Multicointegration in Stock-Flow Models

Tom Engsted^{∗} Department of Finance

Aarhus School of Business, DK-8210 Aarhus V, Denmark

Niels Haldrup Department of Economics University of Aarhus, DK-8000 Aarhus C, Denmark

13 October, 1997

Abstract. Multicointegration, in the sense of Granger and Lee (1990), frequently occurs in models of stock-flow adjustment and implies cointegration amongst I(2) variables and their differences (polynomial cointegration). The purpose of this article is two-fold. First, we demonstrate that based on a multicointegrated vector autoregression (VAR) two equivalent error correction model (ECM) representations can be derived; the first is expressed in terms of adjustments in the flows of the variables (the standard I(2) ECM), and the second is expressed in terms of adjustments in both the stocks and the flows. Secondly, we apply I(2) estimation and testing procedures for multicointegrated time series to analyze data for US housing construction. We find that stocks of housing units started and completed exhibit polynomial cointegration (and hence the flows are multicointegrated) and the associated ECM’s are estimated. Lee (1992, 1996) also found multicointegration in this data set but without explicitly exploiting the I(2) property.

Keywords: Cointegration, multicointegration, I(2) processes, housing construction. JEL Codes: C12, C13, C22, C32, C51.

1. Introduction

Granger and Lee (1989, 1990) have defined the notion of multicointegration amongst economic variables integrated of order one: If the cumulation of cointegration (equilibrium) errors cointegrate with the original variables the series are said to be multicointegrated. The phenomenon is most likely to occur in models describing the dynamic interactions between stock and flow variables. Multicointegration implies that in the bivariate I(1) model, for example, there can exist more than just a single cointegration vector. More generally, the number of cointegrating relations and the number of common stochastic trends will not add up to the dimension of the system which is the case in the usual cointegrated I(1) model. In a recent paper Engsted and Johansen (1997) show that when variables are multicointegrated the requirements for the system to be an I(1) system will fail; in fact, an I(1) specification will be misspecified even though the main interest lies in the analysis of the I(1) series. Instead the system should be formulated as an I(2) model where multicointegration can be shown to result in cointegration amongst generated I(2) variables and their first differences (polynomial cointegration)^{1}. Others have been aware of the link between multicointegration and polynomial cointegration, see inter alia Engle and Yoo (1991), and Lee (1992), but to the best of our knowledge there have been no empirical contributions to the multicointegration literature which takes explicitly account of the I(2) property in estimation and hypothesis testing. In this paper we provide an example of how such an analysis can be undertaken.

∗Both authors are affiliated with Centre for Non-Linear Modelling in Economics at the University of Aarhus. Part of this work was carried out while the second author was visiting the European University Institute in Florence during the spring, 1997. We appreciate helpful comments from Søren Johansen and we thank Tae-Hwy Lee for providing us with the US housing construction data set.

We also want to show how multicointegrated variables from a general vector autoregression (VAR) can be given two equivalent error correction model (ECM) representations. The first representation is a direct consequence of the I(2) formulation of the model and follows Johansen’s (1992) version of Granger’s representation theorem for I(2) variables. It describes how the flow variables adjust in response to disequilibrium errors between the stocks and the flows as well as between the flow variables themselves. The second ECM representation describes adjustments in both the stock and the flow variables in response to the equilibrium errors. Although equivalent, the two distinct ways of writing the ECM provide different insights of the complex dynamics characterizing multicointegrated systems.

The plan of the paper is the following. In section 2 we briefly summarize some known results for the relationship between optimum control models and error correction models. We also demonstrate how multicointegrating relationships can be given an I(2) formulation. The next section describes the relationship between multicointegration and cointegrated I(2) VAR models and the section also serves to define the notation to be used subsequently in section 4 where we show how the two different ECM representations for multicointegrated time series can be obtained. In section 5 we demonstrate the I(2) analysis on the US housing construction data set 1968:1-1994:12 which has previously been examined by Lee (1996) (and Lee (1992) for a shorter data period) but without exploiting the I(2) property. This data set is found to nicely characterize a multicointegrating relationship amongst the variables and we calculate the associated ECMs. The final section concludes.

1In the literature the notions of ’multicointegration’ and ’polynomial cointegration’ are frequently used as synonyms. However, we use ’multicointegration’ in the particular sense of Granger and Lee, and hence this notion applies to systems where the basic variables are really I(1) but can be transformed to an I(2) system. On the other hand, ’polynomial cointegration’ refers to cointegrated models where the basic variables are I(2) and the levels and the first differenced variables cointegrate.

