Theoretical Confidence Level Problems with Confidence Intervals for the Spectrum of a Time Series

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 575 December 1996 THEORETICAL CONFIDENCE LEVEL PROBLEMS WITH CONFIDENCE INTERVALS FOR THE SPECTRUM OF A TIME SERIES Jon Faust NOTE:InternationalFinanceDiscussionPapersarepreliminarymaterialscirculated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.

Abstract Textbookapproachesto formingasymptoticallyjusti(cid:12)ed con(cid:12)denceintervalsfor the spectrum under very general assumptions were developed by the mid-1970s. This papershowsthatunderthetextbookassumptions,thetruecon(cid:12)dencelevelforthese intervals does not converge to the asymptotic level, and instead is (cid:12)xed at zero in all sample sizes. The paper explores necessary conditions for solving this problem, mostnotablyshowingthatunderweakconditions,formingvalidcon(cid:12)denceintervals requires that one limit consideration to a (cid:12)nite-dimensional time series model.

Theoretical con(cid:12)dence level problems with con(cid:12)dence intervals for the spectrum of a time series 1 Jon Faust Often we wish to make inferences about the spectrum of a time series and have little a priori basis for restricting the class of possible processes. It is well known that the formation of con(cid:12)dence intervals for points on the spectrum requires various assumptions beyond, say, those implied by stationarity. By the early 1970s, however, there were textbook approaches to forming asymptotically justi(cid:12)ed con(cid:12)dence intervals under very weak restrictions [Hannan, 1970; Anderson, 1971], and these approaches remain standard [Priestley, 1981; and Brockwell and Davis, 1991]. The con(cid:12)dence intervals are based on asymptotically normal point estimates of the spectrum. This paper shows that under the standard assumptions the textbook con(cid:12)dence intervals|and any othercon(cid:12)dence intervals|have con(cid:12)dence level zero in all sample sizes. The formal explanation for the textbook case is that the convergence to normality of the point estimates is not uniform. There are two natural solutions to the con(cid:12)dence level problems: (i) impose further restrictions, or (ii) change the parameter of interest to be, for example, the average of the spectrum over some interval. The paper characterizes necessary conditions for valid con(cid:12)dence intervals to exist under each of these approaches. The most notable result is that under some weak and appealing conditions, meaningful con(cid:12)dence intervals for points on the spectrum exist only if the maintained model is restricted to a (cid:12)nite-dimensional space of Wold (moving average) representations. This rules out valid con(cid:12)dence statements when using in(cid:12)nite- 1 Theauthorisasta(cid:11)economistattheInternationalFinanceDivisionoftheBoardofGovernors of the Federal Reserve System and can be reached at faustj@frb.gov. The author thanks Steve Blough,DavidBowman,TimCogley,NeilEricsson,ChristianGilles,EricLeeper,TomRothenberg, ChrisSims,DougSteigerwald, Harald Uhlig, andseminarparticipantsatNu(cid:14)eld College,U. Cal. Santa Barbara, and U. Cal. Berkeley. The views in this paper are solely the responsibility of the author and should not be interpreted as re(cid:13)ecting the views of the Board of Governors of the FederalReserve System or ofany otherpersonassociatedwith the Federal ReserveSystem. 1

