the econometrics of the hodrick-prescott lter preliminary · 2018-02-12 · the econometrics of the...

The Econometrics of the Hodrick-Prescott filter

Preliminary

Robert M. de Jong∗ Neslihan Sakarya†

May 19, 2013

Abstract

The Hodrick-Prescott (HP) filter is a commonly used tool in macroeconomics, andis used to extract a trend component from a time series. In this paper, we derive anew representation of the transformation of the data that is implied by the HP filter.This representation highlights that the HP filter is a weighted average plus a numberof adjustments that are important near the begin and end of the sample. Using thisrepresentation, we characterize the large T behavior of the HP filter and find conditionsunder which it is asymptotically equivalent to a symmetric weighted average. We findthat the cyclical component of the HP filter possesses weak dependence propertieswhen the HP filter is applied to a stationary mixing process, a linear deterministictrend process and/or a process with a unit root. This justifies the use of the HP filteras a tool to achieve weak dependence in a time series and illustrates that the finding inempirical macro that data series tend to have deterministic trends and/or unit rootsand the practice of using inference procedures based on the cyclical component of theHP filter are not contradictory. In addition, a large bandwidth approximation to theHP filter is derived, and using this approximation we find an alternative justificationfor the procedure given in Ravn and Uhlig (2001) for adjusting the bandwidth for thedata frequency.

∗Department of Economics, Ohio State University, 444 Arps Hall, Columbus, OH 43210, USA, [email protected].†Department of Economics, Ohio State University, 410 Arps Hall, Columbus, OH 43210, USA, Email

[email protected].

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1 Introduction

The Hodrick-Prescott (HP) filter is the standard technique in macroeconomics for separatingthe long run trend in a data series from short run fluctuations. While it seems intuitivelyclear that no smoothing technique should be equally well applicable to all types of trendedmacroeconomic data, the HP filter is universally used in macroeconomics, and while differenttypes of criticism can be found in the literature on the HP filter, as Ravn and Uhlig (2002)state, “the HP-filter has withstood the test of time and the fire of discussion remarkablywell.” Well-cited papers in which the HP filter is applied are for example Kydland andPrescot (1990) and Backus and Kehoe (1992); however, the HP filter is at this time acommonplace tool in macroeconomics.

The HP filter smoothed series τT = (τT1, τT2, . . . , τTT )′ as introduced into Economics inHodrick and Prescott (1980,1993) results from minimizing, over all τ ∈ RT ,

T∑t=1

(yt − τt)2 + λT−1∑t=2

(τt+1 − 2τt + τt−1)2, (1)

where T denotes sample size, λ is a (positive) smoothing parameter that for quarterly datais often chosen to equal 1600, and y = (y1, . . . , yT )′ is the data series to be smoothed. Ithas been pointed out that in Whittaker (1923) a similar filtering technique was introduced.The τTt are typically referred to as the “trend component,” while yt − τTt is referred toas the “cyclical component.” It can be shown that there exists a unique minimizer to theminimization problem of Equation (1), and that for a known positive definite (T ×T ) matrixFT , letting IT denote the (T × T ) identity matrix,

y = (λFT + IT )τT and τT = (λFT + IT )−1y; (2)

see for example Kim (2004). Therefore the trend component τTt and the cyclical componentyt − τTt are both weighted averages of the yt, and in the sequel of this paper, we writeτTt =

∑Ts=1wTtsys; note that the dependence of wTts and τTt on λ is suppressed for notational

convenience. However, the inability to find a simple expression for the elements of (λFT +IT )−1 has prevented researchers from finding a simple expression for the weights that areimplicit in the HP filter. It also follows from the definition of the HP filter that τTt(y1+1, y2+1, . . . , yT + 1) = τTt(y1, y2, . . . , yT ) + 1, and therefore,

∑Ts=1wTts = 1 for t ∈ {1, 2, . . . , T}.

Also from the definition of the HP filter it follows that τTt(1, 2, . . . , T ) = t, implying that∑Ts=1wTtss = t for t ∈ {1, 2, . . . , T}.

The first order condition for τTt, t ∈ {3, . . . , T − 2} is

−2(yt−τTt)−4λ(τT,t+1−2τTt+τT,t−1)+2λ(τTt−2τT,t−1+τT,t−2)+2λ(τT,t+2−2τT,t+1+τTt) = 0

2

(3)

which, letting B denote the forward operator and B the backward operator, simplifies to

yt = (λB2 − 4λB + (1 + 6λ)− 4λB + λB2)τTt, (4)

which can be written as

yt = (λ|1−B|4 + 1)τTt. (5)

Some papers in the literature that analyze the HP filter, such as King and Rebelo (1993),Cogley and Nason (1995), and McElroy (2009) take the above equation as the definition of theHP filter, thereby effectively studying an approximate procedure that intuitively makes sensein large samples for values of t away from the begin and end points of the sample. Therefore,such an approach gives an informal large T approximation to the HP filter without a formallarge T justification; is not able to address end point issues; is unable to address the large Tvalidity of the approximation; and cannot establish formal properties of the resulting cyclicalor trend components, such as the weak dependence properties of the cyclical component. Thelatter is of particular interest, since weak dependence properties of the cyclical componentguearantee that standard asymptotic inference procedures are valid when applied to thecyclical component.

2 The weights of the HP filter

In this section, we derive the exact weights wTts implied by the HP filter. Our analysis startsby noting that we can also equivalently minimize over (θ1, . . . , θT ), for a basis of functionspj(·) : [0, 1]→ R for T, j ∈ N+,

T∑t=1

(yt−T∑j=1

θjpj(t/T ))2+λT−1∑t=2

(T∑j=1

θjpj((t+1)/T )−2T∑j=1

θjpj(t/T )+T∑j=1

θjpj((t−1)/T ))2

=T∑t=1

(yt − θ′pTt)2 + λ

T−1∑t=2

(θ′∆2pT,t+1)2 (6)

where θ′ = (θ1, . . . , θT ) and p′Tt = (p1(t/T ), p2(t/T ), . . . , pT (t/T )). Differentiation with re-

spect to θ now yields for the minimizer θT

0 = −2T∑t=1

ytpTt + 2T∑t=1

pTtp′TtθT + 2λ

T−1∑t=2

(∆2pT,t+1)(∆2pT,t+1)

′θT , (7)

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which, if the inverse exists, is solved for

θT = (T−1T∑t=1

pTtp′Tt + λT−1

T−1∑t=2

(∆2pT,t+1)(∆2pT,t+1)

′)−1T−1T∑t=1

ytpTt. (8)

Now we choose p1(t/T ) = 1 and pj(t/T ) =√

2 cos(π(j − 1)(t− 1/2)/T ), j = 2, . . . , T . Notethat this is a minor abuse of notation, since formally, pj(·) depends on T . We can then derivethe following result. Below, IT denotes the identity matrix of dimension (T × T ).

Lemma 1. Let pTt = (p1(t/T ), . . . , pT (t/T ))′, where p1(t/T ) = 1 and pj(t/T ) =√

2 cos(π(j−1)(t− 1/2)/T ), j = 2, . . . , T . Then

T−1T∑t=1

pTtp′Tt = IT (9)

and

T−1T−1∑t=2

(∆2pT,t+1)(∆2pT,t+1)

′

= diag({16 sin(π(j − 1)/(2T ))4, j = 1, . . . , T})− 32T−1qT1q′T1 − 32T−1qT2q

′T2, (10)

and for j ∈ {1, ..., T}, qT1j = sin(π(j − 1)/(2T ))2 cos(π(j − 1)/(2T )) and qT2j = sin(π(j −1)/(2T ))2 cos(π(j − 1)(T − 1/2)/T ).

