**UNSW** **Business** **School**

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**UNSW** **Business** **School** Research **Paper** No. 2016 ECON 18

Testing for Symmetry in Weakly Dependent Time Series

Luke Hartigan

This paper can be downloaded without charge from

The Social Science Research Network Electronic **Paper** Collection:

http://ssrn.com/abstract=2860403

business.unsw.edu.au

Last Updated 29 July 2014

CRICOS Code 00098G

Testing for Symmetry in Weakly Dependent Time Series

Luke Hartigan ∗

University of New South Wales

October 31, 2016

Abstract

I propose a test of symmetry for a stationary time series based on the difference between

the dispersion above the central tendency of the series with that below it. The test

has many attractive features: it is applicable to dependent processes, it has a familiar

form, it can be implemented using regression, and it has a standard Gaussian limiting

distribution under the null hypothesis of symmetry. The finite sample properties of

the test are examined via Monte Carlo simulation and suggest that it is more powerful

than competing tests in the literature for the DGPs considered. I apply the test to investigate

business cycle asymmetry in sectoral data and confirm previous findings that

asymmetry is more often detected in goods producing sectors than service producing

sectors.

Keywords: Symmetry; Weak dependence; Hypothesis testing; Monte Carlo simulation;

**Business** cycle asymmetry.

JEL Classification: C12; C15; C22; C52; E32.

∗ Correspondence: Luke Hartigan, PhD Candidate, **School** of Economics, University of New South Wales,

Sydney, 2052, NSW, Australia. Email: luke.hartigan@unsw.edu.au. I am grateful to Yunjong Eu, James

Morley, Timothy Neal, and Philip Rothman for helpful comments and suggestions. I also thank William

Dunsmuir and Graham Elliott for helpful comments on preliminary work related to this paper. All remaining

errors are solely my own.

1 Introduction

Understanding whether a variate originates from a symmetric or asymmetric distribution has

been a topic of interest in statistics since Pearson (1895). In economics, the business cycle

and the question about its symmetry, or lack there of, has been of interest for nearly as long a

period (see Mitchell 1927 and Keynes 1936). In finance, symmetry is of interest because it is

often an underlying assumption in asset pricing models. Meanwhile, the concept of symmetry

of a distribution is also important during the initial step of a model selection procedure. For

instance, if a time series process were found to be asymmetric, then a standard ARMA-type

model with the assumption of a symmetric innovation term would be misspecified and some

other, potentially nonlinear, model could provide a better fit to the data.

Within economics, Ramsey and Rothman (1996) characterise asymmetry into two different

forms: longitudinal or transversal. Longitudinal asymmetry is related to differences in the

duration and slope of upswings of a time series that are not mirrored by downswings in

the same series. Transversal asymmetry is asymmetry related to differences in the vertical

displacement of a (stationary) time series from its mean. It relates to the ‘deepness’ form

of asymmetry proposed by Sichel (1993) and is most similar to the original description of

symmetry found in Pearson (1895).

Many of the existing symmetry tests in the statistics and economics literature make use of

the sample skewness coefficient (standardised third moment) which can be very sensitive to

outliers (Kim and White 2004), or are only applicable to iid data such as the triples test of

Randles et al. (1980). Only a relatively small number of tests are also applicable to weakly

dependent data (see Psaradakis and Vávra 2015). However, simulation evidence indicates

that the power of these symmetry tests can be very low when considering processes with

sample sizes of around 100–200 observations and moderate amounts of dependence as is

usually encountered in macroeconomics.

In this paper I propose a new test which is suitable for weakly dependent time series processes.

The test uses the original Pearson (1895) definition by comparing the difference in the

2

dispersion of the process above its central tendency to that below it. If the difference is zero,

then the process is deemed to be symmetric. If the difference is non zero, then the process is

considered asymmetric. The test has the appearance of the familiar ‘paired difference’ t-test

and has a standard Gaussian limiting distribution under the null hypothesis of symmetry.

One benefit of my test compared to other tests which use the sample skewness coefficient, is

that it only requires finite fourth moments. This makes my test is less restrictive than the

skewness-base alternatives which require finite sixth moments. Importantly, the test can be

implemented easily using standard regression techniques combined with Heteroskedasticity

and Autocorrelation Consistent (HAC) standard error estimates.

In a range of Monte Carlo simulations the symmetry test is shown to have good power

properties relative to the competing tests of Bai and Ng (2005) and Psaradakis and Vávra

(2015) for the Data Generating Processes (DGPs) investigated. As an empirical application

I use the test developed in this paper to investigate the prevalence of asymmetry in monthly

U.S. nonfarm payroll data by sector. My findings confirm previous results that asymmetry

is detected more often in durable goods manufacturing-related sectors than service-related

sectors.

The remainder of the paper is organised as follows: Section 2 introduces the symmetry

test and its limiting distribution under the null hypothesis of symmetry. Section 3 provides

simulation evidence on the sampling properties of the test statistic as well as its empirical

size and power for a range of DGPs. An empirical application is discussed in Section 4, before

concluding in Section 5. Auxiliary information and results are provided in the Appendix.

2 Testing for Symmetry

2.1 The Test Statistic

In this Section and throughout the remainder of the paper I will use the following definition

to describe a stationary time series which is symmetric:

3

Definition 1. A stationary time series {X t , t ∈ Z} with central tendency given by ζ, is

symmetric if Var [X t | X t > ζ] = Var [X t | X t < ζ].

Definition 1 states that a time series is symmetric if the dispersion of the subsequence of

values strictly greater than its central tendency is equal to the dispersion of the subsequence

of values strictly less than its central tendency.

This definition follows from the original

description provided by Pearson (1895). It is also similar to the descriptions provided by

DeLong and Summers (1984), as well as Sichel (1993) in the context of describing the property

‘deepness’ as it relates to asymmetry and Ramsey and Rothman (1996) in relation to

transversal asymmetry. However, I have specified the variance as the measure of ‘deviation’

and the more general term, ‘central tendency’ in place of any reference to the mean of the

time series. As a result, central tendency can also include other statistics such as the median

which is equivalent to the mean for a symmetric and stationary time series.

