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 5, 2013 Hackl, Econometrics 2, Lecture 4 2 The Lüdeke Model for Germany April 5, 2013 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 5, 2013 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 n Dynamic Models: Examples April 5, 2013 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 The Dynamic of Processes April 5, 2013 Hackl, Econometrics 2, Lecture 4 6 Static processes: immediate reaction to changes in regressors, the adjustment of the dependent variables 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; modeling the expectation may be based on past development Elements of Dynamic Models April 5, 2013 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 5, 2013 Hackl, Econometrics 2, Lecture 4 8 Example: Demand Functions April 5, 2013 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-1 + … + εt with εt = ut + γvt, α0 = α + γθ0, α1 = γδ, β0 = β, βi = γθi, i = 1, 2, … Models with Lag Structures April 5, 2013 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 5, 2013 Hackl, Econometrics 2, Lecture 4 12 Multiplier nDescribes the effect of a change in explanatory variable Xt by ΔX = 1 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 5, 2013 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 5, 2013 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 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 5, 2013 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 5, 2013 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 5, 2013 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 5, 2013 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 5, 2013 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 5, 2013 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 5, 2013 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 5, 2013 Koyck Model: Estimation Problems April 5, 2013 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) 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 5, 2013 Hackl, Econometrics 2, Lecture 4 23 The ADL(1,1) Model April 5, 2013 Hackl, Econometrics 2, Lecture 4 24 nThe autoregressive distributed lag (ADL) model: autoregressive model with lag structure, 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 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 5, 2013 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 (φ0 + φ1)/(1 – θ) April 5, 2013 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 < E{Yt-1} = α + βXt-1, i.e., a negative equilibrium error, increases Yt by – (1 – θ)(Yt-1 – α – βXt-1) nAdjustment parameter: (1 – θ) qDetermines speed of adjustment April 5, 2013 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) = 1 April 5, 2013 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 5, 2013 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 5, 2013 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; modeling the expectation may be based on past development nInvestments depend upon expected profits nInterest rates depend upon expected development of the financial market nEtc. nExpectations ncannot be observed, but ncan be modeled using assumptions on the mechanism of adapting expectations April 5, 2013 Hackl, Econometrics 2, Lecture 4 32 Models for Adapting Expectations nNaive model of adapting expectations: 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 5, 2013 Hackl, Econometrics 2, Lecture 4 33 Adaptive Expectation: 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 are a function of the expected profits nConcepts for Xet+1: nNaive expectation: Xet+1 = Xt nMore realistic is 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 5, 2013 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 5, 2013 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 + … + 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 5, 2013 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 adapted 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: the estimated model is n Ĉ = 0.004 + 0.286C-1 + 0.556Y nadj.R2 = 0.388, AIC = -5.29, DW = 1.91 April 5, 2013 Hackl, Econometrics 2, Lecture 4 37 Partial Adjustment Model nDescribes the process of adapting 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 5, 2013 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 from K*t: K*t – Kt-1 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 5, 2013 Hackl, Econometrics 2, Lecture 4 39 Models in AR Form 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 5, 2013 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 5, 2013 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 5, 2013 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 5, 2013 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 nR2 indicates 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; causes an apparent relationship n n April 5, 2013 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, are cointegrated: equilibrium relation, error-correction models n April 5, 2013 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 nThe error terms are stationary if θ =1, φ0 = φ1 = 0 q εt = Yt – (δ + θYt-1 + φ0Xt + φ1Xt-1) ~ I(0) 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 n April 5, 2013 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 5, 2013 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 nCointegration: n xt–xt-1 = ut+c(yt-1–xt-1) n yt–yt-1 = wt+d(xt-1–yt-1) April 5, 2013 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 5, 2013 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 5, 2013 Hackl, Econometrics 2, Lecture 4 50 Long-run Equilibrium nEquilibrium defined by n Yt = α + βXt 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), Yt, 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 5, 2013 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 (1/1981-6/1996) nVariables: LNIT (log price index Italy), LNFR (log price index France), LNX (log exchange rate France/Italy) nLNIT, LNFR, LNX non-stationary (DF-test) nLNPt = LNITt – LNFRt, i.e., log of price index ratio, non-stationary 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 April 5, 2013 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 France/Italy nResults from DF tests: n April 5, 2013 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 εt ~ I(0) April 5, 2013 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 5, 2013 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 Estimation 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 5, 2013 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: add lagged regressors, e.g., Yt-1 nFor Yt = δ + θYt-1 + φ0Xt + φ1Xt-1 + εt, parameter values can be found such that εt ~ I(0) nConsistent estimates n April 5, 2013 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 5, 2013 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 5, 2013 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 5, 2013 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 qRejection of H0 suggests that Xt and Yt are cointegrated April 5, 2013 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 elements of cointegrating vector (including left-hand side), K qnumber of observations T qsignificance level q nsome asymptotic n critical values for the DF- n test with constant term q April 5, 2013 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 April 5, 2013 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 plus 1 (dependent variable) qNumber of observations T qSignificance level q n some 5% critical values n for the CRDW- test q April 5, 2013 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 (constant): -1.90, p-value: 0.33 n H0 cannot be rejected: no evidence for cointegration n April 5, 2013 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 qDF 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 5, 2013 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 5, 2013 Hackl, Econometrics 2, Lecture 4 67 Cointegration Test in GRETL nModel > Time series > Cointegration tests > Engle-Granger n Performs the qDL 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 5, 2013 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 5, 2013 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 5, 2013 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) = 1, 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 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 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) April 5, 2013 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 5, 2013 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 5, 2013 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 (< 8), using (i) R2 and (ii) BIC. b.Assuming that DL(4) is an appropriate lag structure, calculate (i) the short run and (ii) the long run multiplier as well as (iii) the average and (iv) the median lag time. c.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 (i) whether CONSUM and INCOME are I(1); (ii) estimate the simple linear regression of CONSUM on INCOME and test (iii) whether this is an equilibrium relation; show (iv) the corresponding time series plots. n April 5, 2013 Hackl, Econometrics 2, Lecture 4 74 Your Homework, cont’d 2.Generate 500 random numbers (a) from a random walk with trend: xt = 0.1 +xt-1 + εt; and (b) from an AR(1) process: yt = 0.2 + 0.7yt-1 + ηt; for εt and ηt use Monte Carlo random numbers from N(0,1). Estimate regressions of xt and yt on t; report the values for R2. n n April 5, 2013