The proof of this result and all proofs for this paper can be found in the MathematicalAppendix.

The importance of the above result is that the matrix to be inverted is now “close” toan easily invertible diagonal matrix (in the sense that two matrices of rank 1 have beenadded to the diagonal matrix), and therefore allows for a more tractable expression. Whenminimizing the criterion of Equation (1) over τ , no such structure occurs. It is well-knownthat explicit formulas can be obtained for the inverse of the sum of a matrix plus anothermatrix of rank 1, and such results can be adapted to deal with the inverse of a matrix plusa matrix of rank 2 as well. Our strategy will be to use such a result, as obtained in Miller(1981), in order to obtain a tractable expression for τTt.

For the theorem below in which we derive the exact weights of the HP filter, we need todefine, for m ∈ Z,

fTλ(m) = 1/(2T ) + (−1)m(2T )−1(1 + 16λ)−1

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+T−1T∑j=2

cos(π(j − 1)m/T )(1 + 16λ sin(π(j − 1)/(2T ))4)−1 (11)

and

gTλ(m) = T−1T∑j=1

√2 cos(π(j− 1)(m− 1/2)/T )qT1j(1 + 16λ sin(π(j− 1)/(2T ))4)−1. (12)

In addition, define the sequences

δTλ = T−1q′T1(IT + λDT )−1qT1, (13)

ηTλ = T−1q′T1(IT + λDT )−1qT2, (14)

ξTλ = 32λ(1− 64λδTλ)(1− 64λδTλ + 322λ2(δ2Tλ − η2Tλ))−1

+322λ2(1− 64λδTλ + 322λ2(δ2Tλ − η2Tλ))−1δTλ, (15)

and

φTλ = 322λ2(1− 64λδTλ + 322λ2(δ2Tλ − η2Tλ))−1ηTλ. (16)

Note that fTλ(m) = fTλ(−m) for all m ∈ Z. With these definitions in place, we can nowobtain the following result.

Theorem 1. τTt =∑T

s=1wTtsys, where

wTts = fTλ(t− s) + fTλ(T )I(t+ s− 1 = T )

+ fTλ(s+ t− 1)I(t+ s− 1 < T ) + fTλ(2T − t− s+ 1)I(t+ s− 1 > T )

+ ξTλgTλ(t)gTλ(s) + φTλgTλ(T − t+ 1)gTλ(s)

+ φTλgTλ(t)gTλ(T − s+ 1) + ξTλgTλ(T − t+ 1)gTλ(T − s+ 1)

=8∑j=1

wjT ts (17)

and for m ∈ {1, 2, . . . , T}, |fTλ(m)| ≤ Cm−3 and |gTλ(m)| ≤ Cm−3 for some constant Cnot depending on T .

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A simple computer program to verify the correctness of the formula of Equation (17), aswell as other material related to this paper, can be found at http://dl.dropbox.com/u/

2159931/hp.html.The above formula highlights that a key feature of the HP filter is the creation of a

symmetric weighted average of the data with w1Tts. w

2Tts is of little consequence since it is

bounded by CT−3 by the bound on fTλ(·). w3Tts is small as long as t is away from the begin

point of the sample, since it is bounded by C(t + s − 1)−3. Similarly, w4Tts is small as long

as t is away from T . By the bound on gTλ(·), w5Tts and w7

Tts are also small for t away from1, and w6

Tts and w8Tts are small for t away from T . This can be formalized by noting that for

any γ ∈ (0, 1/2) (writing “t ∈ [γT, (1− γ)T ]” to mean “t ∈ [γT, (1− γ)T ]∩Z” as t will takeinteger values throughout this paper)

supT≥1,s∈{1,...,T},t∈[γT,(1−γ)T ]

|T 3

8∑j=2

wjT ts| <∞. (18)

In conclusion, from the above theorem it follows that the HP filter behaves like a symmetricweighted average with several correction terms that are important near the begin and endof the sample.

Note that since cos(π(j − 1)(2T − t− s+ 1)/T ) = cos(π(j − 1)(t+ s− 1)/T ), it followsthat fTλ(s+ t− 1) = fTλ(2T − t− s+ 1), and therefore,

fTλ(s+ t− 1) = fTλ(2T − t− s+ 1) = fTλ(T )I(t+ s− 1 = T )

+fTλ(s+ t− 1)I(t+ s− 1 < T ) + fTλ(2T − t− s+ 1)I(t+ s− 1 > T ); (19)

however, the formula of Theorem 1 brings out the fact that the HP filter is invariant toreversing the time order, i.e., makes it clear that

τTt(y1, y2, . . . , yT−1, yT ) = τT,T−t+1(yT , yT−1, . . . , y2, y1),

and is more useful when considering large T and large λ approximations to the HP filter.By noting that limλ→∞ fTλ(m) = (2T )−1 and that limλ→∞ gTλ(m) = 0, it follows thatlimλ→∞ τTt = T−1

∑Tt=1 yt, which can also be observed directly from the definition of the HP

filter.In Tables 1 and 2 in Appendix 1, values for ξTλ and φTλ for various values of λ and T are

listed. For both sequences, it appears that as T → ∞, limit values are reached, sometimeswith values near the limit reached for sample sizes as small as T = 50, and this phenomenon

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http://dl.dropbox.com/u/2159931/hp.html


will be discussed in Section 3. Appendix 1 also contains graphs (Figure 1 and Figure 2) ofthe fTλ(m) and gTλ(m) functions. It is notable that for λ = 1600, we found that

maxm=−50,...,50

|f50,1600(m)− f10000,1600(m)| = 1.033 · 10−4

and

maxm=−50,...,50

|g50,1600(m)− g10000,1600(m)| = 2.428 · 10−7

while f50,1600(0) = 0.05607 and g50,1600(0) = 4.4041 · 10−5, and therefore we only providedone graph for λ = 1600 and T ≥ 50. The Matlab programs used for arriving at thisconclusion and used for creating Tables 1 and 2 and Figure 1 and Figure 2 can be found athttp://dl.dropbox.com/u/2159931/hp.html.

3 Large T results

In this section, we gather some large T results that will be used later on and are also of inde-pendent interest, as they show that it is possible to formulate weighted average procedureswith weights that do not depend on T that are asymptotically equivalent to the HP filter.First, we prove that the constants present in Theorem 1 achieve limit values:

Theorem 2. For all λ > 0, limT→∞ ηTλ = 0, limT→∞ φTλ = 0,

limT→∞

δTλ =

∫ 1

0

sin(πr/2)4 cos(πr/2)2(1 + 16λ sin(πr/2)4)−1dr = δλ, (20)

and

limT→∞

ξTλ =32λ

1− 32λδλ= ξλ. (21)

Part of the assertion of Theorem 2 is that 1− 32λδλ 6= 0. Together with Theorem 1, theabove result implies

supT≥1,1≤t≤T

T∑s=1

|wTts| <∞. (22)

It can also be shown that the functions fTλ(·) and gTλ(·) converge pointwise to limit functions:

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Theorem 3. Pointwise in (λ,m),

limT→∞

fT,λ(m) = fλ(m) =

∫ 1

0

cos(πrm)(1 + 16λ sin(πr/2)4)−1dr (23)

and

limT→∞

gT,λ(m) = gλ(m) = 21/2

∫ 1

0

cos(πr(m−1/2)) sin(πr/2)2 cos(πr/2)(1+16λ sin(πr/2)4)−1dr.

