Econometrics 2 - Lecture 6 Multivariate Time Series Models (part 2) Contents nNon-stationary Time Series nCointegration nVAR Models nVAR Models and Cointegration nVEC Model: Cointegration Tests nVEC Model: Specification and Estimation n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 2 Hackl, Econometrics 2, Lecture 6 3 Regression and Time Series nStationary variables are a crucial prerequisite for qestimation methods qtesting procedures n applied to regression models nSpecifying a relation between non-stationary variables may result in a nonsense or spurious regression May 6, 2011 Hackl, Econometrics 2, Lecture 6 4 Spurious Regression: Example nGeneration of Yt by n Yt = Yt-1 + εt n i.e., Yt is a random walk, Yt ~ I(1); similarly Xt ~ I(1) nModel to be estimated: n Yt = α + βXt + εt n it follows (in general) that εt ~ I(1), i.e., the error terms are non- stationary n(Asymptotic) distributions of t- and F -statistics are different from those for stationarity nR2 indicates explanatory potential nDW statistic converges for growing N to zero n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 5 Avoiding Spurious Regression nIdentification of non-stationarity: unit-root tests nModels for non-stationary variables qElimination of stochastic trends: differencing, specifying the model for differences qInclusion of lagged variables may result in stationary error terms qExplained and explanatory variables may have a common stochastic trend, are cointegrated: equilibrium relation, error-correction models nExample: ADL(1,1) model with Yt ~ I(1), Xt ~ I(1) n Yt = δ + θYt-1 + φ0Xt + φ1Xt-1 + εt qThe error terms are stationary if θ =1, φ0 = φ1 = 0 q εt = Yt – (δ + θYt-1 + φ0Xt + φ1Xt-1) ~ I(0) qCommon trend implies an equilibrium relation, i.e.,Yt-1 – βXt-1 ~ I(0); the ADL(1,1) model has an error-correction form q ΔYt = φ0ΔXt – (1 – θ)(Yt-1 – α – βXt-1) + εt May 6, 2011 Contents nNon-stationary Time Series nCointegration nVAR Models nVAR Models and Cointegration nVEC Model: Cointegration Tests nVEC Model: Specification and Estimation n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 6 Hackl, Econometrics 2, Lecture 6 7 Cointegration nNon-stationary variables X, Y: n Xt ~ I(1), Yt ~ I(1) n if a β exists such that n Zt = Yt - βXt ~ I(0) nXt and Yt have a common stochastic trend nXt and Yt are called “cointegrated” nβ: cointegration parameter n(1, - β)’: cointegration vector nCointegration implies a long-run equilibrium; cf. Granger’s representation theorem n May 6, 2011 Hackl, Econometrics 2, Lecture 6 8 Error-correction Model nGranger’s Representation Theorem (Engle & Granger, 1987): If a set of variables is cointegrated, then an error-correction (EC) relation of the variables exists n non-stationary processes Yt ~ I(1), Xt ~ I(1) with cointegrating vector (1, -β)’: error-correction representation n θ(L)ΔYt = δ + Φ(L)ΔXt-1 - γ(Yt-1 – βXt-1) + α(L)εt n with white noise εt, lag polynomials θ(L) (with θ0=1), Φ(L), and α(L) nError-correction model: describes qthe short-run behavior qconsistently with the long-run equilibrium nConverse statement: if Yt ~ I(1), Xt ~ I(1) have an error-correction representation, then they are cointegrated May 6, 2011 Hackl, Econometrics 2, Lecture 6 9 An Example nThe model n ΔYt = δ + φ1ΔXt-1 - γ(Yt-1 – βXt-1) + εt n is a special case of n θ(L)ΔYt = δ + Φ(L)ΔXt-1 - γ(Yt-1 – βXt-1) + α(L)εt n with θ(L) = 1, Φ(L) = φ1L, and α(L) = 1 nNo change steady state equilibrium for ΔYt = ΔXt-1 = 0: n Yt – βXt = δ/γ or Yt = α + βXt if α = δ/γ n the EC model can be written as n ΔYt = φ1ΔXt-1 – γ(Yt-1 – α – βXt-1) + εt nIf α = δ/γ + λ, λ ≠ 0: n ΔYt = λ + φ1ΔXt-1 – γ(Yt-1 – α – βXt-1) + εt n deterministic trends for Yt and Xt, long run equilibrium corresponding to growth paths ΔYt = ΔXt-1 = λ/(1 - φ1) May 6, 2011 Hackl, Econometrics 2, Lecture 6 10 EC Model: Estimation nModel n ΔYt = δ + φ1ΔXt-1 - γ(Yt-1 – βXt-1) + εt (A) n with cointegrating relation n Yt-1 = βXt-1 + ut (B) nCointegration vector (1, - β)’ known: OLS estimation of δ, φ1, and γ from (A), standard properties nUnknown cointegration vector: qParameter β from (B) superconsistently estimated by OLS qOLS estimation of δ, φ1, and γ from (A) is not affected by using the estimate for β nModel specification nChoice of orders of lag polynomials nTheory is symmetric in treating Xt andYt May 6, 2011 Hackl, Econometrics 2, Lecture 6 11 Example: Purchasing Power Parity nVerbeek’s dataset ppp: price indices and exchange rates for France and Italy, T = 186 (1/1981-6/1996) nVariables: LNIT (log price index Italy), LNFR (log price index France), LNX (log exchange rate France/Italy) nPurchasing power parity (PPP): exchange rate between the currencies (Franc, Lira) equals the ratio of price levels of the countries nRelative PPP: equality fulfilled only in the long run; equilibrium or cointegrating relation n LNXt = α + β LNPt + εt n with LNPt = LNITt – LNFRt, i.e., the log of the price index ratio France/Italy May 6, 2011 Hackl, Econometrics 2, Lecture 6 12 Purchasing Power Parity nTest for unit roots (non- n stationarity) of nLNX (log exchange rate n France/Italy) nLNP = LNIT – LNFR, i.e., n the log of the price n index ratio France/Italy nResults from DF tests: n May 6, 2011 const. +trend LNP DF stat -0.99 -2.96 p-value 0.76 0.14 LNX DF stat -0.33 -1.90 p-value 0.92 0.65 DF test indicates: LNX ~ I(1), LNP ~ I(1) PPP: Equilibrium Relations nAs discussed by Verbeek: 1.If PPP holds in long run, real exchange rate is stationary n LNXt – (LNITt – LNFRt) = εt 2.Change of relative prices correspond to the change of exchange rate, i.e., short run deviations are stationary n LNXt – β (LNITt – LNFRt) = εt 3.Generalization of case 2: n LNXt = α + β1 LNITt – β2 LNFRt + εt n with εt ~ I(0) May 6, 2011 Hackl, Econometrics 2, Lecture 6 13 Equilibrium Relation 2 nOLS estimation of n LNXt = α + β LNPt + εt n n n n n n n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 14 Model 2: OLS, using observations 1981:01-1996:06 (T = 186) Dependent variable: LNX coefficient std. error t-ratio p-value --------------------------------------------------------- const 5,48720 0,00677678 809,7 0,0000 *** LNP 0,982213 0,0513277 19,14 1,24e-045 *** Mean dependent var 5,439818 S.D. dependent var 0,148368 Sum squared resid 1,361936 S.E. of regression 0,086034 R-squared 0,665570 Adjusted R-squared 0,663753 F(1, 184) 366,1905 P-value(F) 1,24e-45 Log-likelihood 193,3435 Akaike criterion -382,6870 Schwarz criterion -376,2355 Hannan-Quinn -380,0726 rho 0,967239 Durbin-Watson 0,055469 Equilibrium Relation 2 nResiduals = LNXt – (a + b LNPt) with OLS estimates a, b n n n n n n n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 15 Hackl, Econometrics 2, Lecture 6 16 PPP: Tests for Cointegration nResiduals from LNXt = α + β LNPt + εt: nTime series plot indicates non-stationary residuals nTests for cointegration, H0: residuals have unit root, no cointegration qDF test statistic (with constant): -1.90, 5% critical value: -3.37 qCRDW test: DW statistic: 0.055 < 0.20, the 5% critical value for two variables, 200 observations nBoth tests suggest: H0 cannot be rejected, no evidence for cointegration nSame result for equilibrium relations 1 and 3; reasons could be: nTime series too short nNo PPP between France and Italy nAttention: equilibrium relation 3 has three variables; two cointegration relations are possible May 6, 2011 Contents nNon-stationary Time Series nCointegration nVAR Models nVAR Models and Cointegration nVEC Model: Cointegration Tests nVEC Model: Specification and Estimation n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 17 Hackl, Econometrics 2, Lecture 6 18 Example: Income and Consumption nModel for income (Y) and consumption (C) n Yt = δ1 + θ11Yt-1 + θ12Ct-1 + ε1t n Ct = δ2 + θ21Ct-1 + θ22Yt-1 + ε2t n with (possibly correlated) white noises ε1t and ε1t nNotation: Zt = (Yt, Ct)‘, 2-vectors δ and ε, and (2x2)-matrix Θ = (θij), the model is n n n n in matrix notation n Zt = δ + ΘZt-1 + εt nRepresents each component of Z as a linear combination of lagged variables nExtension of the AR-model to the 2-vector Zt: vector autoregressive model of order 1, VAR(1) model May 6, 2011 Hackl, Econometrics 2, Lecture 6 19 The VAR(p) Model nVAR(p) model: generalization of the AR(p) model for k-vectors Yt n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt n with k-vectors Yt, δ, and εt and kxk-matrices Θ1, …, Θp nUsing the lag-operator L: n Θ(L)Yt = δ + εt n with matrix lag polynomial Θ(L) = I – Θ1L - … - ΘpLp qΘ(L) is a kxk-matrix qEach matrix element of Θ(L) is a lag polynomial of order p nError terms εt qhave covariance matrix Σ; allows for contemporaneous correlation qare independent of Yt-j, j > 0, i.e., of the past of the components of Yt May 6, 2011 Hackl, Econometrics 2, Lecture 6 20 The VAR(p) Model, cont’d nVAR(p) model for the k-vector Yt n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt nVector of expectations of Yt: assuming stationarity n E{Yt} = δ + Θ1 E{Yt} + … + Θp E{Yt} n gives n E{Yt} = μ = (Ik – Θ1 - … - Θp)-1δ = Θ(1)-1δ n i.e., stationarity requires that the kxk-matrix Θ(1) is invertible nIn deviations yt = Yt – μ, the VAR(p) model is n Θ(L)yt = εt nMA representation of the VAR(p) model, given that Θ(L) is invertible n Yt = μ + Θ(L)-1εt = μ + εt + A1εt-1 + A2εt-2 + … nVARMA(p,q) Model: Extension of the VAR(p) model by multiplying εt (from the left) with a matrix lag polynomial of order q n May 6, 2011 Hackl, Econometrics 2, Lecture 6 21 Reasons for Using a VAR Model nVAR model represents a set of univariate ARMA models, one for each component nReformulation of simultaneous equation models as dynamic models nTo be used instead of simultaneous equation models: qNo need to distinct a priori endogenous and exogenous variables qNo need for a priori identifying restrictions on model parameters nSimultaneous analysis of the components: More parsimonious, fewer lags, simultaneous consideration of the history of all included variables nAllows for non-stationarity and cointegration nAttention: the number of parameters to be estimated increases with k and p n Number of parameters n of Θ(L) May 6, 2011 p 1 2 3 k=2 4 8 12 k=4 16 32 48 Hackl, Econometrics 2, Lecture 6 22 Simultaneous Equation Models in VAR Form nModel with m endogenous variables (and equations), K regressors n Ayt = Γzt + εt = Γ1 yt-1 + Γ2 xt + εt n with m-vectors yt and εt, K-vector zt, (mxm)-matrix A, (mxK)-matrix Γ, and (mxm)-matrix Σ = V{εt}; nzt contains lagged endogenous variables yt-1 and exogenous variables xt nRearranging gives n yt = Θ yt-1 + δt + vt n with Θ = A-1 Γ1, δt = A-1 Γ2 xt, and vt = A-1 εt nExtension of yt by regressors xt: the matrix δt becomes a vector of intercepts May 6, 2011 Hackl, Econometrics 2, Lecture 6 23 Example: Income and Consumption nModel for income (Yt) and consumption (Ct) n Yt = δ1 + θ11Yt-1 + θ12Ct-1 + ε1t n Ct = δ2 + θ21Ct-1 + θ22Yt-1 + ε2t n with (possibly correlated) white noises ε1t and ε1t nMatrix form of the simultaneous equation model: n A (Yt, Ct)‘ = Γ (1, Yt-1, Ct-1)‘ + (ε1t, ε2t)’ n with n n n nVAR(1) form: Zt = δ + ΘZt-1 + εt or n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 24 VAR Model: Estimation nVAR(p) model for the k-vector Yt n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt, V{εt} = Σ nEach