition can be made for the case ofn variables, as follows:

Definition 2 IfZ_tdenotes ann×1 vector of seriesZ_1t,Z_2t,Z_3t,. . .,Z_ntand (a) each Z_it isI(d); and (b) there exists an n×1 vectorβ such thatZ_tβ ∼ I(d−b), thenZ_t˜CI(d,b).

For empirical econometrics, the most interesting case is where the series transformed with the use of the cointegrating vector become stationary; that is, whend=b, and the cointegrating coefficients can be identified as parameters in the long-run relationship between the variables. The next sections of this chapter will deal with these cases.

Cointegration and the error-correction mechanism (ECM): a general approach The problem

As noted earlier, when there are non-stationary variables in a regression model we may get results that are spurious. So ifY_tandX_tare bothI(1), if we regress:

Y_t =β₁+β₂X_t+u_t (17.6)

we will not generally get satisfactory estimates ofβˆ₁andβˆ₂.

One way of resolving this is to difference the data to ensure stationarity of our variables. After doing this, Y_t ∼ I(0) and X_t ∼ I(0), and the regression model will be:

Y_t=a₁+a₂X_t+u_t (17.7)

In this case, the regression model may give us correct estimates of theaˆ₁andaˆ₂parame- ters and the spurious equation problem has been resolved. However, what we have from Equation (17.7) is only the short-run relationship between the two variables. Remember that, in the long-run relationship:

Y_t^∗=β₁+β₂X_t (17.8)

soY_tis bound to give us no information about the long-run behaviour of our model.

Knowing that economists are interested mainly in long-run relationships, this consti- tutes a big problem, and the concept of cointegration and the ECM are very useful to resolve this.

Cointegration (again)

We noted earlier thatY_t andX_t are bothI(1). In the special case that there is a linear combination ofY_t andX_t (that is,I(0)), thenY_t andX_t are cointegrated. Thus, if this is the case, the regression of Equation (17.6) is no longer spurious, and it also provides us with the linear combination:

ˆ

u_t =Y_t− ˆβ1− ˆβ2X_t (17.9) which connectsY_tandX_tin the long run.

The error-correction model (ECM)

If, then,Y_t andX_tare cointegrated, by definitionuˆ_t ∼ I(0). Thus we can express the relationship betweenY_tandX_twith an ECM specification as:

Yt=a₀+b₁Xt−πuˆ_t−1+e_t (17.10) which will now have the advantage of including both long-run and short-run infor- mation. In this model,b₁is the impact multiplier (the short-run effect) that measures the immediate impact a change inX_twill have on a change inY_t. On the other hand, π is the feedback effect, or the adjustment effect, and shows how much of the dise- quilibrium is being corrected – that is the extent to which any disequilibrium in the previous period affects any adjustment inY_t. Of courseuˆ_t−1=Y_t−1− ˆβ1− ˆβ2X_t−1, and therefore from this equationβ2is also the long-run response (note that it is estimated by Equation (17.7)).

Equation (17.10) now emphasizes the basic approach of the cointegration and error-correction models. The spurious regression problem arises because we are using non-stationary data, but in Equation (17.10) everything is stationary, the change in XandY is stationary because they are assumed to beI(1)variables, and the residual from the levels regression (17.9) is also stationary, by the assumption of cointegration.

So Equation (17.10) fully conforms to our set of assumptions about the classic linear regression model and OLS should perform well.

Advantages of the ECM

The ECM is important and popular for many reasons:

1 First, it is a convenient model measuring the correction from disequilibrium of the previous period, which has a very good economic implication.

2 Second, if we have cointegration, ECMs are formulated in terms of first differences, which typically eliminate trends from the variables involved, and they resolve the problem of spurious regressions.

3 A third, very important, advantage of ECMs is the ease with which they can fit into the general to specific approach to econometric modelling, which is in fact a search for the most parsimonious ECM model that best fits the given data sets.

4 Finally, the fourth and most important feature of the ECM comes from the fact that the disequilibrium error term is a stationary variable (by definition of cointegration).

Because of this, the ECM has important implications: the fact that the two variables are cointegrated implies that there is some adjustment process preventing the errors in the long-run relationship from becoming larger and larger.

