Econometrics 2 - Lecture 4 Lag Structures, Cointegration Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 2 The Lüdeke Model April 15, 2016 Hackl, Econometrics 2, Lecture 4 3 1.Consumption function Ct = α1 + α2Yt + α3Ct-1 + ε1t 2.Investment function It = β1 + β2Yt + β3Pt-1 + ε2t 3.Import function Mt = γ1 + γ2Yt + γ3 Mt-1 + ε3t 4.Identity relation Yt = Ct + It - Mt-1 + Gt with C: private consumption, Y: GDP, I: investments, P: profits, M: imports, G: governmental spending Variables: nEndogenous: C, Y, I, M nExogenous, predetermined: G, P-1, C-1, M-1 n n Econometric Models April 15, 2016 Hackl, Econometrics 2, Lecture 4 4 Basis is the multiple linear regression model Model extensions nDynamic models, i.e., contain lagged variables nSystems of regression relations, i.e., models describe more than one dependent variable Example: Lüdeke Model nfour dynamic equations (with lagged variables P-1, C-1, M-1) nfor the four dependent variables C, Y, I, M Dynamic Models: Examples April 15, 2016 Hackl, Econometrics 2, Lecture 4 5 Demand model: describes the quantity Q demanded of a product as a function of its price P and the income Y of households Demand is determined by nCurrent price and current income (static model): Qt = β1 + β2Pt + β3Yt + εt nCurrent price and income of the previous period (dynamic model): Qt = β1 + β2Pt + β3Yt-1 + εt nCurrent price and demand of the previous period (dynamic autoregressive model): Qt = β1 + β2Pt + β3Qt-1 + εt n The Dynamic of Processes April 15, 2016 Hackl, Econometrics 2, Lecture 4 6 Static processes: immediate reaction to changes in regressors, the adjustment of the dependent variable to the realizations of the independent variables will be completed within the current period, the process seems to be always in equilibrium Static models are often inappropriate nSome processes are determined by the past, e.g., energy consumption depends on past investments into energy-consuming systems and equipment nActors in economic processes may respond delayed, e.g., time for decision-making and procurement processes exceeds the observation period nExpectations: e.g., consumption depends not only on current income but also on the income expectations; modelling the expectation may be based on past development Elements of Dynamic Models April 15, 2016 Hackl, Econometrics 2, Lecture 4 7 nLag structures, distributed lags: linear combinations of current and past values of a variable nModels for expectations: based on lag structures, e.g., adaptive expectation model, partial adjustment model nAutoregressive distributed lag (ADL) model: a simple but widely applicable model consisting of an autoregressive part and of a finite lag structure of the independent variables Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 8 Example: Demand Functions April 15, 2016 Hackl, Econometrics 2, Lecture 4 9 nDemand for durable consumer goods: demand Q depends on the price P and on the income Y of the current and two previous periods: Qt = α + β0Yt + β1Yt-1 + β2Yt-2 + γPt + εt nDemand for energy: Qt = α + βPt + γKt + ut with P: price of energy, K: energy-related capital stock Kt = θ0 + θ1Pt-1 + θ2Pt-2 + … + δYt + vt with Y: income; substitution of K results in Qt = α0 + α1Yt + β0Pt + β1Pt-1 + β2Pt-2 + … + εt with εt = ut + γvt, α0 = α + γθ0, α1 = γδ, β0 = β, βi = γθi, i = 1, 2, … Models with Lag Structures April 15, 2016 Hackl, Econometrics 2, Lecture 4 10 Distributed lag model: describes the delayed effect of one or more regressors on the dependent variable; e.g., nDL(s) model Yt = δ + Σsi=0 φiXt-i + εt distributed lag of order s model Topics of interest qEstimation of coefficients qInterpretation of parameters n Hackl, Econometrics 