2. Optimum control ECM’s and the definition of stock variables Granger and Lee (1990) have shown how the ECM for multicointegrated time series can be derived as the solution to a particular control problem^{2}. In a slightly more general context Lee (1996) addresses the infinite horizon linear quadratic adjustment cost model

L_{t} = E_{t} ∑_{j=0}^{∞} δ^{j}[(x_{t+j} − y_{t+j})^{2} + λ_{1}(Q_{t+j} − κy_{t+j})^{2} + λ_{2}(x_{t+j} − x_{t+j−1})^{2}]  (1)

where, assuming a zero initial condition, Q_{t} = Q_{t−1} + (x_{t} − y_{t}) = ∑_{j=0}^{t}(x_{j} − y_{j}). The variable x_{t} is frequently referred to as the control variable whereas y_{t} is the target series one is attempting to track by choice of x_{t}. The basic I(1) flow variables are x_{t} and y_{t} while Q_{t} is the I(1) stock variable which measures the cumulated discrepancies between the control and the target series. δ is the discount rate and λ_{1}, λ_{2} are cost parameters. Note that x_{t} − y_{t} and Q_{t} − κy_{t} are cointegrating relations and hence are I(0). Lee (1996) shows how the minimization of the loss function (1) leads to a stock adjustment model which essentially is of the form:

ΔQ_{t} = α + β(Q_{t−1} − κy_{t−1}) + γ(x_{t−1} − y_{t−1}) + μ(y_{t} − y_{t−1})  (2)

where β, γ, and μ are complicated non-linear functions of δ, λ_{1}, and λ_{2}. This type of models is familiar from the control engineering literature and belongs to the general class of feedback controllers with proportional, integral, and derivative corrections, see Kwakernaak and Sivan (1972) and Phillips (1954). It also introduces the idea of an inventory accelerator mechanism, see Metzler (1941) and Holt et al. (1960). Notice that the ECM (2) is defined in terms of adjustments in the stock variable Q_{t}.

The adjustment rule derived in Granger and Lee (1990) is based on a one period decision problem and corresponds to an ECM for the flow variables, e.g.

Δx_{t} = α̃ + β̃(Q_{t−1} − κ̃y_{t−1}) + γ̃(x_{t−1} − y_{t−1}) + μ̃(y_{t−1} − y_{t−2}).  (3)

The decision rules (2) and (3) provide the motivation for the present paper. The idea is that from an optimization point of view it is reasonable to consider adjustment mechanisms for both the stock and the flow variables in this class of models. As such, (2) and (3) motivate the different ECM mechanisms that can be derived within a general framework for multicointegrated VAR models. We want to show how the two sorts of representations are indeed equivalent and hence will capture the basic features of the above adjustment mechanisms^{3}.

2Hendry and von Ungern-Sternberg (1981) report similar results but without using the notion of cointegration because this concept was not developed until the work of Granger (1983).

There are many examples of stock-flow models that can be considered as particular cases of the above decision rules. Granger and Lee (1989, 1990) address the case where x_{t} and y_{t} are production and sales of some commodity, and ΔQ_{t} = x_{t} − y_{t} is the change of inventories. The Hendry and von Ungern-Sternberg (1981) study of the consumption function belongs to the same class of models.

Consider also the following bivariate example which has been analyzed by Lee (1992) in a BULLETIN article and in Lee (1996) for an extended sample period: We focus on housing market data and let x_{t} and y_{t} be the flows of housing units started and completed, respectively. For the current period the increment in the stock of housing units under construction is given by x_{t} − y_{t} and the entire stock (assuming a zero initial stock) is Q_{t} ≡ ∑_{j=1}^{t}(x_{j} −y_{j}). Also in this case the potentially cointegrating relations are between the variables x_{t} and y_{t} (with an obvious cointegration vector given by (1, −1)), and between the variables Q_{t} and y_{t} (or possibly x_{t}). However, the identity defining the increment in the stock of housing units under construction ΔQ_{t} needs not always be given simply as x_{t} − y_{t}.

”It could be [that] some building companies go bankrupt and some starts are therefore not completed, and it could also be due to recording errors. The possibility that dwellings disappear by demolition, fire or conversion to other uses would also affect the identity”, (Lee, 1992, p. 423).

The percentage of never completed starts is naturally an unknown figure. Lee solves this problem by using a smoothing estimator and for US data he finds that approximately 2 % of all housing units started are never completed and hence the appropriate stock variable is redefined as Q_{t} ≡ ∑_{j=1}^{t}(.98x_{j} − y_{j}).

The I(2) representation of the model suggests a different way of estimating this parameter. The idea is to consider the generated I(2) variables Y_{t} = ∑_{j=1}^{t} y_{j}, X_{t} = ∑_{j=1}^{t} x_{j}, and the I(1) variables ΔY_{t} = y_{t}, ΔX_{t} = x_{t} (upper case letters represent the cumulated series) and the associated relation

Y_{t} = κ_{0}X_{t} + κ_{1}ΔY_{t} + κ_{2}ΔX_{t} + u_{t}  (4)

which is potentially a multicointegrating (polynomially cointegrating) relation^{4} when u_{t} ∼ I(0). The parameter κ_{0} is the estimated percentage of starts that are eventually completed. Note that when κ_{0} = 1, (4) simplifies to Q_{t} = X_{t}−Y_{t} = −(κ_{1}y_{t}+κ_{2}x_{t}+u_{t}). In the present context, given that the model is a satisfactory description of the data, κ_{0} can be estimated at the super-super consistent rate, O_{p}(T^{−2}) where T is the sample size and hence, from an estimation point of view, there are very good statistical reasons for formulating the model in terms of an I(2) model. Observe that since the generated I(2) series Y_{t}, and X_{t} generally will have a linear trend it may also be necessary to allow for deterministic components in (4).