Definition 1 could form the basis of a statistical test for equality in the variance of the

subsequence of X t above the central tendency to the subsequence of X t below the central

tendency. This would involve using a test for equal variances such as Levene (1960) or one

of the alternative versions proposed by Brown and Forsythe (1974). One problem with this

approach is that the two subsequences might not be independent from each other.

The

potential lack of independence can be avoided by constructing a new sequence, {ˆδ n , n ∈ Z},

representing the difference between squared deviations from the two subsequences. More

formally, let {X 1,ti } be the subsequence of {X t } defined by the indices {t i ∈ Z : X t > ˆθ},

and {X 2,tj } be the second subsequence of {X t } defined by the indices {t j ∈ Z : X t < ˆθ} and

let ˆθ be the sample median of {X t }. Then {ˆδ n } can be defined as:

ˆδ n = (X 1,ti − ˆµ X1 ) 2 − ( X 2,tj − ˆµ X2

) 2

(1)

Where ˆµ Xk k ∈ {1, 2} is an estimate of the mean for that particular subsequence. Defining the

central tendency of a time series in terms of the median has an important implication: if the

median is unique, which it is for a continuous random variable, then the two subsequences X 1

and X 2 will be equal in length (exactly half the length of the original sequence X t ), thereby

4

allowing us to compute the differences between the two subsequence square deviations.

Using this sequence of differences, we can state the following proposition:

Proposition 2. If {X t } is a stationary and symmetric time series with a continuous distribution

function, then E

] [ˆδn = 0.

The proof follows immediately from Definition 1 and the linearity of the expectation operator.

Using Proposition 2 we can state the following theorem:

Theorem 3. Assuming {ˆδ n }, is a sequence of N mixing random scalars such that either

φ (m) or α (m) = O ( m −λ) ∣

∣ ∣∣

r

for λ > −r/2 (r − 1), or λ > −r/ (r − 1), r ≥ 2, and E ∣ˆδ n <

∞ and where φ (m) and α (m) are the uniform or strong mixing coefficients as defined in

Definition 3.45 of White (2000). Then by application of Theorem 5.20 of White (2000):

√

N ¯δ

d

−→ N ( )

0, σδ

2

(2)

where ¯δ = N ∑ −1 N ˆδ

]

n=1 n and σδ 2 = lim N−→∞ NVar

[ˆδn

variance of the difference sequence, ˆδ n .

are the sample mean and long run

From Equation (2), I define the symmetry test statistic S as follows:

Theorem 4. Assuming the previous assumptions all hold and let ˆσ 2 δ

be a consistent estimate

of σδ 2 , then under the null hypothesis of symmetry:

S =

√

N ¯δ

ˆσ δ

d

−→ N (0, 1) (3)

This result leads directly from Theorem 3 and Theorem 5.20 of White (2000). Serial dependence

in ˆδ n is explicitly taken into account through σδ 2 . As such, Theorem 4 permits

testing for symmetry in weakly dependent time series. The test statistic S is conceptually

just a standard ‘paired difference’ t-test statistic (see Casella and Berger 2002) and it could

be computed using linear regression. This would involve regressing the estimated difference

sequence, ˆδ n , on a constant and then compute the HAC-corrected standard error for this

term. The estimate of the constant is identical to ¯δ and the standard error is a consistent

estimator for σ δ even in the presence of heteroskedasticity and/or serial dependence (see

5

Newey and West 1987 and den Haan and Levin 1997). The test would reject the null hypothesis

of symmetry at a given significance level α if |S| > Z α/2 where Z i is the upper i th

quantile of the standard Gaussian distribution.

The method of computing the test statistic S in this way is not new and is the basis for

alternative symmetry-type tests such as those proposed by Sichel (1993) using the sample

skewness coefficient, Ramsey and Rothman (1996) and their test of time irreversibility, as well

other tests such as the Diebold and Mariano (1995) test for comparing predictive accuracy.

Remarks:

1. The pairing of {X 1,ti } and {X 2,tj } in Equation (1) is arbitrary and the computation of

the test statistic S is invariant to permutations in the ordering of either subsequence.

2. The sign of the test statistic S provides information on the form of symmetry. A

negative value indicates left skewness, while a positive value indicates right skewness.

3. The symmetry test only requires {ˆδ n } have finite second moments and by implication

that {X t } have finite fourth moments. This requirement is less restrictive than tests

based on the sample skewness coefficient which requires finite six moments (for example

the skewness tests of DeLong and Summers 1984, Sichel 1993, Bai and Ng 2001, or Bai

and Ng 2005).

4. To implement the symmetry test the sample median must be unique otherwise the test

cannot be computed since we are unable to construct the sequences of differences. 1

5. The sequence {X t } need not have mean equal to zero to use the symmetry test.

6. The symmetry test can be applied to the residuals from an OLS regression using

Theorem 5 of Bai and Ng (2005).

1 In the Monte Carlo simulations to be discussed in the Section 3, this was never a problem (with 1240

difference experiments); however, in the empirical application in Section 4, there was one series for which

the test could not be computed. This issue is most likely to occur with floating point numbers which have

only a small number of significant figures after the decimal place and periods of no change.

6

3 Monte Carlo Simulation Analysis

3.1 Simulation Design

Two Monte Carlo experiments were undertaken to investigate the finite sample properties of

the symmetry test S. The first one is to ascertain the sampling distribution of the S under

the null hypothesis of symmetry based on variates generated from a Gaussian distribution

and using sample sizes T = 100 to represent a small sample and T = 1000 to represent a

large sample. This is also helpful to judge how quickly the sample distribution of the test

statistic approaches that of the limiting Gaussian distribution as T −→ ∞. The second

experiment is to assess the empirical size and power properties of the proposed symmetry

test applying various well known and frequently used distributions in the symmetry testing

literature including: Gaussian and Student-t as examples of symmetric distributions as

well as Chi-squared to represent an asymmetric distribution. Additionally, the Generalised

Lambda Distribution (GLD) of Ramberg and Schmeiser (1974) is also considered. This is

a distribution which can be used to generate both symmetric and asymmetric variates depending

on the values chosen for the four parameters which describe it. The inverse function

of the GLD is defined as:

[

]

F −1 (u) = λ 1 + λ2

−1 u λ 3

− (1 − u) λ 4

, 0 ≤ u ≤ 1 (4)

The skewness and peakedness of the GLD are determined by the two parameters λ 3 and λ 4 .

