Explaining the gold price after the Bretton Woods Agreement using
independent variables: an ARIMA model approach
Stefan Frank Heini
Subject Area: Finance & Economics
Supervisor: Dr. Panayiotis Savvas
Submitted: March 2014
Dissertation submitted to the University of Leicester in partial fulfilment of the requirements of
the degree of Master of Science in Finance
1
Table of Contents
Table of Contents
2
List of Tables
4
List of Figures
4
Executive Summary
5
Chapter 1 – Introduction
6
1.1 Background
6
1.2 Gold is different
7
1.3 The gold price since the end of Bretton Woods
7
1.4 Research questions
9
Chapter 2 – Literature Review and Theory
11
2.1 Theoretical framework: Explaining the movements of the gold price
11
2.2 Empirical findings: Independent variables correlating with the gold price 13
2.3 Conclusion
20
Chapter 3 – Data and Methods
21
3.1 The ARIMA model
22
3.2 Assumptions of an ARIMA model
23
3.3 Data collection and sources
24
3.4 Defining an ARIMA model to fit the gold price
25
3.5 Evaluation of the ARIMA model
26
3.6 Conclusion
28
Chapter 4 – Analysis and Results
29
4.1 Data description
29
4.2 The best fitting ARIMA model
33
4.3 ARIMA model fit during normal times and crises
37
4.4 Explaining divergences of the model fit during normal times and crises
38
4.5 Conclusion
39
Chapter 5 – Discussion and Conclusions
41
2
5.1 Summary
41
5.2 Implications
43
5.3 Limitations
44
5.4 Direction for Future Research
45
5.5 Reflections
46
References
47
Appendices
54
Project Proposal
56
3
List of Tables
Table 1: Data sources ................................................................................................................ 25
Table 2: Descriptive statistics of dependent and independent variables: minimum, maximum,
mean and standard deviation. .................................................................................................... 30
Table 3: Correlation between the gold price and the independent variables .............................. 32
Table 4: ARIMA model description ............................................................................................. 33
Table 5: ARIMA model statistics ................................................................................................ 35
Table 6: ARIMA model parameters contributing to the fit of the monthly price of gold ............... 36
Table 7: Independent variables contributing to the quality of an ARIMA model during the
complete period (March 1973 to December 2011), during normal times and during crises
(January 1978 to January 1981) ................................................................................................ 37
Table 8: ARIMA model statistics during crises (January 1978 to January 1981 and August 2007
to December 2011). ................................................................................................................... 54
Table 9: ARIMA model statistics during normal times (March 1973 to December 1977 and from
February 1981 to July 2007). ..................................................................................................... 54
Table 10: ARIMA model parameters contributing to the fit of the monthly price of gold during
crises (January 1978 to January 1981 and August 2007 to December 2011)............................ 54
Table 11: ARIMA model parameters contributing to the fit of the monthly price of gold during
normal times (March 1973 to December 1977 and from February 1981 to July 2007). ............. 55
List of Figures
Figure 1: Average gold prices in US dollars from 1973 to 2011 (rounded). Sources: Data for
1973 to 2008 taken from World Gold Council 2008 and for 2008 to 2011 taken from Kitco.com . 9
Figure 2: Gold time-series made stationary by differencing once .............................................. 33
Figure 3: Partial autocorrelation function (PACF) and autocorrelation function (ACF) of the
model ......................................................................................................................................... 34
Figure 4: Observed gold price and gold price model fit (MAPE for the model: 2.861%) ............ 36
4
Executive Summary
To date, nobody has formulated a comprehensive theorem to determine gold valuation or
precious metal prices. Until fairly recently, Eugene Fama’s Efficient Market Hypothesis was the
predominant paradigm explaining asset markets but today it is widely acknowledged that
markets can be irrational and investors are prone to act irrationally. When trying to explain gold
market anomalies, behavioural science approaches can be useful. Phenomena such as
herding (‘group think’), ‘safe value bias’ and investors’ ‘excessive extrapolation’ can help explain
positive price performance over a certain time.
In this dissertation, the author investigates the applicability of a multivariate ARIMA (autoregressive, integrated, moving average) model to help explain gold price movements from 1973
to 2011. This model uses the gold price and independent variables such as inflation, real
interest rates, silver prices, the US dollar money supply (M2), oil prices, the MSCI World index
and the S&P 500 as these are linked to gold and/or highly correlated with the gold price. The
evaluation criteria were defined as R-squared, mean absolute percentage error (MAPE) and
BIC. The model was calculated over so-called ‘normal times’ and times of crises (one political,
one financial). The researcher used SPSS’ Expert Modeler to find the best-fitting ARIMA model
and to identify the independent variables significantly contributing to the fit of the model.