2, Lecture 4 11 Example: Consumption Function nData for Austria (1990:1 – 2009:2), logarithmic differences (relative changes): n Ĉ = 0.009 + 0.621Y n with t(Y) = 2.288, R2 = 0.335 nDL(2) model, same data: n Ĉ = 0.006 + 0.504Y – 0.026Y-1 + 0.274Y-2 n with t(Y) = 3.79, t(Y-1) = – 0.18, t(Y-2) = 2.11, R2 = 0.370 nEffect of income on consumption: nShort term effect, i.e., effect in the current period: n ΔC = 0.504, given a change in income ΔY = 1 nOverall effect, i.e., cumulative current and future effects n ΔC = 0.504 – 0.026 + 0.274 = 0.752, given a change ΔY = 1 n April 15, 2016 Hackl, Econometrics 2, Lecture 4 12 Multiplier nDescribes the effect of a change ΔX = 1 in explanatory variable X on current and future values of the dependent variable Y nDL(s) model: Yt = δ + φ0Xt + φ1Xt-1 + … + φsXt-s + εt nShort run or impact multiplier q q q effect of the change in the same period, immediate effect of ΔX = 1 on Y: ΔY = φ0 nLong run multiplier q Effect of ΔX = 1 after 1, …, s periods: q q q Cumulated effect of ΔX = 1 at t over all future on Y: ΔY = φ0 + … + φs April 15, 2016 Hackl, Econometrics 2, Lecture 4 13 Equilibrium Multiplier nIf after a change ΔX an equilibrium occurs within a finite time: Long run multiplier is called equilibrium multiplier nDL(s) model n Yt = δ + φ0Xt + φ1Xt-1 + … + φsXt-s + εt n equilibrium after s periods nNo equilibrium for models with an infinite lag structure April 15, 2016 Hackl, Econometrics 2, Lecture 4 14 Average Lag Time nCharacteristics of lag structure φ0Xt + φ1Xt-1 + … + φsXt-s nPortion of equilibrium effect in the adaptation process qAt the end of the current period t: q w0 = φ0/(φ0 + φ1 + … + φs) qAt the end of the period t +1: q w0 + w1 = (φ0 + φ1)/(φ0 + φ1 + … + φs) qEtc. nWith weights wi = φi/(φ0 + φ1 + … + φs) nAverage lag time: Si i wi nMedian lag time: time till 50% of the equilibrium effect is reached, i.e., minimal s* with n w0 + … ws* ≥ 0.5 April 15, 2016 Hackl, Econometrics 2, Lecture 4 15 Consumption Function nFor ΔY = 1, the function n Ĉ = 0.006 + 0.504Y – 0.026Y-1 + 0.274Y-2 n gives nShort run effect: 0.504 nOverall effect: 0.752 nEquilibrium effect : 0.752 nAverage lag time: 0.694 quarters, i.e., ~ 2.3 months nMedian lag time: s* = 0; cumulative sums of weights are 0.671, 0.636, 1.000 n April 15, 2016 Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 16 Hackl, Econometrics 2, Lecture 4 17 Lag Structures: Estimation nDL(s) model: Problems with OLS estimation nLoss of observations: For a sample size N, only N-s observations are available for estimation; infinite lag structure! nMulticollinearity nOrder s (mostly) not known nConsequences: nMisspecification nLarge standard errors of estimates nLow power of tests nIssues: nChoice of s nModels for the lag structure with smaller number of parameters, e.g., polynomial structure n April 15, 2016 Hackl, Econometrics 2, Lecture 4 18 Consumption Function nFitted function n Ĉ = 0.006 + 0.504Y – 0.026Y-1 + 0.274Y-2 n with p-value for coefficient ofY-2: 0.039, adj.R2 = 0.342, AIC = -5.204 n n April 15, 2016 s AIC p-Wert adj.R2 1 -5.179 0.333 0.316 2 -5.204 0.039 0.342 3 -5.190 0.231 0.344 4 -5.303 0.271 0.370 5 -5.264 0.476 0.364 6 -5.241 0.536 0.356 7 -5.205 0.884 0.342 Models for s ≤ 7 Koyck’s Lag Structure April 15, 2016 Hackl, Econometrics 2, Lecture 4 19 Specifies the lag structure of the DL(s) model Yt = δ + Σsi=0 φiXt-i + εt as an infinite, geometric series (geometric lag structure) φi = λ0(1 - λ)λi nFor 0 < l < 1 Σsi=0 φi = λ0 nShort run multiplier: λ0(1 - λ) nEquilibrium effect: λ0 nAverage lag time: λ/(1 - λ) nStability condition 0 < l < 1 for l > 1, the φi and the contributions to the multiplier are exponentially growing l 0.1 0.3 0.5 0.7 