serial correlation coefficient - manfred mudelsee

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Investigating a new estimator of the serial correlation coefficient Manfred Mudelsee Institute of Mathematics and Statistics University of Kent at Canterbury Canterbury Kent CT2 7NF United Kingdom 8 May 1998 Abstract A new estimator of the lag-1 serial correlation coefficient ρ, n-1 i=1 3 i i+1 / n-1 i=1 4 i , is compared with the old estimator, n-1 i=1 i i+1 / n-1 i=1 2 i , for a stationary AR(1) process with known mean. The mean and the variance of both estimators are calculated using the second or- der Taylor expansion of a ratio. No further approximation is used. In case of the mean of the old estimator, we derive Marriott and Pope’s (1954) formula, with (n - 1) -1 instead of (n) -1 , and an additional term (n - 1) -2 . In case of the variance of the old estimator, Bartlett’s (1946) formula results, with (n - 1) -1 instead of (n) -1 . The theoretical expressions are corroborated with simulation experiments. The main results are as follows. (1) The new esti- mator has a larger negative bias and a larger variance than the old estimator. (2) The theoretical results for the mean and the variance of the old estimator describe the principal behaviours over the entire range of ρ, in particular, the decline to zero negative bias as ρ approaches unity.

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Page 1: Serial correlation coefficient - Manfred Mudelsee

Investigating a new estimator of theserial correlation coefficient

Manfred Mudelsee

Institute of Mathematics and StatisticsUniversity of Kent at Canterbury

CanterburyKent CT2 7NFUnited Kingdom

8 May 1998

Abstract

A new estimator of the lag-1 serial correlation coefficient ρ,∑n−1i=1 ε3

i εi+1 /∑n−1

i=1 ε4i , is compared with the old estimator,∑n−1

i=1 εiεi+1 /∑n−1

i=1 ε2i , for a stationary AR(1) process with known mean. The

mean and the variance of both estimators are calculated using the second or-

der Taylor expansion of a ratio. No further approximation is used. In case of

the mean of the old estimator, we derive Marriott and Pope’s (1954) formula,

with (n− 1)−1 instead of (n)−1, and an additional term ∝ (n− 1)−2. In case

of the variance of the old estimator, Bartlett’s (1946) formula results, with

(n− 1)−1 instead of (n)−1. The theoretical expressions are corroborated with

simulation experiments. The main results are as follows. (1) The new esti-

mator has a larger negative bias and a larger variance than the old estimator.

(2) The theoretical results for the mean and the variance of the old estimator

describe the principal behaviours over the entire range of ρ, in particular, the

decline to zero negative bias as ρ approaches unity.

Mudelsee
Text Box
Mudelsee M (1998) Investigating a new estimator of the serial correlation coefficient. Institute of Mathematics and Statistics, University of Kent, Canterbury, IMS Technical Report UKC/IMS/98/15, Canterbury, United Kingdom, 22 pp.
Page 2: Serial correlation coefficient - Manfred Mudelsee

1 Introduction

Consider the following estimators of the serial correlation coefficient ρ oflag 1 from a time series εi (i = 1, . . . , n) sampled from a process Ei withknown (zero) mean:

ρ̂ =

∑n−1i=1 εiεi+1∑n−1

i=1 ε2i

, (1)

ρ̂? =

∑n−1i=1 ε3

i εi+1∑n−1i=1 ε4

i

. (2)

Estimators of type (1) are well-known (e. g. Bartlett 1946, Marriott andPope 1954), whereas (2) is new to my best knowledge.

Let Ei be the stationary AR(1) process,

E1 ∼ N(0, 1),

Ei = ρ Ei−1 + Ui, i = 2, . . . , n, (3)

with Ui i. i. d. ∼ N(0, σ2) and σ2 = 1− ρ2. The following misjudgement ofmine led to the present study. ρ̂? minimises

n−1∑i=1

(εi+1 − ρεi)2 ε2

i ,

which is a weighted sum of squares. However, since Ei has constant varianceunity, no weighting should be necessary.

Nevertheless, we compare estimators (1) and (2) for process (3). We re-strict ourselves to 0 < ρ < 1. We first calculate their variances, respectivelymeans, up to the second order of the Taylor expansion of a ratio, similarlyto Bartlett (1946), respectively Marriott and Pope (1954). Since we concen-trate on the lag-1 estimators and process (3), no further approximation isnecessary. These theoretical expressions and also those of White (1961) forestimator (1) are then examined with simulation experiments.