dimensional models common in both parametric and nonparametric work. The results are shown using the topological approach of Sims [1971,1972] Oftenineconometricswebaseasymptoticinferencestatementsonpointwiseconvergence of estimates, rather than uniform convergence (see, e.g., Hanson [1996] for a recent example and discussion). In some cases, this may be justi(cid:12)edby the knowledge that, say, convergence is uniform on compact subsets of the stated parameter space. Sweeting [1980] states a useful result of this sort. In such cases, there may be many di(cid:11)erent sets of assumptions that are su(cid:14)cient for compactness, but no known additional necessary restrictions. In contrast, this paper shows that under weak conditions, there are additional simply interpretable necessary conditions for forming con(cid:12)dence intervals on the spectrum. The practical importance of these results is discussed in the (cid:12)nal section. 1 Standard approaches to spectral inference Beginwith asketch of thetextbook approach to forming con(cid:12)dence intervalsfor the spectrum[e.g., Anderson,1971; BrockwellandDavis,1991;Hannan,1970; Priestley, 2 1981]. Thestandardapproachistospecifyasmoothed-periodogrampointestimate, establish its asymptotic normality, and form con(cid:12)dence intervals around the point estimatebasedontheasymptoticstandarderror. Theproofofasymptoticnormality 3 of the smoothed periodogram estimates is based on assumptions like the following: Take the set of univariate, real, time series processes, yt satisfying, f g 1 A 1 yt = j=0aj"t(cid:0)j. P A 2 i) The "t are independent and identically distributed and parameterized by 2 4 2 (cid:9), ii) E["t] = 0, iii) E["t] = 1, iv) E["t] < , iv) The probability measure for "t 1 4 is absolutely continuous with respect to the Lebesgue measure. 2 Whatfollowsisnottheweakestsetofassumptionsthathavebeenassertedtosupportasymptotically valid con(cid:12)dence intervals. The paper shows that these assumptions|and, hence, any weakerassumptions|aretoo weak fortheexistenceof validcon(cid:12)denceintervals. 3 Theapproachesinthetextbooksdi(cid:11)erinsomedetails,e.g.,HannantreatsA4di(cid:11)erently,and Brockwelland Davis give a di(cid:11)erent form for the con(cid:12)dence intervals based on the asymptotically normal point estimates in (1). None ofthese di(cid:11)erences matter forthe issues discussedhere. 4 Since a0 is not normalized, settingthe variance of"t to oneimposes noloss of generality. 2

1 A 3 j=0 aj < . j j 1 P Given the parameterization for "t , A1 suggests a natural parameterization for f g yt processes in terms of (cid:18) = (A; ), where A = a0;a1;::: . Throughout the f g f g paper, (cid:9) is (cid:12)xed, and the discussion focuses on restrictions on the A parameter space, which is called . There are no restrictions across the and A parameter A spaces: (cid:18) (cid:2) = (cid:9). 2 A(cid:2) 1 Under A3, the A parameter space is a subset of ‘ , the space of summable sequences. If the bias of the smoothed periodogram estimator is to vanish must A satisfy some further assumption like, 1 p A 4 There is a p > 0 such that for each A , j=(cid:0)1 j (cid:27)(j) < where (cid:27)(j) is the j th autocovariance under A. 2 A P j j j j 1 The (cid:12)nal assumption is speci(cid:12)ed in terms of the spectrum of yt at frequency !, f g which is de(cid:12)ned as, 2 s(!;A)= f(!;A) ; j j where x means the modulus of x, ! [ (cid:25);(cid:25)], and f(!;A)is the Fourier transform j j 2 (cid:0) of the A parameter, 1 (cid:0)1=2 (cid:0)i!j f(!;A)= (2(cid:25)) aje ; jX=0 with i= p 1. When f and s are treated as functions of A for (cid:12)xed !, I will write (cid:0) f!(A) and s!(A). The (cid:12)nal assumption is required only because the distribution theory is di(cid:11)erent when s!(A) = 0: A 5 s!(A)>s! > 0 for all A . 2 A Under these assumptions, smoothed periodogram estimates of the spectrum at T T !, s!(y ), will be asymptotically normal and centered on the true value: T T 2 2 T=KT s!(y ) s!(A) N 0; (1+(cid:31)0)s!(A) v ; (1) q (cid:16) (cid:0) (cid:17) ) (cid:16) (cid:17) T where y = y1;:::;yT is a random sample of size T under the chosen process, f g means converges in distribution to, v depends on the particular window chosen ) 3