component of Yt: a linear combination of lagged variables nError terms: Possibly contemporaneously correlated, covariance matrix Σ, uncorrelated over time nSUR model nEstimation, given the order p of the VAR model nOLS estimates of parameters in Θ(L) are consistent nEstimation of Σ based on residual vectors et = (e1t, …, ekt)’: n n nGLS estimator coincides with OLS estimator: same explanatory variables for all equations May 6, 2011 Hackl, Econometrics 2, Lecture 6 25 VAR Model: Estimation, cont’d nChoice of the order p of the VAR model nEstimation of VAR models for various orders p nChoice of p based on Akaike or Schwarz information criterion May 6, 2011 Hackl, Econometrics 2, Lecture 6 26 Income and Consumption nAWM data base, 1970:1-2003:4: PCR (real private consumption), PYR (real disposable income of households); respective annual growth rates: C, Y nFitting Zt = δ + ΘZt-1 + εt with Z = (Y, C)‘ gives n n n n n n n n n with AIC = -14.40; for the VAR(2) model: AIC = -14.35 nIn GRETL: OLS equation-wise, VAR estimation, SUR estimation give very similar results May 6, 2011 δ Y-1 C-1 adj.R2 Y θij 0.001 0.825 0.082 0.80 t(θij) 0.91 12.09 1.07 C Θij 0.003 0.061 0.826 0.78 t(θij) 2.36 0.97 11.69 Hackl, Econometrics 2, Lecture 6 27 Impulse-response Function nMA representation of the VAR(p) model n Yt = Θ(1)-1δ + εt + A1εt-1 + A2εt-2 + … nInterpretation of As: the (i,j)-element of As represents the effect of a one unit increase of εjt upon the i-th variable Yi,t+s in Yt+s nDynamic effects of a one unit increase of εjt upon the i-th component of Yt are corresponding to the (i,j)-th elements of Ik, A1, A2, … nThe plot of these elements over s represents the impulse-response function of the i-th variable in Yt+s on a unit shock to εjt May 6, 2011 Hackl, Econometrics 2, Lecture 6 28 Stationarity nAR(1) process Yt = θYt-1 + εt nis stationary, if the root z of the characteristic polynomial n Θ(z) = 1 - θz = 0 n fulfills |z| > 1, i.e., |θ| < 1; qΘ(z) is invertible, i.e., Θ(z)-1 can derived such that Θ(z)-1Θ(z) = 1 qYt can be represented by a MA(∞) process: Yt = Θ(z)-1εt nis non-stationary, if n z = 1 or θ = 1 n i.e.,Yt ~ I(1), Yt has a stochastic trend May 6, 2011 Hackl, Econometrics 2, Lecture 6 29 VAR Models, Stationarity, and Cointegration nVAR(1) model for the k-vector Yt n Yt = δ + Θ1Yt-1 + εt nIf Θ(L) is invertible, n Yt = Θ(1)-1δ + Θ(L)-1εt = μ + εt + A1εt-1 + A2εt-2 + … n i.e., each variable in Yt is a linear combination of white noises, is a stationary I(0) variable nIf Θ(L) is not invertible, not all variables in Yt can be stationary I(0) variables: at least one variable must have a stochastic trend qIf all k variables have independent stochastic trends, all k variables are I(1) and no cointegrating relation exists; e.g., for k = 2: q q q i.e., θ11 = θ22 = 1, θ12 = θ21 = 0 qThe more interesting case: at least one cointegrating relation; number of cointegrating relations equals the rank r{Θ(1)} of matrix Θ(1) May 6, 2011 Hackl, Econometrics 2, Lecture 6 30 Example: A VAR(1) Model nVAR(1) model for k-vector Y in differences with Θ(L) = I - Θ1L n ΔYt = - Θ(1)Yt-1 + δ + εt n r = r{Θ(1)}: rank of (kxk) matrix Θ(1) = Ik - Θ1 1.r = 0: then ΔYt = δ + εt, i.e., Y is a k-dimensional random walk, each component is I(1), no cointegrating relationship 2.r < k: (k - r)-fold unit root, (kxr)-matrices γ and β can be found, both of rank r, with Θ(1) = γβ' the r columns of β are the cointegrating vectors of r cointegrating relations (β in normalized form, i.e., the main diagonal elements of β being ones) 3.r = k: VAR(1) process is stationary, all components of Y are I(0) n May 6, 2011 Hackl, Econometrics 2, Lecture 6 31 Cointegration Space nGiven a set of k variables, the components of the k-vector Yt ~ I(1) nCointegration