3The decision rules (2) and (3) entail certain parameter restrictions for the underlying optimizing model to be valid. It is beyond the purpose of the present paper to address this particular issue but merely we want to emphasize how the basic control mechanisms will arise in a multicointegrated VAR model.

4Observe that in (4) the cumulated variables X_{t} and Y_{t} are allowed to cointegrate with both of

Another example where the I(2) representation is useful in estimating unknown parameters defining the stock variables is the life cycle hypothesis of consumption. According to this model consumption cointegrates with income and wealth where the latter, by definition, equals accumulated savings, i.e. the accumulated difference between income and consumption. Hence, there is multicointegration between consumption and income. However, in order to find an appropriate measure of savings, it is frequently necessary in this type of models to estimate the relationship between income (decomposed into labour and capital income) and the value of non-durables consumption and service flows, see e.g. Campbell (1987, section 4) for details. An O_{p}(T^{−2}) consistent estimate of this relationship can be obtained using the I(2) cointegration analysis. In contrast, Campbell (1987) estimates this relationship in a standard I(1) analysis and hence only obtains an O_{p}(T^{−1}) consistent estimate.

3. Multicointegration and I(2) cointegration

The cointegrated VAR model for I(1) models can be written in the well-known vector error correction model (VECM ) form, see Johansen (1991),

Δx_{t} = Πx_{t−1} + ∑_{i=1}^{k−1} Γ_{i}Δx_{t−i} + ε_{t}  (5)

where x_{t} is a p dimensional vector I(1) time series (here assumed to consist of flow variables) and Π is a matrix determining the cointegration rank of the VAR. ε_{t} is a sequence of i.i.d. zero mean errors with non singular covariance matrix Ω. If rankΠ =r the x_{t} series has r cointegrating relations and p − r common stochastic I(1) trends driving the system. In this case the matrix Π can be decomposed as Π = αβ′ where α and β are both p × r matrices. The cointegrating relations are given by β′x_{t} and the associated adjustment coefficients summarized in α. Engsted and Johansen (1997) show that if x_{t} is multicointegrated such that the cumulated equilibrium errors ∑_{i=1}^{t} β′x_{j} cointegrate with x_{t}, then the representation (5) is invalid. In fact, the (longrun) covariance matrix defined for the cointegration relations and the differenced variables will be singular in this case but alternatively the model can be formulated as a VAR for cointegrated I(2) variables. If we, as we did in section 2, let upper case letters denote cumulated (lower case) series then X_{t} ≡ ∑_{j=1}^{t} x_{j} follows a p dimensional VAR for I(2) variables with the VECM representation

the variables x_{t} and y_{t}. Strictly speaking only one of series x_{t} or y_{t} is needed for the cointegration property to hold, compare the adjustment models (2) and (3). However, as we shall see it follows as a natural output from the statistical analysis that the specification (4) is useful when the variables are considered from a systems approach using the Johansen ML procedure.

Δ^{2}X_{t} = ΠX_{t−1} − ΓΔX_{t−1} + ∑_{i=1}^{k−2} Ψ_{i}Δ^{2}X_{t−i} + ν_{t}  (6)

where Γ = I− ∑_{i=1}^{k−1} Γ_{i} and ν_{t} is an error having non-singular covariance matrix. The I(2) VAR is introduced in Johansen (1992, 1995) and he shows that the parameters satisfy

Π = αβ′, with α, β p × r, r < p  (7)
α′_{⊥}Γβ_{⊥} = ξη′, with ξ, η (p − r) × s, s < (p − r)  (8)

where α_{⊥} denotes the orthogonal complement of α and hence has dimension p×(p−r) and satisfies α′_{⊥}α = 0. Paruolo (1996) denotes the numbers r, s, and p − r − s the integration indeces of the I(2) VAR model: r relations cointegrate to I(0) level, possibly by including the first differences of X_{t}, s relations are I(1) and constitute the I(1) common stochastic trends, and finally, p − r − s relations are I(2) which we refer to as the I(2) common trends. Associated with these three cases it is possible to find mutually orthogonal matrices (β, β_{1}, β_{2}) where each component individually provides a basis for the I(0), I(1), and I(2) directions, respectively.