l/(1-l) 0.10 0.43 1.00 2.33 The Koyck Model April 15, 2016 Hackl, Econometrics 2, Lecture 4 20 nThe DL (distributed lag) or MA (moving average) form of the Koyck model Yt = δ + λ0(1 – λ) Σi λiXt-i + εt nAR (autoregressive) form Yt = δ(1 – λ) + λYt-1 + λ0(1 – λ)Xt + ut with ut = εt – λεt-1 Hackl, Econometrics 2, Lecture 4 21 Consumption Function nModel with smallest AIC: n Ĉ = 0.003 + 0.595Y – 0.016Y-1 + 0.107Y-2 + 0.003Y-3 n + 0.148Y-4 n with adj.R2 = 0.370, AIC = -5.303, DW = 1.41 nKoyck model in AR form n Ĉ = 0.004 + 0.286 C-1 + 0.556Y n with adj.R2 = 0.388, AIC = -5.290, DW = 1.91 n April 15, 2016 Koyck Model: Estimation Problems April 15, 2016 Hackl, Econometrics 2, Lecture 4 22 Parameters to be estimated: δ, λ0, and λ; problems are nDL form [Yt = δ + λ0(1 – λ) Σi λiXt-i + εt] qHistorical values X0, X-1, … are unknown qNon-linear estimation problem nAR form [Yt = δ(1 – λ) + λYt-1 + λ0(1 – λ)Xt + ut with ut = εt – λεt-1] qNon-linear estimation problem qLagged, endogenous variable used as regressor qCorrelated error terms Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 23 The ADL(1,1) Model April 15, 2016 Hackl, Econometrics 2, Lecture 4 24 nThe autoregressive distributed lag (ADL) model: autoregressive model with lag structure of regressor, e.g., the ADL(1,1) model Yt = δ + θYt-1 + φ0Xt + φ1Xt-1 + εt nThe error correction model: ΔYt = – (1 – θ)(Yt-1 – α – βXt-1) + φ0 ΔXt + εt obtained from the ADL(1,1) model with α = δ/(1 – θ) β = (φ0+φ1)/(1 – θ) Example: nSales St are determined qby advertising amounts At and At-1, but also qby St-1: St = μ + θSt-1 + β0At + β1At-1 + εt ΔSt = – (1 – θ)[St-1 – μ/(1 – θ) – (β0+β01)/(1 – θ)At-1] + β0ΔAt + εt Hackl, Econometrics 2, Lecture 4 25 Multiplier nADL(1,1) model: Yt = δ + θYt-1 + φ0Xt + φ1Xt-1 + εt nEffect of a change ΔX = 1 at time t nImpact multiplier: ΔY = φ0; see the DL(s) model nLong run multiplier qEffect after one period n n qEffect after two periods n n qCumulated effect over all future on Y q φ0 + (θφ0 + φ1) + θ(θφ0 + φ1) + … = (φ0 + φ1)/(1 – θ) q decreasing effects requires |θ|<1, stability condition q April 15, 2016 Hackl, Econometrics 2, Lecture 4 26 ADL(1,1) Model: Equilibrium nEquilibrium relation of the ADL(1,1) model: nEquilibrium at time t means: E{Yt} = E{Yt-1}, E{Xt } = E{Xt-1} n E{Yt} = δ + θ E{Yt} + φ0 E{Xt} + φ1 E{Xt} n or, given the stability condition |θ|<1, n n nEquilibrium relation: n E{Yt} = α + β E{Xt} n with α = δ/(1 – θ), β = (φ0 + φ1)/(1 – θ) nLong run multiplier: change ΔX = 1 of the equilibrium value of X increases the equilibrium value of Y by β or (φ0 + φ1)/(1 – θ) April 15, 2016 Hackl, Econometrics 2, Lecture 4 27 The Error Correction Model nADL(1,1) model, written as error correction model n ΔYt = φ0 ΔXt – (1 – θ)(Yt-1 – α – βXt-1) + εt nEffects on ΔY qdue to changes ΔX qdue to equilibrium error, i.e., Yt-1 – α – βXt-1 nNegative adjustment: Yt-1 < α + βXt-1 = E{Yt-1}, i.e., a negative equilibrium error, increases Yt by – (1 – θ)(Yt-1 – α – βXt-1) [> 0] nAdjustment parameter: (1 – θ) qDetermines speed of adjustment April 15, 2016 Hackl, Econometrics 2, Lecture 4 28 The ADL(p,q) Model nADL(p,q): generalizes the ADL(1,1) model n θ(L)Yt = δ + Φ(L)Xt + εt n with lag polynomials n θ(L) = 1 - θ1L - … - θpLp , Φ(L) = φ0 + φ1L + … + φqLq nGiven invertibility of θ(L), i.e., θ1 + … + θp < 1, n Yt = θ(1)-1δ + θ(L)-1Φ(L)Xt + θ(L)-1εt nThe coefficients of θ(L)-1Φ(L) describe the dynamic effects of X on current and future values of Y nequilibrium multiplier n n nADL(0,q): coincides with the DL(q) model; θ(L) = I April 15, 2016 Hackl, Econometrics 2, Lecture 4 29 ADL Model: Estimation nADL(p,q) model nerror terms εt: white noise, independent of Xt, …, Xt-q and Yt-1, …, Xt-p