DI IORIO, FRANCESCA

R. Lucchetti

S. Fachin

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Testing hypothesis in VARs with I(2) andnear-I(2) latent stochastic trendsFrancesca Di Iorio, Stefano Fachin, Riccardo LucchettiAbstract We review the I(2) model in an empirical perspective, and report the re-sults of some Monte Carlo simulations on the small sample performance of asymp-totic tests on the long-run coefficients in I(2) and near-I(2) systems. The resultsshow that, although moderate size bias tend to be present in both cases, the I(2)model may provide an suitable tool for analysing near-I(2) systemsKey words: VAR, I(2), latent trend1 IntroductionVector Autoregressions (VAR), introduced by Sims (1980), are one of the mostpowerful and popular ways of modelling the interactions between economic vari-ables. Assume we are interested in a set of I(d) variables such that ∆ dXit = εit ,i = 1, . . . , p, t = 1, . . . ,T and εit ∼ IID(0,σ2i ). When d = 1 and the variables arelinked by some long-run equilibrium (cointegration) relationship we know that theVector Error Correction Mechanism (VECM) ∆Xt = ΠXt−1 +∑kj=1Γj∆Xt− j + εtholds, where the matrix Π satisfies the reduced rank restriction Π = αβ ′ with αand β p× r matrices (Johansen, 1988). Of particular interest here is that in this casethe variables of the system are driven by a small number of latent stochastic trends.From the VECM representation is easy to derive the VMA form; setting the initialFrancesca Di IorioDipartimento di Scienze Politiche, Universita` di Napoli Federico II, e-mail: fdiiorio@unina.itStefano FachinDipartimento di Scienze Statistiche, ”La Sapienza” Universita` di Roma e-mail: ste-fano.fachin@uniroma1.itRiccardo LucchettiDipartimento di Scienze Economiche e Sociali, Universita` Politecnica delle Marche e-mail:r.lucchetti@univpm.it12 Francesca Di Iorio, Stefano Fachin, Riccardo Lucchettivalues to zero and denoting by α⊥,β⊥ the orthogonal complements of α and β ,this is Xt = β⊥(α ′⊥β⊥)−1α ′⊥∑ts=1 εs = β˜⊥(α ′⊥∑ts=1 εs)(see, e.g., Juselius, 2006).This representation shows clearly that the pushing forces of the system are the latentstochastic trends α ′⊥∑ts=1 εs, linear combinations of the cumulated shocks to eachequation of the VAR, which are loaded onto the X ′s through the loadings β˜⊥. Boththe coefficients of the linear combinations and the loadings can be recovered fromthe estimates of the VAR obtained under the reduced rank restriction.The cointegrated I(1) VAR has proved to be immensely popular and to be able toshed considerable light on the long-run dynamics of economic systems. However,some recent contributions (e.g., Johansen, Juselius Frydman and Goldberg, 2010)pointed out that in empirically important cases (for instance, when modelling ex-change rates) the I(1) model is not adequate, as the variables of interest may be I(2),or nearly so. That this could be the case is empirically suggested by estimated I(1)cointegrated VARs having unrestricted characteristic roots very close to 1, so thatvery long swings in the data are not accounted for. The statistical analysis of I(2)systems, started by Johansen (1992, 1997), is now two decades old and fairly welldeveloped. However, much still remains to be done: for instance, Johansen (2006)showed that likelihood ratio tests of many hypothesis on the long-run structure areasymptotically χ2, but nothing is known on the validity of the approximation (i) insmall samples, an empirically crucial point, and (ii), in near-I(2) systems. Our aimis thus (i) to review the I(2) model in an empirical perspective, and (ii), to report theresults of some Monte Carlo simulations on the small sample performance of testson the long-run coefficients.2 The I(2) and near-I(2) VARConsider for simplicity a VAR(2)∆ 2Xt = αβ ′Xt−1 +Γ∆Xt−1 + εt (1)where εt=(ε1t ε2t ε3)′ andΠ =αβ ′. If further α ′⊥Γβ⊥= ξη′ for some matrices ξ ,ηof dimensions (p− r)× s, s< p− r, the X ′s are I(2). The common latent stochastictrends representation is thenXt =C2∑ts=1∑si=1 εi+C1∑ti=1 εi+A (2)whereC2 = β⊥2(α ′⊥2θβ⊥2)−1α ′⊥2, β⊥2 = β⊥η⊥ and α⊥2 = α⊥ξ⊥. For convenienceof notation the deterministic and stationary parts are both included in the term A .Thus, analogously what seen above, the I(2) common trends α ′⊥2∑ts=1∑si=1 εi arelinear combinations of the twice cumulated shocks of the various equations of theVAR that load onto the variables through the loadings β⊥2(α ′⊥2θβ⊥2)−1. Paruoloand Mosconi (2011) recently showed that the following reparametrisation of (1)holds:Testing hypothesis in VARs with I(2) and near-I(2) latent stochastic trends 3∆ 2Xt = α(β ′Xt−1 +υ ′∆Xt−1)+(ξγ ′+ ςβ ′)∆Xt−1 + εt (3)where the matrices υ and γ are respectively p× r and p× s. The terms (β ′Xt−1 +υ ′∆Xt−1) and (γ ′+ β ′)∆Xt−1 are both stationary; the former is known as multi-cointegration relation, or integral control term, the latter as mediun run relation, orproportional control term. As we will see, for testing purposes it is convenient torewrite (3) as∆ 2Xt = α(β ′ : υ ′) Xt−1∆Xt−1+(ξ : ς)γ ′β ′∆Xt−1 + εt (4)= (α : ξ : ς)β ′ υ ′0 γ ′0 β ′ Xt−1∆Xt−1+ εt (5)= ηζ ′(X′t−1 : ∆X′t−1)+ εt (6)Estimation of (4) can be carried out by a two-step switching algorithm similar tothat proposed by Johansen (1997), see Paruolo and Mosconi (2011).The matrix ζ = (ζ 1 : ζ 2), which collects the cointegration parameters, is dividedin two 2r× p blocks, with the second block a linear combination of the first. Hence,any hypothesis on β and υ can be expressed as an hypothesis on ζ 1. More precisely,a convenient formulation is H0 : vec(ζ 1) = BΦ , with Φ = (φβ : φυ)′ the (mβ +mυ)×1 vector collecting the unconstrained coefficients of the both vectors and B a2pr× (mβ +mυ) block matrix given byB= Bβ 0Bβυ Bυ .Here Bβ is the pr×mβ matrix projecting the mβ unconstrained elements of β ontothe full pr vector, Bυ fulfills the same purpose for υ , and Bβυ defines possiblecross-restrictions between the two vectors. Although this formulation allows jointhypothesis on β and υ as well hypothesis on each of these two vectors, Johansen(2006) showed that in the case of hypothesis on all elements of the multicointegra-tion relation inference is not LAMN. Hence, in our experiments we will concentrateon hypothesis on β , expressed as H0 : vec(ζ 1) = BΦ with Bβυ = 0, mυ = pr andBυ = Ipr.3 Simulation experimentsOur experiments will be based on a Data Generating Process (DGP) derived by Jo-hansen (1992) from Hendry and Von Urgen-Stenberg’s (1981) famous specification4 Francesca Di Iorio, Stefano Fachin, Riccardo Lucchettiof the consumption function, namely:∆ct = ν∆yt−1 +a11(yt−1− ct−1)+a12(zt−1− lt−1)+ ε1t (7a)∆ lt = a21(yt−1− ct−1)+ ε2t (7b)∆yt = ρ∆yt + ε3t (7c)where c is consumption, l liquid assetts, and y disposable income. This DGP isparticularly convenient for our purposes, as setting ρ = 1 we have an I(2) system,whereas setting ρ = 1− ε for some arbitrarily small ε we have a near-I(2) one.Further, since r = 2 and s= 0 the proportional control term is simply β ′∆Xt−1, andα ′ =a11 a21 0a12 0 0 ,β ′ =−1 0 10 −1 1 .Defining Xt = (ct lt yt)′, εt=(ε1t ε2t ε3)′, and assuming ρ = 1 the DGP can becompactly written as in equation (4) as:∆ 2Xt = (α : ς)β ′ 0υ ′ β ′ Xt−1∆Xt−1+ εt (8)= ηζ ′(X′t−1 : ∆X′t−1)+ εt (9)To evaluate Type I errors of the likelihood ratio tests we shall letBβ =−1 00 −11 1 ,φβ =φ1φ2To evaluate the performances of the test in both correctly and misspecified VARswe considered ρ = 1,0.90, while the loadings ai j have been chosen so to ensurecontrol of the roots of the VAR(2) in levels which can be derived from (7a)-(7c).The following combinations, covering a wide range of adjustment speeds, have beenchosen:1. a11 = k0.20,a21 =−k0.20,a12 = k0.20,k = 1,2,3; in this case all loadings havethe same absolute value;2. a11 = h0.20,a21 =−k0.20,a12 = k0.20,h= 2,3,k= 