If λ 3 = λ 4 the the GLD will be symmetric and it will be asymmetric when λ 3 ≠ λ 4 . Given

these two values, the variance is determined by λ 2 , while the mean can be shifted to any

value by the appropriate choice of λ 1 . Note, if the GLD is asymmetric, then its mean will

not be equal to λ 1 which is the case when it is symmetric.

The data used in both experiments are generated as an AR(1) model: y t = φy t−1 + ɛ t were

ɛ is generated as iid from one of a total of 12 different distributions with six representing

symmetry and the remaining six representing asymmetry. In all cases the initial observation

is drawn from the process’s stationary distribution. I adopt, with some modifications to

7

e explained, the distribution parameter values from Bai and Ng (2005). These parameter

values were also used by Psaradakis and Vávra (2015).

To help with comparison across the different distributions examined, each process is constructed

so that its mean is zero and its variance is unity. For the Student-t and Chi-squared

distributions this was achieved as follows: Let ν represent the degrees of freedom parameter,

then for the Student-t variates, which already have a zero mean, I divided the variates by

ν/ (ν − 2). With the Chi-squared variates it was necessary to also demean the variates by

subtracting ν before dividing them by 2ν to get a process with zero mean and unit variance.

GLD-generated variates are not likely to have zero mean and unit variance for all parameter

values investigated in Bai and Ng (2005). As was mentioned previously, if λ 3 ≠ λ 4 then

the mean of the process will not be λ 1 . This might have implications when constructing the

various DGPs and could impact the statistical properties of the symmetry test. To avoid this

potential issue, I deviate from both Bai and Ng (2005) and Psaradakis and Vávra (2015) and

instead follow the guidance of Ramberg and Schmeiser (1974). to ensure the GLD-generated

variates do have the desired mean and variance they recommend the following: To set the

variance of the GLD to be equal to σ let:

[ ]

λ 2 = ±(

1/ (2λ 3 + 1) − 2β (λ 3 + 1, λ 4 + 1) + 1/ (2λ 4 + 1)

[

) 2 1/2

− 1/ (λ 3 + 1) − 1/ (λ 4 + 1)]

σ −1 (5)

Where the sign of λ 2 depends on the sign of both λ 3 and λ 4 and β is the beta function.

Then, given this value of λ 2 , the mean of the GLD can be set to be equal to µ by letting:

[

]

λ 1 = µ − λ −1

2 1/ (λ 3 + 1) − 1/ (λ 4 + 1)

(6)

Table A1 in the Appendix lists the set of parameter values used for each distribution employed.

In the two experiments ten evenly spaced values from 0.0 to 0.9 were used for the AR

8

parameter, with results presented for three cases: the best situation of iid data, (i.e., φ =

0.0), the worst case of very strong dependence (i.e., φ = 0.9), and somewhere in between

with a modest amount of dependence (i.e., φ = 0.5). In the second experiment a grid of ten

evenly spaced sample sizes from 100 to 1000 were used (additional results are available from

the author upon request).

In both experiments, the symmetry test was computed using the previously described DGPs

and for the first experiment a sequence of symmetry test statistics were estimated and saved,

while for the second experiment the number of rejections of the null hypothesis of symmetry

using a nominal size of α = 0.05 was recorded. When estimating the standard error in the

regression used to compute the test statistic I adopt the Quadratic Spectral density kernel

and the automatic bandwidth method described by Andrews (1991). Furthermore, I apply

the OLS residual adjustment of Hartigan (2016) to improve the finite sample properties of the

test statistic. In each case the number of repetitions was set to 10000. All experiments were

undertaken in C++ using the Armadillo Template Linear Algebra library (Sanderson and

Curtin 2016) and the Mersenne twister pseudo random number engine (std::mt19937 64)

from the C++ standard library. The default seed (5489) was used in each experiment to aid

in comparisons.

3.2 Simulation Results

Figure 1 plots the finite sampling distribution for the symmetry test statistic for an AR(1)

process and Gaussian innovation term with T = 100 and T = 1000. The top two panels

show the estimated density of the test statistic overlaid with a Gaussian probability density

function (PDF) for comparison.

The bottom two panels show the empirical cumulative

distribution function (CDF) for the test statistic compared to the true Gaussian distribution

function. Additional plots for φ = 0.0 and φ = 0.9 are presented in Figure A1 and Figure A2

in the Appendix.

9

0.5

φ = 0.5 and T = 100

φ = 0.5 and T = 1000

Gaussian PDF

0.5

0.4

0.4

Density

0.3

0.2

0.3

0.2

0.1

0.1

0.0

0.0

1.2

φ = 0.5 and T = 100

φ = 0.5 and T = 1000

1.2

1.0

1.0

Distribution

0.8

0.6

0.4

0.8

0.6

0.4

0.2

0.0

Empirical CDF

Gaussian CDF

0.2

0.0

−4 −2 0 2 4

−4 −2 0 2 4

Figure 1: Symmetry Test Sampling Distribution – AR(1) Process: φ = 0.5

For both the iid and moderate dependence cases, the symmetry test statistic displays a

reasonable approximation to the limiting Gaussian distribution at the smaller sample size

of T = 100. In the strong dependence case; however, the statistic’s sampling distribution

is clearly different to the standard Gaussian distribution with much more mass located in

the tails than is the case with a Gaussian distributed variate. Nonetheless, in the second

example with T = 1000 all three cases show a much closer approximation to the limiting

Gaussian distribution, although the very persistent AR(1) process with φ = 0.9 still shows

a slight difference which is more evident in the empirical CDF plot.

