ECON 3350/7350: Applied Econometrics for Macroeconomics and Finance
Tutorial 5: Deterministic and Stochastic Trends The specification for a general ARIMA(p, d, q) model is
pq
∆dyt =πi∆dyt−i +αjεt +δt
i=1 j=0
where α0 = 1.
• If you decide there is a constant only then
δt = a0
• If you decide there is a constant and a trend then
δt = a0 + a2t
• If you decide there is a constant, a trend and a quadratic trend, then
δt = a0 + a2t + a3t2 • If you decide there are no deterministic trend, then
δt = 0
Testing for Trends Lecture slides provide a procedure to test for unit roots which involves the three testing equations (constant and trend, constant, and no constant or trend). However, this procedure is a bit complicated. We will be using a simplified version.
1. WeconducttheADFtests(ττ,τμ,τ)usingthethreetestequations,andERStests. (a) ADF
△yt =ao +γyt−1 +a2t+β△yt−1 +εt (1) △yt =ao +γyt−1 +β△yt−1 +εt (2) △yt =γyt−1 +β△yt−1 +εt (3)
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(b) ERS
ytd = yt−aˆ0−aˆ2t (4) or
(5)
ytd = yt−aˆ0
∆yd = γyd +c∆yd +ε
p
t t−1 it−it
i=1
2. If we reject γ = 0 in all cases, we conclude the series has no unit roots.
3. If we reject γ = 0 using equations (1) and (4) we can conclude the series has no unit roots. In this case we do not consider the results from the other equations and proceed to test for a deterministic trend.
4. If we fail to reject γ = 0 in all cases, we can conclude the series has at least one unit root.
5. If the tests on equations (1), (2), (4) and (5) fail to reject γ = 0, we test further for the option of a deterministic trend in addition to the stochastic trend.
We use the following diagram to guide the testing procedure (modified from En- ders’).
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Practical approach to test for trends
Testing Equations:
Δy = 𝑎 + 𝑎 t + γy + ∑p−1 β Δy + ε (1)
t 0 2 t−1 i=1 i t−i t
Δy=𝑎+γy +∑p−1βΔy +ε (2) t 0 t−1p−1 i=1 i t−i t
Δyt=γyt−1 +∑i=1βiΔyt−i+εt (3)
∆𝑦𝑡 = 𝑎𝑜 + 𝛾𝑦𝑡−1 + 𝑎2 𝑡+ ∑𝛽∆𝑦𝑡−1 + 𝜀𝑡 Eq(1)use𝜏𝑡 totest𝛾=0
Is:𝛾=𝑎2 =0
∆𝑦𝑡 = 𝑎𝑜 + 𝛾𝑦𝑡−1 + ∑𝛽∆𝑦𝑡−1 + 𝜀𝑡 Eq(2)use𝜏𝜇 totest𝛾=0
If 𝛾 =0
Estimate the model (manually)
∆𝑦𝑡 =𝑎𝑜+𝛾𝑦𝑡−1 +𝑎2𝑡+ ∑𝛽∆𝑦𝑡−1 +𝜀𝑡 Conduct ‘Wald’ Test : 𝛾 = 𝑎2 = 0, using 𝜙3critical values
No
Yes
Is 𝑎2 = 0 using t distribution
Is γ =0 using t distribution
No
Series is trend stationary
No
Yes
Has unit roots with a quadratic time trend
Is 𝑎0 = 0 using t distribution
No
Is γ =0 using t distribution .
……………………………………………
… …………… ………
… …………… ……… ……………………………………………. .
SeriesisstationaryIfreject 𝛾=0inEq(3)using𝜏totest =0
…………
…………
Yes
No
…………
Yes
…………
Has unit roots with a drift
Yes
Series is stationary around non zero mean
Tutorial Questions The file usdata.csv contains 209 observations on: • rt = the overnight Federal Funds Rate for the US (ffr);
• yt = log real per capita GDP (gdp); and
• pt = the natural log of the CPI for the US (cpi).
1. For rt
(a) Plot the time series plot.
(b) Conduct the appropriate tests for deterministic and stochastic components using ADF (equations (1), (2), and (3)). Transform the series as needed be- fore continuing.
(c) Plot the ACF and PACF to suggest AR and MA terms needed.
(d) Propose three possible models.
(e) Estimate the proposed models.
(f) Which model do you choose? Why?
2. Repeat for yt. 3. Repeat for pt.
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