ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Tutorial 4: Single Equation Models of Multiple Time Series
ARDL and ECM.
1. Derive the ECM representation of the following ARDL(1,1,2) model:
ct =δ+θ1ct−1 +γ0at +γ1at−1 +λ0yt +λ1yt−1 +λ2yt−2 +εt
Which parameter(s) in the resulting ECM are long-run multiplier(s) and adjust-
ment parameter(s)?
2. The file wealth.csv contains observations on: 1
Solution:
• ct = the log of total real per capita expenditures on durables, nondurables and services;
• at = the log of a measure of real per capita household net worth(including all financial and household wealth); and
• yt = the log of after-tax labour income.
The data are from 1952Q2 through 2006Q2 (see Koop, G., S. Potter and R. W. Stra- chan (2008) “Re-examining the consumption-wealth relationship: The role of un- certainty” Journal of Money, Credit and Banking, Vol. 40, No. 2.3, 341-367.
(a) Draw time series plots of ct, at, and yt. Compute and plot the ACF and PACF of ct, at, and yt. Comment on your findings.
(b) Fit ARDL(p, q, m) models to the data with each component order of (p, q, m) up to 2. Use BIC for model selection. Report the best model. Hint: Install the ardl package and use its ardl command.
(c) Estimate the ECM representation of the ARDL model selected in Part (b) and report the estimated model. Hint: Use the ec1 option with the ardl com- mand.
Solution: See the do-file tutorial4.do.
(a) For all these three processes, we can see that
(1) There is an obvious (upward) trend. (2) ACF decays very slowly.
(3) PACF has one peak very close to 1.
These results imply that none of these three processes is stationary and prob- ably the most suitable model for them is an ARMA model with AR coefficient ≈ 1.
(b) Based on BIC, we choose the ARDL(1,2,2) model. The estimated model can be represented as
ct =0.0725 + 0.0002t + 0.9145ct−1 + 0.0558at + 0.0009at−1 − 0.0511at−2 (0.0347) (0.0001) (0.0284) (0.0187) (0.0258) (0.0180)
+ 0.3589yt − 0.1204yt−1 − 0.1972yt−2 (0.0424) (0.0595) (0.0435)
R2 =0.9998
(c) The estimated ECM can be represented as
∆ct =0.0725 + 0.0002t − 0.0855(ct−1 − 0.0651at−1 − 0.4836yt−1 (0.0347) (0.0001) (0.0284) (0.0865) (0.0943)
+ 0.0558∆at + 0.0511∆at−1 + 0.3589∆yt − 0.1972∆yt−1 (0.0187) (0.0180) (0.0424) (0.0436)
R2 =0.4142
2