(24)

Given these results, we can now formulate a weighted average procedure that, away fromthe begin and end of the sample, is asymptotically equivalent to the HP filter procedure. Inorder to be able to formulate our result, we replaced t/T by r in the theorem below. Also,we used a scaling factor of E|yt| to account for the possibly increasing nature of yt (and τTt).

Theorem 4. Assume that γ is a constant in (0, 1/2), and assume r ∈ [γ, 1 − γ]. In ad-dition assume that E|yt|, t ≥ 1, is nondecreasing in t and that for any r ∈ [γ, 1 − γ],lim supT→∞E|yT |(E|y[rT ]|)−1 <∞. Then

lim supT→∞

(E|y[rT ]|)−1E|T∑s=1

ys(wT,[rT ],s − fλ([rT ]− s))| = 0. (25)

Therefore, noting that fλ(m) = fλ(−m), away from the end points, under the conditionsof the theorem the HP filter is asymptotically equivalent to the symmetric weighted average

T∑s=1

ysfλ(t− s)

as long as the regularity conditions hold. These regularity conditions holds for a wide classof processes. For a series yt = α + βt + γzt + ut for which β 6= 0, supt≥1E|ut| < ∞, and

E|zt| = o(t1/2) (such as in the case where zt is a unit root process that satisfies mild regularityconditions), we have

βrT (E|y[rT ]|)−1 → 1 and E|yT |(E|y[rT ]|)−1 → r−1,

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implying that the conditions of the theorem hold. In the case where β = 0 and zt is a unitroot process such that t−1/2E|zt| → c for some constant c > 0, we have

γcT 1/2r1/2(E|y[rT ]|)−1 → 1 and E|yT |(E|y[rT ]|)−1 → r−1/2,

again implying that the conditions of the theorem hold. Similar results can be establishedif the regressors are integrated of finite order and/or polynomials of the time trend.

In conclusion, it appears that away from the end points of the sample for a wide range ofdata-generating processes, the HP filter is effectively a symmetric weighted average withweights fλ(t−s). It should be noted that fλ(·) takes on negative values, and therefore, someobservations are given negative weights, even asymptotically.

4 Weak dependence of the cyclical component

Given the fact that the HP filter is a weighted average of T observations, it follows that thecyclical component is a triangular array. Also, for every value of t, the HP filter necessarilyweighs t−1 earlier observations and T − t later observations; and therefore, it is not possiblefor the cyclical component of the HP filter to be strictly stationary or weakly stationary.However, it is possible to show that the cyclical component of the HP filter has weak depen-dence properties when the HP filter is applied to the sum of a stationary mixing process,a linear trend process, and a unit root process. Such a property then ensures that laws oflarge numbers and central limit theorems can hold for functions of the cyclical component.This provides a justification for the use of the HP filter, as it implies that inference based onthe cyclical component can be asymptotically correct. Therefore, the results below illustratethat the finding in empirical macro that time series tend to possess time trends and/or unitroots, and the practice of using inference procedures based on the cyclical component of theHP filter, are not contradictory.

In addition, the weak dependence of the cyclical component indicates that the unit rootprocess has essentially been absorbed into the trend component. This can be taken to implythat the HP filter is capable of removing unit roots from a data-generating process.

In this section, we consider weak dependence properties of the cyclical component whenthe HP filter has been applied to the sum of a stationary mixing process, a linear trendprocess, and a unit root process; that is, we assume that yt satisfies the following assumption:

Assumption 1.

yt = α + βt+ γzt + ut,

where zt =∑t

j=1 εj, sups≥1 ‖ εs ‖p<∞ and sups≥1 ‖ us ‖p<∞.

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We explicitly allow for the cases us = 0, α = 0, β = 0 and/or εs = 0. Let the cyclicalcomponent of yt after application of the HP filter to yt = α+ βt+ ut + zt be denoted by cTt,i.e. cTt = yt − τTt = yt −

∑Ts=1wTtsys, and define

cmTt = ut −T∑s=1

wTtsusI(|s− t| ≤ m)−T∑j=1

εj(T∑s=j

wTts − I(j ≤ t))I(|j − t| ≤ m).

Note that cmTt is an approximation to cTt uses only (vmax(1,t−m), . . . , vmin(T,t+m)), where vt =(ut, εt)

′. Therefore, cmTt is an approximation to cTt that uses only information that is atmost m time periods away from the current time period. Concepts of closeness to such anapproximation have a long history in statistics and econometrics; see for example Ibragi-mov (1962), Billingsley (1968), McLeish (1975), Gallant and White (1988), Wooldridge andWhite (1988), Andrews (1988), and Potscher and Prucha (1997), among others. Near epochdependence is a simpler condition than mixing to verify, but the “base” (here, vt) needs tosatisfy a mixing condition in order for the condition to be useable for showing laws of largenumbers and/or central limit theorems. Our result is the following:

Theorem 5. Assume that yt satisfies Assumption 1. Then for any γ ∈ (0, 1/2),

supT≥1,t∈[γT,(1−γ)T ]

‖ cTt − cmTt ‖p= O(m−1)

and


‖ cTt ‖p<∞. (26)

Note that if εt = 0, we can improve the O(m−1) to O(m−2) in the above theorem.In Gallant and White (1988) and Potscher and Prucha (1997), the authors demonstrate

how a complete theory of inference can be based on the near epoch dependence property asdemonstrated above.

In the above theorem, we had to limit ourselves to sample values away from the begin andend of the sample. However, it is also possible to choose the γ arbitrarily small, and use anasymptotic negligibility argument for the begin and the end of the sample, and along thoselines, a complete theory of inference based on the near epoch dependence property usingthe entire sample is feasible also. As an example, we prove the following weak law of largenumbers for functions of the cyclical component:

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Theorem 6. Assume that yt satisfies Assumption 1 for p = 2, and assume that (ut, εt)′ is

strong mixing. Let f(·) be a function that is bounded and continuous on R. Then

T−1T∑t=1

(f(cTt)− Ef(cTt))p−→ 0.

5 Results for large T and large λ

While in Section 3 we found that the HP filter for large T is equivalent to a weighted average,the weight function depended on λ. When considering large values of λ, it turns out to alsobe possible to find an approximation to the HP filter using a weighted average that onlyuses λ as a bandwidth type parameter. This result is both of interest in its own right andprovides a route towards a formal result on adjusting the HP filter for the frequency of theobservations. Our result is based on the following theorem:

Theorem 7. Pointwise in m for all m ≥ 0,

limλ→∞

λ1/4fλ(λ1/4m) = f(m) = 2−3/2 exp(−2−1/2m)(sin(2−1/2m) + cos(2−1/2m))

and

limλ→∞

λ3/4gλ(λ1/4m) = g(m) = 2−3 exp(−2−1/2m)(cos(2−1/2m)− sin(2−1/2m)).

Furthermore, for m ≥ 1, fλ(m) ≤ Cλ1/4m−2 for some constant C > 0 not depending on λ,and for all K > 0,

limλ→∞

sup|m|≤K

|λ1/4fλ(λ1/4m)− f(m)| = 0.