nOLS estimators are consistent April 15, 2016 Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model April 15, 2016 Hackl, Econometrics 2, Lecture 4 30 Hackl, Econometrics 2, Lecture 4 31 Expectations in Economic Processes nExpectations play important role in economic processes nExamples: nConsumption depends not only on current income but also on the income expectations nInvestments depend upon expected profits nInterest rates depend upon expected development of the financial market nEtc. nExpectations ncannot be observed, but ncan be modelled using assumptions on the mechanism of adapting expectations; nmodeling the expectation may be based on past development April 15, 2016 Hackl, Econometrics 2, Lecture 4 32 Models for Adapting Expectations nNaive model: the (for the next period) expected value equals the actual value nModel of adaptive expectation nPartial adjustment model nThe latter two models are based on Koyck’s lag structure April 15, 2016 Hackl, Econometrics 2, Lecture 4 33 Adaptive Expectation: The Concept nModels of adaptive expectation: describe the actual value Yt as function of the value Xet+1 of the regressor X that is expected for the next period n Yt = α + βXet+1 + εt nExample: Investments Y are a function of the expected profits Xe nConcepts for modelling Xet+1: nNaive expectation: Xet+1 = Xt nMore realistic: a weighted sum of in the past realized profits n Xet+1 = β0Xt + β1Xt-1 + … qGeometrically decreasing weights βi n βi = (1-λ) λi n with 0 < λ < 1 April 15, 2016 Hackl, Econometrics 2, Lecture 4 34 Adaptive Mechanism for the Expectation nWith βi = (1- λ) λi, the expected value Xet+1 = β0Xt + β1Xt-1 + … results in n Xet+1 = λXet + (1 – λ)Xt n or n Xet+1 - Xet = (1 – λ)(Xt - Xet) nInterpretation: the change of expectation between t and t+1 is proportional to the actual „error in expectation”, i.e., the deviation between the actual expectation and the actually realized value nExtent of the change (adaptation): 100(1 – λ)% of the error nλ: adaptation parameter April 15, 2016 Hackl, Econometrics 2, Lecture 4 35 Models of Adaptive Expectation nAdaptive expectation model (AR form) n Yt = α(1 – λ) + λYt-1 + β(1 – λ)Xt + vt n with vt = εt – λεt-1; an ADL(1,0) model nDL form n Yt = α + β(1 – λ)Xt + β(1 – λ)λ Xt-1 + β(1 – λ)λ2 Xt-2 + … + εt nExample: Investments (I) as function of the expected profits Pet+1 and interest rate (r) n It = α + βPet+1 + γrt + εt nAssumption of adapted expectation for the profits Pet+1: n Pet+1 = λPet + (1 – λ)Pt n with adaptation parameter λ (0 < λ < 1) nAR form of the investment function (vt = εt – λεt-1): n It = α(1 – λ) + λIt-1 + β(1 – λ)Pt + γrt – λγrt-1 + vt April 15, 2016 Hackl, Econometrics 2, Lecture 4 36 Consumption Function nConsumption as function of the expected income n Ct = α + βYet+1 + εt n expected income derived under the assumption of adaptive expectation n Yet+1 = λYet + (1 – λ)Yt nAR form is n Ct = α(1 – λ) + λCt-1 + β(1 – λ)Yt + vt n with vt = εt – λεt-1 nExample: AWM data base, 1970:1-2003:4; the estimated model is n Ĉ = 0.004 + 0.286C-1 + 0.556Y nadj.R2 = 0.388, AIC = -5.29, DW = 1.91 April 15, 2016 Hackl, Econometrics 2, Lecture 4 37 Partial Adjustment Model nDescribes the process of adaptation to a desired or planned value Y*t as a function of regressor Xt n Y*t = α + βXt + ηt n(Partial) adjustment of the actual Yt according to n Yt – Yt-1 = (1 - θ)(Y*t – Yt-1) n adaptation parameter θ with 0 < θ < 1 nActual Yt: weighted average of Y*t and Yt-1 n Yt = (1 - θ)Y*t + θYt-1 nAR form of the model n Yt = (1 - θ)α + θYt-1 + (1 - θ)βXt + (1 – θ)ηt n = δ + θYt-1 + φ0Xt + εt n which is an ADL(1,0) model April 15, 2016 Hackl, Econometrics 2, Lecture 4 38 Example: Desired Stock