Daniel Thomson - Final

An Analysis of South African GDP using Fourier and Periodogram Methods1

UNIVERSITY OF CAPE TOWN

An analysis of South African Gross Domestic Productusing Fourier and periodogram methods

Dissertation submitted

in partial fulfillment of the requirements

for the degree of Bachelor of Commerce (Honours)

in Financial Analysis and Portfolio Management

by

Daniel Thomson (THMDAN008)

Supervisor: Gary van Vuuren

1


ABSTRACT

This paper provides an alternative method of analysis to determine the length of the South

African business cycle, as measured by changes in real gross domestic product. Using the spec-

tral methods of Fourier series and periodogram analysis, the length of this cycle is found to be

7.11 years. We use this cycle to provide a one year forecast of GDP and compare it to observed

data. We find promising forecast potential and demonstrate that forecast accuracy may be im-

proved by including a greatly reduced number of cycle components than contained in the ori-

ginal series. We conclude that Fourier analysis is effective in estimating the length of the busi-

ness cycle, but is vague in determining the current position of the economy on the business

cycle. The study proposes the use of wavelets to analyse macroeconomic data such as GDP as

they do not suffer from many of the limitations of Fourier analysis and provide a representation

of the data in the time and frequency domains simultaneously.

The author would like to thank his dissertation supervisor, Gary van Vuuren, for the ongoing sup -

port, advice and expertise unselfishly given throughout the essay process.

PLAGIARISM DECLARATION

1. I know that plagiarism is wrong. Plagiarism is to use another’s work and pretend

that it is my own.

2. I have used the Harvard convention for citation and referencing. Each

contribution to, and quotation in this tutorial from the work(s) of other people

has been contributed, and has been cited and referenced.

3. This dissertation represents my own work.

4. I have not allowed, and will not allow, anyone to copy my work.

Signature: Daniel Thomson

Date: Sunday, March 15, 2015

2


1 Introduction

1.1 Spectral Analysis Methods

In finance and economics, the predominant method of analysing time-series data is usu-

ally to view these data in the time-domain, i.e., analysing changes of a series as it pro-

gresses through time. The problem in using only this approach to study financial data-

sets is that all realisations are recorded at a predetermined frequency. This frequency

corresponds to whichever period the realisations are recorded at and the implicit as-

sumption is made that the relevant frequency to study the behaviour of the variable

matches with its sampling frequency (Masset, 2008). This can be construed as analysing

inflation figures with a one year time frame and presuming that the cycle will repeat it-

self the following year as the cycle is presumed to be one year long. The realisations of

financial and economic variables often depend on a number of frequency components

rather than just one so the time-domain approach alone will not be able to process the

information in the time-series precisely.

Spectral analysis methods that enable a frequency-domain representation of the data,

such as Fourier series and wavelet methods, are able to identify at which frequencies

the time series-series variable is active. The strength of the activity may be measured

using Fourier analysis to construct a periodogram – a graphic representation of the in-

tensity of a frequency component potted against the period at which it occurs. This

method is particularly attractive for the use of economic variables that exhibit cyclical

behaviour as the cycle length may be identified using the Fourier transform.

1.2 The Business Cycle in South Africa

Understanding the business cycle and having an approximate idea of its current posi-

tion enables participants in the economy to make informed decisions. Because business

cycle information is so valuable, much research has been done to identify its behaviour

and the South African business cycle is no exception.1 In fact, owing to South Africa’s

volatile political and economic history, modelling its behaviour provides a robust test to

structural breaks and regime shifts of any technique.2

1.3 Problem Statement

1 See Du Plessis et. al (2014), Bosch and Ruch (2012) and Venter (2005).2 As explained in Aaron & Muellbauer (2002) and Chevillon (2009).

3


This paper will examine and attempt to forecast South African Gross Domestic Product

(GDP) time-series data by applying Fourier series analysis. The author aims to identify

potentially significant cycles present and quantify the length of these cycles by examin-

ing the data in the frequency-domain rather than the time-domain, thus providing an al-

ternative method of business cycle analysis to those in the literature. Using single fre-

quency components or a combination will also provide an unprecedented perspective in

South African GDP forecasting. A further aim of this paper is that it will be able to pave

the way for further alternative methods of analysis of economic and financial variables,

wavelets in particular, through highlighting potential limitations of pure frequency-do-

main analysis.

The remainder of this paper is structured as follows. In Section 2, a brief literature re-

view providing an overview of spectral methods applied to finance (Section 2.1) and

previous attempts at modelling South African GDP (Section 2.2) is given. Section 3

provides a description of the data used in the analysis (Section 3.1) and outlines the

methodologies employed (Sections 3.2 – 3.5). The results and discussion of the problem

(Sections 4.1 – 4.2) are found in Section 4, where our forecasts are also found (Section

4.3). The conclusions (Section 5.1) and recommendations (Section 5.2) for further study

can be found in Section 5.

4


2 Literature review

Spectral analysis methods have a broad range of applications in the real world. “Indeed,

the Fourier integral formula…is regarded as one of the most fundamental result of mod-

ern mathematical analysis, and it has widespread physical and engineering applica-

tions” (Debnath, 2012) These include circuitry, spectroscopy, crystallography, imaging,

signal processing, communications. Fourier series have more recently gained traction as

a tool in finance and econometrics, from the early works of Granger (1966),

Cunnyngham (1963), Nerlove (1964) and others on simple economic time series in the

1960s to modern day applications in derivative pricing and ground-breaking wavelet

analysis.

This literature review focuses on the application to finance and econometrics. In partic-

ular, literature that describes practical uses of Fourier and Periodogram analysis as ap-

plied to modelling and forecasting economic data is presented.

In a first step, we review literature relating to spectral analysis and its application to

economic time series by way of Fourier and Periodogram analysis. We begin with

Hamilton (1994) and then include work with more emphasis on practical studies such

as Liu et al. (2012) and Omekara, Ekpenyong and Ekrete (2013).

In a second step, we investigate methods to model the South African business cycle and

forecast the GDP.

2.1 Spectral analysis and application to economic time-series

In his seminal work, Hamilton (1994) discussed spectral analysis and introduced the

frequency domain. Hamilton (1994) also explained the concepts of the population spec-

trum, the sample periodogram and estimation based on the population spectrum.3 An

analysis of US manufacturing data demonstrated the use of spectral methods on real

time series. Hamilton (1994) explains why adjustment (in the form of taking natural

logs) of the data must be performed, owing to the assumptions of a covariance-station -

ary process4 implicit in the transform. To rid the data of seasonal effects which showed

up in the periodogram, (Hamilton, 1994) suggested using year-on-year growth rates.