space: nAmong the k variables, r ≤ k-1 independent linear relations βj‘Yt, j = 1, …, r, are possible so that βj‘Yt ~ I(0) nIndividual relations can be combined with others and these are again I(0), i.e., not the individual cointegrating relations are identified but only the r-dimensional space nCointegrating relations should have an economic interpretation nCointegration matrix β: nThe kxr matrix β = (β1, …, βr) of vectors βj that state the cointegrating relations βj‘Yt ~ I(0), j = 1, …, r nCointegrating rank: the rank of matrix β: r{β} = r May 6, 2011 Contents nNon-stationary Time Series nCointegration nVAR Models nVAR Models and Cointegration nVEC Model: Cointegration Tests nVEC Model: Specification and Estimation n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 32 Hackl, Econometrics 2, Lecture 6 33 Granger‘s Representation Theorem nGranger’s Representation Theorem (Engle & Granger, 1987): If a set of I(1) variables is cointegrated, then an error-correction (EC) relation of the variables exists nExtends to VAR models: if the I(1) variables of the k-vector Yt are cointegrated, then an error-correction (EC) relation of the variables exists May 6, 2011 Hackl, Econometrics 2, Lecture 6 34 Granger‘s Representation Theorem for VAR Models nVAR(p) model for the k-vector Yt n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt n transformed into n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + ΠYt-1 + εt (A) qΠ = – Θ(1) = – (Ik – Θ1 – … – Θp): „long-run matrix“, determines the long-run dynamics of Yt qΓ1, …, Γp-1 matrices which are functions of Θ1,…, Θp nΠYt-1 is stationary: ΔYt and εt are I(0) nThree cases 1.r{Π} = r with 0 < r < k: there exist r stationary linear combinations of Yt, i.e., r cointegrating relations 2.r{Π} = 0: then Π = 0, equation (A) is a VAR(p) model for stationary variables ΔYt 3.r{Π} = k: all variables in Yt are stationary, Π = - Θ(1) is invertible n May 6, 2011 Hackl, Econometrics 2, Lecture 6 35 Vector Error-Correction Model nVAR(p) model for the k-vector Yt n Yt = δ + Θ1Yt-1 + … + ΘpYt-p + εt n transformed into n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + ΠYt-1 + εt n with r{Π} = r and Π = γβ' gives n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt (B) nr cointegrating relations β'Yt-1 nAdaptation parameters γ measure the portion or speed of adaptation of Yt in compensation of the equilibrium error Zt-1 = β'Yt-1 nEquation (B) is called the vector error-correction (VEC) model n May 6, 2011 Hackl, Econometrics 2, Lecture 6 36 Example: Bivariate VAR Model nVAR(1) model for the 2-vector Yt = (Y1t, Y2t)’ n Yt = ΘYt-1 + εt nLong-run matrix n n n nΠ = 0, if θ11 = θ22 = 1, θ12 = θ21 = 0, i.e., Y1t, Y2t are random walks nr{Π} < 2, if (θ11 – 1)(θ22 – 1) – θ12 θ21 = 0; cointegrating vector: β‘ = (θ11 – 1, θ12), long-run matrix n n n nThe error-correction form is May 6, 2011 Hackl, Econometrics 2, Lecture 6 37 Deterministic Component nVEC(p) model for the k-vector Yt n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt (B) nThe deterministic component (intercept) δ: 1.E{ΔYt} = 0, i.e., no deterministic trend in any component of Yt: given that Γ = Ik – Γ1 – … – Γp-1 has full rank: qΓ E{ΔYt} = δ + γE{Zt-1} = 0 with equilibrium error Zt-1 = β'Yt-1 qE{Zt-1} corresponds to the intercepts of the cointegrating relations; with r-dimensional vector E{Zt-1} = α q ΔYt = Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γ(- α + β'Yt-1) + εt (C) qIntercepts only in the cointegrating relations, i.e., no deterministic trend in the model May 6, 2011 Hackl, Econometrics 2, Lecture 6 38 Deterministic Component, cont’d nVEC(p) model for the k-vector Yt n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γβ'Yt-1 + εt (B) nThe deterministic component (intercept) δ: 2.Addition of a k-vector λ with identical components to (C) n ΔYt = λ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + γ(- α + β'Yt-1) + εt qLong-run equilibrium: steady state growth with growth rate E{ΔYt} = Γ-1λ qDeterministic trends cancel out in the long run, so that no deterministic trend in the error-correction term; cf. (B) qAddition of k-vector λ can be repeated: up to k-r separate deterministic trends can cancel out in the error-correction term qThe general notation is equation (B) with δ containing r intercepts of the long-run relations and k-r deterministic trends in the variables of Yt May 6, 2011 Hackl, Econometrics 2, Lecture 6 39 The Five Cases nBased on empirical observation and economic reasoning, choice between: 1)Unrestricted constant: variables show deterministic linear trends 2)Restricted constant: variables not trended but mean distance between them not zero; intercept in the error-correction term 3)No constant n Generalization: deterministic component contains intercept and trend 4)Constant + restricted trend: cointegrating relationships include a trend but the first differences of the variables in question do not 5)Constant + unrestricted trend: trend in both the cointegration relationships and the first differences, corresponding to a quadratic trend in the variables (in levels) n n May 6, 2011 Contents nNon-stationary Time Series nCointegration nVAR Models nVAR Models and Cointegration nVEC Model: Cointegration Tests nVEC Model: Specification and Estimation n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 40 Hackl, Econometrics 2, Lecture 6 41 Choice of the Cointegrating Rank nBased on k-vector Yt ~ I(1) nEstimation procedure needs as input the cointegrating rank r nEngle-Granger procedure nJohansen‘s R3 method n May 6, 2011 Hackl, Econometrics 2, Lecture 6 42 Engle-Granger Approach nNon-stationary processes Yt ~ I(1), Xt ~ I(1) ; to be estimated: n Yt = α + βXt + εt nStep 1: OLS-fitting nTests for cointegration based on residuals, e.g., DF test with special critical values; H0: no cointegration nIf H0 is rejected, qOLS fitting in step 1 gives consistent estimates of the cointegrating vector qStep 2: OLS estimation of the EC model based on the cointegrating vector from step 1 nCan be extended to k-vector Yt, given that at most one cointegrating relation exists n May 6, 2011 Hackl, Econometrics 2, Lecture 6 43 Engle-Granger Cointegration Test: Problems nResidual based cointegration tests can be misleading nTest results depend on specification qWhich variables are included qNormalization of the cointegrating vector nTest may be inappropriate due to wrong specification of cointegrating relation nTest power suffers from inefficient use of information (dynamic interactions not taken into account) n May 6, 2011 Hackl, Econometrics 2, Lecture 6 44 Johansen‘s R3 Method nReduced rank regression or R3 method: an iterative method for specifying the cointegrating rank r nAlso called Johansen's test nThe test is based on the k eigenvalues λi (λ1> λ2>…> λk) of n Y1‘Y1 – Y1ΔY(ΔY‘ΔY)-1ΔY‘Y1, n with ΔY: (Txk) matrix of differences ΔYt, Y1: (Txk) matrix of Yt-1 qeigenvalues λi fulfill 0 ≤ λi < 1 qif r{Θ(1)} = r, the k-r smallest eigenvalues obey q log(1- λj) = λj = 0, j = r+1, …, k nIterative test procedures qTrace test qMaximum eigenvalue test or max test May 6, 2011 Hackl, Econometrics 2, Lecture 6 45 Trace and Max Test: The Procedures nLR tests, based on the assumption of normally distributed errors nTrace test: for r0 = 0, 1, …, test of H0: r ≤ r0 (r0 or fewer cointegrating relations) against H1: r0 < r ≤ k n λtrace(r0) = - T Σkj=r0+1log(1- Îj) qÎj: estimator of λj qH0 is rejected for large values of λtrace(r0) qStops when H0 is not rejected for the first time