Remarkably, a multivariate ARIMA model using independent variables explained almost twice
as much of the variability of the gold price as a univariate ARIMA model using only the gold
price. Also, throughout the complete period and during normal times the model explained a
much higher percentage of the variability of the gold price than during crises and comparably
more of the independent variables contributed significantly to the fit of the model (5 vs. 2). This
can be explained by investors’ tendencies to buy gold to preserve their assets (“safe value”), to
follow the crowd (“herding”) and to extrapolate past price chart developments.
The results show that in an attempt to discern the cause of gold price movements, a multivariate
ARIMA model outperforms a univariate ARIMA model significantly. The results of the study
furthermore indicate researchers evaluating different methods to fit a time series should
consider a multivariate ARIMA model, especially if the independent variables are highly
correlated with the dependent variable.
5
Chapter 1 – Introduction
1.1 Background
“Gold – the different asset class” was the title of an article by Baumann and Sullivan (2011)
published on Swiss bank Credit Suisse’s website in November 2011. It expressed the
peculiarity of gold. Gold has been called a “zero-beta asset”, an “inflation hedge” and a
“currency” (Fei and Adibe 2010, p. 1).
This study intends to develop a model to explain the movements in the price of gold since the
end of the Bretton Woods era in March 1973 – and the reintroduction of free floating currencies
– until 2011. It does so by testing those independent variables that researchers have found to
show a strong correlation and/or connection with the gold price in US dollars. Given the data
and the research goal, the research is based on ARIMA (auto-regressive, integrated, moving
average, for model discussion see Chapter 3), the most frequently used model for time series or
longitudinal type data (Adbullah 2012, p. 153). ARIMA models are often used to fit time series
data as they comprise a robust family of models capable of correcting for autocorrelations, nonstationary data, and excessive volatility (SPSS 2012, p.3/4).
While univariate ARIMA models are limited to the information contained in the series itself to
predict future data within the time series, multivariate ARIMA models include explanatory
variables (De Gooijer and Hyndman 2006, p. 447). For the purposes of this dissertation, a
multivariate ARIMA model included the following independent variables designed to illuminate
gold price movements: inflation, the real interest rate, silver prices, the US dollar money supply
(M2), oil prices, the MSCI World index, the S&P 500.
During crises, demand for gold tends to be unusually high as investors favour assets
considered conservative. Therefore, for the purposes of this paper, research will include periods
of so-called ‘normality’ and those of crises (Harmston 1998, p. 6/7). Between 1973 and 2011 the
world experienced two major crises with global implications on the financial markets, oil and
gold prices: a political crisis between 1978 and 1981 and a financial crisis that broke out in
2007. Oil price shocks will be considered by including the independent variable “oil price”; stock
market crashes will be considered by including the independent variables MSCI World Index
and the S&P 500.
To discuss the effect of certain crises on the gold price, the research focuses on two of the most
significant periods (Amey 1998, p. 49/50):
6
•
From January 1978 until the end of January 1981: Civil resistance against the monarchy
in Iran intensified and culminated in the abdication of the Shah on February 11, 1979.
Shortly afterwards, Iran held 52 US citizens hostage and that Christmas the Soviet Union
invaded Afghanistan. The US hostages were released on January 20, 1981
•
From 9th August 2007 until the end of the period under consideration, as the Eurozone
was still struggling and the US was trying to exit a prolonged recession, a global financial
crisis broke out after the French bank BNP Paribas announced it had ceased support for
three hedge funds that specialised in US mortgage debt. This, in turn, was followed by
the bankruptcy of the first major investment bank, Lehman Brothers, on September 15,
2008 (Elliott 2011 and Kingsley 2012).
1.2 Gold is different
Gold is not like other metals because its industrial use is negligible, which makes it different to
other commodities such as zinc, copper or silver. This explains why the price of gold often
moves differently than the price of other commodities during a recession or a depression and
especially during periods of high inflation (World Gold Council 2011, p. 8). The gold supply is
primarily absorbed in the production of jewellery, by central banks, investors and more recently
by financial institutions offering gold ETFs (Shafiee and Topal 2010, p.178). Gold is also special
because of its distinctive place in economic history and its use as a financial asset, in particular
as a hedge against inflation and geopolitical and/or economic risk. Many individuals add gold to
their portfolios as a risk diversifier (Dempster 2008, p. 5).
