Econometrics 2 - Lecture 4 Univariate (part 2) and Multivariate Time Series Models Contents nARCH and GARCH Models nDynamic Models nLag Structures nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration n n n n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 2 Hackl, Econometrics 2, Lecture 4 3 ARCH Processes nAutoregressive Conditional Heteroskedasticity (ARCH): nSpecial case of heteroskedasticity nVolatility (error variance) show autoregressive behavior: large errors induce a period of large volatility nAllows to model successive periods with high, other periods with low volatility nTypical for asset markets like stock markets, in particular for high frequencies like daily data nIdea: The variance of the errors (or innovations) εt is allowed to depend upon its history, follows an autoregressive process nARCH models developed by Robert Engle in the 1980ies; Nobel Memorial Prize in Economic Sciences, 2003 April 1, 2011 Hackl, Econometrics 2, Lecture 4 4 Example: Exchange Rate nVerbeek‘s data set GARCH April 1, 2011 1867 daily observa-tions on exchange rates of the US dollar against the DM Hackl, Econometrics 2, Lecture 4 5 Exchange Rate: A Model nDaily log exchange rate yt of the US dollar against the DM n yt = θ + εt n εt = σtvt with vt ~ NID(0,1) n where σt² follows the ARCH model n σt² = E{εt²|It-1} = ϖ + αεt-1² nError terms εt are uncorrelated nVolatility (error variance) show autoregressive behavior n April 1, 2011 Hackl, Econometrics 2, Lecture 4 6 The ARCH(1) Process nARCH(1) process describes the conditional error variance, i.e., the variance conditional on information dated t-1 and earlier q σt² = E{εt²|It-1} = ϖ + αεt-1² nIt-1 is the information set containing all past including εt-1 nConditions for σt² ≥ 0: ϖ ≥ 0, α ≥ 0 nA big shock at t-1, i.e., a large value |εt-1|, qinduces high volatility, i.e., large σt² qmakes large values |εt| more likely at t (and later) nARCH process does not imply correlation of the errors! nThe unconditional variance of εt is n σ² = E{εt²} = ϖ + αE{εt-1²} = ϖ/(1 - α) n given that 0 ≤ α < 1 nThe εt process is stationary n April 1, 2011 Hackl, Econometrics 2, Lecture 4 7 ARCH-Model: Estimation nModel yt = xt’θ + εt with conditional error variance σt² following an ARCH process, i,.e., εt = σtvt, vt ~ NID(0,1) with n σt² = E{εt²|It-1} = ϖ + αεt-1² nConditional upon It-1, εt ~ N(0, σt²) nContribution of yt to the likelihood function n n n n with εt = yt – xt’θ and σt² = ϖ + αεt-1² nEstimates for θ, α and ϖ by maximizing the log likelihood function April 1, 2011 Exchange Rate: ARCH Model nARCH model for differences DMM of log exchange rate US dollar against DM April 1, 2011 Hackl, Econometrics 2, Lecture 4 8 Model 5: WLS (ARCH), using observations 3-1867 (T = 1865) Dependent variable: DDM Variable used as weight: 1/sigma coefficient std. error t-ratio p-value ----------------------------------------------------------- const -5.487e-05 0.0001769 -0.3101 0.7565 alpha(0) 5.382e-05 3.1787e-06 16.93 6.17e-060 *** alpha(1) 0.108035 0.0230333 4.690 2.93e-06 *** Statistics based on the weighted data: Sum squared resid 1848.942 S.E. of regression 0.995953 R-squared 0.000000 Adjusted R-squared 0.000000 Log-likelihood -2638.257 Akaike criterion 5278.513 Schwarz criterion 5284.044 Hannan-Quinn 5280.552 rho -0.059930 Durbin-Watson 2.119846 Statistics based on the original data: Mean dependent var -0.000020 S.D. dependent var 0.007770 Sum squared resid 0.112543 S.E. of regression 0.007770 yt = θ + εt σt² = α0 + α1εt-1² Hackl, Econometrics 2, Lecture 4 9 3-Step Estimation Procedure nModel yt = xt’θ + εt with conditional error variance σt² following an ARCH process, i,.e., εt = σtvt, vt ~ NID(0,1) with n σt² = E{εt²|It-1} = ϖ + αεt-1² nEstimation of θ, α and ϖ in 3 steps 1.OLS estimation of the regression model, residuals et 2.Auxiliary regression of the squared residuals et² on its own lagged values 3.Weighted least squares estimation; weights are the reciprocals of the fitted error variances from the auxiliary regression April 1, 2011 Hackl, Econometrics 2, Lecture 4 10 More ARCH Processes nVarious generalizations nARCH(p) process nGARCH(p,q) process, Generalized ARCH nEGARCH or exponential GARCH nEtc. nARCH(p) process n σt² = ϖ + α1εt-1² + … αpεt-p2 = ϖ + α(L)εt-1² n with lag polynomial α(L) of order p-1 nConditions for σt² ≥ 0: ϖ ≥ 0; αi ≥ 0, i = 1,…p nCondition for stationarity: α(1) < 1 April 1, 2011 Hackl, Econometrics 2, Lecture 4 11 GARCH Process nGARCH(p,q) process n„Generalized ARCH“ nSimilar to the