qCritical values from simulations nMax test: tests for r0 = 0, 1, …: H0: r = r0 (the eigenvalue λr0+1 is different from zero) against H1: r = r0+1 n λmax(r0) = - T log(1 - Îr0+1) qStops when H0 is not rejected for the first time qCritical values from simulations May 6, 2011 Hackl, Econometrics 2, Lecture 6 46 Trace and Max Test: Critical Limits nCritical limits are shown in Verbeek’s Table 9.9 for both tests nDepend on presence of trends and intercepts qCase 1: no deterministic trends, intercepts in cointegrating relations qCase 2: k unrestricted intercepts in the VAR model, i.e., k - r deterministic trends, r intercepts in cointegrating relations nDepend on k – r nNeed small sample correction, e.g., factor (T-pk)/T for the test statistic: avoids too large values of r n May 6, 2011 Hackl, Econometrics 2, Lecture 6 47 Example: Purchasing Power Parity nVerbeek’s dataset ppp: price indices and exchange rates for France and Italy, T = 186 (1/1981-6/1996) nVariables: LNIT (log price index Italy), LNFR (log price index France), LNX (log exchange rate France/Italy) nPurchasing power parity (PPP): exchange rate between the currencies (Franc, Lira) equals the ratio of price levels of the countries nRelative PPP: equality fulfilled only in the long run; equilibrium or cointegrating relation n LNXt = α + β LNPt + εt n with LNPt = LNITt – LNFRt, i.e., the log of the price index ratio France/Italy nGeneralization: n LNXt = α + β1 LNITt – β2 LNFRt + εt May 6, 2011 PPP: Cointegrating Rank r nAs discussed by Verbeek: Johansen test for k = 3 variables, maximal lag order p = 3 n n n n n nH0 not rejected that smallest eigenvalue equals zero: series are non-stationary nBoth the trace and the max test suggest r = 2 May 6, 2011 Hackl, Econometrics 2, Lecture 6 48 r0 eigen-value λtr(r0) p-value λmax(r0) p-value 0 0.301 93.9 0.0000 65.5 0.0000 1 0.113 28.4 0.0023 22.0 0.0035 2 0.034 6.37 0.169 6.4 0.1690 Contents nNon-stationary Time Series nCointegration nVAR Models nVAR Models and Cointegration nVEC Model: Cointegration Tests nVEC Model: Specification and Estimation n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 49 Hackl, Econometrics 2, Lecture 6 50 Estimation of VEC Models nEstimation of n ΔYt = δ + Γ1ΔYt-1 + … + Γp-1ΔYt-p+1 + ΠYt-1 + εt n requires finding (kxr)-matrices α and β with Π = αβ‘ qβ: matrix of cointegrating vectors qα: matrix of adjustment coefficients nIdentification problem: linear combinations of cointegrating vectors are also cointegrating vectors nUnique solutions for α and β require restrictions nMinimum number of restrictions which guarantee identification is r2 nNormalization qPhillips normalization qManual normalization May 6, 2011 Hackl, Econometrics 2, Lecture 6 51 Phillips Normalization nCointegrating vector n β’ = (β1’, β2’) n β1: (rxr)-matrix with rank r, β2: [(k-r)xr]-matrix nNormalization consists in transforming β into n n n with matrix B of unrestricted coefficients nThe r cointegrating relations express the first r variables as functions of the remaining k - r variables nFulfills the condition that at least r2 restrictions are needed to guarantee identification nResulting equilibrium relations may be difficult to interpret nAlternative: manual normalization May 6, 2011 Hackl, Econometrics 2, Lecture 6 52 Example: Money Demand nVerbeek’s data set “money”: US data 1:54 – 12:1994 (T=164) nm: log of real M1 money stock ninfl: quaterly inflation rate (change in log prices, % per year) ncpr: commercial paper rate (% per year) ny: log real GDP (billions of 1987 dollars) ntbr: treasury bill rate n May 6, 2011 Hackl, Econometrics 2, Lecture 6 53 Money Demand: Cointegrating Vectors nML estimates, lag order p = 6, cointegration rank r = 2, restricted constant nCointegrating vectors β1 