1.3 The gold price since the end of Bretton Woods
Before the introduction of the gold standard in 1900 (it was dropped in 1933), gold had been
traded over the counter in London since the 17th century. In 1944, the Bretton Woods system of
pegged exchange rates was introduced. It was named after an international conference held in
the town of Bretton Woods in the US state of New Hampshire, which was designed to establish
new global commercial rules following World War II. Between 1944 and 1971 the currencies of
43 nations were pegged to the US dollar, which in turn was fixed to the gold price (The Gold
Standard). The price of an ounce of gold was fixed at 35 US Dollars (Hammes and Wills 2005,
p. 504). The International Monetary Fund (IMF) was given the authority to intervene in case of
an imbalance of payments (Karunagaran 2011, p. 5/6). The Bretton Woods system meant US
dollars were fully convertible into gold at this fixed rate. Steadily growing trade volumes and
enhanced world production meant increased demand for US dollars. At the same time gold
holdings – U.S. gold holdings in particular – decreased. This caused the collapse of the system
(Garber 1993, p. 461).
7
In 1968 a two-tier gold market was established, whereby central banks continued to trade gold
among themselves at the official rate while the private sector traded at the market price (World
Gold Council 2008, p. 1). During this time it became apparent the dollar was already
overvalued, particularly given the significant increase in domestic spending resulting from
President Lyndon Johnson’s ‘Great Society’ programmes. The cost of the Vietnam War to the
American exchequer worsened the situation further. “By early 1971, the US dollar liabilities
exceeded 70 billion, backed by only 12 billion US dollars” (Hammes and Wills 2005). In the
same year, the US President Richard Nixon declared a temporary suspension of the dollar-gold
convertibility. The system was finally dissolved in 1973 (IMF 2012). After March 1973, the
former Bretton Woods currencies were floating freely again (Stephey 2008).
After the Bretton Woods system was dissolved, the gold price was floating freely for the first
time in 250 years (Fei and Adibe 2010, p. 1 and Oxford Economics 2011, p. 5). From then on,
the gold price steadily increased from its starting point of 35 US dollars an ounce and quickly
rose to a high of 127 US dollars on July 6th, 1973 (World Gold Council 2008, p. 2). The gold
price kept rising and peaked temporarily (in USD) during the second oil crisis in 1980. After the
collapse of the Shah’s regime and the subsequent withdrawal of Iran’s oil from the world
market, gold hit a record high of 850 US dollars an ounce (Hamilton 2011, p.16). In the following
years, gold prices oscillated according to various economic and political crises but did not reach
its 1979 heights again until 2008. During that decade, the gold price drifted and flattened until a
low of 251.70 US dollars was reached in 1999. In that year, 15 European central banks agreed
on limiting gold sales, which boosted the gold price to a two-year high of 338 US dollars an
ounce in October, 1999. Since then, the gold price has risen inexorably because of a number of
factors, including the 2003 Iraq war, a weakening dollar, relatively high oil prices, political
tensions over Iran’s nuclear ambitions and worries about contagion of debt problems in the
Eurozone. In 2011, it reached a new peak above 1900 US dollars an ounce, after a 650% rally
(Popper 2013).
Since March 1973, gold prices can be viewed in three distinct periods:
(1) A bull market since the introduction of a freely floating gold price until a temporary peak
in 1980
(2) A period of a rather flat to slightly falling gold prices between 1981 and 2000
(3) An increasing gold price since 2001 until the end of 2011 (see Figure 1). By the end of
2011, the price of an ounce of gold was 1575 US dollars (Kollewe 2010).
8
Figure 1: Average gold prices in US dollars from 1973 to 2011 (rounded). Sources: Data for 1973 to 2008 taken
from World Gold Council 2008 and for 2008 to 2011 taken from Kitco.com
Researchers found that the price of gold is influenced by some factors that are easy to quantify
such as currency exchange rates, inflation, the price of crude oil, the price of silver and the US
dollar money supply. However, other factors are far more intangible, such as political risk,
official sector activity and central bank gold reserve sales.
1.4 Research questions
In an attempt to analyse how identifiable independent variables influence the gold price, the
following questions will be addressed:
The central research question is: How well can the gold price since the end of Bretton Woods
be explained using a multivariate ARIMA model that includes the following independent
variables: inflation, real interest rates, silver prices, the US dollar money supply (M2), oil prices,
the MSCI World index and the S&P 500 (evaluation criteria: R-squared, mean absolute
percentage error (MAPE) and BIC)?
To answer the research question, the author will discuss the following:
A. How effective is the model including these independent variables in explaining gold price
variations in times of so-called ‘normality’ and in times of crises?
B. What is the explanation for the differences in variability described by the model during times
of so-called ‘normality’ and times of crises?
9
The following literature review comprises two sections. The first discusses theories explaining
asset price behaviour. The second covers studies that tested determinants (independent
variables) that supposedly have a significant effect on the gold price. If an independent variable
can be expected to increase the quality of an ARIMA to explain the gold price, it will be included
in the analysis.
In the following chapter, the research will discuss data collection, the ARIMA model and its
assumptions. It will then provide analysis, interpretation and contextualisation of the results.