Note that in the above result, f(·) and g(·) are not dependent on λ, and also note that∫∞−∞ f(m)dm = 1 and

∫∞−∞ g(m)dm = 0. Using Theorem 7 as a basis, we can now establish

the following result.

Theorem 8. Assume that γ is a constant in (0, 1/2), and assume r ∈ [γ, 1 − γ]. In ad-dition assume that E|yt|, t ≥ 1, is nondecreasing in t and that for any r ∈ [γ, 1 − γ],lim supT→∞E|yT |(E|y[rT ]|)−1 <∞. Then

lim supλ→∞

lim supT→∞

(E|y[rT ]|)−1E|T∑s=1

ys(wT,[rT ],s − λ−1/4f(λ−1/4([rT ]− s)))| = 0. (27)

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Theorem 8 suggests that for large T and λ, away from the beginning and end of oursample,

τTt ≈T∑s=1

ysλ−1/4f(λ−1/4(t− s)),

which for λ = 1600 simplifies to

τTt ≈ 0.1581T∑s=1

ys f(0.1581(t− s)).

The above theorem also implies that the negative weights remain a feature of the HP filtereven when both T and λ are assumed large.

6 Adjusting the HP filter for the frequency of the ob-

servations

Hodrick and Prescott’s suggestion to use a quarterly smoothing parameter of 1600 raisesthe question as to what smoothing parameter is appropriate for annual or monthly data.For annual data, Backus and Kehoe (1992) suggested a value of 100, and Correia, Neves,and Rebelo (1992) and Cooley and Ohanian (1991) suggested a value of 400, while Baxterand King (1999) suggested a value of 10. Ravn and Uhlig (2002) suggested that a value of6.25 for the annual smoothing parameter corresponds to a quarterly smoothing parameter of1600. They obtain this value as 1600×4−4 = 6.25. In this paragraph, we will give a differentjustification for rescaling the smoothing parameter using a fourth power when adjusting forthe data frequency.

Assume for simplicity that we have 4T observations of quarterly data and apply the HPfilter using a smoothing value λQ, implying that we have observations for T years. Then,letting wTts(λQ) denote the weights corresponding to λQ, the HP filter trend in the tth yearof data in quarter k, t = 1, ..., T , k = 1, 2, 3, 4 would equal

4T∑s=1

ysw4T,4t−4+k,s(λQ),

while using annual data (1/4)∑4

s=1 y4t−4+s and smoothing parameter λA, we would use

T∑i=1

((1/4)4∑s=1

y4i−4+s)wTti(λA).

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Note that this assumes that the quarterly data are measured on a per year basis. Alterna-tively of course, we can assume that our annual data are measured on a quarterly basis. Theasymptotic equivalence of both procedures for

λA = 4−4λQ

is then shown in the following theorem:

Theorem 9. Assume that γ is a constant in (0, 1/2), and assume r ∈ [γ, 1 − γ]. In ad-dition assume that E|yt|, t ≥ 1, is nondecreasing in t and that for any r ∈ [γ, 1 − γ],lim supT→∞E|yT |(E|y[rT ]|)−1 <∞. Then for λA = 4−4λQ and k = 1, 2, 3, or 4,

limλQ→∞

lim supT→∞

(E|y[rT ]|)−1E|4T∑s=1

ysw4T,4t−4+k,s(λQ)−T∑i=1

((1/4)4∑s=1

y4i−4+s)wTti(λA)| = 0. (28)

It should be noted that our asymptotic equivalence argument using λA = 4−4λQ only followsif we use quarterly data yi and annual data (1/4)

∑4s=1 y4i−4+s. This implies that quarterly

and annual data must be measured both on a quarterly or an annual basis for the equivalenceto hold. Therefore, if we have quarterly data on domestic car sales, then the asymptoticequivalence between the HP filter for quarterly data using λ = 1600 and the HP filter forannual data using λ = 6.25 only holds if we measure domestic car sales on a per year ora per quarter basis for both applications of the HP filter. However, when using quarterlysales and employing the HP filter with λ = 1600 and using annual sales and λ = 6.25, noasymptotic equivalence is obtained.

Using a similar argument, going from quarterly to monthly data, we find that the smoothingparameter λM asymptotically corresponds to (1/3)−4λQ = 81λQ, and for λQ = 1600, thisthen suggests λM = 129600. This also corresponds to the suggestion of Ravn and Uhlig(2002). We omit a formal theorem to this effect, as it will be analogous to Theorem 9.Again, for the asymptotic equivalence to hold, we need to measure both monthly and annualdata on either a monthly or an annual basis. Similar statements can of course be made foradjustments to the smoothing parameter when considering different frequency adjustments.

Appendix 1: the constants of Section 2

13

Table 1: Values of ξTλ for various values of λ and T . The Matlab program used to cal-culate these values (and values for the δTλ and ηTλ sequences) can be found at https:

//dl.dropbox.com/u/2159931/hp.html.

T = 25 T = 50 T = 100 T = 1000λ = 100 14491.2 14490.8 14490.8 14490.8λ = 400 81643.7 81461.9 81461.8 81461.8λ = 1600 461398 459391 459380 459380

Table 2: Values of φTλ for various values of λ and T . The Matlab program used to cal-culate these values (and values for the δTλ and ηTλ sequences) can be found at https:

//dl.dropbox.com/u/2159931/hp.html.

T = 25 T = 50 T = 100 T = 1000λ = 100 -8.52750 0.33904 6.37970× 10−6 −5.86296× 10−11

λ = 400 4421.81 -57.7142 0.02290 −4.69732× 10−10

λ = 1600 33352.1 -403.167 10.3726 −3.73839× 10−9

Figure 1: The function fT,λ(m) for T ≥ 50 and λ = 1600.

0.02

0.04

0.06

10 20 30 40 50−10−20−30−40−50

Appendix 2: Mathematical proofs

Lemma 2. Let p1(t/T ) = 1 and pj(t/T ) =√

2 cos(π(j− 1)(t− 1/2)/T ), j = 2, . . . , T . Then

T−1∑T

t=1 pTtp′Tt = IT .

14

https://dl.dropbox.com/u/2159931/hp.html




Figure 2: The function gT,λ(m) for T ≥ 50 and λ = 1600; y axis needs to be multiplied by10−5.

1

2

3

4

5

0 10 20 30 40 50

Proof. This follows from Lemma 3.4 of Eubank (1999, p. 143).

Lemma 3. Assume that h : [0, 1]→ R is continuous on [0, 1]. Then

limT→∞

T−1T∑j=1

h((j − 1)/T ) =

∫ 1

0

h(r)dr.

Proof. This follows because by continuity of h(·) and setting j = rT ,

T−1T∑j=1

h((j − 1)/T ) = T−1∫ T

0

h([j]/T )dj =

∫ 1

0

h([rT ]/T )dr,

and because supr∈[0,1] |h(r)| <∞ by continuity and because r−1/T ≤ [rT ]/T ≤ r, the resultnow follows by the dominated convergence theorem.