The p different relations can also be written as

r : β′X_{t} − δβ′_{2}ΔX_{t} ∼ I(0)  (9)
s : β′_{1}X_{t} ∼ I(1)  (10)
p − r − s : β′_{2}X_{t} ∼ I(2).  (11)

Note that the I(0) relations can be expressed in terms of linear combinations between the levels of β′X_{t} (which generally are I(1)) and the differenced I(2) trends β′_{2}ΔX_{t} (which are also I(1)), and hence include the polynomially cointegrating relations. The matrix δ has dimension r × (p − r − s) and has the orthogonal complement δ_{⊥} of dimension r × (r − (p − r − s)) satisfying δ′_{⊥}δ = 0. More precisely δ is defined as δ =\overline{α}′Γ\overline{β}_{2} where \overline{α} = α(α′α)^{−1} and \overline{β}_{2} is defined in a similar way^{5}.

Not all the I(0) relations in (9) need to be polynomially cointegrating. The linear combinations where the differenced variables are not needed to yield stationarity are given by δ′_{⊥}β′X_{t}. On the other hand, the relations which surely require the differenced variables are the polynomially cointegrating relations and are given by δ′β′X_{t} − δ′δβ′_{2}ΔX_{t}. These are the truely multicointegrating relations. Hence the r I(0) relations can be partitioned into p − r − s multicointegrating relations and the remaining r−(p−r−s) cointegrating relations. Of course this separation requires that r > p − r − s. In practical situations with multicointegration it seems less likely that the directly stationary cointegrating relations δ′_{⊥}β′X_{t} will be present. This would imply that the differenced linear combinations δ′_{⊥}β′x_{t} be I(−1). Hence we believe that it is most common that r = p − r − s when the focus is on multicointegration in the sense of Granger and Lee. Finally, it should be mentioned that the I(1) trends β′_{1}X_{t} will be stationary in first differences, i.e. Δβ′_{1}X_{t} = β′_{1}x_{t} ∼ I(0).

5Note that P_{α} = \overline{α}α′ = α\overline{α}′ denotes the projection onto the space spanned by the columns of α.

Observe that for bivariate systems certain simplifications will arise. In particular, if the series are multicointegrated then (r, s, p − r − s) = (1, 0, 1) and hence no I(1) trends can be present.

4. Error correction representation of multicointegrated variables The VECM for general I(2) systems (6) can be given the following alternative representation due to Paruolo and Rahbek (1996):

Δ^{2}X_{t} = α(β′X_{t−1} − δβ′_{2}ΔX_{t−1}) − (ζ_{1}, ζ_{2})(β, β_{1})′ΔX_{t−1} + ∑_{i=1}^{k−2} Ψ_{i}Δ^{2}X_{t−i} + ν_{t}, (12)

where ζ_{1} = Γ\overline{β} and ζ_{2} = Γ\overline{β}_{1} are adjustment parameters. This way of writing the VECM is perhaps more intuitive compared to (6) since all the cointegration possibilities (9)-(11) are displayed rather straightforwardly. The representation (12) forms the basis for the VECM formulated in terms of the original I(1) flow variables x_{t} :

Theorem 1. Error correction model representation for the flow variables. Let x_{t} be a p-dimensional vector process of flow variables integrated of order one and multicointegrated such that r ≥ p − r − s > 0 and such that X_{t} = ∑_{j=1}^{t} x_{j} is an autoregressive process given by (12). Then Δx_{t} can be given the following representation:

Δx_{t} = ξ_{1} [δ′Q_{t−1} − δ′δβ′_{2}x_{t−1}] + ξ_{2} [δ′_{⊥}Q_{t−1}]  (13)
−ζ_{1}ΔQ_{t−1} − ζ_{2}β_{1}′x_{t−1} + Ψ(L)Δx_{t} + ν_{t}

where ξ_{1} = α\overline{δ} and ξ_{2} = α\overline{δ}_{⊥} are adjustment parameters of dimensions p×(p−r−s) and p × (r − (p − r − s)), respectively, and Q_{t} = ∑_{j=1}^{t} β′x_{j} is the stock of cumulated I(0) errors. Furthermore Ψ(L) = ∑_{i=1}^{k−2} Ψ_{i}L^{i}.

Proof. Consider the ECM representation (12) and in place of α use the identity α = α(P_{δ} + P_{δ_{⊥}}) where P_{δ} = δ(δ′δ)^{−1}δ′ denotes the orthogonal projection matrix on the space spanned by δ. P_{δ_{⊥}} is defined in a similar fashion. The representation now follows straightforwardly by collecting terms and noting that x_{t} = ΔX_{t} and X_{t} = ∑_{j=1}^{t} x_{j} and by using the definition of the stock variables Q_{t}.

In (13) ξ_{1} adjusts for the p − r − s multicointegrating relations which are given by [δ′Q_{t−1} − δ′δβ′_{2}x_{t−1}] and correspond to the ’integral’ control mechanisms. ξ_{2} adjusts for the remaining r − (p − r − s) cointegrating relations amongst the stock variables but these relations do not have the property that they multicointegrate. ζ_{1} and ζ_{2} adjust for the ’proportional’ error correction terms given by ΔQ_{t} and β_{1}′x_{t}. Finally the ’derivative’ control mechanisms are given by Ψ(L)Δx_{t}. Note that, ignoring the error term, the flow ECM given in (3) can be obtained as a special case of (13).