3 Spectral analysis meaning the study of a variable over the frequency spectrum or frequency-domain.4 For a mathematical definition of a covariance-stationary process, see Lindgren, Rootzén and Sandsten

(2013).

5


Shumway and Stoffer (2000) provide a wide range of time series analysis techniques

and applications, covering spectral analysis and filtering. A walk-through example on

real data is provided and the chapter contains detailed explanations and illustrations in-

cluding parametric and nonparametric estimation as mentioned in Masset (2008). For

more general and theoretical work on spectral analysis, see Granger and Hatanaka

(1964), Granger (1966) and Cunnyngham (1963).

Practical studies involving the application of spectral methods to economic data are

considered next. Granger & Morgenstern (1964) used Fourier analysis to study stock

market prices on New York stock price series. Granger and Morgenstern (1964) found

that stock prices followed the random-walk hypothesis in the short term, but long run

components were owed greater consideration than the hypothesis suggests. A flat spec-

trum of share price changes provided a non-parametric test of the random-walk hypo-

thesis. Seasonal variation and the business-cycle components were found to be largely

irrelevant in explaining the evolution of stock market prices.

Praetz (1973) studied Australian share prices and share price indices in the frequency

domain using spectral analysis methods. In contrast to the findings of Granger and Mor-

genstern (1964), they found small departures from the random-walk hypothesis in their

share price series from 1947 to 1968, although not large enough to abandon the hypo-

thesis as ‘a crude first approximation.’ Some clearly defined seasonal patterns appeared

in their study of share price indices and certain sectors were shown to lead or lag the

market as a whole. This evidence contrasted in comparison to Granger and Morgen-

stern’s (1964) and Godfrey, Granger and Morgenstern’s (1964) studies on the New York

Stock Exchange and the London Stock Exchange, where such patterns and lags were far

less significant. The authors gave a tentative conclusion that Australian share markets

were less efficient than their overseas counterparts.

Iacobucci (2003) elaborated on the issues of cross-spectral analysis and filtering, with

typical concepts of coherency and phase spectrum being broached. He applied this ana-

lysis to US inflation and unemployment data and showed that a Phillips relation existed

at typical business cycle components of 14 and six years. In his analysis, he showed how

cross-spectral analysis and filtering can be used to find correlation between the two

factors through the Phillips curve. An interesting result of this paper found that unem-

ployment leads inflation with the lag of inflation being one year.

6


Masset (2008) provided an easy-to-follow introduction to spectral and wavelet methods

of analysis with many practical examples using real economic data. A spectral analysis

on home prices in New York City covering the period January 1987 to May 2008 was

performed using both parametric and non-parametric methods to show the subsequent

difference in the power spectrum. It was found that the spectra from the non-paramet-

ric methods (Periodogram and Welch method) contained more noise than the spectra

obtained from the parametric methods (Yule-Walker and Burg). The Fourier analysis of

the data confirmed that strong seasonalities affected home prices in New York and had

a particular frequency cycle of 12 months. The study then provided a discussion on fil-

tering, before a more detailed exposition on wavelet analysis was put forward as a way

to overcome many of the shortfalls of Fourier transform and filtering methods.

Liu et al. (2012) investigated business and growth cycles in the frequency domain by

running Fourier analyses on several data sources including electricity demand, foreign

currency data, monthly retail sales, quarterly GDP, labour market and productivity stat-

istics from Statistics New Zealand. In their analysis of the GDP data, the data were trans-

formed using natural logarithms and detrended using the Hodrick-Prescott filter before

conducting Fourier analysis of the detrended transformed data. Using a periodogram,

definitive cycles corresponding to eight years and four-and-a-half years were found. Be-

cause of the distance between energy spikes in the periodogram for the aforementioned

cycles, it was proposed that the cycle length varied between four-and-a-half to eight

years. The paper concluded that Fourier analysis could be used to detect cyclical beha-

viour in any type of time series data, although they found no cyclical behaviour in the

majority of the time series data they had tested and omitted from this paper. Liu et al.

(2012) proposed a natural extension to the paper on Fourier analysis, to wavelet ana-

lysis.

Omekara, Ekpenyong and Ekrete (2013) used Fourier series analysis to identify cycles

in the Nigerian all-items monthly inflation rates from 2003 to 2011. A square root trans-

formation was used to increase stability and normality of the inflation rate data. Period-

ogram analysis showed a short term and a long term cycle of 20 months and 51 months

respectively with the long cycle corresponding to the length of two different govern-

ment administrations that existed during the sample period. They then fitted a general

Fourier series model to the data and used the model to make reasonably accurate short

7


term monthly inflation rate forecasts from an out-of-sample period of September 2011

to September 2012.

More recent academic research (Masset, 2008 and Liu et al., 2012) of Fourier series of-

ten leads to a recommendation of wavelet analysis as a natural extension to the limited,

frequency-domain only methods such as Fourier transforms. Masset (2008) states that,

“Both spectral analysis and standard filtering methods have two main drawbacks: (i)

they impose strong restrictions regarding the possible processes underlying the dynam-

ics of the series (e.g. stationarity), and, (ii) they lead to a pure frequency-domain repres-

entation of the data, i.e. all information from the time-domain representation is lost in

the operation.” A large proportion of the literature surrounding spectral analysis re-

volves around the study of wavelets, with Fourier analysis being part of the process of

its development. This paper presents a practical application of Fourier analysis to South

African GDP data and while a recommendation to further investigate these data using

wavelet analysis is made, a thorough investigation and review of wavelet analysis is

beyond the scope of this study.

2.2 Models and forecasting methods applied to the SA business cycle

and GDP

Aron and Muellbauer (2002) developed a GDP forecasting model for South Africa to

measure interest rate effects on output. They preferred multistep forecasting models to

recursive forecasting with vector autoregressive (VAR) models because of the structural

breaks present in the South African economy. The multistep model consisted of a factor

model which was then evolved to a single equation equilibrium correction model with a

built in term for the stochastic trend. The model made forecasts for up to four quarters

and was tested for stability using sample breaks. Tests for normality and heterosce-

dasticity yielded satisfactory results and the authors concluded their model was robust.