Lemma 4. For h(r) = 1/(1 + 16λ sin(πr/2)4),

T−1T∑j=2

cos(π(j − 1)m/T )h((j − 1)/T )

15

= −(2T )−1h(1/T )

−(2T )−1(−1)mh((T − 1)/T )

−T−1(2 sin(πm/(2T ))−2(h(2/T )− h(1/T )) cos(πm/T )

+T−1(2 sin(πm/(2T ))−2(h(1− 1/T )− h(1− 2/T )) cos(πm(1− 1/T ))

+T−1(2 sin(πm/(2T )))−3(h(3/T )− 2h(2/T ) + h(1/T )) sin(π(3/2)m/T )

−T−1(2 sin(πm/(2T )))−3(h(1− 1/T )− 2h(1− 2/T ) + h(1− 3/T )) sin(π(T − 3/2)m/T )

+T−1(2 sin(πm/(2T )))−3T−3∑j=2

∆3h((j + 2)/T ) sin(π(j + 1/2)m/T ).

Proof. The proof of this result is somewhat tedious and can be found at https://dl.

dropbox.com/u/2159931/hp.html.

Proof of Lemma 1: By Lemma 2, T−1∑T

t=1 pTtp′Tt = IT , and therefore it only remains to

prove the second part of Lemma 1. To show this, note that for j ∈ {1, 2, ..., T}, rememberingthat p1(t/T ) = 1 for t ∈ Z,

pj(t/T )− pj((t− 1)/T ) =√

2 cos(π(j − 1)(t− 1/2)/T )−√

2 cos(π(j − 1)(t− 3/2)/T ),

and because

cos(α + β)− cos(α− β) = −2 sin(α) sin(β)

and setting α + β = π(j − 1)(t− 1/2)/T and α− β = π(j − 1)(t− 3/2)/T , we get

pj(t/T )− pj((t− 1)/T ) = −2√

2 sin(π(j − 1)(t− 1)/T )) sin(π(j − 1)/(2T )).

Therefore,

pj((t+ 1)/T )− 2pj(t/T ) + pj((t− 1)/T )

= (pj((t+ 1)/T )− pj(t/T ))− (pj(t/T )− pj((t− 1)/T ))

= 2√

2 sin(π(j − 1)/(2T ))(sin(π(j − 1)(t− 1)/T )− sin(π(j − 1)t/T )).

16



Now because

sin(α + β)− sin(α− β) = 2 sin(β) cos(α)

and setting α + β = π(j − 1)(t− 1)/T and α− β = π(j − 1)t/T ), we have

sin(π(j − 1)(t− 1)/T )− sin(π(j − 1)t/T )

= −2 sin(π(j − 1)/(2T )) cos(π(j − 1)(t− 1/2)/T ).

Therefore, for j ∈ {1, ..., T} and t ∈ Z,

pj((t+1)/T )−2pj(t/T )+pj((t−1)/T ) = −4√

2 sin(π(j−1)/(2T ))2 cos(π(j−1)(t−1/2)/T ),

implying that, for j, k ∈ {1, 2, ..., T},

[T−1∑t=2

(pT,t+1 − 2pTt + pT,t−1)(pT,t+1 − 2pTt + pT,t−1)′]jk

= 32 sin(π(j−1)/(2T ))2 sin(π(k−1)/(2T ))2T−1∑t=2

cos(π(j−1)(t−1/2)/T ) cos(π(k−1)(t−1/2)/T ).

Also, for j, k ∈ {1, 2, . . . , T}, by Lemma 2,

T−1∑t=2

cos(π(j − 1)(t− 1/2)/T ) cos(π(k − 1)(t− 1/2)/T )

= (1/2)TI(j = k)− cos(π(j − 1)/(2T )) cos(π(k − 1)/(2T ))

− cos(π(j − 1)(T − 1/2)/T ) cos(π(k − 1)(T − 1/2)/T ),

implying that

T−1T−1∑t=2

(pT,t+1 − 2pTt + pT,t−1)(pT,t+1 − 2pTt + pT,t−1)′

= diag({16 sin(π(j − 1)/(2T ))4, j = 1, . . . , T})− 32T−1qT1q′T1 − 32T−1qT2q

′T2

= DT − 32T−1qT1q′T1 − 32T−1qT2q

′T2,

where qT1j = sin(π(j−1)/(2T ))2 cos(π(j−1)(1/2)/T ) and qT2j = sin(π(j−1)/(2T ))2 cos(π(j−1)(T − 1/2)/T ).

17

Proof of Theorem 1: It follows from Lemma 1 that

θT = [T−1T∑t=1

pTtp′Tt+λT

−1T−1∑t=2

(pT,t+1−2pTt+pT,t−1)(pT,t+1−2pTt+pT,t−1)′]−1T−1

T∑t=1

ytpTt

= (IT + λDT +HT (IT + λDT )]−1 T−1T∑t=1

ytpTt

= (IT + λDT )−1(IT +HT )−1T−1T∑t=1

ytpTt

where IT is the identity matrix of dimension (T × T ),

DT = diag({16 sin(π(j − 1)/(2T ))4, j = 1, . . . , T})

and

HT = (−32λT−1qT1q′T1 − 32λT−1qT2q

′T2)(IT + λDT )−1.

Now according to Miller (1981), for a matrix HT of rank 0, 1 or 2, we have

(IT +HT )−1 = IT − (aT + bT )−1(aTHT −H2T )

for aT = 1 + tr(HT ) and 2bT = (tr(HT ))2 − tr(H2T ). Therefore,

(IT + λDT +HT (IT + λDT ))−1 = (IT + λDT )−1(IT − (aT + bT )−1(aTHT −H2T ))

= (IT + λDT )−1

+32λaT (aT + bT )−1(IT + λDT )−1(T−1qT1q′T1)(IT + λDT )−1

+32λaT (aT + bT )−1(IT + λDT )−1(T−1qT2q′T2)(IT + λDT )−1

+322λ2(aT + bT )−1(IT + λDT )−1(T−1qT1q′T1)(IT + λDT )−1(T−1qT1q

′T1)(IT + λDT )−1


′T2)(IT + λDT )−1


′T1)(IT + λDT )−1


′T2)(IT + λDT )−1

=7∑i=1

MT i,

18

say, and we can write

τTt = p′TtθT = p′Tt

7∑i=1

MT iT−1

T∑s=1

pTsys

=T∑s=1

ys

7∑i=1

T−1p′TtMT ipTs =T∑s=1

ys

7∑i=1

wiT ts

where

w1Tts = T−1p′TtMT1pTs = T−1p′Tt(IT + λDT )−1pTs,

w2Tts = T−1p′TtMT2pTs

= 32λaT (aT + bT )−1(T−1p′Tt(IT + λDT )−1qT1)(T−1q′T1(IT + λDT )−1pTs),


= 32λaT (aT + bT )−1(T−1p′Tt(IT + λDT )−1qT2)(T−1q′T2(IT + λDT )−1pTs),


= 322λ2(aT+bT )−1(T−1pTt(IT+λDT )−1qT1)(T−1q′T1(IT+λDT )−1qT1)(T

−1q′T1(IT+λDT )−1pTs),







and



−1q′T2(IT+λDT )−1pTs).