Given the nature of the questions, providing conclusive answers to Question B will prove
particularly challenging because it requires interpretation and examination through the
application of a theoretical framework.
The rest of this dissertation is organised as follows: Chapter 2 discusses the existing literature
and theory on the topic with particular emphasis on the connection between this paper and the
existing theoretical framework; Chapter 3 covers data and methods: ARIMA and the approach
applied will be discussed in detail; Chapter 4 presents and discusses the findings of the
analysis; Chapter 5 concludes by summarising and listing separate sections on the implications
and limitations of this study as well as on how the results might be used for further research.
10
Chapter 2 – Literature Review and Theory
The aim of Chapter Two is to review the relevant literature and theory in order to develop a
coherent theoretical framework. This involves the discussion of various theories to explain the
variations in asset prices that cannot be explained by the identified variables. The second part
of the chapter provides examples of identifiable independent variables that are related to gold
and/or correlate significantly with the gold price.
2.1 Theoretical framework: Explaining the movements of the gold price
In 2005, Faugère and van Erlach (p. 99) wrote that “assessing the fair value of gold largely
remains a mystery in finance”. No comprehensive theory of gold valuation exists, they wrote,
that is able to show how factors like inflation, exchange rates or other asset classes influence its
value. This finding holds true given there is no widely accepted theory that explains the price of
precious metals. Instead, the efficiency of the gold market will be discussed: how the gold price
moves in times of crises and investor psychology that helps explain the gold price in normal
times and times of crises.
Eugene Fama’s Efficient Market Hypothesis (EMH) was long considered to be the best
description of securities’ price movements and was widely accepted. The main concept of EMH
is that markets are efficient, that prices are unpredictable and that prices of securities reflect all
available information at any time (Fama 1970, p.383). This view is no longer dominant. Many
securities professionals and academics now agree markets are irrational at times and
anomalies and irrational investors are at the root of mania and panics (Dieupart-Ruel et al.
2013, p. 129 and Yalcin 2010, p. 24). Kindleberger and Aliber (2005, p. 38) say the assumption
of the always rational investor as defined by the EMH is unrealistic, citing the frequency of
speculative manias and irrational exuberance.
The common understanding today is that the EMH fails to recognise that psychology plays an
important part in our investment decisions (Dreman 1998, p. 4). In this sense, the gold market
seems to be no exception. Solt and Swanson (1981) looked at the price of gold from January
1971 until the end of the decade. They found positive autocorrelations, including considerable
heteroscedasticity in the variance and that the means of the price change is non-zero and nonstationary (p. 470). Overall, they conclude, the results are not consistent with gold market
efficiency (p. 476/477).
11
Well-known phenomena postulated to explain market anomalies and manias are ‘herding’
(group think) and excessive extrapolation, which is the tendency of investors to extrapolate
recent positive news and price developments into the future without the fundamentals changing.
Positive price performance over a certain time and “excessive extrapolation” often lead to
market participants giving too rosy market forecasts (Utkus 2011, p. 6). Baur and Glover (2011,
p. 7) argue the rapid price increase of gold from 400 US dollars to 1600 US dollars an ounce
between 2005 and 2011 cannot be explained solely by a change in the fundamental value of the
metal. This explanation is based on the assumption that the gold price primarily rises if the
expected inflation increases, given the fact that gold pays neither interest nor dividends.
Therefore, the somewhat speculative investing of chartists based on extrapolating past price
trends must drive the price upwards. Utkus (2011, p. 3/7) writes that if buying based on the
positive price development of the recent past goes on long enough and an ever larger
percentage of investors start to join in the group-think, it can lead to the development of a
bubble: understood as a situation when the current price of an asset substantially differs from its
intrinsic value.
Demand for gold is particularly high during times of crises as investors are looking for means to
protect their assets. The main triggers of such demand include financial instability, the decisions
of central banks (Dieupart-Ruel et al. 2013, p. 129), political tensions, wars or distrust in the
policies and prospects of nations (Nadler 2006, p. 56). The latter is linked to the expectation of
rising inflation – as experienced during the Euro crisis and in the current era of monetary
inflation – which encourages investors to invest in gold as they try to preserve their wealth
(Dempster and Artigas 2010, p. 69). Times of crises also cause investors to become uncertain
regarding the capital markets, which triggers the purchase of gold as a substitute investment
(Cohen and Qadan 2010, p. 43). The extent to which fear can drive stock prices down and at
the same time increase demand for gold can be observed during the Eurozone crisis. On
November 16th 2011, prominent commentator Matthew Lynn wrote on marketwatch.com: “Gold
is the only winner from the Euro crisis” as the Euro falls and “equities have struggled to make
any progress all year”. On June 25th 2012 the New York Times reported: “Wall Street drops on
Euro pessimism” as media commentators and the public doubted whether Europe could solve
its debt crisis.