Chevillon (2009) draws on the research of Aron and Muellbauer (2002) and establishes

whether direct multi-step estimation can improve the accuracy of forecasts. They set up

779 different models and applied them to South African GDP data to see which gave the

most accurate forecasts and coped best with the large number of regime changes and

structural breaks. They found that Aron and Muellbauer’s direct multi-step model per-

formed best within short time horizons and that multivariate and univariate models,

with DMS, worked well with intermediate to long term time horizons.

8


Du Plessis, Smit and Steinbach (2014) developed a dynamic stochastic general equilib-

rium (DSGE) model for the South African economy. The model uses Bayesian techniques

to incorporate prior information about the economy into the parameter estimates. Its

forecasting capability extends up to seven quarters and was tested against a panel of

professional forecasters and a random walk. It was found to outperform the profes-

sional forecasters and the random walk, especially over longer time horizons, when

used to predict CPI inflation and GDP growth.

Venter (2005) discusses the methodology used by the South African Reserve Bank to

identify business cycle turning points. He explains that this methodology includes the

use of three composite business cycle indicators and two diffusion indexes. Leading, lag-

ging and coincident indicators make up the composites while movement in historic and

current diffusion indexes help to confirm whether changes in the economy are localised

or all-encompassing.

Bosch and Ruch (2012) provided an alternative methodology to dating business cycle

turning points in South Africa. They used a Markov switching model and Bry-Boschan

method to date the turning points and found that the model estimates generally coin-

cided with the business cycle turning points determined by the SARB. They applied the

model to GDP data but also to 114 of the 186 stationary variables the SARB uses to date

the business cycle.5 Using Principle Component Analysis (PCA) on these variables

provided the authors with correlation data that enabled a more accurate measure of the

business cycle turning points than using GDP data alone.

5

9


3 Data Description and Methodology

3.1 Data Description

The data used are the GDP, measured in 2005 constant market prices, with monthly

periodicity and seasonally adjusted at an annual rate. The period used for the in-sample

Fourier analysis is 28 February 1969 to 31 October 2011 and the period used for the

out-of sample forecast is 30 November 2011 to 31 October 2014. These data were ob-

tained from the South African Reserve Bank (SARB).6

GDP is defined by the SARB as “…the total value of all final goods and services produced

within the boundaries of a country in a particular period.” This study seeks a simple and

readily available proxy for South African economic activity from which to identify po-

tentially meaningful cycles. GDP, although not a perfect measure of the business cycle

(see Boehm and Summers, 1999), provides a reasonable measure of economic activity

and the business cycle, over a satisfactory sample period.

The Fourier analysis tool in Excel performs a Fast Fourier Transform (FFT) on data.

This version of the Fourier Transform enables much faster computing, but restricts the

number of data points in the input to a power of two. Thus, the data selected are in

monthly terms instead of quarterly points, so that the sample period may cover an ap -

propriate time period of 512 months.

6 Available at: http://wwwrs.resbank.co.za/webindicators/SDDSDetail.aspx?DataItem=NRI6006D.

10

http://wwwrs.resbank.co.za/webindicators/SDDSDetail.aspx?DataItem=NRI6006D


4 Methodology

4.1 Remarks

This section outlines the methodology used to generate the study results. The Hodrick-

Prescott filter and the Baxter-King filter are described as time-domain methods to ex-

tract the trend and cycle components of a time-series. The results of these are used for

comparison to the frequency-domain approach used in this paper. We then show how

the GDP time-series is stationarised and de-trended by taking their natural logarithms.

The Fourier and periodogram analysis methods are explained with the defining equa-

tions shown. Lastly, we describe the forecasting method employed to measure the ac-

curacy of certain frequency components in predicting GDP returns out to 12 months.

4.2 Time series filtering methods

4.2.1 The Hodrick-Prescott Filter

Hodrick and Prescott (1997) showed a procedure for representing a time series X tas

the sum of a smoothly-varying trend component τ t, and a cyclical component c t, where,

X t=τ t+c t t=1,2 ,…,T .

They find the trend component τ t by choosing a positive value of λ and solving for min

{∑t=1

T

( y t−τ t )2+ λ∑

t=2

T

[ (τ t+1−τ t )−( τ t−τ t−1 )2}. The parameter λ is a smoothing parameter

which “penalises variability in the growth (trend) component series” (Hodrick and

Prescott, 1997). The larger the value of λ ,the smoother is the output series.

The HP filter has been criticised for a number of limitations and undesirable properties

(Ravn & Uhlig, 2002). Canova (1994 and 1998) found reason to use the HP filter to ex-

tract business cycles from macroeconomic data of average length of four to six years,

but was sceptical of the methodology used to determine key parameter inputs. Spurious

cycles and distorted estimates of the cyclical component when using the HP filter were

obtained by Harvey and Jaeger (1993). Cogley and Nason (1995) also found spurious

cycles when using the HP filter on difference-stationary input data. Application of the

HP filter to US time series data was found to alter measures of persistence, variability

and co-movement dramatically (King and Rebelo, 1993). Many of these critiques do not

provide sufficient compelling evidence to discourage use of the HP filter (van Vuuren,

2012). As a result, it remains widely-used among macroeconomists for detrending data

11


which exhibit short term fluctuations superimposed on business cycle-like trends (Ravn

& Uhlig, 2002).

4.2.2 The Baxter-King Filter

In contrast to the HP filter, the Baxter and King (1999) filter introduced a third variable,

‘noise,’ to the time series equation.

Consider a time series function X (t ) consisting of 3 components: a trend component τ ,

cyclical component γ , and a ‘noise’ (random) component, ϵ such that

X❑t=τ t+γ t+ϵt t=1,2 ,…,T .

The Baxter-King filter removes the trend and ‘noise’ components, leaving the cycle

component. That is:

γt=X t−τ t−ϵt t=1,2 ,…,T .

Guay and St-Amant (2005) estimated the ability of the HP and BK filters to extract the

business cycle component of macroeconomic time series using two different definitions

of the business cycle component. They first defined the duration of a business cycle to

be between six and 32 quarters, which is the definition of business cycle frequencies

used by Baxter and King. The second definition is made by discerning between perman-

ent and transitory components. Guay and St-Amant (2005) concluded that in both cases,

the filters performed adequately when the spectrum of the original series had a peak at

business-cycle frequencies. Low frequencies dominant in the spectrum were found to

provide a distorted business cycle. Their results suggest that the use of HP and BK filters

on series resembling the Granger shape of an economic variable may be problematic

(Guay and St-Amant, 2005).

We perform Fourier analysis on the cyclical components of the HP filter and the BK fil -

ter, c t∧¿ γt, for comparison and discuss the results in the next section.