ARMA representation of levels n σt² = ϖ + α1εt-1² + … αpεt-p2 + β1σt-1² + … + βqσt-q² = n = ϖ + α(L)εt-1² + β(L)σt-1² nExample: GARCH(1,1) n σt² = ϖ + αεt-1² + βσt-1² n“surprises“ vt = εt-1² - σt² n εt² = ϖ + (α + β)εt-1² + vt - βvt-1 ni.e. εt² follow ARMA(1,1) nvt: uncorrelated, but heteroskedastic n April 1, 2011 Hackl, Econometrics 2, Lecture 4 12 EGARCH Process nEGARCH or exponential GARCH n log σt² = ϖ + β log σt-1² + γεt-1/σt-1 + α|εt-1|/σt-1 nAsymmetric: for γ < 0 qpositive shocks reduce volatility qnegative shocks incease volatility nAllows for larger impacts on volatility qof drops in price („bad news“) than qincreases in price („good news“) n q April 1, 2011 Hackl, Econometrics 2, Lecture 4 13 Test for ARCH Processes nNull hypothesis of homoskedasticity, to be tested against the alternative ARCH(p) 1.Estimate the model of interest using OLS: residuals et 2.Auxiliary regression of squared residuals et2 on a constant and p lagged et2 3.Test statistic TRe2 with Re2 from the auxiliary regression, p-value from the chi squared distribution with p df April 1, 2011 Exchange Rate: Test for Homoskedasticity nAuxiliary regression of squared residuals et2 on a constant and et-12; residuals are differences of DMM from their mean (yt = θ + εt) n n n n n n n n n n n nTRe2 = (1865)x(0.011671) = 21.77; p-value = 3.08E-6 April 1, 2011 Hackl, Econometrics 2, Lecture 4 14 Model 11: OLS, using observations 3-1867 (T = 1865) Dependent variable: usq10 coefficient std. error t-ratio p-value ---------------------------------------------------------- const 5.382e-05 3.17866e-06 16.93 6.20e-060 *** usq10_1 0.108035 0.0230333 4.690 2.93e-06 *** Mean dependent var 0.000060 S.D. dependent var 0.000124 Sum squared resid 0.000028 S.E. of regression 0.000123 R-squared 0.011671 Adjusted R-squared 0.011140 F(1, 1863) 21.99959 P-value(F) 2.93e-06 Log-likelihood 14139.08 Akaike criterion -28274.16 Schwarz criterion -28263.10 Hannan-Quinn -28270.09 rho -0.008175 Durbin's h -3.352472 yt = θ + εt σt² = α0 + α1εt-1² Hackl, Econometrics 2, Lecture 4 15 GARCH Models in GRETL nModel > Time Series > ARCH nEstimates the specified model allowing for ARCH: (1) model estimated via OLS, (2) auxiliary regression of the squared residuals on its own lagged values, (3) weighted least squares estimation nModel > Time Series > GARCH nEstimates a GARCH model, with or without exogenous regressors n April 1, 2011 Contents nARCH and GARCH Models nDynamic Models nLag Structures nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration n n n n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 16 The Lüdeke Model for Germany April 1, 2011 Hackl, Econometrics 2, Lecture 4 17 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 n n Econometric Models April 1, 2011 Hackl, Econometrics 2, Lecture 4 18 Basis is the multiple linear regression model Model extensions nDynamic models, i.e., models which contain lagged variables nSystems of regression relations, i.e., models which describe more than one dependent variable Example: Lüdeke Model nConsists of four dynamic equations nfor the four dependent variables C, Y, I, M n Dynamic Models: Examples April 1, 2011 Hackl, Econometrics 2, Lecture 4 19 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 1, 2011 Hackl, Econometrics 2, Lecture 4 20 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 1, 2011 Hackl, Econometrics 2, Lecture 4 21 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 nARCH and GARCH Models nDynamic Models nLag Structures nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration n n n n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 22 Example: Demand Functions April 1, 2011 Hackl, Econometrics 2, Lecture 4 23 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 + γ Pt + 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 + γδ, β0 = β, and βi = γθi, i = 1, 2, … Models with Lag Structures April 1, 2011 Hackl, Econometrics 2, Lecture 4 24 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 25 Example: Consumption Function nData for Austria (1976:1 – 1995:2), logarithmic differences: 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 in income ΔY = 1 n April 1, 2011 Hackl, Econometrics 2, Lecture 4 26 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 qeffect of the change in the same period, immediate effect of ΔX = 1 on Y: ΔY = φ0 nLong run multiplier qEffect of ΔX = 1 after 1, …, s periods: q q qCumulated effect of ΔX = 1 at t over all future on Y: ΔY = φ0 + … + φs April 1, 2011 Hackl, Econometrics 