Dieupart-Ruel et al. (2013) explored the extent of investor rationality and availability of
information when making decisions. They evaluated if what they termed ‘the cognitive biases’ –
’anchoring bias’ and ‘safe value bias’ – influenced gold investors. They analysed the gold price
between Q4 2003 and the end of 2012 and assumed three classes of investors:
12
(1) Rational informed agents (RIA) who take into account the fundamentals
(2) Irrational informed agents (IIA) who are informed and are prone to cognitive biases
(3) Non-informed agents (NIA) who take their information from observing the market. NIAs
are influenced by the propositions of informed agents and by behavioural biases (p.
129/130).
Dieupart-Ruel et al. looked at the difference between a gold price they calculated based on
global demand, which they termed the “fundamental price” and the actual gold price. The
difference between the two – they assumed – would be accounted for by the influence of
investor biases (p. 132). By considering anchoring bias and safe haven bias in their
calculations, the estimation of the actual gold price could be improved significantly. To test the
anchoring bias the researchers evaluated whether investors under-reacted to analysts’
predictions but found the effect to be rather small; to test the safe value bias they used the
volatility indicator VIX. VIX measures the volatility of the S&P 500 and is based on the
calculation of the average of calls and puts on the S&P 500 (p.130). Their analysis shows the
safe value bias is much more important when trying to explain the gold price than the anchoring
bias (p. 132). Interestingly, Dieupart-Ruel et al. found the importance of the safe value bias
differs between uncertain times and “normal” times – just as could be expected. From 2004 and
the latest financial crisis of 2008, volatility of the S&P 500 was weak and so was the effect of the
safe value bias on the gold price. The calculated price during this period hardly differed from the
real gold price. But from the beginning of the financial crisis onwards, volatility increased and
the influence of the safe heaven bias kept pace with it (p. 132).
Having discussed various theories dealing with (commodities) markets and investor psychology,
the discussion will now turn to the quantitative variables connected with the gold price.
2.2 Empirical findings: Independent variables correlating with the gold price
Researchers have found a number of variables that correlate with the gold price. First, gold is
said to protect in times of inflation and/or serve as a hedge against future inflation, measured by
the Consumer Price Index or the Producer Price Index (Dimson, Marsh and Staunton 2012, S.
9). Inflation erodes cash values but gold is generally considered a safe haven as its value tends
to increase during periods of inflation. Inflation might drive up the gold price because investors
expect prices to go even higher and therefore see the purchase power of their dollars
deteriorate (Koutsoyiannis 1983, p. 571). O’Connor and Lucey (2012, p. 16) argue that because
gold is primarily traded in dollars, if the dollar weakens then gold becomes cheaper when paid
for in a foreign currency. This drives up its demand. If this logic holds true, we can expect a
13
positive correlation between inflation and the gold price: the higher the inflation, the higher the
gold price.
Several researchers have found a positive correlation between (expected) inflation and the gold
price. Ghosh et al. (2004) write that because the dollar constantly lost purchasing power
between January 1982 and December 1999 while during the same period gold lost 59% of its
value, it did not serve as an inflation hedge in the short run (p. 2). In their own analysis,
however, the researchers found that gold can be regarded as an efficient long-run inflation
hedge. They analysed the monthly average spot dollar price for an ounce of gold between
January 1976 and December 1999 and used cointegration regression techniques to analyse the
relationship between and gold and inflation. When testing for the cointegration of the Retail
Price Index and the price of gold, the hypothesis that these two variables are cointegrated could
not be rejected. In the short run, they concluded, fluctuations in the price of gold are based on
short-run influences such as changes in the gold lease rate, the real interest rate, default risk
and the exchange rate of the dollar (Ghosh et al. 2004, p. 1/9/10/18).
Capie et al. (2005) agree that gold serves as a hedge over the long-run. They analysed the
weekly gold prices between 1971 and 2004 in relation to the Sterling-Dollar and the Yen-Dollar
exchange rates. Although the gold price peaked in 1980 and started increasing again from
2001, the yen-dollar exchange rate decreased steadily (thanks to a strengthening yen) during
the entire period. The Sterling-Dollar exchange rate peaked in 1985 and was subsequently
relatively stable. In 2004 it was slightly higher than 34 years earlier (p. 347). The scatterplots of
the logarithm of the gold price against the logarithms of the two exchange rates led them to
conclude that gold was a hedge against the dollar, but with several caveats. The researchers
concluded that gold served as a hedge mainly because it is a homogenous asset that can easily
be traded and because it cannot be produced by the same authorities that have control over the
currencies. The researchers see three possible reasons for the varying quality of the hedge:
(1) Actors expected exchange-rate fluctuations to be temporary and decided to ride them out
rather than rearrange their portfolios.
