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.,c)Test the residuals for unit roots using an ADF testtttXYεβμ++=Σ−=−−+Δ+=Δ111pjtjtjttvεδρεεioncointegratstationary is 0:ioncointegrat noarynonstation is 0:10⇒⇒<⇒⇒=ερερHHNo trend or interceptis required in this ADF test equation.The alternative is that εis stationary around a zero mean (the residuals have a zero mean by construction if the long-runequation includes a constant)Warwick Business School 23
Engle and Granger Two Step EstimatorStep 2 (Estimate the short run dynamics)If ε~I(0) then we can estimate an ECM using OLS e.g.,ΣΣ==−−−++Δ+Δ+=ΔmipjttjtjititvYXY011ˆεαγδαOnly include ΔXtin the model if X is ‘weakly exogenous for the short run parameters’⇒cov(ΔXt,vt)=0Sometimes a model with just lagged values of ΔX(the ‘reduced form’) is estimated due to endogeneityissues.Lagged long-run residuals from Step 1.α<0 is the speed of adjustmentto dis-equilibrium in the previousperiod.The short-run dynamics are obtained fromthese parametersInclude an intercept in the ECMif there are trends in the data(analogous to including a drift termin a random walk).Warwick Business School 24
Comments on EG 2 step
The OLS estimators of the long-run parameters are super-consistent (see slide 11).
However do notuse t and F tests for inferences –they are invalid for inferences due to the nonstationarityof the variables.
Also standard DF critical values are invalid for the cointegrationtest
–The test involves estimated residuals rather than raw data.
OLS (which minimizes the residual variance) will tend to make these residuals appear stationaryeven if there is no cointegration.
–Standard DF critical values will tend to reject the null (no cointegration) too often.
–Use alternative critical values which take this issue into account (see Seminar 7).Warwick Business School 25
Conclusions
Distinguishing between spurious regressionsand cointegratingrelationshipsis crucialin multivariate analyses involving nonstationaryvariables.
Only in the presence of cointegrationis there a meaningful relationship between the variables.
Therefore cointegrationanalysis (testing for long-run relationships, estimating ECMs…) forms a fundamental (and commonplace) part of modern time-series econometrics.
In this context the EG 2 Step estimator has intuitive appeal and is easily implemented (see Seminar 7).
However there are major drawbacks with this approach which we will address in the final lecture.Warwick Business School 26
References
Brooks (2002), Introductory econometrics for finance, CUP: Cambridge. Chps7.4-7.8**.
Cuthbertsonand Nitzsche(2004) Quantitative financial economics: stocks, bonds and foreign exchange, Wiley: Chichester. Chp20.1 and 22.1
Verbeek(2004) A Guide to Modern Econometrics, 2nd Edition, Wiley: Chichester. Chps9.1-9.3.本论文由英语论文网提供整理，提供论文代写，英语论文代写，代写论文，代写英语论文，代写留学生论文，代写英文论文，留学生论文代写相关核心关键词搜索。