2 Series expansions

2.1 Old estimator

2.1.1 Variance

We write (1) as

ρ̂ =1n

∑n−1i=1 εiεi+1

1n

∑n−1i=1 ε2

i

(4)

=c

v, say,

1

Page 3: Serial correlation coefficient - Manfred Mudelsee

and have, up to the second order of deviations from the means of c and v,the variance

var(ρ̂) = var(c/v) ' var(c)

E2(v)− 2 E(c)

cov(v, c)

E3(v)+ E2(c)

var(v)

E4(v). (5)

We now apply a standard result for quadravariate standard Gaussian distri-butions with serial correlations ρj (e. g. Priestley 1981:325),

cov(εaεa+s, εbεb+s+t) = ρb−a ρb−a+t + ρb−a+s+t ρb−a−s,

from which we derive, for εi following process (3):

cov

(1

n

n−1∑a=1

εaεa+s,1

n

n−1∑b=1

εbεb+s+t

)=

=1

n2

n−1∑a,b=1

(ρ|b−a| ρ|b−a+t| + ρ|b−a+s+t| ρ|b−a−s|). (6)

Without further approximation we can directly derive the various constituentsof the right-hand side of (5) by the means of (6).

var(c) = var

(1

n

n−1∑i=1

εiεi+1

)= (put s = 1 and t = 0 in (6))

=1

n2

n−1∑a,b=1

ρ2|b−a| +n−1∑

a,b=1

ρ|b−a+1| ρ|b−a−1|

.

We find, by summing over the points of the a-b lattice in a ‘diagonal’ manner,

n−1∑a,b=1

ρ2|b−a| = (n− 1) + 2n−2∑i=1

i ρ2(n−i−1)

= (n− 1)

(2

1− ρ2− 1

)+ 2

(ρ2n−4 − ρ−2)

(1− ρ−2)2(7)

(at the last step we have used the arithmetic-geometric progression), and

n−1∑a,b=1

ρ|b−a+1| ρ|b−a−1| = (n− 1) ρ2 + 2n−2∑i=1

i ρ2(n−i−1)

= (n− 1)

(2

1− ρ2+ ρ2 − 2

)+ 2

(ρ2n−4 − ρ−2)

(1− ρ−2)2. (8)

(7) and (8) give var(c). Further,

cov(v, c) = cov

(1

n

n−1∑i=1

ε2i ,

1

n

n−1∑i=1

εiεi+1

)

2

Page 4: Serial correlation coefficient - Manfred Mudelsee

= (put s = 0 and t = 1 in (6))

=2

n2

n−1∑a,b=1

ρ|b−a| ρ|b−a+1|.

We find

n−1∑a,b=1

ρ|b−a| ρ|b−a+1| = ρ2n−3 +n−2∑i=1

(2 i + 1) ρ2(n−i)−3

= (n− 1)2

ρ

(1

1− ρ2− 1

)

+2(ρ2n−5 − ρ−3)

(1− ρ−2)2+

(ρ2n−3 − ρ−1)

(1− ρ−2). (9)

(At the last step we have used the arithmetic-geometric and also the geomet-ric progression.) (9) gives cov(v, c). Further,

var(v) = var

(1

n

n−1∑i=1

ε2i

)= (put s = t = 0 in (6))

=2

n2

n−1∑a,b=1

ρ2|b−a|,

which is given from (7). We further need

E(v) =n− 1

n(10)

and

E(c) =ρ(n− 1)

n. (11)

All three terms contributing to var(ρ̂) have a part ∝ (n− 1)−2. However,these parts cancel out:

var(ρ̂) ' (1− ρ2)

(n− 1). (12)

Bartlett (1946) already gave, up to the order (n)−1,

var(ρ̂) ' (1− ρ2)

n, (13)

which cannot be distinguished from our result.

3

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2.1.2 Mean

We have, up to the second order of deviations from the means of c andv, the mean

E(ρ̂) = E(c/v) ' E(c)

E(v)− cov(v, c)

E2(v)+ E(c)

var(v)

E3(v). (14)

As above, we derive cov(v, c) and var(v) from (9) and (7), respectively, andhave E(v) and E(c) given by (10) and (11), respectively, yielding

E(ρ̂) ' ρ− 2 ρ

(n− 1)+

2

(n− 1)2

(ρ− ρ2n−1)

(1− ρ2). (15)

Marriott and Pope (1954) investigated the estimator

ρ̂′ =1

n−1

∑n−1i=1 εiεi+1

1n

∑n−1i=1 ε2

i

and gave, up to the order (n)−1,

E(ρ̂′) ' ρ− 2 ρ

n,

which cannot be distinguished from the first two terms of our result.

2.1.3 Unknown mean of the process

I have also tried to calculate up to the same order of approximation themean and the variance of the estimator∑n−1

i=1

(εi − 1

n−1

∑n−1j=1 εj

) (εi+1 − 1

n−1

∑n−1j=1 εj+1

)∑n−1

i=1

(εi − 1

n−1

∑n−1j=1 εj

)2 ,

for the case when the mean of the process is unknown. That would requireto calculate the following sums over the points of a cubic lattice:

n−1∑a,b,c=1

ρ|b−a| ρ|c−a−s|

for s = 0 and 1. However, I was unable to obtain exact formulas.