(2) Private sector investors might have been influenced by problems in gold-producing
countries, i.e. they expected these problems to influence the gold supply in the future.
(3) States may constantly change their attitudes towards their gold holdings.
Given these insecurities, the authors cautioned that while gold has served as a dollar hedge, its
price development remains highly insecure as it is influenced by the actions of individuals as
well as political attitudes and unpredictable events (p. 351/352). While the exposure to political
14
attitudes and unpredictable events is certainly true, it is not unique to gold but is also true for
many, especially multinational corporations as they, too, are influenced by political decisions
and unpredictable events such as natural disasters and wars. However, as states have direct
control over their gold reserves and influence various factors that are claimed to influence the
gold price (interest rates, foreign policy, and money supply), the influence of states on the gold
price can be expected to be greater – bar exceptions like the nationalisation of corporations –
than on securities.
Worthington and Pahlavani (2006) analysed two data sets, comprised of monthly data of the
gold price in US dollars per ounce and the monthly US inflation rate. The researchers analysed
two subsamples encompassing January 1945 to February 2006 and from January 1973 to
February 2006. The second subsample starts after the Bretton Woods’ System of fixed
exchange rates was dismantled. They used unit root tests for their analysis and had to consider
two structural breaks in the gold price due to the oil crises in January 1973 and in December
1978 and two structural breaks in the inflation rate in February 1973 and January 1979. They
concluded that gold served as a useful inflation hedge between 1945 and 1973 as in the period
following the dismantling of the fixed exchange rates. They said a “strong cointegrating
relationship exists” between gold and inflation from 1945 to 2006 (p. 260/261). Joy (2011)
agreed with Chua and Woodward (1982) that gold serves as an inflation hedge. Joy applied a
multivariate GARCH model using weekly data of the gold price and 16 exchange-rate pairings
and found that gold served as an inflation hedge from 1986 to 2008 (p. 124/129); Chua and
Woodward (1982) analysed whether gold served as an inflation hedge against the currencies of
Canada, Germany, Japan, Switzerland, the United Kingdom and the United States between
1975 and 1980. They collected monthly data comprising gold prices and domestic consumer
price indices (CPI).Inflation was computed by calculating the percentage change in the CPI. A
simple regression model was used. The return from gold was positive solely for the US – and
the US dollar – and the result significant, based on the data collected.
Not all researchers agree on the strong connection between inflation and the price of gold.
Lawrence (2003, p. 2) denies any statistical correlation between the gold price and inflation (as
well as between gold and GDP and gold and interest rates). His research is based on a time
series analysis using quarterly data from January 1975 to December 2001. However, as his
study was undertaken under the patronage of the World Gold Council it has to be taken with a
pinch of salt. It must be expected that the sponsor is interested in positioning its “product” as
attractively as possible and as a “safe haven”. Fisher (2011), a chartist, has also challenged the
significant correlation of gold and the inflation rate. He concluded there is only a weak
15
correlation, if any, between inflation and the gold price, claiming that the price of gold rises
independently from inflation. He gives three indicators to underline his claim: From 1976 until
January 1980, the gold price rose 523% while inflation increased 167%; between 1980 and April
2001 gold decreased 67% while the consumer price index advanced 226%; in the next bull
market from 2001 until February 2011 the gold price went up 530%. In the same period the
inflation rate was 125%.
Second, inflation in US dollar terms is very closely related to the US dollar money supply (M2),
which is why – if the inflation rate is strongly connected to the gold price – it can be expected
that the US dollar money supply (M2) is also strongly interrelated with the gold price. In
November 2013, as the Federal Reserve was still buying 85 billion dollars’ worth of Treasuries
and mortgage-bonds each month, thus steadily expanding the monetary base (the sum of US
currency in circulation and bank reserves), so-called experts argued whether the expansion in
the monetary base would someday inevitably cause inflation or not. While the Federal Reserve
is convinced everything is under control as the “quantity of currency in circulation is entirely
determined by demand from people and businesses” (Williams 2012), others like Huberts
(2013, p. 3) argue that as soon as the velocity of money increases, so will inflation. As inflation
is said to have an influence on the gold price, it is not surprising that many researchers
investigated whether the US dollar supply influences the gold price. The logic behind this
reasoning is thus: The US dollar is the most important reserve currency; many individuals and
institutions are invested in US dollars; gold is traded against the US dollar. If the US dollar
weakens, US dollar holders lose money. The weaker the US dollar, the bigger is the incentive to
invest in another “reserve currency” – such as gold: If the dollar weakens, the demand for gold
rises (Fei and Adibe 2010, p. 25).
