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 constantterm to the model:yt = β + yt-1 + εt , εt ~ iidThis is called a random walk with drift.Note that- E(Δyt) = βSo, on average the series will begrowing over time if β > 0- yt = y0 + βt + (ε1 + … + εt) (Derivation?)SoE(yt) = E(y0) + βtE(yt│yt-1,yt-2,…) = yt-1 + βE(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 thatshifts 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 randomwalk (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 meaningfulcyclical components by replacing εt in the random walk model with ut, where ut is a zero-mean stationary (vs. iid) process.C. The more general I(1) Processyt = β + yt-1 + ut ut ~ I(0) with mean zero and finite variance.- E(Δyt) = βSo, like the random walk with drift, on average the series will be growing over time if β > 0. - Cov(Δyt, Δyt-s) = cov(ut,ut-s)So, unlike the random walk with drift, changes in yt can be serially correlated.- yt = y0 + βt + (u1 + … + ut)SoE(yt) = E(y0) + βtE(yt│yt-1,yt-2,…) = yt-1+β+E(ut│ut-1,…)Suppose, e.g., that ut = εt – δεt-1 , 0 <δ<1Thenyt = 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 = 1dyt+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:Δyt = β + a(L)εtwhere εt is a w.n. process and0)(ijjLaLa, ,10a 02ijaFact: dyt+s/dεt = sja0It follows that lims→∞dyt+s/dεt = a(1) = 0ja{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 trendcomponent 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)εtNote 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 thatyt+s = yt+s + (yt+s-1-yt+s-1) +…+ (yt-yt)= yt + Δyt+s + … + Δyt+1Soτt = lims→∞Etyt+s = yt + lims→∞Et(Δyt+s + … + Δyt+1) ≈ yt + Et(Δyt+s + … + Δyt+1), for large sHow do we compute the EtΔyt+j’s?-Assume that Δyt has an AR(p) (or MA(q) or ARMA(p,q)) form (with an intercept)-Fit the AR(p) model to Δy1,…,ΔyT-For each t, use the AR(p) model to forecast Δyt+1,…, Δyt+sOnce the trend component has been estimated, the cyclical component is simply yt-τt.Notes- How to choose s? Large enough thatEt(Δyt+s + … + Δyt+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 maynot be appealing. At the other extreme, Peter Clark (1987, QJE) suggested an approach that also begins with the idea that the generic DS stationary processΔyt = β + a(L)εtcan 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 ofthe BN or Clark type can be constructed for any specified correlation between vt and wt!The decomposition of a DS time series into a randomwalk plus stationary component is not identified until the correlation between the innovations driving each component is