Cointegration and the error-correction

mechanism: a more mathematical approach A simple model for only one lagged term of X and Y

The concepts of cointegration and the error-correction mechanism (ECM) are very closely related. To understand the ECM it is better to think of it first as a convenient reparametrization of the general linear autoregressive distributed lag (ARDL) model.

Consider the very simple dynamic ARDL model describing the behaviour ofY in terms ofX, as follows:

Y_t=a₀+a₁Y_t−1+γ₀X_t+γ₁X_t−1+u_t (17.11) where the residualu_t∼iid(0,σ²).

In this model the parameterγ₀denotes the short-run reaction ofY_tafter a change in X_t. The long-run effect is given when the model is in equilibrium, where:

Y_t^∗=β0+β1X_t^∗ (17.12)

and for simplicity assume that

X^∗_t =X_t=X_t−1= · · · =X_t−p (17.13) Thus, it is given by:

Y_t^∗=a₀+a₁Y_t^∗+γ₀X_t^∗+γ₁X^∗_t +u_t Y_t^∗(1−a₁)=a₀+(γ₀+γ₁)X^∗_t +u_t

Y_t^∗= a₀

1−a₁+γ0+γ1 1−a₁X_t^∗+u_t

Y_t^∗=β₀+β₁X^∗_t +u_t (17.14) So the long-run elasticity betweenYandXis captured byβ1=(γ0+γ1)/(1−a₁). Here, we need to make the assumption thata₁<1 so that the short-run model in Equation (17.11) converges to a long-run solution.

We can then derive the ECM, which is a reparametrization of the original Equation (17.11) model:

Y_t=γ₀X_t−(1−a)[Y_t−1−β₀−β₁X_t−1] +u_t (17.15) Y_t=γ0X_t−π[Y_t−1−β0−β1X_t−1] +u_t (17.16)

Proof that the ECM is a reparametrization of the ARDL

To show that this is the same as the original model, substitute the long-run solutions forβ0=a₀/(1−a₁)andβ1=(γ0+γ1)/(1−a₁)to give:

Y_t=γ0X_t−(1−a)

Y_t−1− a₀

1−a₁−γ0+γ1 1−a₁X_t−1

+u_t (17.17) Y_t=γ0X_t−(1−a)Y_t−1−a₀+(γ0+γ1)X_t−1+u_t (17.18) Y_t−Y_t−1=γ0X_t−γ0X_t−1−Y_t−1+aY_t−1−a₀−γ0X_t−1−γ1X_t−1+u_t

(17.19) and by rearranging and cancelling out terms that are added and subtracted at the same time we get:

Y_t=a₀+a₁Y_t−1+γ0X_t+γ1X_t−1+u_t (17.20) which is the same as for the original model.

What is of importance here is that when the two variablesYandXare cointegrated, the ECM incorporates not only short-run but also long-run effects. This is because the long-run equilibriumY_t−1−β0−β1X_t−1is included in the model together with the short-run dynamics captured by the differenced term. Another important advantage is that all the terms in the ECM model are stationary, and standard OLS is therefore valid.

This is because ifYandXareI(1), thenYandXareI(0), and by definition ifYand Xare cointegrated then their linear combination(Y_t−1−β₀−β₁X_t−1)∼I(0).

A final, very important, point is that the coefficientπ = (1−a₁)provides us with information about the speed of adjustment in cases of disequilibrium. To understand this better, consider the long-run condition. When equilibrium holds, then(Yt−1− β0−β1X_t−1)=0. However, during periods of disequilibrium, this term will no longer be zero and measures the distance the system is away from equilibrium. For example, suppose that because of a series of negative shocks in the economy (captured by the error termu_t)Y_t increases less rapidly than is consistent with Equation (17.14). This causes(Yt−1−β0−β1X_t−1)to be negative, becauseY_t−1has moved below its long-run steady-state growth path. However, sinceπ =(1−a₁)is positive (and because of the minus sign in front ofπ) the overall effect is to boostY_tback towards its long-run path as determined byX_t in Equation (17.14). The speed of this adjustment to equilibrium is dependent on the magnitude of(1−a₁). The magnitude ofπwill be discussed in the next section.