Simplification of w1Tts can be achieved by noting that

cos(α) cos(β) = (1/2) cos(α + β) + (1/2) cos(α− β),

which implies that for j ≥ 2,

pj(t/T )pj(s/T ) = 2 cos(π(j − 1)(t− 1/2)/T ) cos(π(j − 1)(s− 1/2)/T )

19

= cos(π(j − 1)(t+ s− 1)/T ) + cos(π(j − 1)(t− s)/T ),

and therefore, remembering that p1(t/T ) = 1 for t ∈ {1, 2, . . . , T},

w1Tts = T−1p′Tt(IT + λDT )−1pTs

= T−1T∑j=2

cos(π(j − 1)(t+ s− 1)/T )(1 + 16λ sin(π(j − 1)/(2T ))4)−1

+T−1T∑j=2

cos(π(j − 1)(t− s)/T )(1 + 16λ sin(π(j − 1)/(2T ))4)−1 + T−1

= fTλ(t+ s− 1)− (2T )−1 − (2T )−1(−1)t+s−1(1 + 16λ)−1

+fTλ(t− s)− (2T )−1 − (2T )−1(−1)t−s(1 + 16λ)−1 + T−1

= fTλ(t+ s− 1) + fTλ(t− s)

because (−1)t+s−1+(−1)t−s = (−1)t((−1)s−1+(−1)s) = 0. Note that by symmetry ofDT andIT , T−1q′T1(IT + λDT )−1qT2 = T−1q′T2(IT + λDT )−1qT1 = ηT . Also, since for j ∈ {1, . . . , T}

qT2j = cos(π(j − 1))qT1j

which is trivial for j = 1 and for j ≥ 2 follows because

qT2j = sin(π(j − 1)/(2T ))2 cos(π(j − 1)) cos(π(j − 1)/(2T )) = cos(π(j − 1))qT1j,

and

pj((T − t)/T ) =√

2 cos(π(j − 1)((T − t− 1/2)/T ))

=√

2 cos(π(j − 1)) cos(π(j − 1)(t+ 1/2)/(2T )) = cos(π(j − 1))pj((t+ 1)/T ),

it follows that

T−1p′Tt(IT + λDT )−1qT2 =T∑j=1

T−1pj(t/T )qT2j(1 + λdjj)−1

= T−1p′T−t+1(IT + λDT )−1qT1 = gT (T − t+ 1),

and

T−1q′T2(IT + λDT )−1qT2 = T−1q′T1(IT + λDT )−1qT1 = δT .

20

Therefore,

w2Tts = 32λaT (aT + bT )−1gT (t)gT (s),

w3Tts = 32λaT (aT + bT )−1gT (T − t+ 1)gT (T − s+ 1),

w4Tts = 322λ2(aT + bT )−1δTgT (t)gT (s),

w5Tts = 322λ2(aT + bT )−1ηTgT (t)gT (T − s+ 1),

w6Tts = 322λ2(aT + bT )−1ηTgT (T − t+ 1)gT (s),

w7Tts = 322λ2(aT + bT )−1δTgT (T − t+ 1)gT (T − s+ 1).

Therefore, the assertion of Equation (17) now follows and

ξTλ = 32λaT (aT + bT )−1 + 322λ2(aT + bT )−1δT ,

φTλ = αT3 = 322λ2(aT + bT )−1ηT ,

and the formulas for ξTλ and φTλ given prior to Theorem 1 now follow by noting that

aT = 1 + tr(HT ) = 1 + tr((−32λT−1qT1q′T1 − 32λT−1qT2q

′T2)(IT + λDT )−1)

= 1− 32λT−1q′T1(IT + λDT )−1qT1 − 32λT−1q′T2(IT + λDT )−1qT2

= 1− 64λδT

and

bT = (1/2)(tr(HT ))2 − (1/2)tr(H2T )

= (1/2)(−64λδT )2 − (1/2)322λ2(2δ2T + 2η2T )

= 322λ2(δ2T − η2T ),

which completes the proof of the representation for wTts. By Lemma 4, it follows that forh(r) = 1/(1 + 16λ sin(πr/2)4),

fTλ(m) = −(2T )−1(h(1/T )− h(0))

−(2T )−1(−1)m(h((T − 1)/T )− h(1))

−T−1(2 sin(πm/(2T ))−2(h(2/T )− h(1/T )) cos(πm/T )

+T−1(2 sin(πm/(2T ))−2(h(1− 1/T )− h(1− 2/T )) cos(πm(1− 1/T ))

+T−1(2 sin(πm/(2T )))−3(h(3/T )− 2h(2/T ) + h(1/T )) sin(π(3/2)m/T )

21

−T−1(2 sin(πm/(2T )))−3(h(1− 1/T )− 2h(1− 2/T ) + h(1− 3/T )) sin(π(T − 3/2)m/T )

+T−1(2 sin(πm/(2T )))−3T−3∑j=2

∆3h((j + 2)/T ) sin(π(j + 1/2)m/T ).

Noting that h′(0) = h′(1) = 0 and that supx∈[0,1] |h′′′(x)| < ∞, it follows that the first andsecond terms are bounded in absolute value by

(1/2)T−3 supx∈[0,1]

|h′′(x)|.

Similarly, the third and fourth terms are bounded by a multiple of

T−3(2 sin(πm/(2T ))−2,

and the fifth, sixth and seventh terms by a multiple of

T−3(2 sin(πm/(2T ))−3.

By noting that

supr∈[0,1]

|r/ sin(πr/2)| = 1

and that m ≤ T , it now follows that |fTλ(m)| ≤ Cm−3. The proof of the bound for gTλ(m)is analogous.

Proof of Theorem 2: By Lemma 3,

δTλ = T−1q′T1(IT + λDT )−1qT1

= T−1T∑j=1

sin(π(j − 1)/(2T ))4 cos(π(j − 1)/(2T ))2(1 + 16λ sin(π(j − 1)/(2T ))4)−1

→∫ 1

0

sin(πr/2)4 cos(πr/2)2(1 + 16λ sin(πr/2)4)−1dr = δλ

and

ηTλ = T−1q′T1(IT + λDT )−1qT2

22

= T−1T∑j=1

qT1jqT2j(1 + 16λ sin(π(j − 1)/(2T ))4)−1,

and because qT2j = cos(π(j − 1))qT1j,

ηTλ = T−1T∑j=1

cos(π(j − 1))q2T1j(1 + 16λ sin(π(j − 1)/(2T ))4)−1

= T−1T∑

j=1,j odd

q2T1j(1 + 16λ sin(π(j − 1)/(2T ))4)−1

−T−1T∑

j=1,j even

q2T1j(1 + 16λ sin(π(j − 1)/(2T ))4)−1

= T−1[(T−1)/2]∑j=0

sin(πj/T )4 cos(πj/T )2(1 + 16λ sin(πj/T )4)−1

−T−1[T/2]∑j=1

sin(π(2j − 1)/(2T ))4 cos(π(2j − 1)/(2T ))2(1 + 16λ sin(π(2j − 1)/(2T ))4)−1,

and by an argument similar to that of Lemma 3, it now follows that both terms con-verge to the same number, implying that limT→∞ ηTλ = 0. Note that sin(πr/2)4λ(1 +16λ sin(πr/2)4)−1 is increasing in λ for λ ≥ 0 for any r 6= 0, and therefore bounded by 1/16,it follows that

λδλ ≤ (1/16)

∫ 1

0

cos(πr/2)2dr = 1/32.