4.3 Data Stationarity

Masset (2009) states that spectral methods such as Fourier transforms “…require the

data under investigation to be stationary.”7 In the literature, stationarity usually means

weak stationarity (covariance-stationary), unless otherwise specified. An augmented

Dickey-Fuller (ADF) test, used widely in statistics and econometrics, can be used to

7 For a comprehensive definition of strict stationarity and weak stationarity, see (Pelagatti, 2013)

12


check for this condition. The results obtained from the ADF test on nominal GDP data

used failed to reject the null hypothesis that the index levels series is non-stationary. We

therefore take the natural logarithms of the time-series:

ln (x t )−ln (x t−1)

where

x t and x t−1 are consecutive months in the series.

This transformation means we are studying the returns of the monthly GDP data, a sta-

tionary series.

4.4 Fourier Analysis

The basic idea of spectral analysis is to re-express the original time-series x (t ) as a new

sequenceX ( f ), which determines the importance of each new frequency component f in

the dynamics of the original series (Masset, 2008). This is achieved by using the discrete

version of the Fourier transform, which decomposes a periodic signal into its constitu-

ent frequencies. Time series data that comprise periodic components can be written as

a sum of simple waves (that is oscillations of a single frequency) represented by sine

and cosine functions (Brown and Churchill, 1993). A Fourier series is an expansion of a

periodic function in terms of an infinite sum of sines and cosines by making use of the

orthogonality relationships of the sine and cosine functions (Askey and Haimo, 1996).

The generalised Fourier series, obtained using the functions f 1 ( x )= cos xand f 2 ( x ) = sin x

(which form a complete orthogonal system over [−π , π]) gives the Fourier series of a

functionf ( x ):

f ( x )=12a❑0+∑

n=1

∞

an cos (nx )+∑n=1

∞

bnsin (nx )

where

a0=1π ∫−π

π

f ( x ) dx

an=1π ∫−π

π

f ( x ) cos (nx )dx

bn=1π ∫−π

π

f ( x ) sin (nx )dx

13


For a function f (x) periodic on an interval [0,2L] instead of [−π ,π ¿ ,a simple change of

variables can be used to transform the interval of integration from [−π ,π ¿ to [0,2L]. Let

x=π x'

L and dx= πd x '

L

Solving for x ' and substituting into Equation 3 gives (Krantz, 1999):

f ( x ' )=12a❑0+∑

n=1

∞

an cos ( nπ x 'L )+∑n=1

∞

bnsin ( nπ x'

L)

where

a0=1L∫0

2L

f ( x ' )dx

an=1L∫0

2L

f ( x ' )cos ( nπx 'L )dx

bn=1L∫0

2L

f ( x ' )sin ( nπ x 'L )dx .A periodogram plotting those frequency components with the greatest intensity or amp-

litude against the period shows which components bear significant meaning and which

components are random ‘noise.’ In cyclical data, it may be found that a few frequencies

are able to model the behaviour of the series relatively accurately. “The ‘noise’ (low

amplitude) frequencies may be discarded and a new, ‘cleaner’ time-series –free of noise

and comprising only time-series signals characterised by the dominant frequencies-may

thus be constructed (van Vuuren, 2014).”

4.5 A forecast of GDP

We make a forecast using the most important (highest amplitude) frequency compon-

ents to test the fit of these components to out-of-sample GDP data. A 12 month forecast

is shown in the results section. We used a fan chart with bounds equal to the standard

deviation scaled with the square root of time out to 12 months. Both one standard devi-

ation and two standard deviations are used to model the error bands.

Forecasts using (a) the highest amplitude wave, (b) highest and second highest amp-

litude waves and (c) first five highest amplitude waves are constructed.

14


5 Results and Discussion

5.1 Results

5.1.1 Data

The data are a time-series of seasonally adjusted, nominal GDP in millions of Rands,

taken monthly from February 1971 to September 2013. In Figure 1 it is clear that an up-

ward trend exists, although cyclical variations are difficult to discern. The discrete Four-

ier transform assumes that the input signal (GDP time-series) is statistically stationary,

i.e. it has a constant mean through time. This is a fair assumption, because if the data

were taken as is (due to the convex growth curve), vastly more weight would be given

to more recent fluctuations as the scale has increased greatly in the latter years, relative

to the initial years. This would not be an accurate representation of the time series and

the Fourier analysis would not effectively identify cycles.

Figure 1: Nominal GDP in Millions of Rands, seasonally adjusted from February 1971 to

September 2013.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Jan-70 Jan-74 Jan-78 Jan-82 Jan-86 Jan-90 Jan-94 Jan-98 Jan-02 Jan-06 Jan-10 Jan-14

Rand

(Tril

lion)

Monthly nominal GDPFebruary 1971 - September 2013

Source: South African Reserve Bank8

We performed an ADF test shown in Table 1 to confirm the data were non-stationary.

The ADF test examines the inputs for the existence of a unit-root in the context of a hy -

pothesis test. If a unit-root exists, we reject the null hypothesis and accept the alternat-

ive hypothesis that this root exists. The results showed that our assumption was correct

and that we needed to perform a transformation to stationarise the data.

To stationarise the data, the natural logarithm difference from month to month was cal-

culated to produce the percentage returns series shown in Figure 2. These returns do

8 Available at: http://wwwrs.resbank.co.za/webindicators/SDDSDetail.aspx?DataItem=NRI6006.

15

http://wwwrs.resbank.co.za/webindicators/SDDSDetail.aspx?DataItem=NRI6006


not scale with time and have a non-trending mean, so these are suitable for use in the

Fourier analysis framework.

Figure 2: De-trended GDP returns series using first differences.

-2%

-1%

0%

1%

2%

3%

4%

5%


Retu

rns (

%)

Log Returns of Monthly GDPFebruary 1971 - September 2013

Source: Author’s calculations

In the case of returns over the sample period from February 1971 to September 2013,

the mean monthly return is 1.05%. This positive average produces the upward ‘trend’

observed in the monthly GDP level in Figure 1. We are interested in identifying the cyc-

lical changes around this trend.

Because of the volatility in the returns series, we apply two filtering methods which ex-

tract the trend and cycle components from the series and produce a smoother returns

series. These are the Hodrick-Prescott Filter and the Baxter-King Filter. The log returns

series is plotted alongside the filtered series in Figure 3. A table of summary statistics il-

lustrating the effectiveness of the filters in capturing the trend and filtering through the

noise is shown in Table 2 below.