The second ECM representation for multicointegrated time series can be stated as follows:

Theorem 2. Error correction model representation for the stock variables. Let x_{t} be a p-dimensional vector process of flow variables integrated of order one and multicointegrated such that r ≥ p − r − s > 0 and such that X_{t} = ∑_{j=1}^{t} x_{j} is an autoregressive process given by (12). Then the process Δx̃_{t} = (ΔQ′_{t}, β′_{1}x_{t}, Δx′_{t}β_{2})′ where Q_{t} = ∑_{j=1}^{t} β′x_{j} can be given the following representation:

Δx̃_{t} = ξ̃_{1} [δ′Q_{t−1} − δ′δβ′_{2}x_{t−1}] + ξ̃_{2} [δ′_{⊥}Q_{t−1}]  (14)
+ζ̃_{1}ΔQ_{t−1} + ζ̃_{2}β′_{1}x_{t−1}+MΨ(L)\overline{β}_{2}Δβ′_{2}x_{t−1}
+Ψ̃(L)Δ^{2}x̃_{t} + ν̃_{t}

where the parameters and symbols are defined in the following way: M = (β, β_{1}, β_{2})′, ξ̃_{1} = Mξ_{1}, ξ̃_{2} = Mξ_{2}, ζ̃_{1} = (ι_{r} − Mζ_{1}), ζ̃_{2} = (ι_{s} − Mζ_{2}), ι_{r} = (I_{r}, 0, 0)′, ι_{s} = (0, I_{s}, 0)′, Ψ̃(L) = MΨ(L)M^{−1}D_{⊥}(1), and ν̃_{t} = Mν_{t}. The dimensions of the ’∼’ parameters are the same as the parameters without the ’∼’.

Proof. We adopt the same technique as in Mellander et al. (1992) and define the matrices

M = ( \begin{matrix}β & ′ \\ β & {′ \atop 1} \\ β & {′ \atop 2} \end{matrix} ) , D(L) = ( \begin{matrix}Δ_{r} \\ & Δ_{s} \\ & & I_{p−r−s} \end{matrix} ) , D_{⊥}(L) = ( \begin{matrix}I_{r} \\ & I_{s} \\ & & Δ_{p−r−s} \end{matrix} ) .

Observe that M^{−1} = (\overline{β}′, \overline{β}′_{1}, \overline{β}′_{2}) and D(L)D_{⊥}(L) = Δ_{p}.

We want to transform the variables in (12) such that error correction adjustments are defined in terms of β′X_{t}, β′_{1}X_{t}, and β′_{2}ΔX_{t}, which are all I(1), c.f. (9)-(11). Eqn.

(12) can be written as follows:

ΔD(L)D_{⊥}(L)MX_{t} = Mα [I, 0, −δ] D_{⊥}(L)MX_{t−1}  (15)
−M(ζ_{1}, ζ_{2}, 0)ΔD_{⊥}(L)MX_{t−1}
+MΨ(L)ΔM^{−1}D(L)D_{⊥}(L)MX_{t−1} + Mν_{t}.

Now, defining x̃_{t} = (X′_{t}β, X′_{t}β_{1}, ΔX′_{t}β_{2})′ and rewriting (15) yields

Δx̃_{t} = Mα [I, 0, −δ] x̃_{t−1}  (16)
+ {D_{⊥}(1) − M(ζ_{1}, ζ_{2}, 0) + MΨ(L)M^{−1}D(1)} Δx̃_{t−1}
+MΨ(L)M^{−1}D_{⊥}(1)Δ^{2}x̃_{t−1} + Mν_{t}.

Alternatively, by letting ι_{r} = (I_{r}, 0, 0)′ and ι_{s} = (0, I_{s}, 0)′ and collecting terms we obtain

Δx̃_{t} = Mα [I, 0, −δ] x̃_{t−1}  (17)
+ \{(ι_{r} − Mζ_{1}) , (ι_{s} − Mζ_{2}) , (MΨ(L)\overline{β}_{2})\} Δx̃_{t−1}
+MΨ(L)M^{−1}D_{⊥}(1)Δ^{2}x̃_{t−1} + Mν_{t}.

With the definitions of symbols given in the Theorem and by using the identity α = α(P_{δ} + P_{δ_{⊥}}), like in the proof of Theorem 1, the representation (14) results.