2.2 New estimator

2.2.1 Variance

We write (2) as

ρ̂? =1n

∑n−1i=1 ε3

i εi+11n

∑n−1i=1 ε4

i

=d

w, say.

4

Page 6: Serial correlation coefficient - Manfred Mudelsee

var(ρ̂?), up to the second order of the Taylor expansion, is given by (5),substituting d for c and w for v, respectively. We need the following resultfor octavariate standard Gaussian distributions with serial correlations ρj

(Appendix),

cov(ε3aεa+s, ε

3bεb+s+t) = 9 ρb−a ρb−a+t + 9 ρb−a+s+t ρb−a−s

+ 18 ρs ρb−a ρb−a+s+t + 18 ρb−a ρb−a−s ρs+t

+ 18 ρs ρ2b−a ρs+t + 18 ρ2

b−a ρb−a+s+t ρb−a−s

+ 6 ρ3b−a ρb−a+t, (16)

from which we derive, for εi following process (3):

cov

(1

n

n−1∑a=1

ε3aεa+s,

1

n

n−1∑b=1

ε3bεb+s+t

)=

=1

n2

n−1∑a,b=1

( 9 ρ|b−a| ρ|b−a+t| + 9 ρ|b−a+s+t| ρ|b−a−s|

+ 18 ρ|s| ρ|b−a| ρ|b−a+s+t| + 18 ρ|b−a| ρ|b−a−s| ρ|s+t|

+ 18 ρ|s| ρ2|b−a| ρ|s+t| + 18 ρ2|b−a| ρ|b−a+s+t| ρ|b−a−s|

+ 6 ρ3|b−a| ρ|b−a+t| ) . (17)

Without further simplification we can directly obtain the various constituentsof var(ρ̂?) by the means of (17).

var(d) = var

(1

n

n−1∑i=1

ε3i εi+1

)= (put s = 1 and t = 0 in (17))

=1

n2

9n−1∑

a,b=1

ρ2|b−a| + 9n−1∑

a,b=1

ρ|b−a+1| ρ|b−a−1|

+ 18 ρn−1∑

a,b=1

ρ|b−a| ρ|b−a+1| + 18 ρn−1∑

a,b=1

ρ|b−a| ρ|b−a−1|

+ 18 ρ2n−1∑

a,b=1

ρ2|b−a| + 18n−1∑

a,b=1

ρ2|b−a| ρ|b−a+1| ρ|b−a−1|

+ 6n−1∑

a,b=1

ρ4|b−a|

.

The first and the fifth sum is given by (7), the second sum by (8) and thethird by (9). We find,

n−1∑a,b=1

ρ|b−a| ρ|b−a−1| =n−1∑

a,b=1

ρ|b−a| ρ|b−a+1|,

5

Page 7: Serial correlation coefficient - Manfred Mudelsee

which is given by (9),

n−1∑a,b=1

ρ2|b−a| ρ|b−a+1| ρ|b−a−1| = (n− 1) ρ2 + 2n−2∑i=1

i ρ4(n−i−1)

= (n− 1)

(2

1− ρ4+ ρ2 − 2

)+ 2

(ρ4n−8 − ρ−4)

(1− ρ−4)2(18)

and

n−1∑a,b=1

ρ4|b−a| = (n− 1) + 2n−2∑i=1

i ρ4(n−i−1)

= (n− 1)

(2

1− ρ4− 1

)+ 2

(ρ4n−8 − ρ−4)

(1− ρ−4)2. (19)

(7), (8), (9), (18) and (19) give var(d). Further,

cov(w, d) = cov

(1

n

n−1∑i=1

ε4i ,

1

n

n−1∑i=1

ε3i εi+1

)

= (put s = 0 and t = 1 in (17))

=1

n2

36n−1∑

a,b=1

ρ|b−a| ρ|b−a+1| + 36 ρn−1∑

a,b=1

ρ2|b−a|

+ 24n−1∑

a,b=1

ρ3|b−a| ρ|b−a+1|

.

The last sum of this expression,

n−1∑a,b=1

ρ3|b−a| ρ|b−a+1| = (n− 1) ρ +n−2∑i=1

i ρ4(n−i)−3 +n−2∑i=1

i ρ4(n−i)−5

= (n− 1)

(1

ρ−1 − ρ3+

1

ρ− ρ5− 1

ρ

)

+(ρ4n−7 + ρ4n−9) (1− ρ−4n+4)

(1− ρ−4)2. (20)

(7), (9) and (20) give cov(w, d). Further,

var(w) = var

(1

n

n−1∑i=1

ε4i

)= (put s = t = 0 in (17))

=1

n2

72n−1∑

a,b=1

ρ2|b−a| + 24n−1∑

a,b=1

ρ4|b−a|

,

6

Page 8: Serial correlation coefficient - Manfred Mudelsee

which is given from (7) and (19), respectively. We further need

E(w) =3 (n− 1)

n(21)

and

E(d) =3 ρ (n− 1)

n. (22)