Level nStock level K and revenues S nThe desired (optimal) stock level K* depends of the revenues S n K*t = α + βSt + ηt nActual stock level Kt-1 in period t-1: deviates by K*t – Kt-1 from K*t n(Partial) adjustment strategy according to n Kt – Kt-1 = (1 – θ)(K*t – Kt-1) n adaptation parameter θ with 0 < θ < 1 nSubstitution for K*t gives the AR form of the model n Kt = Kt-1 + (1 – θ)α + (1 – θ)βSt – (1 – θ)Kt-1 + (1 – θ)ηt n = δ + θKt-1 + φ0St + εt n δ = (1 – θ)α, φ0 = (1 – θ)β, εt = (1 – θ)ηt nModel for Kt is an ADL(1,0) model April 15, 2016 Hackl, Econometrics 2, Lecture 4 39 ADL Models nModels in ADL(1,0) form 1.Koyck’s model n Yt = α (1 – λ) + λYt-1 + β(1 – λ)Xt + vt n with vt = εt – λεt-1 2.Model of adaptive expectation n Yt = α(1 – λ) + λYt-1 + β(1 – λ)Xt + vt n with vt = εt – λεt-1 3.Partial adjustment model n Yt = (1 - θ)α + θYt-1 + (1 - θ)βXt + εt nError terms are nWhite noise for partial adjustment model nAutocorrelated for the other two models April 15, 2016 Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 40 Hackl, Econometrics 2, Lecture 4 41 Regression and Time Series nStationarity of variables is 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 April 15, 2016 Hackl, Econometrics 2, Lecture 4 42 An Illustration nIndependent random walks: Yt = Yt-1 + εyt, Xt = Xt-1 + εxt n εyt, εxt: independent white noises with variances σy² = 2, σx² = 1 nFitting the model n Yt = α + βXt + εt n gives n Ŷt = - 8.18 + 0.68Xt nt-statistic for X: t = 17.1 n p-value = 1.2 E-40 nR2 = 0.50, DW = 0.11 April 15, 2016 Hackl, Econometrics 2, Lecture 4 43 Models in Non-stationary Time Series nGiven that Xt ~ I(1), Yt ~ I(1) and the model n Yt = α + βXt + εt n it follows – in general – that εt ~ I(1), i.e., the error terms are non- stationary nConsequences for OLS estimation of α and β n(Asymptotic) distributions of t- and F-statistics are different from those under stationarity nt-statistic, R2 indicate explanatory potential nHighly autocorrelated residuals, DW statistic converges for growing N to zero nNonsense or spurious regression (Granger & Newbold, 1974) nNon-stationary time series are trended; non-stationarity causes an apparent relationship n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 44 Avoiding Spurious Regression nIdentification of non-stationarity: unit-root tests nModels for non-stationary variables qElimination of stochastic trends: 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, may be “cointegrated”: equilibrium relation, error-correction models n April 15, 2016 Hackl, Econometrics 2, Lecture 4 45 An Example: ADL(1,1) Model nADL(1,1) model with Yt ~ I(1), Xt ~ I(1) n Yt = δ + θYt-1 + φ0Xt + φ1Xt-1 + εt nCommon trend implies an equilibrium relation, i.e., n Yt-1 – βXt-1 ~ I(0) n error-correction form of the ADL(1,1) model q ΔYt = φ0ΔXt – (1 – θ)(Yt-1 – α – βXt-1) + εt nInclusion of lagged variables Yt-1 and Xt-1 allows a solution (θ =1, φ0 = φ1 = 0) such that εt is I(0): n εt = Yt – (δ + θYt-1 + φ0Xt + φ1Xt-1) ~ I(0) n OLS estimates are consistent for all parameters April 15, 2016 Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 46 Hackl, Econometrics 2, Lecture 4 47 The Drunk and her Dog nM. P. Murray, A drunk and her dog: An illustration of cointegration and error correction. The American Statistician, 48 (1997), 37-39 ndrunk: xt – xt-1 = ut ndog: yt – yt-1 = wt n white noises ut, wt nCointegration: n xt–xt-1 = ut+c(yt-1–xt-1) n yt–yt-1 = wt+d(xt-1–yt-1) April 15, 2016 C:\Users\PHackl\Documents\O'trie\_Brno\Lecture_6\A drunk and her dog An illustration of cointegration and error correction. - Powered by Google Text & Tabellen_files\viewer(3) Hackl, Econometrics 2, Lecture 4 48 Cointegrated Variables 