The simple random walk does not have a
tendency to increase or decrease over time since
its changes are serially uncorrelated and have
zero mean. Starting from an initial value y0,
yt = y0 + (ε1 + … + εt)
E(yt) = E(y0)
E(yt│yt-1,yt-2,…) = yt-1 (a rw is a martingale)
E(yt+s│yt-1,yt-2,…) = yt-1
We can modify the random walk to create a
tendency to grow over time by adding a constant
term to the model:
yt = β + yt-1 + εt , εt ~ iid
This is called a random walk with drift.
Note that
• E(∆yt) = β
So, on average the series will be
growing over time if β > 0
• yt = y0 + βt + (ε1 + … + εt)
(Derivation?)
So
E(yt) = E(y0) + βt
E(yt│yt-1,yt-2,…) = yt-1 + β
E(yt+s│yt,yt-1,…) = yt+ βs
• It is still the case that each innovation
has a permanent effect on the (dyt+s/dεt
=1 for s = 0,1,…).
• We can think of the trend of this series
as being made up of the sum of two
parts – a deterministic component, βt,
and a stochastic component, ε1+…+εt.
We sometimes refer to this process as
having a stochastic trend.
In effect, {yt}follows a linear trend that
shifts up/down with each new ε; the
slope of the trend line is nonstochastic
but the intercept is a zero mean iid
sequence.
• In fact all I(1) processes have
stochastic trends made up of a
deterministic function of t and a
stochastic (though not necessarily i.i.d.)
intercept because at least part of each
period’s innovation will have a
permanent effect on the level of the
series, i.e., lims→∞dyt+s/dεt≠0.
There is no cyclical component to the random
walk (with or without drift) since each
innovation changes both the trend and actual
level of the series by the same amount. That is,
yt is always equal to its trend level. There are no
temporary or transitory deviations from the
trend.
We can construct I(1) processes with meaningful
cyclical components by replacing εt in the
random walk model with ut, where ut is a zeromean stationary (vs. iid) process.
C. The more general I(1) Process
yt = β + yt-1 + ut
ut ~ I(0) with mean zero and finite variance.
• E(∆yt) = β
So, like the random walk with drift, on
average the series will be growing over
time if β > 0.
• Cov(∆yt, ∆yt-s) = cov(ut,ut-s)
So, unlike the random walk with drift,
changes in yt can be serially correlated.
• yt = y0 + βt + (u1 + … + ut)
So
E(yt) = E(y0) + βt
E(yt│yt-1,yt-2,…) = yt-1+β+E(ut│ut-1,…)
Suppose, e.g., that ut = εt – δεt-1 , 0 <δ<1
Then
yt = y0 + βt + (u1 + … + ut)
= y0 + βt +[(ε1-δε0)+(ε2-δε1)+…+(εt-δεt-1)]
= y0 + βt +[-δε0 +(1-δ)ε1+…+(1-δ)εt-1+εt]
So
dyt/dεt = 1
dyt+s/dεt = 1-δ for s = 1,2,…
• The long-run effect of a shock at t is a
fraction of the (short-run effect of the)
shock; the fraction decreases as δ
increases.
• So at time t, the intercept of the trend
line jumps up/down by (1-δ)εt but yt
will be above/below the new trend line
in period t. In period t+1, in the
absence of any new shocks, the series
will return to trend.
• Since there are periods when the
system is above/below trend, we can
talk about the series having a cyclical
and trend components.
More generally, if yt is a DS process then:
∆yt = β + a(L)εt
where εt is a w.n. process and
∞
∞
a ( L) = ∑ a j L
i =0
j
a 2j < ∞
, a0 = 1, ∑
i =0
Fact:
s
aj
dyt+s/dεt = ∑
0
It follows that
∞
aj
lims→∞dyt+s/dεt = a(1) = ∑
0
{Example: a(L)= a0 = 1, then a(1)= 1; rw
a(L) = 1- δL, a(1) = 1- δ}
So, the general I(1) process will have a
stochastic trend but will, in any given period t,
lie above or below that trend. That is, it will
have a permanent (trend) and a transitory
(cyclical) component. How do we decompose a
general I(1) series into these two component?
That is, how do we detrend an I(1) series?
D. The Beveridge-Nelson (BN) Decomposition
It seems natural to define the change in the trend
component of an I(1) process yt as the sum of β,
the drift parameter, which is the deterministic
change in the trend) and the long-run effect of
the current innovation, εt. That is:
τt = τt-1 + β + a(1)εt
Note that τt is a random walk with drift.
The cyclical component of yt is
ct = yt-τt.
Assume that the economy is on its trend path in
period t-1 and a positive ε shock occurs in
period t. The trend line shifts up by a(1)εt,
assuming a(1)>0. yt itself increases by β+εt. If,
e.g., 0 < a(1)<1, then yt < τt and ct < 0.
Notice that if ct < 0, then y will be expected to
grow faster than the normal rate (β) in the short
run toward trend. If ct > 0, then y will be falling
in the short run toward trend. So ct < 0 =
expansion and ct > 0 = recession.
The key to making this operational is to
recognize that if the trend in yt is a random walk
then, setting aside the drift term,
τt = lims→∞Etyt+s .
Then, note that
So
yt+s = yt+s + (yt+s-1-yt+s-1) +…+ (yt-yt)
= yt + ∆yt+s + … + ∆yt+1
τt = lims→∞Etyt+s
= yt + lims→∞Et(∆yt+s + … + ∆yt+1)
≈ yt + Et(∆yt+s + … + ∆yt+1), for large s
How do we compute the Et∆yt+j’s?
• Assume that ∆yt has an AR(p) (or MA(q) or
ARMA(p,q)) form (with an intercept)
• Fit the AR(p) model to ∆y1,…,∆yT
• For each t, use the AR(p) model to forecast
∆yt+1,…, ∆yt+s
Once the trend component has been estimated,
the cyclical component is simply yt-τt.
Notes
• How to choose s? Large enough that
Et(∆yt+s + … + ∆yt+1) appears to converge.
(In the applied part of their paper BN used
s=100 with quarterly data.)
• A number of algorithms have been
developed that simplify the computation of
the BN decomposition of a series.
The BN decomposition assumes that the trend
and cyclical components are generated by
exactly the same innovations, which may or may
not be appealing.
At the other extreme, Peter Clark (1987, QJE)
suggested an approach that also begins with the
idea that the generic DS stationary process
∆yt = β + a(L)εt
can be decomposed into the sum of a random
walk trend component and a stationary cyclical
component. But in Clark’s framework εt can be
decomposed into two uncorrelated w.n.
components, say vt and wt, one of which drives
the trend component and the other drives the
cyclical component.
In fact it has been shown that a decomposition of
the BN or Clark type can be constructed for any
specified correlation between vt and wt!The
decomposition of a DS time series into a random
walk plus stationary component is not identified
until the correlation between the innovations
driving each component is specified.