Empirical size for the six different symmetric DGPs are displayed in Figure 2. Generally, the

symmetry test statistic is modestly over-sized, especially for the Gaussian process (S1) and

the smaller sample sizes . In contrast, the test shows more correctly sized test statistics for

the more heavy-tailed distributions such as the Student-t distributed variates with degrees of

freedom equal to 5 (S2). This finding is interesting because previous research has highlighted

how leptokutic (i.e., heavy-tailed) distributions can adversely impact measures of symmetry

based on the use of the skewness coefficient (see for example Kim and White 2004 and

10

Premaratne and Bera 2005).

The performance of the symmetry test deteriorates when

testing the more highly persistent processes and the test statistic is over-sized for all six

DGPs. Although, as the number of observations grows, the performance of the symmetry

test improves as would be expected.

0.25

0.20

S1

S2

0.25

0.20

0.15

0.15

0.10

0.10

0.05

0.05

Rejection frequency

0.00

0.25

0.20

0.15

0.10

0.05

0.00

0.25

0.20

S3

S5

S4

S6

0.00

0.25

0.20

0.15

0.10

0.05

0.00

0.25

0.20

0.15

0.15

0.10

0.10

0.05

0.05

0.00

0.00

200 400 600 800 1000 200 400 600 800 1000

Alternative sample sizes

φ = 0 φ = 0.5 φ = 0.9 α

Figure 2: Symmetry Test Empirical Size

Figure 3 illustrates the empirical power for the six asymmetric DGPs. Generally, the test

statistic shows good power properties, predominately for iid and moderately dependent data.

However, the test’s power deteriorate’s significantly for the highly persistent DGPs, but this

does improve marginally as T increases with the notable exception of DGP A2 which shows

no real improvement at all as the sample size increases.

11

1.2

1.0

A1

A2

1.2

1.0

0.8

0.8

0.6

0.6

0.4

0.4

Rejection frequency

0.2

1.2

1.0

0.8

0.6

0.4

0.2

1.2

1.0

A3

A5

A4

A6

0.2

1.2

1.0

0.8

0.6

0.4

0.2

1.2

1.0

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

200 400 600 800 1000 200 400 600 800 1000

Alternative sample sizes

φ = 0 φ = 0.5 φ = 0.9

Figure 3: Symmetry Test Empirical Power

The notable loss in power of the symmetry test for very persistent processes is something

which has been previously highlighted in the literature (see Kim et al. 1996, Bai and Ng

2001, and Bai and Ng 2005). Indeed, Bai and Ng (2001) noted that with an ARMA model

even if the innovation term ɛ t is asymmetric, it does not necessarily imply that the process

y t will also be asymmetric as well.

This is supported by Figure 4 which plots T = 250 iid standardised Chi-squared variates

with 2 degrees of freedom (top left) as well as an AR(1) process with φ = 0.9 generated

using the same 250 standardised Chi-squared variates (top right). Below each of these are

histograms for the two processes. As can be seen, the histogram for the very dependent

process is much more symmetrical relative to the iid variates. Furthermore, when tested

using the symmetry test, the test statistic for the iid variates is 3.69 with a p–value of

0.0002, rejecting the null of symmetry at the α = 0.05 significance level. Whereas, the test

statistic for the highly dependent process is 1.08 with a p–value of 0.28, indicating a lack of

evidence against the null of symmetry.

12

8

Independent process

AR(1) process: φ = 0.9

10.0

6

7.5

4

5.0

Value

2

0

−2

2.5

0.0

−2.5

−4

−5.0

0.90

Independent process

AR(1) process: φ = 0.9

0.30

0.75

0.25

Density

0.60

0.45

0.30

0.15

0.20

0.15

0.10

0.05

0.00

0.00

−1 0 1 2 3 4 5 6

−4 −2 0 2 4 6 8

Figure 4: Asymmetry & Dependence – Standardised χ 2 (2) Process

Finally, it is important to compare the finite sample performance of the symmetry test

proposed in this paper with alternative tests in the literature. The tests I use for comparison

are the two-sided skewness test of Bai and Ng (2005) and the quantile-based symmetry test

of Psaradakis and Vávra (2015). These two tests are relatively recent developments in the

symmetry testing literature and, most importantly, are applicable to weakly dependent time

series data. I take the published results from each paper and compare them with those of

the symmetry test statistic S using the same DGPs for comparison. The comparisons for

empirical size are in Table A2, while the comparisons for empirical power are in Table A3,

both of which are in the Appendix.

From the size comparisons, the symmetry test S shows a tendency to reject more frequently

than the other two tests, especially at the smaller sample sizes and more persistent DGPs.

However, as the sample size increases, the differences between the three tests become smaller.

When considering a sample size of T = 500 and a DGP from an AR(1) process with φ = 0.5,

then the test of Bai and Ng (2005) shows the best empirical size properties, followed closely

by S and the test of Psaradakis and Vávra (2015).

13

However, when considering empirical power for these three tests, the results are very much

in favour of S. The S test statistic displays relatively good power, at the smallest sample

size considered for the three different AR(1) processes examined. The results for the most

dependent process are particularly favourable for S with the lowest empirical power of 0.22

for DGP A2, while the highest figure for the test of Bai and Ng (2005) is 0.06 for the same

DGP. When considering a sample size of T = 500 and a DGP from an AR(1) process with

φ = 0.5, then S shows the best empirical power properties followed closely by the test of

Psaradakis and Vávra (2015) and then the test of Bai and Ng (2005).

4 Sectoral Evidence of **Business** Cycle Asymmetry

In this section I employ the previously developed symmetry test in an empirical application

to investigate business cycle asymmetry using sectoral data. The existence of business cycle

asymmetry has been an area of active research since the seminal work of Neftçi (1984). The

idea has it origins in the works of Mitchell (1927) and Keynes (1936) who both characterised

downturns in activity as being distinctly different to upturns in activity and therefore are

likely to result in time series which are not symmetric. Other notable studies using both U.S.

and international data include: DeLong and Summers (1984), Falk (1986), Rothman (1991),

Sichel (1993), Ramsey and Rothman (1996), Verbrugge (1997) Bidarkota (1998), Razzak

(2001), and Korenok et al. (2009). Generally, these papers have tended to use data related

to activity such as growth rates of GDP/GNP or industrial production or the unemployment

rate as a measure of the business cycle.