The interesting thing to note about Theorem 2 compared to Theorem 1 is that although the same equilibrium relations appear in the ECM, the model is formulated in terms of stock and flow variables rather than in terms of the original flow variables x_{t}. But of course the adjustment coefficients will be different. The stock variables in the representation (14) are given by the variables Q_{t} = ∑_{j=1}^{t} β′x_{j} and Q_{t}^{∗} = ∑_{j=1}^{t} β′_{1}x_{j} whereas the flow variables enter through the differenced I(2) trends, β′_{2}x_{t}. Note, however, that the stock variables Q_{t}^{∗} only enter through their derivatives, the flow counterparts ΔQ_{t}^{∗}. The ECM given in (2) can be obtained as a special case of (14).

Of course the above representations can be extended to the case where deterministics are included in the models. However, since this complicates the analysis considerably without gaining much additional insight it is excluded in the present exposition. The way that deterministics affect the representation of the I(2) VAR and the statistical analysis is discussed in Paruolo (1996) and Jørgensen et al. (1996).

5. An application to US housing construction data

In this section we use I(2) techniques to analyze multicointegration in the US housing data set previously examined by Lee (1992, 1996). The series are from Citibase and are: new privately owned housing units started, x_{t} = HSF R, and new privately owned housing units completed, y_{t} = HCP. Data are monthly seasonally adjusted and covers the period 1968:1-1994:12. In figure 1 x_{t} and y_{t} are displayed. Unit root tests (not reported) strongly indicate the presence of a single unit root in both series. Since the I(2) analysis is based on the cumulated series X_{t} = ∑_{j=1}^{t} x_{j} and Y_{t} = ∑_{j=1}^{t} y_{j} the graphs of these time series are displayed as well in figure 1. Note that assuming x_{t} and y_{t} to be I(1), X_{t} and Y_{t} will be I(2) by their construction and will measure the stocks of started and completed housing units, respectively.

This is a bivariate example and naturally this simplifies the analysis somewhat. If x_{t} and y_{t} are multicointegrated the theory predicts that s = 0 (no I(1) common trends), r = 1, (one multicointegrating relation), and p − r − s = 1 (a single common I(2) trend).

Figure 1 about here

5.1. Testing for multicointegration. Lee (1992, 1996) defined the stock of housing units under construction as Q_{t} = ∑_{j=1}^{t}(.98x_{j} − y_{j}) = .98X_{t} − Y_{t} and he found strong support for Q_{t} being I(1) and cointegrating with y_{t}, that is x_{t}, y_{t} are multicointegrated. However, since Q_{t} is a constructed variable a priori his analysis did not explicitly involve I(2) variables, c.f. the discussion in section 2. Here we adopt the I(2) analysis both by single equation tests and by tests based on the VAR.

Single equation tests. The single equation procedure has recently been suggested by Engsted et al. (1997) as an extension of the Engle and Granger (1987) two-step procedure to the case with I(2) variables, see also Haldrup (1994). The idea is in the first step to conduct the regression

Y_{t} = κ̂_{0}X_{t} + κ̂_{1}ΔY_{t} + δ̂_{0} + δ̂_{1}t + û_{t}  (18)

and in the second step to test the integration order of û_{t}. For instance, an augmented Dickey Fuller test (or Phillips (1987) Z−test) can be conducted. Given that y_{t} and x_{t} cointegrate to I(0) level the critical values for the ADF test for the null of no multicointegration are reported in Engsted et al. (1997) for the case where an intercept plus a trend are included in the first step regression as in (18). For m_{1} = 1 I(1) regressor and m_{2} = 1 I(2) regressor the asymptotic 1% critical value is given by -4.73. Note that the reason for including a trend regressor in (18) is to account for the fact that the generated series X_{t} and Y_{t} are likely to drift when x_{t} and y_{t} have a non-zero mean. The ADF test gives a test value of -4.82 when 3 lags are included in the auxiliary DF regression^{6}. Hence there is rather strong evidence in favour of multicointegration.

6The single equation analyses were conducted by using PcGive 9.0, see Hendry and Doornik

The estimated cointegrating relation (apart from deterministics) is given by

Y_{t} = .947X_{t} − 8.153ΔY_{t},  (19)

so the estimate of κ_{0}, the percentage of housing constuction starts that are eventually completed, is given by 94.7%.

Alternatively, since cointegration has already been found it is also possible to estimate κ_{0} from a more complex dynamic regression including lags of ΔY_{t} and ΔX_{t}. The implied ’static’ long run solution (see Hendry and Doornik, 1996, p. 234), from a model including 4 lags of ΔY_{t} and ΔX_{t} reads:

Y_{t} = .947X_{t} − 5.581ΔY_{t} − 2.651ΔX_{t}  (20)

and hence it is seen that the estimate of κ_{0} happens to be identical to the estimate based on the regression (19).

Tests based on the I(2) VAR. Single equation analysis is not the most efficient way of estimating κ_{0}. The I(2) VAR model for the variables Y_{t} and X_{t} is an alternative and more comprehensive benchmark for analyzing the multicointegrating properties of the data. The statistical analysis for the I(2) model in VAR systems is discussed in Johansen (1995, 1997). The basic model we consider is of the form (6) but we also want to allow for an intercept and a deterministic trend in the model. However, since we do not want the model to generate quadratic trends we restrict the trend to lie in the cointegration space. Essentially this corresponds to a multicointegrating relation with a trend similar to (18). The statistical analysis for this situation is discussed in Jørgensen et al. (1996). In the subsequent analysis this restriction has been imposed in the estimation^{7}.