Tully and Lucey (2006) agree on the relationship of the US dollar and gold and write that “gold
appears to be the anti-dollar” (p. 317). Pukthuangthong and Roll (2011, p. 2070) share the
same view but claim that the inverse relationship between a currency and the gold price holds
not only for the US dollar but for any currency. In their own analysis, Tully and Lucey used a
generalised autoregressive conditional heteroskedasticity model (GARCH) to investigate the
macroeconomic influences on the gold price for the period 1983-2003. They researched such
macroeconomic factors as the US dollar supply, the Pound Sterling supply, the British stock
index FTSE 100, the UK consumer price index and US interest rates. The researchers came to
the conclusion that among the variables considered, the US dollar money supply had the
biggest significant impact on the gold price (p. 322/323).
16
Ismail, Yahya and Shabri (2009) found that the US dollar money supply (M1) was positively
correlated with the gold price. The researchers developed a multiple linear regression model
and used independent variables such as the inflation rate, the Commodity Research Bureau
future price index, the US Dollar/Euro exchange rate, US dollar money supply (M1) and the
NYSE and S&P’s 500 stock indices, employed SPSS and took the mean square error as the
measure for the quality of the model’s forecast accuracy. Around 70% of the variance could be
explained by a model using the variables that significantly influence the Commodity Research
Bureau future price index, US Dollar/Euro exchange rates, the inflation rate and the US dollar
money supply (M1). “M2 contains M1 plus certain other financial assets” such as savings, small
denomination time deposits at all depository institutions, mutual funds, overnight Eurodollars
and overnight repurchase agreements at commercial banks (Batten and Thornton 1983, p. 40).
Given that M1 and M2 are very closely linked, research results that are found for M1 can also
be expected to be valid for M2.
An interesting analysis on the connection between the US dollar money supply and the gold
price comes from Artigas (2010). Using a multiple regression model, he analysed the
correlations between the independent variables’ year-on-year growth in money supply (M1) of
the US dollar, the Euro, the Indian Rupee and the Turkish Lira and the dependent variable yearon-year percentage changes in the price of gold for a given month. He found that an increase in
the money supply of the US dollar does increase the gold price as much as an increase in the
money supply of the other currencies. This confirms the findings of Pukthuangthong and Roll
that increases in the money supply of other currencies have an effect on the gold price, too.
According to Artigas, the highest correlation between supply increase and an effect on the gold
price can be witnessed six months later (p. 8).
Third, the demand for gold is also influenced by the opportunity costs of capital. The higher the
interest rates on government bonds and in bank accounts, the higher the opportunity cost of
holding gold that pays no interest (Oxford Economics 2011, p. 7). If the nominal interest rate is
lower than inflation, the real interest rate is negative. In such a situation gold is attractive for
investors and demand is high, some researchers say. The inverse relationship between the real
interest rate and the gold price seems to be widely confirmed by the findings of researchers and
market analysts: Mickey (2009), Chief Investment Strategist of “Q1 Publishing”, an investment
newsletter, calls the real interest rate the “main driver for gold prices”. Mitra (2011), another
market analyst from Axis Bank, claims the existence of a negative relationship between real
interest rates and the gold price: the more negative the real rate of interest, the higher the gold
price. He found this tendency to be true for India, the US, Japan and China from 1998 through
17
2008. Academics like Barsky and Summers (1988) agree with this. They used an ARIMA model
to analyse the relationship between the real interest rate in the US and the gold price between
1973 and 1984 and found a strong, significant correlation (pp. 543-545).
Fourth, several researchers and market analysts claim a positive correlation between gold and
silver. According to the World Gold Council (2011), gold and silver showed a correlation of
+0.67 between January 1991 and December 2010. Baur and Tran 2012 (p. 2) wrote that “gold
and silver were substitutes for thousands of years suggesting that there is a long-run
relationship between the two precious metals”. However, they also mention other factors that
uncouple the prices of gold and silver from each other such as the industrial uses for silver and
the use of gold for jewellery as well as central bank demand. Klapwijk (2011) writes that silver
benefits from the attraction of gold as “for some, silver is a more economical alternative to gold”.
The analysis of Tully and Lucey (2006), mentioned already on page 14, also discussed the
gold-silver relationship from 1978 to 2002 and found that while the positive correlation between
gold and silver holds in the long run, the relationship is weak or even broken in certain periods,
in particular during the 1990s when it had been unstable.
Fifth, gold and the price of oil are also said to have a positive relationship and both tend to
increase in price whenever there is a global (political) crisis, when there are tensions between
nations or war breaks out. The correlation of oil and gold prices during the last 40 years was
around 85% (Shafiee and Topal 2010, p. 180/181); Laidi (2008, p. 42) wrote that since 1972 the
gold-oil-relationship has remained generally robust. The strong relationship between oil and
gold is also confirmed by Simakova (2011), who analysed the relationship between oil and gold
prices for the period from 1970 to 2010 and undertook a simple correlation analysis by using
monthly data. The researcher confirmed the strong positive correlation of oil and gold over the
entire period. However, during the financial crisis of 2008 the price of gold rose steeply while the
price of oil fell along with the stock market (p. 656/657).