2, Lecture 4 27 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 1, 2011 Hackl, Econometrics 2, Lecture 4 28 Average Lag Time nCharacteristic of lag structure 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. n with 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 1, 2011 Hackl, Econometrics 2, Lecture 4 29 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 1, 2011 Hackl, Econometrics 2, Lecture 4 30 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: 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 1, 2011 Hackl, Econometrics 2, Lecture 4 31 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 1, 2011 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 1, 2011 Hackl, Econometrics 2, Lecture 4 32 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 1, 2011 Hackl, Econometrics 2, Lecture 4 33 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 34 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 1, 2011 Koyck Model: Estimation Problems April 1, 2011 Hackl, Econometrics 2, Lecture 4 35 Parameters to be estimated: δ, λ0, and λ; problems are nDL form: qHistorical values X0, X-1, … are unknown qNon-linear estimation problem nAR form qNon-linear estimation problem qLagged, endogenous variable used as regressor qCorrelated error terms Contents nARCH and GARCH Models nDynamic Models nLag Structures nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration n n n n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 36 The ADL(1,1) Model April 1, 2011 Hackl, Econometrics 2, Lecture 4 37 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 38 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 1, 2011 Hackl, Econometrics 2, Lecture 4 39 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 1, 2011 Hackl, Econometrics 2, Lecture 4 40 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 1, 2011 Hackl, Econometrics 2, Lecture 4 41 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 1, 2011 Hackl, Econometrics 2, Lecture 4 42 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 1, 2011 Contents nARCH and GARCH Models nDynamic Models nLag Structures nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration n n n n n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 43 Hackl, Econometrics 2, Lecture 4 44 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 1, 2011 Hackl, Econometrics 2, Lecture 4 45 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 1, 2011 Hackl, Econometrics 2, Lecture 4 46 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 1, 2011 Hackl, Econometrics 2, Lecture 4 47 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 1, 2011 Hackl, Econometrics 2, Lecture 4 48 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 1, 2011 Hackl, Econometrics 2, Lecture 4 49 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 1, 2011 Hackl, Econometrics 2, Lecture 4 50 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 1, 2011 Hackl, Econometrics 2, Lecture 4 51 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 1, 2011 Hackl, Econometrics 2, Lecture 4 52 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 1, 2011 Contents nARCH and GARCH Models nDynamic Models nLag Structures nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration n n n n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 53 Hackl, Econometrics 2, Lecture 4 54 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: n t = 17.1 (p-value n = 1.2 E-40) nR2 = 0.50, DW = 0.11 April 1, 2011 Hackl, Econometrics 2, Lecture 4 55 Spurious Regression nRegression model n Yt = α + βXt + εt n with two independent random walks n Yt = Yt-1 + ε1t, ε1t ~ IIDN(0, σ1² ) n Xt = Xt-1 + ε2t, ε2t ~ IIDN(0, σ2² ) n ε1t, ε2t mutually independent nConsequences for OLS estimators for α and β nt-statistic for β indicate explanatory power of Xt nR2 indicates explanatory potential nHighly autocorrelated residuals nNonsense or spurious regression (Granger & Newbold, 1974) nNon-stationary time series are trended; this causes an apparent relationship n April 1, 2011 Hackl, Econometrics 2, Lecture 4 56 Models in Non-stationary Time Series nNon-stationary time series are trended nExample: random walk with trend n Yt = δ + Yt-1 + εt or n Yt = Y0 + δt + Σi≤t εi n Yt‘s are correlated, show stochastic trend (even for δ = 0) 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 nR2 indicates explanatory potential n(Asymptotic) distributions of t- and F -statistics are different from those for stationarity nDW statistic converges for growing N to zero