Palgrave Handbook of Econometrics: Applied Econometrics

252 Investigating Economic Trends and Cycles 0 0.05 0.10 0.15 0.20 0 π/ 4 π/ 2 3 π/ 4 π Figure 6.6 The periodogram of the residu ...

D.S.G. Pollock 253 For matrix representations of these transforms, one may define: U=T−^1 /^2 [exp{−i2πtj/T};t,j=0,...,T− 1 ], U ...

254 Investigating Economic Trends and Cycles must hold for allj. For (6.25) and (6.26) to hold, it is sufficient that: E(αjβk)=0 ...

D.S.G. Pollock 255 The assumptions regardingdA(ω)anddB(ω)are analogous to those regarding the random variablesαjandβj, which are ...

256 Investigating Economic Trends and Cycles In the case of a continuous spectral distribution function, we may writedF(ω)= f(ω) ...

D.S.G. Pollock 257 The ARMA(2, 2) process has been formed by the additive combination of a second-order autoregressive AR(2) pro ...

258 Investigating Economic Trends and Cycles On writing ∑ ψje−iωj=ψ(ω), which is the frequency response function of the filter, ...

D.S.G. Pollock 259 0 1 2 3 4 0 π/ 4 π/ 2 3 π/ 4 π D W N Figure 6.8 The squared gain of the difference operator, labeledD, and th ...

260 Investigating Economic Trends and Cycles section 6.3, we have encountered a process with a limited bandwidth. Later, in sect ...

D.S.G. Pollock 261 from which it is manifest that the necessary and sufficient condition for stability is thatλ<0. That is to ...

262 Investigating Economic Trends and Cycles Here,(λ 1 −λ 2 )( 1 −κ 1 L)( 1 −κ 2 L)= 1 +φ 1 L+φ 2 L, and we have defined(θ 0 +θ ...

D.S.G. Pollock 263 The covariance of the changesZ(j)(t)−Z(j)(t−h)andZ(k)(t)−Z(k)(t−h)of the jth and thekth integrated processes ...

264 Investigating Economic Trends and Cycles On the right-hand side of this equation is a sum of stationary stochastic processes ...

D.S.G. Pollock 265 0 5 10 15 20 25 –5 0 50 100 150 200 250 Figure 6.9 The graph of 256 observations on a simulated series genera ...

266 Investigating Economic Trends and Cycles 0 500 1000 1500 2000 0 50 100 150 200 250 Figure 6.10 The graph of 256 observations ...

D.S.G. Pollock 267 The usual recourse in the face of this problem is scrupulously to avoid the use of the cumulation operator∇−^ ...

268 Investigating Economic Trends and Cycles applying a filterθ(z)/α(z)to a white-noise sequenceε(t)of independently and identic ...

D.S.G. Pollock 269 are respectively the trend component and the transitory component. This is the so-called Beveridge–Nelson dec ...

270 Investigating Economic Trends and Cycles To enhance the smoothness of the trend and of the seasonal component yet further, a ...

D.S.G. Pollock 271 information set. Thus, the following condition applies for allk: 0 =E { yt−k(ξt−xt) } =E(yt−kξt)− ∑ j βjE(yt− ...