16


Figure 3: De-trended GDP series comparison with Hodrick-Prescott filtered series and

Baxter-King filtered series.

-2%

-1%

0%

1%

2%

3%

4%

5%


Mon

thly

Ret

urns

Returns SeriesFebruary 1971 - September 2013

Log Returns HP Filter BK Filter

Source: Authors calculations using NumXL software.

Table 2: Summary statistics illustrating the effects of filtering

Log Return HP Filter BK Filter

Mean 1.05% 1.05% 1.05%Standard Deviation 0.64% 0.33% 0.36%Skewness 1.21 -0.62 -0.13Excess Kurtosis 6.44 0.92 0.94

Summary Statistics

The standard deviation of the log returns series is 0.64%, whilst the HP and BK filtered

series produce less variation with standard deviations of 0.33% and 0.36% respectively.

The filtered data also contain less excess kurtosis than the unfiltered returns series

while there is a positive skew in the unfiltered series and negative skew of less mag -

nitude in the filtered series. A Bera-Jarque test of normality (Table 3) confirmed that the

data were not normally distributed. Severe excess-kurtosis and a positive skew, shown

in a histogram plot with a normal distribution curve for comparison in Figure 4 below,

characterise the monthly GDP returns series.

17


Figure 4: Frequency distribution of log returns series compared with the normal distri-

bution curve.

0%

5%

10%

15%

20%

25%

-1.6% -1.1% -0.5% 0.1% 0.6% 1.2% 1.8% 2.3% 2.9% 3.4% 4.0%

Freq

uenc

y

Return (%)

Distribution of ReturnsFebruary 1971 - September 2013

Frequency

Normal

Source: Authors calculation using NumXL software.

Value intervals on the horizontal axis are determined by the Freedman-Diaconis choice bin rule in NumXL.

5.1.2 Fourier Analysis

Using the discrete version of the Fourier transform, the time-series of GDP returns is

transformed from a representation in the time-domain into the frequency-domain. The

time-series is decomposed into a series of sine and cosine waves occurring at different

frequencies with different intensities,9 which in summation are able to exactly mimic the

behaviour of the original signal. The power or amplitude of each frequency component

(which explains the importance of the particular frequency in making up the original

signal) is plotted against its period in Figure 5 below. The period is defined as 1

frequency

and is shown in months. The filtered series show much less static than the unfiltered

series as they rid the data of random deviations or ‘noise.' The periodogram omits peri -

ods longer than 180 months, which distort our analysis.

9 Intensity, power and amplitude are used inter-changeably.

18


Figure 5: Periodogram plotting power against period of transformed returns data, HP

filtered data and BK filtered data

0.00%

0.02%

0.04%

0.06%

0.08%

0.10%

0.12%

0.14%

0 20 40 60 80 100 120 140 160 180

Pow

er (A

mpl

itude

)

Period (months)

Periodogram of ReturnsLog Returns HP filteredBK filtered

Source: Authors calculations.

In Figure 6 below, we consider the 15 frequencies with the highest intensities. Beyond

these and including the frequencies with lesser amplitude below, the remaining fre-

quencies are pure random ‘noise’ and do not help in explaining the business cycle.

Figure 6: Bar graph showing the 15 components with the highest amplitude.

512

85.3

64 13.8 170.7 102.4 24.417.7 6.1 12.5 34.1 16 25.6 73.1 256

0.00%

0.05%

0.10%

0.15%

0.20%

0.25%

0.30%

0.35%

Ampl

itude

(%)

Cycle period (months)

Components ranked by amplitude

Source: van Vuuren (2014).

There are two clearly dominant frequencies above, which differ substantially from the

others. These are at a period of 512.0 and 85.3 months, or frequencies of 0.00195 and

0.01172 cycles per month respectively. In simple terms, the periodogram states that

one cycle occurs per given period. For the period of 85.3 months, this means that 1 cycle

occurs every 85.3 months. Similarly, the model states that 1 cycle occurs every 512.0

months. The entire length of the sample data set is made up of 512 months however, so

19


naturally, one cycle would occur that covers this period and this cycle is not necessarily

repetitive. This all-encompassing cycle has harmonic waves of 2n and includes cycles oc-

curring at periods of 2, 4, 8, 16, 32, 64, 128, 256 and 512. In Figure 6, the harmonics oc -

curring at 16, 64, 256 and 512 months are all included in the 15 highest power compon-

ents. There is no literature regarding the South African business cycle as having a period

of 512 , 256 or 64 months, but Botha (2004) found evidence that a cycle exists and that

cycle lasted 7.00 years. Her result serves as evidence for us to reject the cycles at the

harmonics of 2n periods and focus on the period of 85.33 months, corresponding to a

period of 7.11 years.

5.1.3 Forecast Potential

Using the frequencies with the highest relative power in the dataset, we produce a 12

month forecast and compare it to out-of-sample data of the same period.

Figure 7 shows a 12 month forecast from October 2013 to October 2014 using the single

frequency component with period of 85.33 months or 7.11 years. This frequency

showed the highest amplitude (0.3%) in the Fourier analysis. Our model error bands

are produced by multiplying the standard deviation of the reconstructed input signal

with the square root of time from the last in-sample data point. In this case, September

2013 is the last data point. This product is then added and subtracted from the input

signal return to produce a ‘fan-like’ forecast zone, implying that the accuracy of the fore-

cast decreases as one projects further into the future. Lastly, out-of-sample observed

data from October 2013 to October 2014 are plotted alongside the forecast framework.

20


Figure 7: Fan chart showing the 85.33 period reconstructed signal with error bands of

one standard deviation scaled with the square root of time against observed out-of-

sample GDP returns from October 2013 to October 2014.

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%

Jun-13 Aug-13 Oct-13 Dec-13 Feb-14 Apr-14 Jun-14 Aug-14 Oct-14

Mon

thly

Ret

urns

Forecast using 1 cycle component and 1 standard deviation error bands

Observed GDP±1ߪ

Source: van Vuuren (2014).

Figure 7 clearly shows that the observed data do not fall within the forecast zone for the

majority of the 12 month time horizon forecast. The standard deviation of returns from

the reconstructed input signal to September 2013 is 0.18%, whilst the standard devi-

ation of the observed data is 0.64% and 0.54% in the out-of-sample 12-month period.

This volatility distorts the effectiveness of the forecast and for this reason we recom-

mend using quarterly GDP data and moving averages in future studies to smooth the ef-

fects of short-term fluctuations, in line with Botha (2004) who states that “Statistics on

a daily, weekly or monthly basis tend to contain too much static.” Sherman and Kolk

(1996) assert that the best time interval to use in cyclical analysis is quarterly data.