for smoothing the periodogram, KT is the width of the window and rises to in(cid:12)nity more slowly than T, and (cid:31)0 is 1 if ! = 0, 0 otherwise. Based on (1), the textbooks suggest that con(cid:12)dence intervals of the form (cid:0)1 T T s!(y ) 1 c((cid:11))v (1+(cid:31)0)KT=T (2) (cid:18) (cid:6) q (cid:19) will asymptotically have con(cid:12)dence level 100(1 (cid:11)) percent when c((cid:11)) is the two- (cid:0) sided100(cid:11)percentpointofthestandardnormaldistribution. Thenexttwosections demonstrate thatthese con(cid:12)denceintervalswillnotbeoftheassertedsizeunderthe assumptions and give conditions on for meaningful con(cid:12)dence intervals to exist. A 2 Con(cid:12)dence intervals and con(cid:12)dence levels In general, given a sample of size T, a con(cid:12)dence interval on the real, scalar param- T 1 eter s!(A) is a mapping, WT, from to intervals of . The con(cid:12)dence interval R R attains the 1 (cid:11) level if and only if, (cid:0) T pr(cid:18) s!(A) WT(y ) 1 (cid:11) for all (cid:18) (cid:2); (3) (cid:16) 2 (cid:17)(cid:21) (cid:0) 2 where pr(cid:18)(x) means the probability of event x under the process parameterized by (cid:18) [e.g., Lehmann, 1986]. A sequence of such con(cid:12)dence intervals, WT , asymptotf g ically attains the con(cid:12)dence level (1 (cid:11)) if (3) holds in the limit as T increases. (cid:0) The intuition for why the textbook con(cid:12)dence intervals for s! do not have the propersizeissimple: iftherearetwoprocessesthathavearbitrarilysimilarempirical properties, but arbitrarily di(cid:11)erent s!, then observation will not help pin down the value of s!, and no (cid:12)nite-length con(cid:12)dence intervals exist. Three propositions apply this reasoning in the current context. Prop. 1: if the A parameters for two 2 5 processes are su(cid:14)ciently close in the ‘ norm, then the processes will be nearly 2 indistinguishable empirically. Prop. 2: if s!(A) is discontinuous under the ‘ -norm topologyon ,thenprocesseswitharbitrarilysimilarAcanhavearbitrarilydi(cid:11)erent A s!(A) and con(cid:12)dence intervals will not exist. Prop. 3: s! is discontinuous under standard assumptions. 5 The ‘p distance between AandA 0 is denoted kA(cid:0)A 0 kp andequals ( 1 j=0jaj(cid:0)a 0 jj p ) 1=p . P 4

Henceforth, we consider the existence of meaningful con(cid:12)dence intervals: con- (cid:12)dence intervals that are of (cid:12)nite-length with probability one and for which the con(cid:12)dence level is greater than zero and known, at least asymptotically. Con(cid:12)dence intervals with unknown level, level zero, or that cover the entire parameter space exist trivially, but are uninteresting. For Proposition 1, take a model satisfying A1{A3, with parameter space (cid:2) = T (cid:9) and (cid:12)x any T and (cid:18) = (A; ) (cid:2). De(cid:12)ne y as a random variable A (cid:2) 2 T with parameter (cid:18), and take any sequence of random variables yk where yk has f g parameter (cid:18)k = (Ak; ) for some Ak . 2 A T T Proposition 1 If Ak A 2 0, then yk converges in distribution to y . k (cid:0) k ! ThispropositionisastraightforwardgeneralizationofBernstein’sLemma[e.g.,Hannan, 1970] and is stated without proof; remaining proofs are in the Appendix. The second proposition uses the notion of unbounded discontinuity. A function, g, from the normed vector space to the normed vector space has an unbounded A B discontinuityat Aif g(A) is(cid:12)nite and iffor every (cid:14) > 0, every open neighborhood k k 0 0 A contains an A such that g(A) > (cid:14). k k For Proposition 2, take any con(cid:12)dence intervals, WT , for s! that are of (cid:12)nite f g length withprobabilityone. Take any(cid:12)niteT, andanymaintainedmodelsatisfying A1{A3. 2 Proposition 2 If s! has unbounded discontinuities in the ‘ -norm topology on , A the con(cid:12)dence intervals have con(cid:12)dence level zero. 2 This result mirrors the central point of Sims [1971, 1972]: inference about ‘ discontinuous functions of A is beset with problems. For concreteness, it is now useful explicitly to specify an parameter space A satisfying A3{A5. Begin with the parameter space for the (cid:12)nite-order, movingaverage processes, 1 = A l aj = 0 for all but a (cid:12)nite number of j. ; F f 2 j g 5

