ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Tutorial 11: Cointegration – The Multiple Equation Case
The data file term structure.csv contains data of a system of four Australian in- terest rates: the 5 year (i5y) and 3 year (i3y) Treasury Bond (capital market) rates, along with the 180 day (i180d) and 90 (i90d) day Bank Accepted Bill (money market) rates. The data are annualized monthly rates for the period June 1992 to August 2010 (T = 219).
(a) Consider the system of four rates as a VAR system. Test for the appropriate lag length.
(b) Using the chosen lag length, test for the cointegration rank and most suitable VECM specification. Note that Stata implements a method called the Johansen test for this purpose. Given the outcome of the test, choose the appropriate model to continue the analysis. What is the cointegration rank r?
(c) Given r, fit the VECM to the data using the following identifying restrictions: i90dt
1β12 −1β14i3y t,
0 1 β23 β24 i180dt i5yt
and plot the cointegrating vectors.
(e) Produce impulse response functions based on the VECM. Comment on your findings.
Solution: See the do-file tutorial11.do.
(a) The VAR(2) model suggested by BIC seems to fit the data well. Although the AIC favours the VAR(5) the VAR(2) yields acceptable residuals, and we con- tinue to work with this lower-order VAR (while keeping in mind the VAR(5) as a robustness check). Observe that estimates indicate both the VAR(2) and the VAR(5) to be stable, but with at least one eigenvalue very close to one. As we will see below, this suggests that standard information criteria based on unrestricted VARs may be less reliable in terms of inference regarding the lag length.
(b)
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(b) The general specification for VECM in Stata is
p−1
xt =γ+τt+α(μ+ρt+βxt−1)+Γj∆xt−j +εt.
j=1
Restricted specifications are obtained with the option trend(), which re-
sults in imposing various restrictions on γ, τ, μ, and/or ρ. Specifically,
• trend: no restrictions;
• rtrend: τ = 0;
• constant:ρ=τ =0;
• rconstant:ρ=γ=τ=0; • none:μ=ρ=γ=τ=0.
Given p = 2, the cointegration rank r = 2 seems to be chosen by all criteria and tests. The BIC chooses the model with no unrestricted intercept or de- terministic trend, and no deterministic trend in the cointegrating vector. In what follows, our analysis employs this specification.
(c) See the log file for the estimation results. We can use the estimated coeffi- cients on the cointegrating vectors to calculate the “error correction” terms (i.e., residuals of the regressions for the long-run relationship). The time se- ries plots of the saved residuals show that the two long-run relationships are mean-reverting, as expected (i.e., they appear to be stationary). The speed of adjustments are within the expected range of [0, 1) in absolute value. How- ever, it is usually difficult to interpret what these long-run relationships are capturing.
Note that the VECM residuals appear to exhibit significant autocorrelations, suggesting that perhaps p = 2 is not sufficiently large. Recall that the VECM imposes certain non-linear restrictions on the parameters of the VAR, which (assume the restrictions are correct) improves the accuracy of estimated auto- correlations in the residuals as well as information criteria. Consequently, we may wish to re-assess the lag length of the VAR using the VECM specification directly.
(e) From the IRFs, we can conclude that
• An unexpected shock to i90d induces a permanent negative response in all other rates.
• An unexpected shock to i180d induces a more significant positive per- manent response in money market rates than in capital market rates.
• An unexpected shock to i3y induces a significant positive, permanent response in money market rates and a small negative but permanent response in i5y. However, it seems that an unexpected shock to i3y dis- sipates over time.
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• An unexpected shock to i5y induces a significant negative, permanent response in the money market, while in the capital market, it induces a more significant (positive) permanent response in i5y than in i3y.
Notice that different identifying restrictions may lead to different IRF results. Feel free to try other identifying restrictions.
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