The main findings have been that there is some evidence for both longitudinal and transversal

asymmetry in activity-based output measures of the U.S. economy. This evidence is stronger

for industrial production than it is for GDP/GNP. Meanwhile, the results for the unemployment

rate are mixed. Neftçi (1984) found support for asymmetry in the U.S. unemployment

rate; however, Sichel (1989) identified an error in Neftçi (1984)’s calculations which when

corrected weakened the original result. Rothman (1991) did find evidence for asymmetry

when using a corrected version of Neftçi (1984)’s method, but in a more recent study and

14

with a longer data set, Rothman (2008) found less compelling evidence and this depended

on the form of differencing employed and whether asymptotic or estimated p–values were

used. Some authors have found asymmetry when looking at subsectors of the economy using

disaggregated data. In the cases when asymmetry is detected, it appears to originate more

from sectors connected to durable goods manufacturing compared to the more service-based

sectors.

4.1 Testing for Asymmetry in U.S. Sectoral Nonfarm Payroll Data

To investigate this hypothesis I applied the symmetry test to the Bureau of Labor Statistics’s

monthly nonfarm payrolls by industry sector time series data set which has not previously

been used in the asymmetric business cycle testing literature. Another reason for using this

particular data set is that is a relatively long data set by macroeconomic standards with a

broad coverage of many different sectors of the U.S. economy. The time period examined is

1960M1–2016:M8, providing 680 observations for each of the 18 different series considered.

This sample size should give the symmetry test good power properties given the findings of

the simulation experiments in Section 3.

Before computing the test statistic, I convert the data to growth rates by taking the first

difference of the logarithm of the data. In addition, I also compute the sample skewness

coefficient as well as the Bowley coefficient (BC) for the data set. The BC is a nonparametric

measure of skewness bounded between ±1 that has been shown to be robust to outliers

(Groeneveld and Meeden 1984) and Kim and White (2004) recommended that robust measures

such as it should be routinely reported. The BC can be defined as:

BC = Q (0.75) + Q (0.25) − 2Q (0.5)

Q (0.75) − Q (0.25)

(7)

where Q (i) is the i th quantile of the series examined. A value of -1 indicates extreme left

skewness, a value of 1 indicates extreme right skewness, while a value near 0 indicates a

lack of skewness.

The results for the different nonfarm payroll sectors are presented in

Table 1. The method for computing the HAC-corrected standard error in the regression

15

used to estimate the test statistic is the same as that used in the Monte Carlo simulations

in Section 3.

Table 1: Selected U.S. Nonfarm Payroll Data

Symmetry Test

Bai and Ng (2005) Tests

Sector Skewness BC S p–value ˆπ 3 p–value ˆµ 35 p–value

Goods producing -1.11 -0.11 -2.51 0.01 2.18 0.03 5.40 0.07

Mining & logging (A%) -0.03 0.01 -0.69 0.49 -0.08 0.94 2.53 0.28

Construction 0.06 -0.19 -0.35 0.73 0.16 0.88 0.05 0.98

Manufacturing -1.30 -0.01 -2.51 0.01 -2.33 0.02 7.01 0.03

Durable goods -1.22 -0.05 -2.43 0.02 -2.01 0.04 7.09 0.03

Nondurable goods -1.03 -0.11 -1.64 0.10 -1.57 0.12 4.42 0.11

Service producing -0.14 -0.01 -0.94 0.35 -0.35 0.73 3.22 0.20

Trade, transport & utilities -0.34 -0.07 -2.12 0.03 -1.40 0.16 3.78 0.15

Wholesale trade -0.52 -0.10 -1.99 0.05 -1.54 0.12 5.62 0.06

Retail trade -0.32 0.02 -1.21 0.23 -1.13 0.26 1.99 0.37

Information 7.40 -0.15 0.41 0.68 0.93 0.35 1.10 0.58

Financial services -0.56 -0.03 -1.19 0.24 -1.17 0.24 1.38 0.50

Professional & business services -1.24 -0.14 -2.06 0.04 -1.70 0.09 5.59 0.06

Education & health services 0.30 0.10 1.58 0.11 1.63 0.10 2.68 0.26

Leisure & hospitality 0.04 -0.14 -0.80 0.42 0.14 0.89 4.41 0.11

Other services -0.13 0.02 -0.74 0.46 -0.58 0.56 0.35 0.84

Government 1.00 0.14 1.38 0.17 1.48 0.14 1.86 0.40

Total nonfarm -0.48 -0.08 -1.97 0.05 -1.48 0.14 4.84 0.09

Note: The time series are sourced from the Bureau of Labor Statistics, Table B-1a Employees on nonfarm payrolls by industry

sector and selected industry detail, seasonally adjusted (in thousands). Each series is converted to monthly compounding growth

rates, defined as ln (y t/y t−1 ). The sample size is 680 observations for each series covering the time period: 1960:M1 – 2016:M8.

‘A%’ denotes that series was converted to annual compounding growth rates, defined as ln (y t/y t−12 ). The skewness statistic is

computed using a bias correction for the standard deviation (i.e., T − 1). ‘BC’ is the Bowley coefficient measure of skewness.

Values close to -1 indicate extreme left skewness, values close to 1 indicate extreme right skewness, and values close to 0 indicate

an absence of skewness. ‘ˆπ 3 ’ and ‘ˆµ 35 ’ are the two-sided skewness test and joint test of the third and fifth moments which are

distributed as N (0, 1) and χ 2 respectively and were developed by Bai and Ng (2005).

(2)

Note, one series, ‘Mining & logging’ did not have a uniquely defined median for the time

period considered when converted to monthly compounding growth rates. As a result, the

series cannot be tested using the symmetry test. However, the problem could be overcome

by using an alternative detrending mechanism. As long as the resultant series is stationary

the test is still applicable. Hence, I use the test on the Mining & logging series converted to

annual compounding growth rates (denoted by ‘A%’ in Table 1).