6 and exclude the As that don’t satisfy A5. In particular, for any (cid:14) < , de(cid:12)ne 1 (cid:14) = A s!(A) > (cid:14) for all ! [ (cid:25);(cid:25)] : F f 2 Fj 2 (cid:0) g We can now state, Proposition 3 If (cid:14), then for every ! [ (cid:25);(cid:25)], s! has unbounded disconti- A2(cid:19) F 2 (cid:0) nuities on in the ‘ -norm topology. A Of course, the standard assumptions guarantee that for (cid:12)xed A, s(!;A) is bounded and continuous, but since s!(A) is discontinuous in A, the textbook con(cid:12)dence intervals have con(cid:12)dence level zero under standard assumptions. Further, no meaningful con(cid:12)dence intervals exist. This does not contradict the asymptotic normalityofthepointestimates: the resultarisesbecausetheconvergencetonormality in (1) is not uniform in A. 3 Necessary and su(cid:14)cient conditions for solving the problem Prop. 3 gives a necessary condition for existence of meaningful con(cid:12)dence intervals on estimates of the spectrum at !: s! must be continuous in A. Under one further assumption we can give a necessary and su(cid:14)cient condition for continuity of s!; under a second further assumption, continuity of s! requires that be (cid:12)nite A dimensional. 1 A 6 The A parameter space is (cid:14) where is a linear subspace of ‘ . A A Many familiar sarelinear,such as . Whenwework within(cid:12)nite-dimensional A F spaces, we usually hope to cover a broader range of cases than possible when considering just (cid:12)nite-order moving averages ( ); thus, most spaces used in practice F 1 contain a linear subspace of ‘ . While the results require more cumbersome language to state, a version of the results obviously holds when the parameter space is 1 7 (cid:14), where contains a linear subspace of ‘ . A A 6 Each A2F clearly satis(cid:12)es A3 and A4, since A and the associated autocorrelation function have a (cid:12)nitenumber of nonzero elements. 7 Forexample,ifthecon(cid:12)denceleveliszeroonthelinearsubspace,itmustalsobe zeroforany parameterspace containingthe linear subspace. 6

2 When is a linear space, then under the ‘ -norm topology we have a normed A linear space. Since f! is a linear function on this space, a necessary and su(cid:14)cient condition for f! to be continuous is that it have (cid:12)nite norm [e.g., Berberian, 1976]: F(!)= sup f!(A) = A 2 < : (4) A2Aj j k k 1 Giventhe linearityof ,whenf! isnotboundedinthisway, f! and,hence, s! have A unbounded discontinuities; when f! is continuous, so is s!. Of course, the value of 2 8 F(!) is the maximum height of the (normalized) spectral density at ! under the maintained model. Thus, under A1{A6, a necessary and su(cid:14)cient condition for s! to be continuous is that the normalized spectral density have a bound that is uniform in A . This result is not too surprising. Without such a bound, there 2 A can be processes for which the normalized density has an arbitrarily high peak at ! with arbitrarily little mass under it. Contributing arbitrarily little to the variance of the process, such a peak is hard to detect. What is more surprising is that under a reasonable assumption on the natureofthe uniformbound, meaningfulcon(cid:12)dence intervals exist only under (cid:12)nite-dimensional . A Whenever we are willing to accept that the normalized spectrum is bounded at all !, so that con(cid:12)dence intervals can be formed at arbitrary !, the assumed bound probablyissimilaracrossnearbyfrequencies. Aweakrestrictionofthistypeisgiven by A 7 (cid:14) is such that F(!) is upper semi-continuous for ! [ (cid:25);(cid:25)]. A 2 (cid:0) Whilethere isnothingincoherentaboutmodelsthat violate A7, modelsthat violate the assumption have the property that the maximumnormalizedspectral density at some frequencyisdiscontinuouslylower than the boundforneighboringfrequencies. Such a model would certainly call for some justi(cid:12)cation. Proposition 4 If the parameter space (cid:14) satis(cid:12)es A6 then (cid:12)nite-dimensionality A of is su(cid:14)cient for s! to be continuous for all !. If (cid:14) also satis(cid:12)es A7 then A A (cid:12)nite-dimensionality of is also necessary for s! to be continuous. A 8 2 The normalized spectraldensity of the fytgprocess iss! dividedbythevariance ofyt,kAk2. 7