As for the old estimator, the terms ∝ (n−1)−2 which contribute to var(ρ̂?)cancel out:

var(ρ̂?) ' 5

3

(1− ρ2)

(n− 1). (23)

2.2.2 Mean

We use (14) with d instead of c and w instead of v, respectively. Wederive cov(w, d) from (7), (9) and (20), further, var(w) from (7) and (19),and we have E(w) and E(d) given by (21) and (22), respectively. This yields,up to the second order of deviations from the means of d and w,

E(ρ̂?) ' ρ − 1

(n− 1)

4

(5− 2

1 + ρ2

)

+1

(n− 1)2

[4

(ρ− ρ2n−1)

(1− ρ2)+

8

3

(ρ3 − ρ4n−1)

(1− ρ2) (1 + ρ2)2

]. (24)

2.3 Remark

ρ̂? is found to have a larger negative bias and also a larger variance than ρ̂.In the simulation experiment, we intend not only to prove that. We are alsointerested to examine the significance of the term ∝ (n−1)−2 in (15). In thecomparison we include the theoretical results of White (1961) who studiedthe old estimator (1) for process (3). He calculated the kth moment of (c/v)via the joint moment generating function m(c, v), with c and v defined as in(4) without the factors 1/n,

E(ρ̂k) =∫ 0

−∞

∫ vk

−∞. . .∫ v2

−∞

∂km(c, v)

∂ck

∣∣∣∣∣c =0

dv1 dv2 . . . dvk.

He expanded the integrand to terms of order n−3 and ρ4 and gave the fol-lowing results (the index ‘W ’ refers to his study):

var(ρ̂)W '(

1

n− 1

n2+

5

n3

)−(

1

n− 9

n2+

53

n3

)ρ2 − 12

n3ρ4, (25)

E(ρ̂)W '(1− 2

n+

4

n2− 2

n3

)ρ +

2

n2ρ3 +

2

n2ρ5. (26)

7

Page 9: Serial correlation coefficient - Manfred Mudelsee

3 Simulation experiments

In the first experiment, for each combination of n and ρ listed in Table1, we generated a set of 250 000 time series from process (3). For every timeseries, ρ̂? and ρ̂ have been calculated. The sample means, µρ̂? sim and µρ̂ sim,and the sample standard deviations, σρ̂? sim and σρ̂ sim, over the simulationsare listed in Table 1. Therein, our theoretical values—from (12), (15), (23)and (24)—are included. In case of the means, these are additionally splittedinto terms ∝ (n− 1)−2 and terms of lower order. Also the theoretical valuesfrom White’s (1961) formula, herein (25) and (26), are listed.

We further investigated whether the distribution of ρ̂? approaches Gaus-sianity as fast as that of ρ̂. Fig. 1 shows histograms of ρ̂?, respectively ρ̂,from the first simulation experiment in comparison with Gaussian distribu-tions N(µρ̂? sim, σ2

ρ̂? sim), respectively N(µρ̂ sim, σ2ρ̂ sim).

In the second simulation experiment (Fig. 2), formulas (12) versus (25)and (15) versus (26) are examined over the entire range of ρ, with particularinterest on ρ large. We plot the negative bias, ρ − E(ρ̂), and

√var(ρ̂). For

each combination of ρ and n, the number of simulated time series after (3)is 250 000.

The error due to the limited number of simulations is small enough fornot influencing any intended comparison.

4 Results and discussion

The first simulation experiment (Table 1) confirms, over a broad range ofρ and n, that ρ̂? has a larger bias and a larger variance, respectively, than ρ̂.

In general, the deviations between the theoretical results and the simula-tion results decrease with n increased, as is to be expected.

For any combination of ρ and n listed in Table 1, our terms ∝ (n−1)−2 in(15), respectively (24), bring these theoretical results closer to the simulationresults, with the exception of E(ρ̂) for ρ = 0.9 and n = 800 where thesimulation noise prevents that comparison. For ρ large, respectively n small,these second order terms contribute heavier.

The frequency distribution of ρ̂? (Fig. 1) has a similar shape as that ofρ̂. It is shifted to smaller values against that and also broader, reflecting thelarger negative bias, respectively the larger variance. The functional formof the distribution of ρ̂ is approximately the Leipnik distribution, which isheavier skewed for ρ large and tends to Gaussianity with n increased. Bothestimators seem to approach Gaussianity equally fast (Fig. 1).

The second simulation experiment (Fig. 2) compares White’s (1961) the-oretical results for ρ̂, (25) and (26), with ours, (12) and (15). It shows thathis are better performing up to a certain value of ρ.

Above, our formulas describe better the simulation, particularly, the de-

8

Page 10: Serial correlation coefficient - Manfred Mudelsee

cline to zero negative bias, respectively zero variance, as ρ approaches unity.The decline to zero negative bias is caused by the term ∝ (n− 1)−2 in (15).Those declines are reasonable, since for ρ = 1 all time series points haveequal value and c = v in (4).