In order to determine the integration indeces an I(2) VAR model with 12 lags and a trend restricted to lie in the cointegration space was estimated for the variables Y_{t} and X_{t}. In table 1 various univariate and multivariate diagnostics of the VAR are reported. As seen there are generally no problems about autocorrelation but due to heteroscedasticity and excess kurtosis there are problems with the normality tests. A possible remedy of this problem is to log transform the variables. However, this is not strictly permitted when the focus is on the stock-flow relations. Since Gonzalo (1994) has found that the Johansen ML procedure is rather robust to discrepancies from the model assumptions due to heteroscedasticity we ignore this problem. In figure 2 the roots of the companion matrix of the VAR(12) are displayed for the case where no I(2) trends are allowed for in the estimation. As seen there is evidence of at Table 1. Misspecification tests of the VAR(12).

7All I(2) VAR computations have been calculated from the the I(2) extension of the CATS in RATS programme. The programme is written by Clara M. Jørgensen and is downloadable from http://www.estima.com/procs/i2index.htm.

Multivariate tests:

Residual autocorrelation LM(1)  χ^{2}(4) = 2.79
Residual autocorrelation LM(4)  χ^{2}(4) = 7.31
Normality  LM  χ^{2}(4) = 19.31^{∗∗∗}
Univariate tests:  ΔY_{t}  ΔX_{t}
ARCH(12)  5.53^{∗∗∗} 33.33^{∗∗∗}
Jarque-Bera, Normality  12.56^{∗∗∗} 7.19^{∗∗∗}
Skewness  .47  -.13
Excess kurtosis  3.91  3.69
R^{2}  .95  .91

Note: (*), (**), and (***) denote significance at 1,5, and 10 % levels, respectively.

least two unit roots in the data. Formal testing for the number of unit roots and the integration indeces can be based on the S_{r,s} trace statistics discussed by Johansen (1995, 1997). The results are reported in table 2. This table should be read from the upper left corner (the most restricted model, r = 0, p − r − s = 2) to the right, and down, until the first hypothesis that cannot be rejected is identified. That is, the hypotheses should be tested successively less and less restricted.

Figure 2 about here

As seen the first hypothesis that cannot be rejected corresponds to the hypothesis where r = 1 and p − r − s = 1. Hence the tests indicate that there is one common I(2) trend in the data and one polynomially cointegrating (or multicointegrating) relation. Note that s = 0, so there are no I(1) trends in the model.

The error correction models. The ECM’s presented in theorems 1 and 2 become somewhat simplified in the present situation when there are no I(1) trends and r = p − r − s. In this case the I(0) relation is also the multicointegrating relation. The estimates of the model parameters are tabulated in table 3. Note that since δ̂_{⊥} = 0 and β̂_{1} = 0 many of the error correction terms will vanish.

As seen the estimated stock of housing units under construction is given by Q_{t} = −β′(Y_{t}, X_{t}) = ∑_{j=1}^{t}(.972x_{j} − y_{j}). This is only slightly different from Lee’s (1992) estimate that 2% of housing starts were never completed. In figure 3 two different measures of the stock variable are displayed and it is apparent that the estimated Table 2. Joint tests of the integration indeces. US Housing data set, 1968:1-1994:12.

p − r  r  S_{r,s}  Q_{r}
2  0 76.90^{∗∗∗} 32.72^{∗}  22.55^{∗}
47.60  34.36  25.43
1  1  13.90  4.20
19.87  12.49
p − r − s  2  1  0

Note: (*), (**), and (***) denote significance at 1,5, and 10 % levels, respectively. Numbers in italics are 95 per cent quantiles, (Jørgensen et al. (1996), table 4). r and p − r − s are the number of I(0) and I(2) components.

Table 3. Parameter estimates of the I(2) VAR for r = 1, s = 0, and p − r − s = 1.