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 April 15, 2016 Hackl, Econometrics 2, Lecture 4 49 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 behaviour qconsistently with the long-run equilibrium nLong-run equilibrium: Yt = βXt, deviations from equilibrium: Yt – βXt nConverse statement: if Yt ~ I(1), Xt ~ I(1) have an error-correction representation, then they are cointegrated April 15, 2016 Hackl, Econometrics 2, Lecture 4 50 I(1) Variables and Equilibrium nEquilibrium between Y and X with Yt ~ I(1), Xt ~ I(1): defined by n Y = α + βX nEquilibrium error: zt = Yt - βXt - α = Zt - α nTwo cases: 1.zt ~ I(0): equilibrium error stationary, fluctuating around zero qYt, βXt cointegrated qYt = α + βXt describes an equilibrium 2.zt ~ I(1) qYt, βXt not integrated qzt ~ I(1) non-stationary process qYt = α + βXt does not describe an equilibrium, spurious regression nCointegration, i.e., existence of an equilibrium vector, implies a long-run equilibrium relation April 15, 2016 Hackl, Econometrics 2, Lecture 4 51 Example: Purchasing Power Parity (PPP) nVerbeek’s dataset PPP: Price indices and exchange rates for France and Italy, monthly, T = 186 (1981:1 – 1996:6) nVariables: LNIT (log price index of Italy), LNFR (log price index of France), LNX (log exchange rate France/Italy) nLNIT, LNFR, LNX non-stationary (DF-test) nLNPt = LNITt – LNFRt, i.e., log of price index ratio Italy/France, non-stationary nPurchasing power parity (PPP): exchange rate between the currencies (Franc, Lira) equals the ratio of price levels of the countries n LNXt = LNPt nRelative PPP: equality fulfilled only in the long run; equilibrium or cointegrating relation n LNXt = α + β LNPt n April 15, 2016 Hackl, Econometrics 2, Lecture 4 52 PPP: The Variables 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 Italy/France nResults from DF tests: n April 15, 2016 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 corresponds 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 nwith white noise εt ~ I(0) April 15, 2016 Hackl, Econometrics 2, Lecture 4 53 PPP: Equilibrium Relation 2 nOLS estimation of n LNXt = α + β LNPt + εt n n n n n n n n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 54 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 Hackl, Econometrics 2, Lecture 4 55 OLS-estimates of Cointegration Parameter nCointegrating relation, Xt ~ I(1), Yt ~ I(1), εt ~ I(0) n Yt = βXt + εt nOLS estimate b of β nEstimate b is super consistent qConverges faster to β than standard asymptotic theory says qConverges to β in spite of omission of relevant regressors (short-term dynamics) qFor b ≠ β: non-stationary OLS residuals with much larger variance than for b close to β qBias of b may be substantial! nNon-standard theory qAsymptotic distribution of √T(b- β) degenerate, not normal (cf. standard theory) qt-statistic may be misleading April 15, 2016 Hackl, Econometrics 2, Lecture 4 56 Estimation of Spurious Regression Parameter nNon-stationary processes Xt ~ I(1), Yt ~ I(1) n Yt = βXt + εt n Spurious regression, εt ~ I(1) nOLS estimate b of β nNon-standard distribution nLarge values of R2, t-statistic nHighly autocorrelated residuals nDW statistic close to zero nRemedy nUse changes ΔXt, ΔYt instead of Xt,Yt (Granger, Newbold, 1974) nAdd lagged regressors, e.g., Yt-1: for Yt = δ + θYt-1 + φ0Xt + φ1Xt-1 + εt, parameter values can be found such that εt ~ I(0) nModel in differences misses error-correction term! n April 15, 2016 Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 57 Hackl, Econometrics 2, Lecture 4 58 Identification of Cointegration nInformation about cointegration nEconomic theory nVisual inspection of data nStatistical tests April 15, 2016 