For comparison purposes I also include the results from using the two alternative skewness

tests developed in Bai and Ng (2005). As was shown in Table A2, Bai and Ng (2005)’s

test based on the two-sided skewness test (ˆπ 3 ) had the overall best empirical size properties

of the three symmetry tests compared, but it also had the worst power (Table A3). As a

16

obustness measure, I also include the results using their second test, a joint test that the

third and fifth moments are jointly zero (ˆµ 35 ). Bai and Ng (2005) state that this test should

have better power properties relative to the skewness-based test. 2

Using S the null hypothesis of symmetry is rejected at the standard α = 0.05 level for five of

the 18 different series tested. The sectors were: Goods producing, Manufacturing, Durable

goods, Trade, transport & utilities, and Professional & business services. Furthermore, the

results are borderline for Total nonfarm as well as Wholesale trade.

In comparison, the

two tests of Bai and Ng (2005) only detect asymmetry in the Goods producing (only ˆπ 3 ),

Manufacturing, and Durable goods sectors. Finally, the conclusions reached for the Mining

& logging series using annual compounding growth rates confirm those obtained via the two

tests of Bai and Ng (2005).

This result is in accordance with the simulation results from Section 3 which indicated that

the symmetry test S was likely to be more powerful relative to Bai and Ng (2005)’s twosided

skewness test at least. Generally, the results confirm findings of previous studies (see

Rothman 1991, Sichel 1993, and Korenok et al. 2009) and indicate business cycle asymmetry

is more likely to be concentrated in goods producing sectors than service producing sectors.

Furthermore, the test results also indicate that transversal (or ‘depth’) asymmetry is not

as prevalent in the U.S. economy as claimed by Verbrugge (1997), at least when using the

BLS nonfarm payroll data set, and when using first differences as the detrending mechanism.

This conclusion is supported by both the symmetry test proposed in this paper as well as

the two tests proposed in Bai and Ng (2005).

Finally, it is worth comparing the test statistic S, which provides and indicator of the

direction of skewness, to both the sample skewness coefficient and the BC. Both S and BC

agree on all but two cases on the direction of skewness (Information and Education & health

services), while there are five disagreements between the skewness statistic and the BC,

indicating some possible distortion coming from the presence of outliers. This is especially

2 Both tests were estimated using the GAUSS codes obtained from Serena Ng’s web site: http://

www.columbia.edu/~sn2294/. I used the Quadratic Spectral density and the Andrews (1991) automatic

bandwidth in estimation.

17

evident for the Information category. Visual inspection of this series in first differences reveals

two very large outliers (1983:M8 -25%, 1983:M9 +35%) while being otherwise relatively

symmetric.

The effect of these two outliers is apparent in the sample skewness statistic

having a relatively large value of 7.4, while having no real impact on the BC or S.

5 Conclusion

A test for symmetry suitable for weakly dependent time series is developed. It is based on

comparing the dispersion of a process above its central tendency to that below it; equality

between the two measures indicates symmetry while any difference suggests asymmetry.

The test statistic is similar to the familiar t-test for paired differences and has a standard

Gaussian limiting distribution under the null hypothesis of symmetry. In simulation experiments

it was shown to converge to this limiting distribution relatively quickly and with

processes containing different amounts of dependence. In comparisons with competing symmetry

tests in the literature applicable for dependent time series, it was shown to be slightly

over-sized at smaller sample sizes, but provided a noticeable improvement in power relative

to those alternative tests for the DGPs considered.

An empirical application to monthly U.S. nonfarm payroll data by sector, suggests that when

asymmetry is present it appears to be detected more often in durable goods manufacturingrelated

sectors than service-related sectors.

18

References

Andrews, D. W. K. (1991). Heteroskedasticity and Autocorrelation Consistent Covariance

Matrix Estimation. Econometrica 59 (3), 817–858.

Bai, J. and S. Ng (2001).

A Consistent Test for Conditional Symmetry in Time Series

Models. Journal of Econometrics 103 (1–2), 225–258.

Bai, J. and S. Ng (2005). Tests for Skewness, Kurtosis, and Normality for Time Series Data.

Journal of **Business** & Economic Statistics 23 (1), 49–60.

Bidarkota, P. V. (1998).

Sectoral Investigation of Asymmetries in the Condition Mean

Dynamics of the Real U.S. GDP. Studies in Nonlinear Dynamics & Econometrics 3 (4),

191–200.

Brown, M. B. and A. B. Forsythe (1974). Robust Tests for the Equality of Variances. Journal

of the American Statistical Association 69 (346), 364–367.

Casella, G. and R. L. Berger (2002). Statistical Inference (Second ed.). Pacific Gove, CA:

Duxbury/Thomson Learning.

DeLong, J. B. and L. H. Summers (1984). Are **Business** Cycles Symmetric? **Working** **Paper**

1444, NBER, 1050 Massachusetts Avenue, Cambridge, MA 02138.

den Haan, W. J. and A. T. Levin (1997). A Practitioner’s Guide to Robust Covariance

Matrix Estimation.

In G. S. Maddala and C. R. Rao (Eds.), Handbook of Statistics,

Volume 15 of Robust Inference, Chapter 12, pp. 299–342. Amsterdam: Elsevier.

Diebold, F. X. and R. S. Mariano (1995). Comparing Predictive Accuracy. Journal of

**Business** & Economic Statistics 13 (3), 253–263.

Falk, B. (1986). Further Evidence on the Asymmetric Behavior of Economic Time Series

over the **Business** Cycle. Journal of Political Economy 94 (5), 1096–1109.

19

Groeneveld, R. A. and G. Meeden (1984). Measuring Skewness and Kurtosis. Journal of the

Royal Statistical Society. Series D (The Statistician) 33 (4), 391–399.

Hartigan, L. (2016). Alternative HAC Covariance Matrix Estimators with Improved Finite

Sample Properties. Research **Paper** 2016-06, **UNSW** **Business** **School**.