Because our log transformation of the data rendered the returns time-series approxim-

ately normal, we operate on the assumption that about two-thirds (0.683) of the

monthly returns should fall between one standard deviation above and below the mean

value of our reconstructed signal. In Figure 8 below, we use error bands based on two

standard deviations to be 95% confident of our forecast model values.

Figure 8: Fan chart showing the 85.33 period reconstructed signal with error bands of

two standard deviations scaled with the square root of time against observed out-of-

sample GDP returns from October 2013 to October 2014.

21


-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%


Mon

thly

Ret

urns

Forecast using 1 cycle component and 2 standard deviation error bands

Observed GDP1 component forecast

ߪ±2


At the 95% level of confidence, the forecast using the 7.11 year, single-period frequency

still fails to encapsulate the values for the 12 months of real data.

We now show the effect of adding a further sine wave with the second highest amp-

litude to the forecast model in Figure 9. This sine wave has a much shorter wavelength

of period 13.84 months (1.15 years) and a higher frequency (0.072 cycles per month). It

acts to convolute the original sine wave and error bands. Figure 9(a) shows error bands

of one standard deviation and (b) shows error bands of two standard deviations, scaled

with the square root of time, as before.

Figure 9: Fan chart showing reconstructed signal using first two components plotted

against observed out-of-sample GDP returns from October 2013 to October 2014: (a)

shows error bands of one standard deviation, (b) shows bands of two standard devi-

ations.

(a)

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%


Mon

thly

Ret

urns

Forecast using 2 cycle components and 1 standard deviation error bands

Observed GDP2 component forecast±1ߪ

22



(b)

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%


Mon

thly

Ret

urns



ߪ±2


Including every one of the 512 sinusoidal waves will fully replicate or reconstruct the

observed signal. By adding consecutive waves, the reconstructed signal resembles the

observed signal more closely. Thus, the sine wave of period 1.15 years added to the

wave of 7.11 years produces a forecast signal which mimic the in-sample data more ac -

curately. Whilst in (a) it is difficult to see any improvement in the observed data falling

within the forecast zone, the result in (b) encapsulates a greater proportion of the ob-

served signal than when only the first sine wave was used.

Finally, we show the forecast based on the first five significant frequencies in Figure 10

and see that there is little improvement from including a further three frequency com-

ponents. Each additional wave added has less amplitude or significance in explaining

the original time-series, so the marginal gains from including more waves diminishes

the more waves we add:

23


Figure 10: Fan chart showing reconstructed signal using first five components plotted

against observed out-of-sample GDP returns from October 2013 to October 2014: (a)

shows error bands of one standard deviation, (b) shows two standard deviations.

(a)

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%


Mon

thly

Ret

urns

Forecast using 5 cycle components and 1 standard deviation error bandsObserved GDP5 component forecast±1ߪ


(b)

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

2.5%

3.0%

3.5%


Mon

thly

Ret

urns



ߪ±2

Source: Authors calculations

5.2 Discussion

5.2.1 Summary of Results

Spectral analysis provides an alternative method of, in this case, business cycle analysis,

to the traditional method used by the South African Reserve Bank and Venter (2005) of

micro and macro-economic indicators which lead, coincide with and lag the economy.

The result of this study, from 512 months of GDP data, showed that a significant cycle

exists with a period of 7.11 years over the period February 1971 to September 2013.

24


We found the 3rd highest power frequency component to have a period twice that of the

component with the highest frequency and put forward that this is a harmonic of the

85.33 month cycle and not a significant cycle of its own.

5.2.2 Forecast Accuracy

Forecasting using the established seven year cycle with a frequency of 0.01172 cycles

per month and including up to five frequencies returned somewhat disappointing res-

ults. Examining the data in Figure 11 below, it is clear that the majority of returns (81%)

are between one standard deviation above and below the mean value of 1.06%. How-

ever, in the time period of the forecast, returns were out of this range as shown in Fig -

ure 12.

Figure 11: De-trended GDP returns series showing the historic mean with one standard

deviation above and below the mean from February 1971 to October 2014.

-1.75%

-0.75%

0.25%

1.25%

2.25%

3.25%

4.25%

5.25%


Mon

thly

Ret

urns

Historic Standard Deviation of Returns

Observed GDP Mean

ߪ±1


25


Figure 12: De-trended GDP returns series showing the historic mean with one standard

deviation above and below the mean from the 12 months before the forecast period be-

gins in October 2013 to October 2014.

-1.0%

-0.5%

0.0%

0.5%

1.0%

1.5%

2.0%

Sep-12 Jan-13 May-13 Sep-13 Jan-14 May-14 Sep-14

Mon

thly

Ret

urns

Historic Standard Deviation of ReturnsOctober 2012-October 2014

Observed GDP

Meanߪ±1


Viewing the time series as being made up of a general trend, cyclical component and er-

ror component, one may investigate the behaviour of each of these as in Omekara

(2013). They assessed each component separately before putting them together to form

a forecast equation. Specifically, they estimated the error component by assessing the

autocorrelation function of the residual for randomness. If the residual is not random,

as in their case, a first order autoregressive model may be fitted to the error values

(Omekara, 2013). The Baxter-King filter separates a time-series into the three compon-

ents mentioned, but it is critical to use the appropriate parameters in the filter to pro -

duce meaningful results. Further studies may be able to establish these parameters and

use the BK filtered data to run Fourier analysis. If serial autocorrelation is present in the

error component and can be taken into account via an autoregressive model as part of

the forecast equation, the forecast accuracy may improve.

5.2.3 Dating the Cycle

Fourier analysis does not identify the start and end points of a cycle. Liu et al. (2012)

contend that with the length of a cycle identified, one can infer the start and end points.

This is the reverse of the method used in Bry and Boschan (1971), where the length of a

cycle is inferred by finding the start and end points first. This paper does not attempt to

provide turning point dates and business cycle phase changes as in Botha (2004), al-

though further study over a longer time period using the first few ‘important’ frequency

26


components may provide useful insight in dating previous business cycles. Use of

quarterly data and moving average or filtering methods would help to control the volat-

ility that characterises the monthly data in this study.