Hackl, Econometrics 2, Lecture 4 59 Testing for Cointegration nNon-stationary variables Xt ~ I(1), Yt ~ I(1) n Yt = α + βXt + εt nXt and Yt are cointegrated: εt ~ I(0) nXt and Yt are not cointegrated: εt ~ I(1) nTests for cointegration: nIf β is known, unit root test based on differences Yt – βXt nTest procedures qUnit root test (DF or ADF) based on residuals et qCointegrating regression Durbin-Watson (CRDW) test: DW statistic qJohansen technique: extends the cointegration technique to the multivariate case n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 60 DF Test for Cointegration nNon-stationary variables Xt ~ I(1), Yt ~ I(1) n Yt = α + βXt + εt nXt and Yt are cointegrated: εt ~ I(0) nResiduals et show pattern similar to εt, et ~ I(0), residuals are stationary nTests for cointegration based on residuals et n Δet = γ0 + γ1et-1 + ut nH0: γ1 = 0, i.e., residuals have a unit root, et ~ I(1) nH0 implies qXt and Yt are not cointegrated! nRejection of H0 suggests that Xt and Yt are cointegrated April 15, 2016 Hackl, Econometrics 2, Lecture 4 61 DF Test for Cointegration, cont’d nCritical values of DF test for residuals nare smaller than those of DF test for observations ndepend upon (see Verbeek, Tab. 9.2) qnumber of components of cointegrating vector (including left-hand side), K qnumber of observations T qsignificance level q nsome asymptotic (T=∞) criti- n cal values for the DF-test n with constant term for ob- n servations and for residuals n (see Verbeek, Tab. 8.1 n and 9.2) q April 15, 2016 DF-test for 1% 5% observations -3.43 -2.86 residuals, K=2 -3.90 -3.34 Hackl, Econometrics 2, Lecture 4 62 Cointegrating Regression Durbin-Watson (CRDW) Test nNon-stationary variables Xt ~ I(1), Yt ~ I(1) n Yt = α + βXt + εt nCointegrating regression Durbin-Watson (CRDW) test: DW statistic from OLS-fitting Yt = α + βXt + εt nH0: residuals et have a unit root, i.e., et ~ I(1), i.e., Xt and Yt are not cointegrated nDW statistic converges with growing T to zero for not cointegrated variables, i.e., under H0 April 15, 2016 Hackl, Econometrics 2, Lecture 4 63 CRDW Test, cont’d nRule of thumb qIf CRDW < R2, cointegration likely to be false; do not reject H0 qIf CRDW > R2, cointegration may occur; reject H0 nCritical values from Monte Carlo simulations, which depend upon (see Verbeek, Tab. 9.3) qNumber of regressors K +1 (plus 1 for the dependent variable) qNumber of observations T qSignificance level q n some 5% critical values n for the CRDW- test q April 15, 2016 K+1 T = 50 T = 100 2 0.72 0.38 3 0.89 0.48 4 1.05 0.58 PPP: Equilibrium Relation 2 nOLS estimation of n LNXt = α + β LNPt + εt n n n n n n n n n n nDF test statistic for residuals (with constant): -1.90, p-value: 0.33 n H0 cannot be rejected: no evidence for cointegration n April 15, 2016 Hackl, Econometrics 2, Lecture 4 64 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 Hackl, Econometrics 2, Lecture 4 65 Testing for Cointegration, cont’d nResiduals from LNXt = α + β LNPt + εt: 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 qCRDW test, rule of thump: 0.055 < 0.665 = R2 nBoth tests suggest: H0 cannot be rejected, no evidence for cointegration nTime series plot indicates non-stationary residuals (see next slide) 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 April 15, 2016 Hackl, Econometrics 2, Lecture 4 66 Testing for Cointegration nResiduals from LNXt = α + β LNPt + εt: nTime series plot indicates non-stationarity of residuals q q q Time series plot q of residuals April 15, 2016 Hackl, Econometrics 2, Lecture 4 67 Cointegration Test in GRETL nModel > Time series > Cointegration test > Engle-Granger n Performs the qDF test for each of the variables qEstimation of the cointegrating