A more general model for large numbers of lagged terms

Consider the following two-variableY_t andX_t ARDL:

Y_t=µ+ n i=1

a_iY_t−i+ m i=0

γ_iX_t−i+u_t (17.21)

Y_t=µ+a₁Y_t−1+ · · · +anY_t−n+γ0X_t+γ1X_t−1+ · · · +γmX_t−m+u_t (17.22) We want to obtain a long-run solution of the model, which would be defined as the point whereY_t andX_tsettle down to constant steady-state levelsY^∗andX^∗, or more simply when:

Y^∗=β₀+β₁X^∗ (17.23)

and again assumeX^∗is constant

X^∗=X_t=X_t−1= · · · =X_t−m

So, putting this condition into Equation (17.21), we get the long-run solution, as:

Y^∗= µ 1−

a_i + γ_i 1−

a_iX^∗

Y^∗= µ

1−a₁−a₂− · · · −a_n+ (γ₁+γ₂+ · · · +γm)

1−a₁−a₂− · · · −a_nX^∗ (17.24) or:

Y^∗=B₀+B₁X^∗ (17.25)

which means we can defineY^∗conditional on a constant value ofXat timet as:

Y^∗=B₀+B₁X_t (17.26)

Here there is an obvious link to the discussion of cointegration in the previous section.

Defininge_tas the equilibrium error as in Equation (17.4), we get:

e_t≡Y_t−Y^∗= Y_t−B₀−B₁X_t (17.27) Therefore, what we need is to be able to estimate the parametersB₀andB₁. Clearly, B₀andB₁can be derived by estimating Equation (17.21) by OLS and then calculating A=µ/(1−

a_i)andB=

γ_i/(1−

a_i). However, the results obtained by this method are not transparent, and calculating the standard errors will be very difficult. However, the ECM specification cuts through all these difficulties.

Take the following model, which (although it looks quite different) is a reparametriza- tion of Equation (17.21):

Y_t =µ+ n−1 i=1

a_iY_t−i+ m−1

i=0

γ_iX_t−i+θ₁Y_t−1+θ₂X_t−1+u_t (17.28)

Note: forn=1 the second term on the left-hand side of Equation (17.28) disappears.

From this equation we can see, with a bit of mathematics, that:

θ₂= m i=1

γ_i (17.29)

which is the numerator of the long-run parameter,B₁, and that:

θ1= −



1− n i=1

a_i



 (17.30)

So the long-run parameterB₀is given byB₀=1/θ1and the long-run parameterB₁=

−θ2/θ1. Therefore the level terms of Y_t andX_t in the ECM tell us exclusively about the long-run parameters. Given this, the most informative way to write the ECM is as follows:

Y_t =µ+ⁿ⁻¹ i=1

a_iY_t−i+^m−1 i=0

γ_iX_t−i+θ₁ Y_t−1− 1 θ1−θ2

θ1 X_t−1

+u_t (17.31)

Y_t =µ+ⁿ⁻¹ i=1

a_iY_t−i+^m−1 i=0

γ_iX_t−i−θ₁(Y_t−1− ˆβ₀− ˆβ₁x_t−1)+u_t (17.32)

whereθ1= 0. Furthermore, knowing that Y_t−1− ˆβ0− ˆβ1x_t−1 = e_t, our equilibrium error, we can rewrite Equation (17.31) as:

Yt =µ+ n−1 i=1

a_iY_t−i+ m−1

i=0

γ_iX_t−i−πeˆ_t−1+εt (17.33)

What is of major importance here is the interpretation ofπ.πis the error-correction coefficient and is also called the adjustment coefficient. In fact,π tells us how much of the adjustment to equilibrium takes place in each period, or how much of the equilibrium error is corrected. Consider the following cases:

(a) Ifπ = 1 then 100% of the adjustment takes place within a given period, or the adjustment is instantaneous and full.

(b) Ifπ=0.5 then 50% of the adjustment takes place in each period.

(c) Ifπ =0 then there is no adjustment, and to claim thatY_t^∗is the long-run part of Y_tno longer makes sense.

We need to connect this with the concept of cointegration. Because of co- integration,ˆe_t ∼I(0)and therefore alsoeˆ_t−1∼I(0). Thus, in Equation (17.33), which is the ECM representation, we have a regression that contains onlyI(0)variables and allows us to use both long-run information and short-run disequilibrium dynamics, which is the most important feature of the ECM.