For small ρ, his formulas perform better since they are more accuratewith respect to powers of (1/n) than ours. For larger ρ his approximationbecomes less accurate. In particular, for n ≥ 3 and ρ > 0, ρ− E(ρ̂)W cannotbecome zero. For n ≥ 10, d

dρ(ρ− E(ρ̂)W ) cannot become negative. For n ≥ 8,

var(ρ̂)W cannot become zero. That means, for those cases White’s (1961)formulas cannot produce the decline to zero negative bias and zero variance,respectively.

The fact that Bartlett’s (1946) formula for the variance, (13), describesthe simulation better than (12) for ρ → 0, is regarded as spurious.

These results mean that the expansion formulas for var(ρ̂) and E(ρ̂), (5)and (14), respectively, are sufficient to derive the principal behaviours forρ → 1. In case of the mean, however, no further approximation is allowedwhich would lead to the term ∝ (n− 1)−2 in (15) be neglected.

5 Conclusions

1. In case of the stationary AR(1) process (3) with known mean, the newestimator, ρ̂?, has a larger negative bias and a larger variance than theold estimator, ρ̂. Its distribution tends to Gaussianity about equallyfast.

2. Our formula (15) for E(ρ̂) describes the expected decline to zero nega-tive bias for ρ → 1.

3. The formula (12) for var(ρ̂) describes the expected decline to zero vari-ance for ρ → 1.

4. The second order Taylor expansion of a ratio is sufficient to derivethese two principal behaviours for ρ → 1, if no further approximationsare made. This condition could be fulfilled since we had concentratedourselves on the lag-1 estimator and process (3).

5. For unknown mean of the process, it is more complicated to derive theequations with the same accuracy.

Acknowledgements

Q. Yao is appreciated for comments on the manuscript. The present studywas carried out while the author was Marie Curie Research Fellow (EU-TMRgrant ERBFMBICT971919).

9

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References

Bartlett, M. S., 1946, On the theoretical specification and sampling prop-erties of autocorrelated time-series: Journal of the Royal Statistical SocietySupplement, v. 8, p. 27-41 (Corrigenda: 1948, Journal of the Royal Statis-tical Society Series B, v. 10, no. 1).

Marriott, F. H. C., and Pope, J. A., 1954, Bias in the estimation of au-tocorrelations: Biometrika, v. 41, p. 390-402.

Priestley, M. B., 1981, Spectral Analysis and Time Series: London, Aca-demic Press, 890 p.

Scott, D. W., 1979, On optimal and data-based histograms: Biometrika,v. 66, p. 605-610.

White, J. S., 1961, Asymptotic expansions for the mean and variance ofthe serial correlation coefficient: Biometrika, v. 48, p. 85-94.

Appendix

We assume that εi is drawn from a standard Gaussian distribution withserial correlations ρj. We derive (16) by means of the moment generatingfunction. Write

cov(ε3aεa+s, ε

3bεb+s+t) = E(ε3

aεa+sε3bεb+s+t) − E(ε3

aεa+s) E(ε3bεb+s+t)

= E(ε3aεa+sε

3bεb+s+t) − 9 ρs ρs+t.

Now, E(ε3aεa+sε

3bεb+s+t) is the coefficient of (t31 t2 t33 t4)/(3! 1! 3! 1!) in the mo-

ment generating function

m(t1, t2, t3, t4) = exp

1

2

4∑i=1

4∑j=1

σijtitj

,

with

σ11 = σ22 = σ33 = σ44 = 1,

σ12 = σ21 = ρs,

σ13 = σ31 = ρb−a,

σ14 = σ41 = ρb−a+s+t,

σ23 = σ32 = ρb−a−s,

10

Page 12: Serial correlation coefficient - Manfred Mudelsee

σ24 = σ42 = ρb−a+t,

σ34 = σ43 = ρs+t.

Terms ∝ (t31 t2 t33 t4) in m(t1, t2, t3, t4) can only be within the fourth order ofthe series expansion of the exponential function, i. e., within

1

4!

1

2

4∑i=1

4∑j=1

σijtitj

4

,

further restricting, within

1

384(t21 + t23 + 2 σ12t1t2 + 2 σ13t1t3

+ 2 σ14t1t4 + 2 σ23t2t3 + 2 σ24t2t4 + 2 σ34t3t4)4,

further restricting, within

1

384(4 σ2

13t21t

23 + 2 t21t

23 + 4 σ12t

31t2 + 4 σ13t

31t3 + 4 σ14t

31t4 + 4 σ23t

21t2t3

+ 4 σ24t21t2t4 + 4 σ34t

21t3t4 + 4 σ12t1t2t

23 + 4 σ13t1t

33 + 4 σ14t1t

23t4

+ 4 σ23t2t33 + 4 σ24t2t

23t4 + 4 σ34t

33t4 + 8 σ12σ13t

21t2t3 + 8 σ12σ14t

21t2t4

+ 8 σ12σ34t1t2t3t4 + 8 σ13σ14t21t3t4 + 8 σ13σ23t1t2t

23 + 8 σ13σ24t1t2t3t4

+ 8 σ13σ34t1t23t4 + 8 σ14σ23t1t2t3t4 + 8 σ23σ34t2t

23t4)

2.