5.2.4 Comments on the data

The Discrete Fourier Transform tool in Excel (Cooley-Tukey version) requires 2n data

inputs up to a maximum of 512 entries. It is for this reason that we chose monthly fig-

ures and not quarterly figures, as 512 quarters of GDP data are not easily available. The

problem statement of this paper was to measure and forecast South African GDP cycles

using Fourier analysis. Over the period used in the analysis, many structural breaks,

sanctions imposed and lifted, and regime shifts have taken place as noted in Aaron &

Muellbauer (2002) and Chevillon (2009). These have had destabilising effects on the

economy of South Africa and are reflected in the GDP data under review. Fourier ana-

lysis identified a clear seven year cycle but much noise was present, making forecasting

using the single cycle rather inaccurate and ineffective in coping with volatile breaks in

the data.

27


6 Conclusion and Recommendations

6.1 Conclusion

Using Fourier and periodogram analysis to study South African GDP from February

1971 to September 2013, this paper showed that a clear individual cycle existed, lasting

approximately seven years. This result, though obtained using an alternative quantitat-

ive approach, compares favourably with the result of Botha (2004), who found a seven

year cycle in quarterly South African GDP from 1961 to 2003. We found that a simple

log transformation enables the use of Fourier analysis methods and that filtering using

the Hodrick-Prescott and Baxter-King filters helps to get rid of the majority of random

‘noise’ whilst still preserving the integrity of explanatory cycles present in the data.

Forecasting using solely the frequency component corresponding to the seven year

period showed limited usefulness when compared to observed data in the forecast

period. We showed that the conviction of a forecast can be improved by increasing the

error bands of the forecast from one standard deviation to two standard deviations.

The results also showed that forecast accuracy improves as successive frequency com-

ponents with the next highest amplitudes are added to the seven year base wavelength.

These improvements show diminishing marginal utility as lower amplitudes are used.

6.2 Recommendations for further study

6.2.1 Business Cycle Data Representation

This paper used nominal GDP as a proxy for the SA business cycle. GDP growth is

termed as “the most natural indicator” of an economy’s aggregate business cycle by the

Basel Committee for Banking Supervision (BCBS), however, a number of macroeco-

nomic indicators with useful business cycle information could have been used. These in-

clude aggregate real credit growth, the credit-to-real GDP growth ratio and the leading,

lagging and co-incident indicators used by the SARB to measure the business cycle. The

BCBS considers real credit growth as a “natural measure of supply” since boom periods

leading to a peak in the business cycle are characterized by rapid credit expansion and

credit contraction has typically heralded credit crunches (van Vuuren, 2012). For a

comprehensive study of the use of Fourier series methods in detecting the length and

timing of South african business cycles, the above data should also be analysed.

28


6.2.2 Data Selection

We found that monthly data contained too much variation. Quarterly data combined

with the use of weighted moving averages and filtering would help to smooth volatility

in the short term and should be considered in further studies. Using the Discrete Four-

ier Transform tool in Excel requires that the number of inputs be a multiple of 2n. We re-

commend, in the South African case, using 256 quarters (768 months) to ensure the re -

liability of the results, with the caveat that economic, environmental and political

factors may have changed considerably over this period.

6.2.3 Method of Return Calculation

This paper calculated returns by taking the natural logarithms of month-to-month GDP

levels and not month-on-month returns. Our method produces monthly changes with

less autocorrelation than using month-on-month data, which gives an annual change.

Post study, the author generated a periodogram by the same methods, using month-on-

month data and found the seven year cycle with the same intensity as a result, although

the 13.8 month cycle which had the second highest amplitude was not present as a rel -

evant frequency component.

6.2.4 Power Spectrum Estimation

6.2.5 This paper made use of the non-parametric, periodogram method to calculate the

power of a frequency component. Masset (2008) compared the parametric meth-

ods of Yule-Walker and Burg and the non-parametric methods of the periodo-

gram and Welch method. The non-parametric methods contained more ‘noise’

over the spectrum and the difference between the periodogram and Welch

method was much greater than the difference in the parametric methods. Pre-

ceding Fourier analysis, one should ensure the most suitable method to estimate

the power spectrum is used or provide a comparison of the results before further

analysis. Forecasting Issues

Separate modelling of the components of the time-series

This paper made use of unfiltered returns data as the Fourier tool used was able to sep -

arate the cyclical components from the general trend internally. However, using the

Baxter-King filtered returns or any method that can separate the trend, cycle and error

components of a time series will allow future studies to attempt to model the error com-

29


ponent identified if it is not random, as in Omekara (2013). This may improve the accur-

acy of forecasts.

Determining the current position of the economy on the business cycle

The methods used to determine the length of the SA business cycle provided satisfact-

ory results in this paper, but ascertaining the approximate current position of the eco-

nomy on the business cycle was not addressed in full. For this purpose, further studies

should include Fourier analysis and forecasting of various measures of the business

cycle, not only the rate of GDP growth. Phase differences between leading, lagging and

co-incident cycles may provide valuable information to pin-point the current position of

the economy and a composite cycle of such indicators may provide a more accurate

forecast.

Statistical Measurement of Forecast Accuracy

The forecasts in this paper were not tested by any statistical measure of significance

and thus cannot be compared to other forecast tools. Although testing may become rig-

orous, at a minimum, we suggest testing for correlation and performing a simple regres-

sion to predict future GDP returns based on the path of the reconstructed signal. A mul-

tiple regression hypothesis test may yield helpful results regarding the significance of

each frequency component in predicting future values. The author found that the peri-

odogram performed this function by ordering components according to amplitude, but

a multiple regression could prove this statistically.

6.2.6 Wavelets

Wavelets are relatively new tools in economics and finance. They are a very attractive

way of analysing financial datasets as they are able to represent data series from both

the time and frequency perspectives simultaneously. Hence, they permit to break down

the activity of the market into different frequency components and to study the dynam-

ics of each of these components separately. They do not suffer from some of the limita-

tions of standard frequency-domain methods, like Fourier anlaysis used in this paper,

and can be employed to study a financial variable, whose evolution through time is dic-

tated by the interaction of a variety of different frequency components. These compon-

ents may behave according to non-trivial (non-cyclical) dynamics – e.g., regime shifts,

30


jumps, long-term trends (Masset, 2008).10 Due to the complex nature and presence of

such non-trivial dynamics in South African GDP data under study, the author strongly

suggests using wavelet analysis on South African and other similar GDP datasets which

are characterised by non-cyclical dynamics.

10 For a non-technical introduction to wavelets and their benefits compared to pure-frequency domain analysis, see Masset (2008).