Thus, ifwe wish to form con(cid:12)dence intervalsfor the spectrum at arbitrary !, we must give up one of three things: 1) A parameter space that contains a linear sub- 1 spaceof‘ ,2)smoothnessoftheboundonthespectrum,or3)in(cid:12)nite-dimensionality of the linear subspace. Up to now we have considered what new restrictions we need in order to form con(cid:12)dence intervals on s!. An alternative approach is to change the parameter for which we seek con(cid:12)dence intervals. The most natural approach is to consider some average of the spectrum’s value over an interval, rather than at a single point. Consider the parameter, (cid:25) Sh(A) = h(!)s(!;A)d!: (5) Z(cid:0)(cid:25) Of course, the variance of the process is an example of such a parameter, with h constant. An interesting class of hs is that comprised by the h!;(cid:1) that are positive, integrate to one and have support ! (cid:1), so that sh!;(cid:1)(A) is a weighted average (cid:6) of s(w;A) in the neighborhood of !. Con(cid:12)dence intervals on Sh require only one restriction on h: 1 Proposition 5 For ‘ , if h(!) is bounded, then Sh is continuous in A under 2 A (cid:18) the ‘ -norm topology. 4 Discussion One can avoid the problems demonstrated here by limiting the A parameter space p to, e.g., for some (cid:12)nite, large p. Alternatively, one can change the parameter of R interesttobeaweightedaverageofthespectruminsomeinterval,! (cid:1). Giventhese (cid:6) solutions, it might be argued that the problems have few substantive implications: surely there is some p large enough or (cid:1) small enough that the added restriction is not too onerous. This view is not very satisfactory. While any choice of p or (cid:1) solves the asymptotic con(cid:12)dence level problem, in any given sample size, it matters just what choice is made. It is straightforward to show that one can drive the con(cid:12)dence level arbitrarily close to zero by choosing p too large or by choosing (cid:1) too small. Little 8

applied work explicitly states the required restriction, and it seems clear that in many contexts a proper choice of p or (cid:1) is not known. For example, in the context of unit root inference|which is analogous to inference about the spectrum at frequency zero|the correct choices are clearly in question [on this, see Faust, 1996]. It is hoped that the results of this paper provide motivation for and guidance in exploring the restrictions required for approximately valid con(cid:12)dence statements. 9

Appendix Proof of proposition 2: Fix a (cid:18) = A; such that s! has an unbounded f g discontinuity at A and a such that the probability measure for "t is absolutely continuous with respect to the Lebesgue measure. Since the con(cid:12)dence intervals are of (cid:12)nite length, the intervals must have a (cid:12)nite upper bound with probability one. Thus, for each probability p > 0, there must be a W(cid:22) < (depending on (cid:18) and p) 1 such that pr(cid:18)(W(cid:22) WT(y T )) < p. Since s! has an unbounded discontinuity at A, 2 there is a sequence Ak such that Ak A 2 0 and s!(Ak) > W(cid:22) for all k. Thus, f g k (cid:0) k ! taking (cid:18)k = Ak; , and using the fact that under the assumptions convergence in f g distribution implies convergence in probability measure, k l ! im 1 pr(cid:18)k(W(cid:22) 2 WT(y T )) = pr(cid:18)(W(cid:22) 2 WT(y T )) < p Since s!(Ak) > W(cid:22) , there must be a K such that, pr(cid:18)K(s!(AK) WT(y T )) < 2p. 2 Since p was arbitrary, the con(cid:12)dence intervals are of con(cid:12)dence level zero. Q.E.D. Proof of proposition 3: First, show the result for = , then trivially A F 2 extend to (cid:14). Under the ‘ -norm topology, is normed linear space, and A (cid:19) F A f! is discontinuous if its norm, (4), is in(cid:12)nite. We construct a sequence Ak f g such that f!(Ak) is bounded away from zero, but Ak 2 0. Fix (cid:24) > 0 and j j k k ! de(cid:12)ne Ak such that Ak has only k non-zero coe(cid:14)cients: set akj = (cid:24)=k if 2 A cos(!j) > 1=2 and if there are fewer than k elements akh satisfying akh > 0 for (cid:0)1=2 h < j. Set akj = 0 otherwise. Note that f!(Ak) > (2(cid:25)) (1=2)(cid:24) for all k, j j and that Ak 2 = (cid:24)=pk. Thus, f!(Ak)= Ak 2 . Since (cid:24) was arbitrary, k k j j k k ! 1 the discontinuities are unbounded. Since is linear, f! and s! have unbounded A discontinuities at each A , and, hence, at each A (cid:14). Q.E.D. 2 F 2 F Proof proposition 4: Su(cid:14)ciency: when is (cid:12)nite-dimensional and linear, A all linear functions, including f! are continuous [Berberian, 1976, p.96]. Necessity. Take where (cid:14) is any parameter space satisfying the stated assumptions. We A A 2 must show that is (cid:12)nite dimensional. Since is a linear subspace of ‘ , it is a A A separable pre-Hilbert space. The map taking A into the function gA(!) = f(!;A) 10