Keynes, J. M. (1936). The General Theory of Employmewnt, Interest and Money. London:

Macmillan.

Kim, J.-R., S. Mittnik, and S. T. Rachev (1996).

Linear Time Series and Unobserved Disturbances.

Detecting Asymmetries in Observed

Studies in Nonlinear Dynamics &

Econometrics 1 (3), 131–143.

Kim, T.-H. and H. White (2004). On More Robust Estimation of Skewness and Kurtosis.

Finance Research Letters 1 (1), 56–73.

Korenok, O., B. Mirach, and S. Radchenko (2009). A Note on Demand and Supply Factors

in Manufacturing Output Asymmetries. Macroeconomic Dynamics 13 (2), 263–277.

Levene, H. (1960). Robust Tests for Equality of Variance. In I. Olkin, S. G. Ghurye, W. Hoeffding,

W. G. Madow, and H. B. Mann (Eds.), Contributions to Probability and Statistics:

Essays in Honor of Harold Hotelling, Chapter 25, pp. 278–292. Stanford University Press.

Mitchell, W. C. (1927). **Business** Cycles: The Problem and Its Setting. Number 10 in

Publications of the National Bureau of Economic Research. New York: NBER.

Neftçi, S. N. (1984).

Are Economic Time Series Asymmetric over the **Business** Cycle?

Journal of Political Economy 92 (2), 307–328.

Newey, W. K. and K. D. West (1987). A Simple, Positive Semi-definite, Heteroskedasticity

and Autocorrelation Consistent Covariance Matrix. Econometrica 55 (3), 703–708.

Pearson, K. (1895). Contributions to the Mathematical Theory of Evolution. II. Skew Variation

in Homogeneous Material. Philosophical Transactions of the Royal Society of London

A: Mathematical, Physical and Engineering Sciences 186, 343–414.

20

Premaratne, G. and A. Bera (2005). A Test for Symmetry with Leptokutic Financial Data.

Journal of Financial Econometrics 3 (2), 169–187.

Psaradakis, Z. and M. Vávra (2015).

A Quantile-Based Test for Symmetry of Weakly

Dependent Processes. Journal of Time Series Analysis 36 (4), 587–598.

Ramberg, J. S. and B. W. Schmeiser (1974).

An Approximate Method for Generating

Asymmetric Random Variables. Communications of the ACM 17 (2), 78–82.

Ramsey, J. B. and P. Rothman (1996). Time Irreversibility and **Business** Cycle Asymmetry.

Journal of Money, Credit and Banking 28 (1), 1–21.

Randles, R. H., M. A. Fligner, G. E. P. II, and D. A. Wolfe (1980). An Asymmetrically

Distribtuion-Free Test for Symmetry Versus Asymmetry. Journal of the American Statistical

Association 75 (369), 168–172.

Razzak, W. A. (2001). **Business** Cycle Asymmetries: International Evidence. Review of

Economic Dynamics 4 (1), 230–243.

Rothman, P. (1991). Further Evidence on the Asymmetric Behavior of Unemployment Rates

over the **Business** Cycle. Journal of Macroeconomics 13 (2), 291–298.

Rothman, P. (2008). Reconsideration of the Markov Chain Evidence on Unemployment Rate

Asymmetry. Studies in Nonlinear Dynamics & Econometrics 12 (3), 1–16.

Sanderson, C. and R. Curtin (2016). Armadillo: A Template-based C++ Library for Linear

Algebra. Journal of Open Source Software 1 (2), 26–32.

Sichel, D. E. (1989). Are **Business** Cycles Asymmetric? A Correction. Journal of Political

Economy 97 (5), 1255–1260.

Sichel, D. E. (1993). **Business** Cycle Asymmetry: A Deeper Look. Economic Inquiry 31 (2),

224–236.

21

Verbrugge, R. (1997). Investigating Cyclical Asymmetries. Studies in Nonlinear Dynamics

& Econometrics 2 (1), 15–22.

White, H. (2000). Asymptotic Theory for Econometricians (Second Revised ed.). Academic

Press.

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Appendix

Data Generating Processes

Table A1: Distributions

Label Distribution Parameter Values

Symmetric Proceses

S1 Gaussian µ = 0.00, σ = 1.00

S2 Student-t ν = 5.00

S3 Generalised Lambda Distribution λ 1 = 0.00000, λ 2 = 0.19745, λ 3 = 0.13492, λ 4 = 0.13492

S4 Generalised Lambda Distribution λ 1 = 0.00000, λ 2 = -0.16809, λ 3 = -0.08000, λ 4 = -0.08000

S5 Generalised Lambda Distribution λ 1 = 0.00000, λ 2 = -0.39791, λ 3 = -0.16000, λ 4 = -0.16000

S6 Generalised Lambda Distribution λ 1 = 0.00000, λ 2 = -0.72959, λ 3 = -0.24000, λ 4 = -0.24000

Asymmetric Processes

A1 Chi-squared ν = 2.00

A2 Generalised Lambda Distribution λ 1 = 0.83503, λ 2 = 0.45906, λ 3 = 1.40000, λ 4 = 0.25000

A3 Generalised Lambda Distribution λ 1 = -0.62900, λ 2 = -0.03716, λ 3 = -0.00750, λ 4 = -0.03000

A4 Generalised Lambda Distribution λ 1 = -0.30656, λ 2 = -0.03536, λ 3 = -0.10000, λ 4 = -0.18000

A5 Generalised Lambda Distribution λ 1 = -0.85162, λ 2 = -0.17429, λ 3 = -0.00100, λ 4 = -0.13000

A6 Generalised Lambda Distribution λ 1 = -0.81183, λ 2 = -0.25217, λ 3 = -0.00010, λ 4 = -0.17000

Note: Each process, x, is generated so that E [x] = 0 and Var [x] = 1.