Finally, these terms are

1

384(2 · 4 σ2

13t21t

23 · 8 σ12σ34t1t2t3t4 + 2 · 4 σ2

13t21t

23 · 8 σ13σ24t1t2t3t4

+ 2 · 4 σ213t

21t

23 · 8 σ14σ23t1t2t3t4 + 2 · 2 t21t

23 · 8 σ12σ34t1t2t3t4

+ 2 · 2 t21t23 · 8 σ13σ24t1t2t3t4 + 2 · 2 t21t

23 · 8 σ14σ23t1t2t3t4

+ 2 · 4 σ12t31t2 · 4 σ34t

33t4 + 2 · 4 σ13t

31t3 · 4 σ24t2t

23t4

+ 2 · 4 σ13t31t3 · 8 σ23σ34t2t

23t4 + 2 · 4 σ14t

31t4 · 4 σ23t2t

33

+ 2 · 4 σ23t21t2t3 · 4 σ14t1t

23t4 + 2 · 4 σ23t

21t2t3 · 8 σ13σ34t1t

23t4

+ 2 · 4 σ24t21t2t4 · 4 σ13t1t

33 + 2 · 4 σ34t

21t3t4 · 4 σ12t1t2t

23

+ 2 · 4 σ34t21t3t4 · 8 σ13σ23t1t2t

23 + 2 · 4 σ12t1t2t

23 · 8 σ13σ14t

21t3t4

+ 2 · 4 σ13t1t33 · 8 σ12σ14t

21t2t4 + 2 · 4 σ14t1t

23t4 · 8 σ12σ13t

21t2t3

+ 2 · 8 σ12σ13t21t2t3 · 8 σ13σ34t1t

23t4 + 2 · 8 σ13σ14t

21t3t4 · 8 σ13σ23t1t2t

23).

11

Page 13: Serial correlation coefficient - Manfred Mudelsee

Thus, we find

E(ε3aεa+sε

3bεb+s+t) =

3! 1! 3! 1!

384(96 σ12σ34 + 96 σ13σ24 + 96 σ14σ23

+ 192 σ12σ13σ14 + 192 σ13σ23σ34

+ 192 σ12σ213σ34 + 192 σ2

13σ214σ23

+ 64 σ313σ24).

This gives the final result

cov(ε3aεa+s, ε

3bεb+s+t) = 9 ρb−a ρb−a+t + 9 ρb−a+s+t ρb−a−s

+ 18 ρs ρb−a ρb−a+s+t + 18 ρb−a ρb−a−s ρs+t

+ 18 ρs ρ2b−a ρs+t + 18 ρ2

b−a ρb−a+s+t ρb−a−s

+ 6 ρ3b−a ρb−a+t.

It should be noted that this result has been checked using the joint cumulantof order eight, in my assessment without less effort.

12

Page 14: Serial correlation coefficient - Manfred Mudelsee

Tab

le1:

Fir

stsi

mul

atio

nex

peri

men

t.T

henu

mbe

rof

sim

ulat

edti

me

seri

esaf

ter

(3)

isin

each

case

250

000.

S.d

.:st

anda

rdde

viat

ion.

The

orde

rof

the

unce

rtai

nty

ofth

em

ean,

resp

ecti

vely

the

stan

dard

devi

atio

n,du

eto

the

limit

ednu

mbe

rof

sim

ulat

ions

,is

S.d

./√

250

000.

Sim

:si

mul

atio

n

(sam

ple

mea

nan

dsa

mpl

est

anda

rdde

viat

ion,

µρ̂

?sim

and

σρ̂

?sim

,re

spec

tive

lyµ

ρ̂sim

and

σρ̂

sim

).T

(24,

23):

theo

reti

calva

lue,

this

stud

y,(2

4),

resp

ective

ly(2

3).

2nd:

term∝

(n−

1)−

2in

(15)

,re

spec

tive

ly(2

4).

1st:

term

sno

t∝

(n−

1)−

2in

(15)

,re

spec

tive

ly(2

4).

ρ̂?

Sim

ρ̂Si

mρ̂

?T

(24)

1st

ρ̂?

