Palgrave Handbook of Econometrics: Applied Econometrics

272 Investigating Economic Trends and Cycles which is obtained fromε(t)by differencingm−dtimes. Then, the filter that is applied ...

D.S.G. Pollock 273 0 0.25 0.50 0.75 1.00 1.25 0 π/ 4 π/ 2 3 π/ 4 π B (^1600) H Figure 6.11 The gain of the HP filterHand of the ...

274 Investigating Economic Trends and Cycles structure of the residual component. More elaborate filters are available that also ...

D.S.G. Pollock 275 the canonical decomposition of the generating function can be represented by: θ(z)θ(z−^1 ) φ(z)φ(z−^1 ) = γT( ...

276 Investigating Economic Trends and Cycles The trend process is usually depicted as the product of two processes that constitu ...

D.S.G. Pollock 277 0 0.25 0.50 0.75 1.00 0 π/ 4 π/ 2 3 π/ 4 π Figure 6.12 The gain function of the trend-extraction filter obtai ...

278 Investigating Economic Trends and Cycles filter has only a short span, it matters little which assumptions are made about th ...

D.S.G. Pollock 279 and that: [ Q′∗ Q′ ][ S∗ S ] = [ Q∗′S∗ Q∗′S Q′S∗ Q′S ] = [ Ip 0 0 IT−p ] . (6.114) Ifg∗is available, thenycan ...

280 Investigating Economic Trends and Cycles A more general method of curve fitting, which embeds polynomial regression as a spe ...

D.S.G. Pollock 281 IfQ′is thepth difference operator, thenQ′φ=μι, withι=[1, 1,...,1]′, will contain a constant sequence of value ...

282 Investigating Economic Trends and Cycles 6.7.3 The polynomial component The formula (6.123) tends to conceal the presence of ...

D.S.G. Pollock 283 The solution is: k∗=−(S′∗−^1 S∗)−^1 S∗′−^1 Sk. (6.137) The starting valuesk∗andd∗can be eliminated from the ...

284 Investigating Economic Trends and Cycles this way, the generating functionγ(z)gives rise to the matrix: ◦=γ(KT) =γ 0 IT+ ∑∞ ...

D.S.G. Pollock 285 wherein the diagonal matricesγδ(D)andγκ(D)contain the ordinates of the spectral density functions of the comp ...

286 Investigating Economic Trends and Cycles 0.0 0.1 0.2 0.3 0.4 0.5 –0.1 –15 –10 –5 00 5 10 15 Figure 6.13 The central coeffici ...

D.S.G. Pollock 287 0.00 0.25 0.50 0.75 1.00 1.25 –0.25 −π −π/ 2 0 π/ 2 π Figure 6.14 The frequency response of the 17-point wrap ...

288 Investigating Economic Trends and Cycles The periodic extension of the linearly trended sequence, which would be gen- erated ...

D.S.G. Pollock 289 10.0 10.5 11.0 11.5 12.0 1960 1970 1980 1990 2000 Figure 6.15 The logarithms of quarterly household expenditu ...

290 Investigating Economic Trends and Cycles The ordinates at timest=0,...,T−1 of the business cycle that is represented in Figu ...

D.S.G. Pollock 291 To demonstrate this result, we must consider the Fourier representation of a real- valued square-integrable f ...