n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 57 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 nExample: ADL(1,1) model: nFor Yt ~ I(1), the error terms are stationary if θ =1 n εt = Yt – (δ + θYt-1 + φ0Xt + φ1Xt-1) ~ I(0) n April 1, 2011 Contents nARCH and GARCH Models nDynamic Models nLag Structures nADL Models nModels for Expectations nModels with Non-stationary Variables nCointegration n n n n n n April 1, 2011 Hackl, Econometrics 2, Lecture 4 58 Hackl, Econometrics 2, Lecture 4 59 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 1, 2011 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 60 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 n April 1, 2011 Hackl, Econometrics 2, Lecture 4 61 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 April 1, 2011 Hackl, Econometrics 2, Lecture 4 62 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 April 1, 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 Relation 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 1, 2011 Hackl, Econometrics 2, Lecture 4 63 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 64 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 nCointegration, i.e., existence of an equilibrium vector, implies a long-run equilibrium relation April 1, 2011 Hackl, Econometrics 2, Lecture 4 65 Identification of Cointegration nInformation about cointegration: nEconomic theory nVisual inspection of data nStatistical tests April 1, 2011 Hackl, Econometrics 2, Lecture 4 66 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 1, 2011 Hackl, Econometrics 2, Lecture 4 67 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 represent εt, show similar pattern, et ~ I(0), residuals are stationary nTests for cointegration based on residuals et n Δet = γ0 + γ0et-1 + ut nH0: γ0 = 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 1, 2011 Hackl, Econometrics 2, Lecture 4 68 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, K+1 qnumber of observations T qsignificance level q nsome asymptotic n critical values for the DF- n test with constant term q April 1, 2011 1% 5% Observations -3.43 -2.86 Residuals, K=1 -3.90 -3.34 Hackl, Econometrics 2, Lecture 4 69 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 nNull hypothesis: residuals et have a unit root, i.e., et ~ I(1), i.e., Xt and Yt are not cointegrated nDW statistic converges with T to zero for not cointegrated variables nCritical values from Monte Carlo simulations, which depend upon (see Verbeek, Tab. 9.4) qNumber of regressors plus 1 (dependent variable) qNumber of observations T qSignificance level April 1, 2011 Hackl, Econometrics 2, Lecture 4 70 PPP: Tests for Cointegration nResiduals from LNXt = α + β LNPt + εt: nTime series plot indicates non-stationarity of residuals nTests for cointegration qDF test statistic for residuals: -1.90, p-value: 0.33, no cointegration qCRDW test: DW statistic: 0.055 < 0.20, the critical value for two variables, 200 observations, significance level 0.05, no cointegration q q q q q q Time series plot q of residuals April 1, 2011 Hackl, Econometrics 2, Lecture 4 71 OLS Estimation of Equilibrium Relation nTo be estimated: n Yt = α + βXt + εt n cointegrated non-stationary processes Yt ~ I(1), Xt ~ I(1) n εt ~ I(0) nOLS estimator b for β nSuper consistent: qT(b – β) converges to zero qIn case of consistency: √T(b – β) converges to zero nRobust against misspecification in stationary part wrt asymptotic distribution of b nNon-standard distribution, non-normal, e.g., t-test misleading nSmall samples: bias April 1, 2011 Hackl, Econometrics 2, Lecture 4 72 OLS Estimation, cont’d nTo be estimated: n Yt = α + βXt + εt n non-stationary processes Yt ~ I(1), Xt ~ I(1) nIf εt ~ I(1), i.e., Yt and Xt not cointegrated: spurious regression nOLS estimator b for β nNon-standard distribution of b nLarge values of R2, t-statistic nHighly autocorrelated residuals nDW statistic close to zero April 1, 2011 Hackl, Econometrics 2, Lecture 4 73 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 1, 2011 Hackl, Econometrics 2, Lecture 4 74 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.Specify a DL(s) model in sd_INCOME (seasonal differences) and choose an appropriate s, 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 adapted expectation. 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 1, 2011 Hackl, Econometrics 2, Lecture 4 75 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 + εt; for εt 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 1, 2011