23

Additional Symmetry Test Sampling Distribution Plots

0.5

φ = 0 and T = 100

φ = 0 and T = 1000

Gaussian PDF

0.5

0.4

0.4

Density

0.3

0.2

0.3

0.2

0.1

0.1

0.0

0.0

1.2

φ = 0 and T = 100

φ = 0 and T = 1000

1.2

1.0

1.0

Distribution

0.8

0.6

0.4

0.8

0.6

0.4

0.2

0.0

Empirical CDF

Gaussian CDF

0.2

0.0

−4 −2 0 2 4

−4 −2 0 2 4

Figure A1: Symmetry Test Sampling Distribution – iid Process

0.5

φ = 0.9 and T = 100

φ = 0.9 and T = 1000

Gaussian PDF

0.5

0.4

0.4

Density

0.3

0.2

0.3

0.2

0.1

0.1

0.0

0.0

1.2

φ = 0.9 and T = 100

φ = 0.9 and T = 1000

1.2

1.0

1.0

Distribution

0.8

0.6

0.4

0.8

0.6

0.4

0.2

0.0

Empirical CDF

Gaussian CDF

0.2

0.0

−4 −2 0 2 4

−4 −2 0 2 4

Figure A2: Symmetry Test Sampling Distribution – AR(1) Process: φ = 0.9

24

Symmetry Test Comparisons

Table A2: Symmetry Test Empirical Size Comparison

T = 100 T = 200 T = 500

DGP S BN PV S BN PV S BN PV

AR(1) Process: φ = 0.0

S1 0.09 0.04 n.a. 0.08 0.05 n.a. 0.08 0.06 n.a.

S2 0.05 0.03 n.a. 0.05 0.04 n.a. 0.05 0.03 n.a.

S3 0.09 0.03 n.a. 0.08 0.04 n.a. 0.07 0.05 n.a.

S4 0.05 0.02 n.a. 0.05 0.04 0.07 0.04 0.04 0.05

S5 0.04 0.02 n.a. 0.04 0.04 0.05 0.04 0.03 0.07

S6 0.04 0.03 n.a. 0.03 0.02 0.06 0.03 0.03 0.07

AR(1) Process: φ = 0.5

S1 0.09 0.03 n.a. 0.07 0.04 n.a. 0.07 0.05 n.a.

S2 0.07 0.03 n.a. 0.06 0.04 n.a. 0.05 0.04 n.a.

S3 0.09 0.03 n.a. 0.07 0.04 n.a. 0.06 0.05 n.a.

S4 0.07 0.03 n.a. 0.05 0.04 0.07 0.05 0.05 0.05

S5 0.06 0.03 n.a. 0.05 0.04 0.05 0.05 0.04 0.05

S6 0.06 0.04 n.a. 0.05 0.02 0.06 0.04 0.03 0.07

AR(1) Process: φ = 0.8

S1 0.14 0.02 n.a. 0.12 0.04 n.a. 0.09 0.03 n.a.

S2 0.14 0.04 n.a. 0.10 0.04 n.a. 0.07 0.04 n.a.

S3 0.15 0.02 n.a. 0.11 0.04 n.a. 0.08 0.05 n.a.

S4 0.14 0.02 n.a. 0.10 0.03 n.a. 0.07 0.04 n.a.

S5 0.13 0.04 n.a. 0.10 0.03 n.a. 0.07 0.06 n.a.

S6 0.13 0.04 n.a. 0.09 0.04 n.a. 0.06 0.04 n.a.

Note: ‘BN’ is the two-sided skewness test ˆπ 3 from Table 1 in Bai and Ng (2005).

‘PV’ refers is the QS k (A) test statistic from Table 2 in Psaradakis and Vávra

(2015). ‘n.a.’ means that particular author did not consider that DGP in their

Monte Carlo simulation experiment.

25

Table A3: Symmetry Test Empirical Power Comparison

T = 100 T = 200 T = 500

DGP S BN PV S BN PV S BN PV

AR(1) Process: φ = 0.0

A1 0.94 0.74 n.a. 0.99 0.90 n.a. 1.00 0.98 n.a.

A2 0.98 0.85 n.a. 1.00 1.00 n.a. 1.00 1.00 n.a.

A3 0.85 0.67 n.a. 0.98 0.85 0.94 1.00 0.98 1.00

A4 0.38 0.23 n.a. 0.66 0.43 0.62 0.93 0.72 0.95

A5 0.84 0.56 n.a. 0.94 0.73 1.00 0.99 0.87 1.00

A6 0.80 0.53 n.a. 0.91 0.67 n.a. 0.98 0.83 n.a.

AR(1) Process: φ = 0.5

A1 0.90 0.62 n.a. 0.98 0.87 n.a. 1.00 0.97 n.a.

A2 0.65 0.30 n.a. 0.89 0.70 n.a. 1.00 0.99 n.a.

A3 0.71 0.44 n.a. 0.93 0.76 0.71 1.00 0.96 0.96

A4 0.32 0.19 n.a. 0.56 0.37 0.40 0.88 0.69 0.76

A5 0.82 0.48 n.a. 0.93 0.67 0.98 0.98 0.86 1.00

A6 0.79 0.46 n.a. 0.90 0.64 n.a. 0.97 0.82 n.a.

AR(1) Process: φ = 0.8

A1 0.53 0.01 n.a. 0.75 0.13 n.a. 0.97 0.73 n.a.

A2 0.22 0.06 n.a. 0.24 0.12 n.a. 0.36 0.32 n.a.

A3 0.36 0.04 n.a. 0.50 0.20 n.a. 0.83 0.64 n.a.

A4 0.25 0.06 n.a. 0.32 0.15 n.a. 0.56 0.45 n.a.

A5 0.57 0.02 n.a. 0.77 0.18 n.a. 0.95 0.72 n.a.

A6 0.58 0.04 n.a. 0.77 0.22 n.a. 0.93 0.72 n.a.

Note: ‘BN’ is the two-sided skewness test ˆπ 3 from Table 1 in Bai and Ng (2005).

‘PV’ refers is the QS k (A) test statistic from Table 2 in Psaradakis and Vávra

(2015). ‘n.a.’ means that particular author did not consider that DGP in their

Monte Carlo simulation experiment.

26