T(2

4)2n

dρ̂

?T

(24,

23)

ρ̂T

(15)

1st

ρ̂T

(15)

2nd

ρ̂T

(15,

12)

ρ̂T

(26,

25)

ρ=

.200

Mea

n.1

5437

433

0.1

6867

697

4.1

0883

190

8.0

1054

171

6.1

1937

362

5.1

5555

555

5.0

0514

403

2.1

6069

958

8.1

6776

640

0

n=

10S.d

..3

2250

070

3.3

0752

412

9.4

2163

702

1.3

2659

863

2.3

0407

367

5

S.d

./

500

.000

645

001

.000

615

048

ρ=

.200

Mea

n.1

7146

424

4.1

8238

719

7.1

5681

511

4.0

0236

531

5.1

5918

043

0.1

7894

736

8.0

0115

420

1.1

8010

156

9.1

8199

160

0

n=

20S.d

..2

3935

337

6.2

1695

536

0.2

9019

050

0.2

2478

059

4.2

1623

505

7

S.d

./

500

.000

478

706

.000

433

910

ρ=

.200

Mea

n.1

8620

844

8.1

9224

108

8.1

8325

484

0.0

0035

563

4.1

8361

047

5.1

9183

673

4.0

0017

353

8.1

9201

027

3.1

9232

345

6

n=

50S.d

..1

6227

341

1.1

3773

871

5.1

8070

158

0.1

3997

084

2.1

3772

031

9

S.d

./

500

.000

324

546

.000

275

477

ρ=

.500

Mea

n.3

8829

477

0.4

2268

803

4.2

4814

814

8.0

3643

334

4.2

8458

149

3.3

8888

888

8.0

1646

084

2.4

0534

973

1.4

2212

500

0

n=

10S.d

..3

0363

569

6.2

9179

935

2.3

7267

799

6.2

8867

513

4.2

8017

851

4

S.d

./

500

.000

607

271

.000

583

598

ρ=

.500

Mea

n.4

6474

052

0.4

8087

103

2.4

5374

149

6.0

0122

911

7.4

5497

061

4.4

7959

183

6.0

0055

532

4.4

8014

716

0.4

8091

700

0

n=

50S.d

..1

4310

087

2.1

2423

156

1.1

5971

914

1.1

2371

791

4.1

2420

950

0

S.d

./

500

.000

286

201

.000

248

463

ρ=

.500

Mea

n.4

8642

804

2.4

9349

091

5.4

8478

747

2.0

0013

292

6.4

8492

039

8.4

9328

859

0.0

0006

005

7.4

9334

864

7.4

9343

581

4

n=

150

S.d

..0

8658

971

1.0

7103

908

3.0

9159

291

3.0

7094

756

5.0

7108

367

5

S.d

./

500

.000

173

179

.000

142

078

13

Page 15: Serial correlation coefficient - Manfred Mudelsee

Tab

le1:

(Con

tinu

ed)

ρ̂?

Sim

ρ̂Si

mρ̂

?T

(24)

1st

ρ̂?

T(2

4)2n

dρ̂

?T

(24,

23)

ρ̂T

(15)

1st

ρ̂T

(15)

2nd

ρ̂T

(15,

12)

ρ̂T

(26,

25)