Multicointegrating relation  β̂′ = (1, −.972), δ̂ = 3.908, (δ̂_{⊥} = 0)
I(2) trend  β̂′_{2} = (1, 1.029)
M−matrix  M̂ = (β̂, β̂_{2})′
Multicointegrating parameters δ̂β̂′_{2} = (3.908, 4.022)
Adjustment coefficients  α̂′ = (−.026, .034)
Gamma matrix  Γ̂ = ( \begin{matrix}.372 & −.158 \\ −.207 & −.064 \end{matrix} )

Derived estimators

Flow ECM  Stock ECM
ξ̂_{1}δ̂ = α̂ˆ\overline{δ}δ̂ = α̂  ˆ  ξ̃_{1}δ̂ = M̂ξ̂_{1}δ̂ = M̂α̂ = ( \begin{matrix}−.059 \\ .009 \end{matrix} )
ξ̂_{2} = α̂ˆ\overline{δ}_{⊥} = 0  ˆ  ξ̃_{2} = M̂ξ̂_{2} = 0
ζ̂_{1} = Γ̂ˆ\overline{β} = ( \begin{matrix}.270 \\ −.074 \end{matrix} )  ˆ  ζ̃_{1} = (( \begin{matrix}1 \\ 0 \end{matrix} ) − M̂ζ̂_{1}) = ( \begin{matrix}.657 \\ −.194 \end{matrix} )
ζ̂_{2} = Γ̂β̂_{1} = 0  ˆ  ζ̃_{2} = −M̂ζ̂_{2} = 0

stock variable rather than the difference Q_{t} = X_{t} − Y_{t} seems to behave more like the original flow variables x_{t} and y_{t}. The variable Q_{t} = X_{t} − Y_{t} tends to drift away compared to the flow series. This drift captures the cumulation of starts that are never completed. It is likely that the drift can be partially corrected for by allowing a trend in the cointegration space as we have done in the present study, however, this will only capture part of the discrepancy since the estimate of κ_{0} is still strictly less than one^{8}.

Figure 3 about here

The multicointegrating relation can be written as follows (apart from the deterministic components), compare (9):

β̂′(Y_{t}, X_{t})′ − δ̂β̂′_{2}(ΔY_{t}, ΔX_{t})′ = Y_{t} − .972X_{t} + 3.908ΔY_{t} + 4.022ΔX_{t} (21)

and the I(2) trend is given by

β′_{2}(Y_{t}, X_{t})′ = Y_{t} + 1.029X_{t}.  (22)

Note that the I(2) trend is close to being the sum (average) of the two series since κ_{0} is not too far away from unity.

The flow ECM can be written as follows:

Δ ( \begin{matrix}y_{t} \\ x_{t} \end{matrix} ) = ( \begin{matrix}−.026 \\ .034 \end{matrix} ) (Q_{t−1} − 3.908y_{t−1} − 4.022x_{t−1})  (23)

+ ( \begin{matrix}−.270 \\ .074 \end{matrix} ) (ΔQ_{t−1}) + lags of \{Δy_{t}, Δx_{t}, \} + constant + trend.

With respect to the stock adjustment ECM this can be written:

Δ ( \begin{matrix}Q_{t} \\ β′_{2}(Y_{t}, X_{t})′ \end{matrix} ) = ( \begin{matrix}−.059 \\ .009 \end{matrix} ) (Q_{t−1} − 3.908y_{t−1} − 4.022x_{t−1})  (24)

+ ( \begin{matrix}.657 \\ −.194 \end{matrix} ) (ΔQ_{t−1}) + lags of \{Δy_{t}, Δx_{t}, \} + constant + trend.

Of course, from an economic perspective the first row in the stock ECM is most interesting. For both ECM’s it is seen that the adjustment coefficients have the correct signs but adjustment also seems to be extremely slow which reflects high costs of adjustment.

8Notwithstanding, it should be mentioned that a LR test for the hypothesis that β′ = (1, −1)

cannot be rejected. The test value is 1.36 which should be compared with the χ^{2}(1) distribution; the p-value is .24. It is likely that the trend in the cointegration space gives rise to this result.

6. Concluding comments

In this paper we have demonstrated how two separate, though equivalent, error correction model representations can be given to time series that are multicointegrated in the sense of Granger and Lee (1989, 1990). The I(1) model is not adequate for describing multicointegrated time series, so much of the analysis has focused on the I(2) model and how this can be reparametrized to describe the particular features of multicointegrated time series. Through an empirical application of US housing construction data we also demonstrated how the I(2) analysis can be useful in practice as a tool for analyzing multicointegrated time series even though the basic economic variables are really I(1). To the best of our knowledge this provides the first empirical example of explicitly adopting I(2) procedures to multicointegrated time series in the Granger-Lee meaning of the concept. What remains is to consider multicointegration in more advanced models where a potentially larger information set is used to describe the stock building and flow decisions of the housing market. At least the I(2) procedures seem to work well for small stylized models.

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7. Figures Figure 1: Housing construction data set for US, 1968:1-1994:12, seasonally adjusted. New privately owned housing units started, x_{t}, and completed, y_{t}. The corresponding stock series have been generated as X_{t} = ∑_{j=1}^{t} x_{j}, and Y_{t} = ∑_{j=1}^{t} y_{j}.

Multicointegration in Stock-Flow Models 19

Figure 2: The roots of the companion matrix of the VAR(12).

Multicointegration in Stock-Flow Models 20

Figure 3: Two different measures of inventories, Q_{t} = X_{t} − Y_{t} and Q_{t} = .972X_{t} − Y_{t} and the associated flow variables x_{t} and y_{t}.