31


Bibliography

Baxter, M., & King, R.G., 1999. Measuring business cycles: approximate band-pass filters for economic time series. The Review of Economics and Statistics, 81(4), 575-593.

Bry, G., & Boschan, C., 1971. Cyclical analysis of time series: selected procedures and com-puter programs. New York: Columbia University Press.

Bosch, A., and Ruch, F., 2013. An Alternative Business Cycle Dating Procedure for South Africa. South African Journal of Economics, 81: 491–516.

Canova, F., 1994. Detrending and turning points. European Economic Review, 38(3-4), 614-623.

Canova, F., 1998. Detrending the business cycle. Journal of Monetary Economics, 41, 475-512.

Chevillon, G., 2009. Multi-step forecasting in emerging economies: An investigation of the South African GDP. International Journal of Forecasting, 25(3), pp.602–628.

Cogley, T., & Nason, J. M., 1995. Effects of the Hodrick-Prescott Filter on trend and differ-ence stationarytime series: implications for business cycle research. JournalofEconomicDy-namicsandControl,19, 253-278.

Cunnyngham, J., 1963. Spectral analysis of time series. Working Paper No. 16, U.S. De-partment of Commerce, Bureau of Census, New York.

Debnath, L., 2011. A short biography of Joseph Fourier and historical development of Fourier series and Fourier transforms. International Journal of Mathematical Education in Science and Technology, 43(5), pp.1–24.

Granger, C.W.J., 1966. The typical spectral shape of an economic variable. Econometrica, 34(1), 150-161

Granger, C.W.J., & Morgenstern., 1964. Spectral Analysis of Stock Market Prices. Kyklos, 34(1), 150-161.

Guay, A., and St-Amant, P., 2005. Do the Hodrick-Prescott and Baxter-King filters Provide a Good Approximation of Business Cycles? Annales d'Economie et de Statistique, No. 77 (Jan. - March., 2005) pp. 133-155

Hamilton, J.D., 1994. Time Series Analysis, Princeton.

Harvey, A. C., & Jaeger, A., 1993. Detrending, stylized facts and the business cycle. JournalofAppliedEconometrics,8, 231-247.

Hodrick, R.J., & Prescott, E.C., 1997. Postwar U.S. business cycles: an empirical investiga-tion. Journal of Money, Credit, and Banking, 29(1), 1-16.

Iacobucci, A., 2003. Spectral Analysis for Economic Time Series, Documents de Travail de I'OFCE.

Aron, J., and Muellbauer, J., 2002. Evidence from a GDP Forecasting Model For South Africa IMF Staff Papers , Vol . 49 , IMF Annual Research Conference ( 2002 ), pp . 185-213

King, R. G., & Rebelo, S. T., 1993. Low frequency filtering and real business cycles. JournalofEconomicDynamicsandControl,17(1-2), 207-231.

32


Lindgren, G., Rootzén, H and Sandsten, M., 2013. Stationary stochastic processes for sci-entists and engineers. Chapman and Hall, USA, 200p.

Liu, L., Paki, E., Stonehouse, J., and You, J.,, 2012. Cycle Identification : An old approach to ( relatively ) new statistics. NZAE Annual Conference, Statistics New Zealand

Masset, P., 2008. Analysis of Financial Time-series using Fourier and Wavelet Methods. … )-Faculty of Economics and Social Science, (October), pp.1–36. Available at: https://psutn2009.googlecode.com/svn-history/r5/trunk/SSRN-id1289420.pdf.

Nerlove, M., 1964. Spectral analysis of seasonal adjustment procedures. Econometrica, 32: 241-286.

Omekara, C. O., Ekpenyong, E., J., and Ekrete, M., P., 20142014. Modelling the Nigerian In-flation Rates Using Periodogram and Fourier Series Analysis .[pdf] Lagos: Nigerian Cent-ral Bank. Plessis, S., Smit, B. & Steinbach, R., 2014. South African Reserve Bank Working Paper Series A medium ‐ sized open economy DSGE model of South Africa. , (July).

Praetz, P.D., 1973. A spectral analysis of Australian share prices. Australian Economic Pa-pers, 12: 70–78. doi: 10.1111/j.1467-8454.1973.tb00296.x

Ravn, M. O., & Uhlig, H., 2002. On Adjusting the Hodrick Prescott Filter for the Frequency of Observations. The Review of Economics and Statistics, 84(2), 371-375.

Shumway, R.H. & Stoffer, D.S., 2000. Time Series Analysis and Its Applications, Available at: http://www.springer.com/statistics/statistical+theory+and+methods/book/978-0-387-29317-2.

Telser, L.G., Granger, C.W.J. & Hatanaka, M., 1967. Spectral Analysis of Economic Time Series. Journal of the American Statistical Association, 62(November), p.687. Available at: http://www.jstor.org/stable/2283996?origin=crossref.

Venter, J.C., 2005. Reference Turning Points in the South African Business Cycle: Recent Developments. SARB Quarterly Bulletin, (September), pp.61–70. Available at: http://www.esaf.org/internet/Publication.nsf/LADV/EE79538CB86F96DA42257083004E8875/$File/ART092005.pdf\nhttp://www.oecd.org/dataoecd/54/26/34898202.pdf.

van Vuuren, G., 2014. Private communication.

van Vuuren, G., 2012. Basel III Countercyclical capital rules: implications for South Africa. South African Journal of Economic and Management Sciences, 15(3): 76 – 89.

33


Appendix

Table 1 (a) Augmented Dickey-Fuller test for stationarity: monthly GDP levels:

Stationary test

TestSta

t P-Value C.V. Stationary? 5.0%

ADF

No Const -2.2 2.5% -1.9 FALSE

Const-Only -5.8 0.1% -2.9 FALSE

Const + Trend -7.1 0.0% -1.6 FALSE

Const+Trend+Trend^2 -7.4 0.0% -1.6 FALSE

Table 1 (b) Augmented Dickey-Fuller test for stationarity: monthly GDP returns:

Stationary test

TestSta

t P-Value C.V. Stationary? 5.0%

ADF

No Const -2.2 2.5% -1.9 TRUE

Const-Only -5.8 0.1% -2.9 TRUE

Const + Trend -7.1 0.0% -1.6 TRUE

Const+Trend+Trend^2 -7.4 0.0% -1.6 TRUE

Table 3 Jarque-Bera test of normality on monthly returns series from February 1971-

September 2013

Normality test

Z-Score Critical Value p-Value Pass? Significance Level=0.05

Jarque-Bera 988.90 5.99 0.0% FALSE

34