ρ=

.900

Mea

n.7

9482

476

5.8

3198

103

0.6

5399

825

5.0

6017

638

2.7

1417

463

7.8

0526

315

7.0

2576

401

2.8

3102

717

0.8

2537

245

0

n=

20S.d

..1

5289

861

7.1

4164

999

8.1

2909

944

4.1

0000

000

0.1

3964

096

8

S.d

./

500

.000

305

797

.000

283

299

ρ=

.900

Mea

n.8

6759

846

6.8

8323

932

7.8

5278

754

3.0

0225

185

8.8

5503

940

2.8

8181

818

1.0

0096

660

3.8

8278

478

5.8

8262

209

8

n=

100

S.d

..0

5614

476

9.0

4955

218

9.0

5655

663

7.0

4380

858

2.0

4983

168

4

S.d

./

500

.000

112

289

.000

099

104

ρ=

.900

Mea

n.8

9464

605

5.8

9775

169

9.8

9415

014

6.0

0003

457

1.8

9418

471

7.8

9774

718

3.0

0001

483

9.8

9776

202

3.8

9775

974

4

n=

800

S.d

..0

1917

615

3.0

1573

253

5.0

1990

800

7.0

1542

067

5.0

1572

382

4

S.d

./

500

.000

038

352

.000

031

465

ρ=

.980

Mea

n.9

0367

096

8.9

2936

735

1.7

0630

147

0.1

8279

971

1.8

8910

118

2.8

7684

210

5.0

7347

766

7.9

5031

977

2.9

0078

056

3

n=

20S.d

..1

1756

580

5.1

0651

587

3.0

5893

796

9.0

4565

315

4.1

1818

543

8

S.d

./

500

.000

235

131

.000

213

031

ρ=

.980

Mea

n.9

6434

177

3.9

7133

571

9.9

5386

797

9.0

0291

532

0.9

5678

329

9.9

7015

075

3.0

0124

943

8.9

7140

019

2.9

7039

001

0

n=

200

S.d

..0

2159

105

1.0

1902

243

0.0

1821

148

7.0

1410

655

7.0

1954

402

2

S.d

./

500

.000

043

182

.000

038

044

ρ=

.980

Mea

n.9

7775

960

2.9

7903

384

1.9

7739

856

3.0

0002

889

9.9

7742

746

2.9

7901

950

9.0

0001

238

6.9

7903

189

5.9

7902

190

2

n=

2000

S.d

..0

0552

619

3.0

0465

273

5.0

0574

599

9.0

0445

083

1.0

0465

873

1

S.d

./

500

.000

011

052

.000

009

305

ρ=

.999

Mea

n.9

9510

599

1.9

9660

456

7.9

8832

531

5.0

0622

666

5.9

9455

198

1.9

9499

599

1.0

0253

513

8.9

9753

113

0.9

9503

590

4

n=

500

S.d

..0

0569

976

4.0

0484

587

7.0

0258

392

8.0

0200

150

2.0

0595

376

0

S.d

./

500

.000

011

399

.000

009

691

ρ=

.999

Mea

n.9

9758

724

0.9

9818

015

5.9

9633

533

4.0

0057

443

4.9

9690

976

8.9

9800

050

0.0

0024

554

3.9

9824

604

4.9

9800

299

4

n=

2000

S.d

..0

0187

081

6.0

0162

235

4.0

0129

099

4.0

0100

000

0.0

0172

844

4

S.d

./

500

.000

003

741

.000

003

244

ρ=

.999

Mea

n.9

9890

792

5.9

9896

063

0.9

9889

346

4.0

0000

093

2.9

9889

439

7.9

9896

003

9.0

0000

039

9.9

9896

043

9.9

9896

004

3

n=

5000

0S.d

..0

0024

757

7.0

0020

794

8.0

0025

813

6.0

0019

995

1.0

0020

777

9

S.d

./

500

.000

000

495

.000

000

415

14

Page 16: Serial correlation coefficient - Manfred Mudelsee

-0.5 0.0 0.5 1.0

ρ = 0.20n = 10

-0.5 0.0 0.5

ρ = 0.20n = 20

-0.2 0.0 0.2 0.4 0.6

ρ = 0.20n = 50

Figure 1: First simulation experiment (cf. Table 1). Histograms of ρ̂?

(thick line), respectively ρ̂ (thin line), compared with Gaussian distributionsN(µρ̂? sim, σ2

ρ̂? sim) (heavy line), respectively N(µρ̂ sim, σ2ρ̂ sim) (light line). The

number of histogram classes follows Scott (1979).

15

Page 17: Serial correlation coefficient - Manfred Mudelsee

ρ = 0.50n = 10

ρ = 0.50n = 50

ρ = 0.50n = 150

-0.5 0.0 0.5 1.0

0.0 0.5

0.3 0.4 0.5 0.6 0.7

Figure 1: (Continued)

16

Page 18: Serial correlation coefficient - Manfred Mudelsee

ρ = 0.90n = 20

ρ = 0.90n = 100

ρ = 0.90n = 800

0.5 1.0

0.7 0.8 0.9 1.0

0.85 0.90 0.95

Figure 1: (Continued)

17

Page 19: Serial correlation coefficient - Manfred Mudelsee

ρ = 0.98n = 20

ρ = 0.98n = 200

ρ = 0.98n = 2000

0.6 0.8 1.0 1.2

0.90 0.95 1.00

0.96 0.97 0.98 0.99

Figure 1: (Continued)

18

Page 20: Serial correlation coefficient - Manfred Mudelsee

ρ = 0.999n = 500

ρ = 0.999n = 2000

ρ = 0.999n = 50000

0.98 0.99 1.00 1.01

0.995 1.000

0.9985 0.9990 0.9995

Figure 1: (Continued)

19

Page 21: Serial correlation coefficient - Manfred Mudelsee

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.05

0.10

ρ - E(ρ^)

n = 10

SQRT[var(ρ^)]

n = 10

0.0 0.2 0.4 0.6 0.8 1.0

ρ

0.00

0.10

0.20

0.30

Figure 2: Second simulation experiment. Above: negative bias, below:standard deviation. Simulation results (dots). Theoretical result, our for-mula, (12) and (15), respectively, (heavy line). Theoretical result, White’s(1961) formula, (25) and (26), respectively, (light line).

20

Page 22: Serial correlation coefficient - Manfred Mudelsee

0.00

0.04

0.080.0 0.2 0.4 0.6 0.8 1.0

ρ - E(ρ^)

n = 20

0.0 0.2 0.4 0.6 0.8 1.0

ρ

0.00

0.10

0.20

SQRT[var(ρ^)]

n = 20

Figure 2: (Continued)

21

Page 23: Serial correlation coefficient - Manfred Mudelsee

0.0 0.2 0.4 0.6 0.8 1.0

0.00

0.01

0.02

ρ - E(ρ^)

n = 100

SQRT[var(ρ^)]

n = 100

0.0 0.2 0.4 0.6 0.8 1.0

ρ

0.00

0.05

0.10

Figure 2: (Continued)

22