and β2 and standard errors (s.e.), Phillips normalization n n n May 6, 2011 m infl cpr y tbr const β1 1.00 0.00 0.61 -0.35 -0.60 -4.27 (s.e.) (0.00) (0.00) (0.12) (0.12) (0.12) (0.91) β2 0.00 1.00 -26.95 -3.28 -27.44 39.25 (s.e.) (0.00) (0.00) (4.66) (4.61) (4.80) (35.5) Hackl, Econometrics 2, Lecture 6 54 Estimation of VEC Models: k=2 nEstimation procedure consists of the following steps 1.Test the variables in the 2-vector Yt for stationarity using the usual ADF tests; VEC models need I(1) variables 2.Determine the order p 3.Specification of qdeterministic trends of the variables in Yt qintercept in the cointegrating relation 4.Cointegration test 5.Estimation of cointegrating relation, normalization 6.Estimation of the VEC model May 6, 2011 Hackl, Econometrics 2, Lecture 6 55 Example: Income and Consumption nModel: n Yt = δ1 + θ11Yt-1 + θ12Ct-1 + ε1t n Ct = δ2 + θ21Ct-1 + θ22Yt-1 + ε2t nWith Z = (Y, C)‘, 2-vectors δ and ε, and (2x2)-matrix Θ, the VAR(1) model is n Zt = δ + ΘZt-1 + εt nRepresents each component of Z as a linear combination of lagged variables May 6, 2011 Hackl, Econometrics 2, Lecture 6 56 Income and Consumption: VEC(1) Model nAWM data base: PCR (real private consumption), PYR (real disposable income of households); logarithms: C, Y 1.Check whether C and Y are non-stationary: n C ~ I(1), Y ~ I(1) 2.Johansen test for cointegration: given that C and Y have no trends and the cointegrating relationship has an intercept: n r = 1 (p < 0.05) n the cointegrating relationship is n C = 8.55 – 1.61Y n with t(Y) = 18.2 n n n n n May 6, 2011 Hackl, Econometrics 2, Lecture 6 57 Income and Consumption: VEC(1) Model, cont’d 3.VEC(1) model (same specification as in 2.) with Z = (Y, C)’ n DZt = - γ(β‘Zt-1 + δ) + ΓDZt-1 + εt n n n n n n n nThe model explains growth rates of PCR and PYR; AIC = -15.41 is smaller than that of the VAR(1)-Modell (AIC = -14.45) May 6, 2011 coint DY-1 DC-1 adj.R2 AIC DY γij 0.029 0.167 0.059 0.14 -7.42 t(γij) 5.02 1.59 0.49 DC γij 0.047 0.226 -0.148 0.18 -7.59 t(γij) 2.36 2.34 1.35 Hackl, Econometrics 2, Lecture 6 58 Estimation of VEC Models nEstimation procedure consists of the following steps 1.Test of the k variables in Yt for stationarity: ADF test 2.Determination of the number p of lags in the cointegration test (order of VAR): AIC or BIC 3.Specification of qdeterministic trends of the variables in Yt qintercept in the cointegrating relations 4.Determination of the number r of cointegrating relations: trace and/or max test 5.Estimation of the coefficients β of the cointegrating relations and the adjustment α coefficients; normalization; assessment of the cointegrating relations 6.Estimation of the VEC model May 6, 2011 Hackl, Econometrics 2, Lecture 6 59 VEC Models in GRETL nModel > Time Series > VAR lag selection… nCalculates information criteria like AIC and BIC from VARs of order 1 to the chosen maximum order of the VAR nModel > Time Series > Cointegration test > Johansen… nCalculates eigenvalues, test statistics for the trace and max tests, and estimates of the matrices α, β, and Π = αβ‘ nModel > Time Series > VECM nEstimates the specified VEC model for a given cointegration rank: (1) cointegrating vectors and standard errors, (2) adjustment vectors, (3) coefficients and various criteria for each of the equations of the VEC model n May 6, 2011 Hackl, Econometrics 2, Lecture 6 60 Your Homework 1.Perform the steps 1 – 6 for estimating a VEC model for Verbeek’s dataset “model”; choose p = 2 and r = 2 for estimating the VEC model. Explain the steps and interpret the results of each step. 2.Derive the VEC form of the VAR(3) model n Yt = δ + Θ1Yt-1 + … + Θ3Yt-3 + εt n assuming a k-vector Yt and appropriate orders of the other vectors and matrices. n n May 6, 2011