Therefore,

ξTλ = 32λ(1−64λδTλ)(1−64λδTλ+322λ2(δ2Tλ−η2Tλ))−1+322λ2(1−64λδTλ+322λ2(δ2Tλ−η2Tλ))−1δTλ

→ 32λ(1− 64λδλ)(1− 64λδλ + 322λ2δ2λ)−1 + 322λ2(1− 64λδλ + 322λ2δ2λ)

−1δλ

=32λ(1− 64λδλ) + 322λ2δλ

(1− 64λδλ + 322λ2δ2λ)=

32λ

1− 32λδλ,

and

φTλ = 322λ2(1− 64λδTλ + 322λ2(δ2Tλ − η2Tλ))−1ηTλ

23

→ 322λ2(1− 32λδλ)−2 × 0 = 0.

Proof of Theorem 3: This is a direct consequence of Lemma 3.

Proof of Theorem 4: First note that, because E|yt| is nondecreasing in t,

E|(E|yt|)−1T∑s=1

ys(wTts − fλ(t− s))|

≤ (E|yt|)−1T∑s=1

E|ys||wTts − fλ(t− s)| ≤T∑s=1

|wTts − fλ(t− s)|

≤T∑s=1

|fTλ(t− s)− fλ(t− s))|+T∑s=1

8∑j=2

|wjT ts|.

≤ 2∞∑m=0

|fTλ(m)− fλ(m)|+T∑s=1

8∑j=2

|wjT ts|.

The second term, by the result of Equation (18), is O(T−1). By Theorem 1, |fTλ(m)| ≤C(|m|+ 1)−3 for a constant C not depending on T , and therefore the result now follows bythe dominated convergence theorem and because |fλ(m)| ≤ C(|m|+ 1)−2 by ... and becausefor all m ≥ 0, fTλ(m)→ fλ(m) by ...

Proof of Theorem 4: First note that

E|(E|y[rT ]|)−1T∑s=1

ys(wTts − fλ(t− s))|

≤ (lim supT→∞

(E|yT |E|(E|y[rT ]|)−1))T∑s=1

|wTts − fλ(t− s)|

24

≤T∑s=1

|fTλ(t− s)− fλ(t− s))|+T∑s=1

8∑j=2

|wjT ts|.

The first term now vanishes by the dominated convergence theorem and because for s 6= t,

|fTλ(t− s)− fλ(t− s)| ≤ C|t− s|−3.

Noting that the second term, by the result of Equation (18), is O(T−1) now completes theargument.

Proof of Theorem 5: Note that, by the properties∑T

s=1wTts = 1 and∑T

s=1wTtss = t thatwere noted in Section 2,

cTt = yt −T∑s=1

wTts(α + βs+ us + zs)

= ut −T∑s=1

wTtsus + zt −T∑s=1

wTtszs

= ut −T∑s=1

wTtsus +t∑

j=1

εj −T∑s=1

s∑j=1

εjwTts

= ut −T∑s=1

wTtsus −T∑j=1

εj(T∑s=j

wTts − I(j ≤ t)).

Since cTt = cmTt for m > T , it therefore suffices to show that for every m ≥ 0,

supT :T≥m

‖T∑s=1

wTtsusI(|t− s| ≥ m) ‖p≤ C(m+ 1)−2

and

supT :T≥m

‖T∑j=1

εj(T∑s=j

wTts − I(j ≤ t))I(|j − t| ≥ m) ‖p≤ C(m+ 1)−1.

To see the first result, note that by the result of Equation (18) and because m ≤ T ,

supT :T≥m

‖T∑s=1

wTtsusI(|t− s| ≥ m) ‖p

25

≤t−m−1∑s=1

C(t− s)−2 ≤ C∞∑

k=m+1

k−2 = O((m+ 1)−1).

For the fourth term, a similar argument holds. Therefore, the conclusion of the theorem nowfollows.

Proof of Theorem 6: Write

T−1T∑t=1

(f(cTt)− Ef(cTt))

= T−1∑

t∈{1,...,T},t/∈[γT,(1−γ)T ]

(f(cTt)− Ef(cTt)),

+T−1∑

t∈[γT,(1−γ)T ]

(f(cTt)− Ef(cTt))

and note that the first term is bounded in absolute value by

4γ supx∈R|f(x)|,

and because γ can be chosen arbitrarily small, it therefore suffices to show a weak law oflarge numbers for the second term. To show this, first note that for all K > 0 and η > 0,


‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m) ‖2

≤ supT≥1,t∈[γT,(1−γ)T ]

‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m)I(|cTt| > K)| ‖2

+ supT≥1,t∈[γT,(1−γ)T ]

‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m)I(|cTt| ≤ K)I(|cTt − cmTt| > η)| ‖2

+ supT≥1,t∈[γT,(1−γ)T ]

‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m)I(|cTt| ≤ K)I(|cTt − cmTt| ≤ η)| ‖2 .

Because f(·) is bounded in absolute value,


‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m)I(|cTt| > K)| ‖22

≤ supx∈R|f(x)| sup

T≥1,t∈[γT,(1−γ)T ]P (|cTt| > K)

27

≤ supx∈R|f(x)|K−2 sup

T≥1,t∈[γT,(1−γ)T ]E|cTt|2,

while similarly,


‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m)I(|cTt| ≤ K)I(|cTt − cmTt| > η)| ‖22

≤ supx∈R|f(x)|η−2 sup

T≥1,t∈[γT,(1−γ)T ]E|cTt − cmTt|2,

and


‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m)I(|cTt| ≤ K)I(|cTt − cmTt| ≤ η)| ‖2

≤ sup|x|≤K

supx′∈R:|x−x′|≤η

|f(x)− f(x′)|.

Therefore, by making first m approach infinity, the making η approach 0, and then Kapproach infinity, it now follows that

limm→∞


‖ f(cTt)− E(f(cTt)|vt, . . . , vt−m) ‖2= 0.

Therefore, f(cTt) is near epoch dependent on vt for t ∈ [γT, (1− γ)T ], and it therefore is abounded L1-mixingale as defined in Andrews (1988). Therefore, Andrews’ weak law of largenumbers applies, and the proof of the theorem is complete.

Proof of Theorem 7: ] Since fλ(m) =∫ 1

0cos(πrm)hλ(r)dr where hλ(r) = l(2λ1/4 sin(πr/2))

for l(x) = (1 + x4)−1, it follows by partial integration that for any integer m

fλ(m) = (πm)−1∫ 1

0

hλ(r)d sin(πrm) = −(πm)−1∫ 1

0

sin(πrm)h′λ(r)dr

= (πm)−2∫ 1

0

h′λ(r)d cos(πrm)

= (πm)−2 h′λ(r) cos(πrm) |10 −(πm)−2∫ 1

0

cos(πrm)dh′λ(r).

Now

h′λ(r) = l′(2λ1/4 sin(πr/2))2λ1/4 cos(πr/2))(π/2),

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and therefore

|(πm)−2 h′λ(r) cos(πrm) |10 | ≤ 2(πm)−2 supr∈[0,1]

|l′(r)|πλ1/4.