2 is an isometric isomorphism between and a linear subspace, callit , of L [ (cid:25);(cid:25)] A B (cid:0) (the space of square integrable functions). The spaces and are of the same A B dimension. The proof shows that is of (cid:12)nite dimension. B The space contains a set of n linearly independent elements for any n less B than or equal to its dimension. Take n < linearly independent elements of . 1 B Orthonormalize the set to form a set uj , j = 1;:::;n, of orthonormal functions f g [Berberian, 1976, p.47]. Since F(!) is an upper semi-continuous mapping on the compact set [ (cid:25);(cid:25)] it has a (cid:12)nite least upper bound, call it F(cid:22), which uniformly (cid:0) bounds uj(!) for all j and !. j j From here, the proof of (cid:12)nite dimensionality is is a standard analysis problem [e.g., Royden, 1968, p. 214] and is completed in two steps. Step one: Show that j n =1 uj(!) 2 < 2F(cid:22) 2 for all !. Fix !. De(cid:12)ne g = j n =1(cid:11)juj, where (cid:11)j = j j P n 2 P rj= k=1rk and rk is the real part of uk(!); mk is the imaginary part. Note q 2 P g 2 = 1, and k k 2 2 2 2 g(!) = ( rj)+( rjmj) = rj; j j X X X 2 2 2 2 2 2 2 implying g(!) rj. Since g(!) = g 2 < F(cid:22) , we have rj(!) < F(cid:22) for each j j (cid:21) j j k k P P !. De(cid:12)ne g~ by replacing r with m in de(cid:12)ning (cid:11), and follow the same steps to show 2 2 2 2 2 mj < F(cid:22) . Thus, rj + mj < 2F(cid:22) , which veri(cid:12)es the step one claim. P P P2 2 Step 2: Show n < 4(cid:25)F(cid:22) . Since the us are orthonormal, n = uj 2 = k k n 2 P j=1 uj 2; thus, k k P n n 2 2 2 2 n = uj(!) d! = uj(!) d! 2F(cid:22) d! = 4(cid:25)F(cid:22) jX=1 Z j j Z jX=1j j (cid:20) Z wheretheinequalitycomesfromthesteponeresult. Thus,theset ut ,oforthonorf g 2 mal elements has cardinality less than or equal to 4(cid:25)F(cid:22) . Since the set of n elements was arbitrary, each such set, and, hence, the dimension of has the same bound. A Q.E.D. Proof proposition 5 Note that s(:;A) is absolutely integrable and that Ak k (cid:0) A 2 0 implies s(!;Ak) s(!;A)d! 0. It follows directly that for any k ! j (cid:0) j ! R bounded function, h, Ak A 2 0 implies Sh(Ak) Sh(A) 0. Q.E.D. k (cid:0) k ! j (cid:0) j! 11

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