regression qDF test for the residuals of the cointegrating regression nModel > Time series > Cointegration tests > Johansen n See next lecture April 15, 2016 Contents nDynamic Models nLag Structures nLag Structure: Estimation nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration nTest for Cointegration nError-correction Model n n n n n April 15, 2016 Hackl, Econometrics 2, Lecture 4 68 Hackl, Econometrics 2, Lecture 4 69 Error-correction Model nGranger’s Representation Theorem (Engle & Granger, 1987): If a set of variables is cointegrated, then an error-correction 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 lag polynomials θ(L) (with θ0=1), Φ(L), and α(L) nE.g., ΔYt = δ + φ1ΔXt-1 - γ(Yt-1 – βXt-1) + εt nError-correction model: describes nthe short-run behavior nconsistently with the long-run equilibrium nConverse statement: if Yt ~ I(1), Xt ~ I(1) have an error-correction representation, then they are cointegrated April 15, 2016 Hackl, Econometrics 2, Lecture 4 70 EC Model and Equilibrium Relation nThe EC 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) = I, Φ(L) = φ1L, and α(L) = 1 n“No change” steady state equilibrium: for ΔYt = ΔXt-1 = 0 n Yt – βXt = δ/γ or Yt = α + βXt if δ = αγ, i.e., α = δ/γ n the EC model can be written as n ΔYt = φ1ΔXt-1 – γ(Yt-1 – α – βXt-1) + εt nSteady state growth: If δ = αγ + 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) April 15, 2016 Hackl, Econometrics 2, Lecture 4 71 Analysis of EC Models nModel specification nUnit-root testing nTesting for cointegration nSpecification of EC-model: choice of orders of lag polynomials, specification analysis nEstimation of model parameters n April 15, 2016 Hackl, Econometrics 2, Lecture 4 72 EC Model: Estimation nModel for cointegrated variables Xt, Yt 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 (1, –β)’: qParameter β from (B) super consistently estimated by OLS qOLS estimation of δ, φ1, and γ from (A) is not affected by use of the estimate for β April 15, 2016 Hackl, Econometrics 2, Lecture 4 73 Your Homework 1.Use Verbeek’s data set INCOME containing quarterly data INCOME (total disposable income) and CONSUM (consumer expenditures) for 1/1971 to 2/1985 in the UK. a.For sd_CONSUM (seasonal difference of CONSUM), specify a DL(s) model in sd_INCOME and choose an appropriate s (≤4), using (i) adj R2 and (ii) BIC. b.Assuming that DL(4) is an appropriate lag structure for sd_INCOME, calculate (i) the short run and (ii) the long run multiplier as well as (iii) the average and (iv) the median lag time. c.Again for seasonal differences, specify a consumption function with the actual expected income as explanatory variable; estimate the AR form of the model under the assumption of adaptive expectation for the income. d.Test whether CONSUM and INCOME are I(1). n April 15, 2016 Hackl, Econometrics 2, Lecture 4 74 Your Homework e.Estimate (i) the simple linear regression of CONSUM on INCOME and test (ii) whether this is an equilibrium relation; show (iii) the corresponding time series plots (CONSUME, INCOME, residuals). 2.Generate 250 random numbers (i) from a random walk with trend: xt = 0.1 + xt-1 + εxt; and (ii) from an AR(1) process: yt = 1 + φ yt-1 + εyt with φ =0.7; for εxt and εyt use Monte Carlo random numbers from N(0,1). a.Produce time series graphs for xt and yt over t. b.Perform DF tests for xt and yt; what are the conclusions? c.Estimate regressions of xt and yt on t; report the values for R2. d.Repeat the generation of yt for φ = 1 and produce again series graphs for xt and yt over t; what indicates the DF test for yt? e.Regress yt on xt; discuss the results as an illustration of spurious regressions. f. n April 15, 2016