Furthermore,

h′′λ(r) = l′′(2λ1/4 sin(πr/2))(2λ1/4 cos(πr/2))(π/2))2+l′(2λ1/4 sin(πr/2))2λ1/4 sin(πr/2))(π/2)2,

and because l′(x) ≤ 0 for 0 ≤ x ≤ 1, l′′(x) = 4x2(5x4 − 3)(x4 + 1)−3 and l′′(x) ≥ 0 for0 ≤ x ≤ (3/5)1/4 and l′′(x) < 0 for (3/5)1/4 < x ≤ 1, it follows that

|(πm)−2∫ 1

0

cos(πrm)dh′λ(r)| ≤ (πm)−2∫ 1

0

|h′′λ(r)|dr

= (πm)−2∫ 1

0

I(2λ1/4 sin(πr/2) ≤ (3/5)1/4)l′′(2λ1/4 sin(πr/2))(2λ1/4 cos(πr/2))(π/2))2dr

−(πm)−2∫ 1

0

I(2λ1/4 sin(πr/2) > (3/5)1/4)l′′(2λ1/4 sin(πr/2))(2λ1/4 cos(πr/2))(π/2))2dr

−(πm)−2∫ 1

0

l′(2λ1/4 sin(πr/2))2λ1/4 sin(πr/2))(π/2)2dr

≤ (πm)−2∫ 1

0

I(2λ1/4 sin(πr/2) ≤ (3/5)1/4)h′′λ(r)dr

−(πm)−2∫ 1

0

I(2λ1/4 sin(πr/2) > (3/5)1/4)h′′λ(r)dr

+3(πm)−2∫ 1

0

|l′(2λ1/4 sin(πr/2))|2λ1/4 sin(πr/2))(π/2)2dr

≤ 4(πm)−2 supr∈[0,1]

|h′λ(r)|+ ...,

and therefore,

|fλ(m)| ≤ Cm−2λ1/4.

From this it follows that

λ1/4|fλ(λ1/4m) ≤ Cλ1/4λ1/4(λ1/4m)−2 = Cm−2.

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Also,

λ1/4fλ(λ1/4m) = λ1/4

∫ 1

0

cos(πrλ1/4m)hλ(r)dr =

∫ λ1/4

0

cos(πym)hλ(λ−1/4y)dy,

so, noting that f(m) =∫∞0

cos(πym)l(πy)dy,

|λ1/4fλ(λ1/4m)− f(m)|

≤ |∫ λ1/14

0

cos(πym)(l(2λ1/4 sin(πλ−1/4y/2))− l(πy))dy|

+|∫ λ1/4

λ1/14cos(πym)l(2λ1/4 sin(πλ−1/4y/2))dy|+ |

∫ ∞λ1/14

cos(πym)l(πy)dy|,

and for y ∈ [0, λ1/14],

|l(2λ1/4 sin(πλ−1/4y/2))− l(πy)| ≤ 2λ1/4| sin(πλ−1/4y/2)− πyλ−1/4/2| supx≥0|l′(x)|

≤ 2λ1/4C(πyλ−1/4/2)2 supx≥0|l′(x)| = (1/2)λ−1/4+1/7Cπ2 sup

x≥0|l′(x)|,

so

|∫ λ1/14

0

cos(πym)(l(2λ1/4 sin(πλ−1/4y/2))− l(πy))dy| ≤ Cλ−1/28.

Furthermore,

|∫ λ1/4

λ1/14cos(πym)l(2λ1/4 sin(πλ−1/4y/2))dy| ≤ λ1/4l(2λ1/4 sin(πλ−1/4λ1/14/2)) ≤ λ1/4l(Cλ1/14),

and because l(x) = O(x−4) as x → ∞, the last term is O(λ1/4−4/14) = O(λ−1/28). Forshowing uniform convergence, note that

λ1/4fλ(λ1/4m) = λ1/4

∫ 1

0

cos(mrπλ1/4)hλ(r)dr =

∫ λ1/4

0

cos(myπ)l(2λ1/4 sin(πλ−1/4y/2))dy,

so, because l(x) is nonincreasing on [0,∞) and sin(πx/2) ≥ x for x ∈ [0, 1],

sup|m|≤K,|m−m′|≤η

|λ1/4fλ(λ1/4m)− λ1/4fλ(λ1/4m′)|

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≤∫ λ1/4

0

supm,|m−m′|≤η

| cos(myπ)− cos(m′yπ)|l(2λ1/4 sin(πλ−1/4y/2))dy

≤∫ ∞0

supm,|m−m′|≤η

| cos(myπ)− cos(m′yπ)|l(2y)dy,

and by the dominated convergence theorem, it now follows that

lim supη↓0

sup|m|≤K,|m−m′|≤η

|λ1/4fλ(λ1/4m)− λ1/4fλ(λ1/4m′)| = 0.

Proof of Theorem 8: Because of the result of Theorem 4, it suffices to show that

E|(E|y[rT ]|)−1T∑s=1

ys(fλ(t− s)− λ−1/4f(λ−1/4(t− s)))| → 0.

Similarly to the proof of Theorem 4,

(E|y[rT ]|)−1T∑s=1

ys(fλ(t− s)− λ−1/4f(λ−1/4(t− s)))

≤ CT∑s=1

|fλ(t− s)− λ−1/4f(λ−1/4(t− s))|

≤ C

∫ T+1

s=1

|fλ(t− [s])− λ−1/4f(λ−1/4(t− [s]))|ds

and setting y = λ−1/4(t − s), which implies that s = t − yλ1/4, it follows that the lastexpression is bounded by

C

∫ ∞−∞|λ1/4fλ(t− [t− yλ1/4])− f(λ−1/4(t− [t− yλ1/4]))|dy.

Because λ1/4|fλ(λ1/4m)| ≤ Cm−2, it follows that

lim supλ→∞

∫|y|>K

|λ1/4fλ(t− [t− yλ1/4])− f(λ−1/4(t− [t− yλ1/4]))|dy

≤ lim supλ→∞

2C

∫|y|>K

(λ−1/4(t− [t− yλ1/4]))−2dy

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≤ lim supλ→∞

2C

∫|y|>K

(|y| − 1)−2dy.

Also, because y − 1 ≤ λ−1/4(t− [t− yλ1/4]) ≤ y + 1 if λ ≥ 1,∫|y|≤K

|λ1/4fλ(t− [t− yλ1/4])− f(λ−1/4(t− [t− yλ1/4]))|dy

≤ sup|x|≤K+1

|λ1/4fλ(λ1/4x)− f(x)| → 0

by the uniform convergence of λ1/4fλ(λ1/4x) to f(x) on compacta as asserted (where?)

Proof of Theorem 9: Note that

4T∑s=1

ysw4T,4t−4+k,s(λQ) =T∑i=1

4∑s=1

y4i−4+sw4T,4t−4+k,4i−4+s(λQ),

and analogously to the proof of Theorem ..., it therefore suffices to show that

limλ→∞

lim supT→∞

T∑i=1

4∑s=1

|wT,4t−4+k,4i−4+s(λ)− (1/4)wTti(4−4λ)| = 0.

Now by Theorem 8,

limλ→∞

lim supT→∞

T∑i=1

4∑s=1

|wT,4t−4+k,4i−4+s(λ)− λ−1/4f(λ−1/4(4t− 4 + k − (4i− 4 + s))| = 0

and

limλ→∞

lim supT→∞

T∑i=1

4∑s=1

|(1/4)wTti(4−4λ)− (1/4)(4−4λ)−1/4f((4−4λ)−1/4(t− s))| = 0.

Therefore, it remains to show that, for k ∈ {1, 2, 3, 4},

limλ→∞

lim supT→∞

T∑i=1

4∑s=1

|λ−1/4f(λ−1/4(4t+ k − (4i+ s))− λ−1/4f(λ−1/4(4t− 4s))| = 0,

and this follows from elementary calculus.

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the econometrics of the hodrick-prescott lter preliminary · 2018-02-12 · the econometrics of the...

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