1,2; in this case the loadingof (yt−1− ct−1), a11, is larger in absolute value than the other two loadings, a12and a21;The lagged effect of income growth on consumption growth, ν , is always fixed at0.5, and the noises εit , i= 1,2,3, are NID(0,1), while sample size is T = 100,200tand the number of Monte Carlo replications has been fixed to 1000. The resultsare reported in Tables 1 and 2 respectively for each case. As it can be immediatelyTesting hypothesis in VARs with I(2) and near-I(2) latent stochastic trends 5appreciated for most parameter combinations the test has some size bias. This gen-erally, but not always, falls with sample size. For the empirical point of view anincovenient feature is the presence of both positive and negative bias, which makesthe interpretation of results very tricky. In near-I(2) systems the tendency to under-reject seems to prevail, but the bias is generall small even for T = 100. Summingup, the message is that in both I(2) and near-I(2) systems asymptotic tests mightbe applied, but the interpretation of results requires some care. Ongoing research isexploring the performances delivered by the bootstrap, already successfully appliedto hypothesis testing in I(1) VARs by Fachin (2000), Omtzigt and Fachin (2006).Table 1Size of Likelihood Ratio tests on long-run coefficientsI(2) near I(2)T = 100 T = 200 T = 100 T = 200a11 α 5.0 10.0 5.0 10.0 5.0 10.0 5.0 10.00.20 6.5 9.1 2.6 4.1 11.4 14.1 6.8 8.60.40 2.9 4.6 0.6 2.6 6.1 8.3 2.5 3.90.60 14.5 15.6 10.5 11.6 6.8 8.8 3.9 5.20.80 19.0 20.2 12.9 14.3 8.3 9.7 5.3 6.5median 10.5 12.4 6.6 7.9 7.6 9.3 4.6 5.9mean 10.7 12.4 6.7 8.2 8.2 10.2 4.6 6.1a21 =−a11,a12 = a11; I(2): ρ = 1; near-I(2): ρ = 0.90.6 Francesca Di Iorio, Stefano Fachin, Riccardo LucchettiTable BSize of Likelihood Ratio tests on long-run coefficientsI(2) near-I(2)T = 100 T = 200 T = 100 T = 200a11 a21 a12 α 5.0 10.0 5.0 10.0 5.0 10.0 5.0 10.00.40 -0.20 0.20 7.2 10.3 4.1 5.9 8.8 12.5 7.1 8.70.60 11.1 15.6 8.8 10.8 7.8 11.5 6.8 9.00.80 15.2 20.1 12.7 14.4 9.8 13.8 8.0 9.40.60 -0.40 0.40 12.6 14.5 9.2 10.2 6.6 8.0 4.5 6.60.80 14.6 16.6 8.3 9.6 12.1 14.0 12.1 13.30.80 -0.60 0.60 11.3 12.9 9.8 10.9 5.5 7.2 2.6 4.1median 12.0 15.6 9.5 10.6 8.3 10.9 6.3 8.0mean 13.4 16.0 10.0 11.3 8.5 10.7 6.8 8.4References1. Fachin, S.: Bootstrap and asymptotic tests of long-run relationships in cointegrated systems.Ox. Bull. Econ. and Stat., 62, 577-585 (2000).2. Johansen, S.: Statistical analysis of cointegration vectors. J Econ Dyn Control. 12, 231–254(1988).3. Johansen, S.: A representation of vector autoregressive processes integrated of order 2.Econom. Th. 8, 188–202 (1992).4. Johansen, S.: Likelihood analysis of the I(2) model. Scand J Stat. 24, 433–462 (1997).5. Johansen, S.: Statistical analysis of hypotheses on the cointegrating relations in the I(2) modelJ Econometrics. 132, 81–115 (2006).6. Johansen, S., Juselius, K., Frydman, R. , Goldberg, M.: Testing hypotheses in an I(2) modelwith piecewise linear trends. An analysis of the persistent long swings in the Dmk/$ rate. JEconometrics. 158, 117–129 (2010)7. Mosconi, R., Paruolo, P., : Identification of cointegrating relations in I(2) vector autoregressivemodels, WP, (2011).8. Omtzigt, P., Fachin, S.: The Size and Power of Bootstrap and Bartlett-Corrected Tests of Hy-potheses on the Cointegrating Vectors. Econometric Rev., 25, 41-60 (2006).9. Juselius, K.: The Cointegrated VAR Model: Methodology and Applications. Oxford Univer-sity Press, Oxford (2006).10. Paruolo, P., Rahbek, A.: Weak exogeneity in I(2) VAR systems, J Econometrics. 93, 281–308(1999).11. Mosconi, R., Paruolo, P., : Identification of cointegrating relations in I(2) vector autoregressivemodels, WP, (2011).12. Sims, C.A.: Macroeconomics and Reality, Econometrica, 48, 1–48 (1980).

Testing hypothesis in VARs with I(2) and near-I(2) latent stochastic trends

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