Le and Chang (2011) use seasonally adjusted monthly averages of oil and gold prices as well
as inflation data from January 1986 to April 2011 to ascertain whether a rise in the oil price
leads to a rise in the gold price. They try to answer this question by testing the following two
hypotheses: A rise in the oil price generates inflation; inflation leads to a rise in the gold price.
They find co-integrating, long-term relationships between the oil price and inflation and also
between inflation and the gold price. A Granger causality analysis supports the suggested
causality of oil and gold prices. They conclude that the oil price can be used to predict the gold
price (pp. 13-19).
18
Sixth, to test how the gold price reacts in a crisis – assuming that the stock markets take a dip in
a crisis – the MSCI World Standard (Large and mid-caps) and the Standard &Poor’s 500 (S&P
500) are included in the analysis, too. The MSCI World is a “common benchmark for global
equity portfolios” that measures the development of equities’ markets of the developed world
(Aon Hewitt 2012); the S&P 500 holds 500 leading US companies and covers roughly 75% of
US equities (Federal Reserve Bank of St. Louis 2013). For the MSCI World as well as for the
S&P 500 analysts and academics found that the correlation is sometimes positive and at other
times negative. The correlation of the MSCI with the gold price fluctuated between 2002 and
2011 between -0.5 and +0.7 (Hindecapital 2012, p. 4). Compared to the previously discussed
predictive variables, these two stock indices seem to be of lesser predictive quality. Other
researchers that evaluated the relationship of the S&P 500 and the gold price found weak
correlations. Duller and Barbee (2012, p. 3) found a positive, weak correlation of 0.12 between
the S&P 500 and the gold price over the years 2007-2012 and an even feebler correlation of 0.06 between 1982-2012. Gault (2012, p. 18), however, used monthly data and looked at the
correlation of the gold price and the S&P 500 and found a correlation of 0.313 from 1990 to the
end of September 2011.
Last but not least, this study also considered the inclusion of the gold supply in the analysis but
it was not possible to gather the required monthly data. Only yearly data is available, which is
why gold supply was not included in the analysis. However, its effects are intuitive because like
in any free market, the price of gold is an equilibrium price set by supply and demand. While
demand is influenced by macroeconomic conditions, politics and special events such as a largescale war (Bapna et al. 2012, p. 1), its supply increases constantly because of production and
the fact it is non-perishable. “Unlike wheat, say, where most of the current supply comes from
this year's crop, gold is storable and most of the supply comes from past production
accumulated over centuries” (Haubrich 1998, p. 1). Shafiee and Topal (2010) claim that in the
long-term, a reduction in gold production was one of three factors that contributed to a rise in
the gold price, the other two being purchases from institutional and retail investors in uncertain
times (“insurance”) and the facilitation of gold purchases through Exchange Traded Funds.
However, Abken (1980, p. 12) claims supply is “relatively insignificant” when it comes to the
price of gold because its annual production is dwarfed by the total amount of gold already on
the market. Given the small academic evidence for a significant correlation of gold production
and price and the fact only a small part of the annual gold on offer is actually newly produced, it
would be no surprise if the gold price was not strongly influenced by variations in its production.
19
2.3 Conclusion
As the gold market — like other markets – does not seem to be efficient as understood by the
EMH, it can be expected that a significant percentage of the variance in the gold price can be
explained by independent variables that correlate with the gold price. The psychology of
investors seems to play an important part – and so does fear. It is likely that herding and groupthink are especially important in this regard.
The following factors were identified as potential contributors to the predictive quality of an
ARIMA model for the gold price – and for which data was available:
•
Inflation Rate
•
Real interest rates
•
The price of silver
•
US dollar money supply (M2)
•
The price of oil
•
The MSCI World Index
•
The S&P 500.
The influence of all of these on the price of gold will be tested but it is highly probable that the
model with the highest predictive quality will include only some of these independent variables
because some - such as the two stock indices MSCI World index and S&P 500 - are highly
correlated. Unfortunately, the VIX is only available from 1990 and tests showed that the data
available does not contribute significantly to the quality of the model. It was therefore omitted.
Psychology seems to be driving the prices to a certain extent and might be the answer to the
question: “What are the reasons for a (potential) difference in the variability the model is able to
explain in “normal” times compared to times of crises?” It might also explain the percentage of
variance that cannot be justified by the identified independent variables. During normal times it
can be expected that investors act more rationally than in times of crises when emotions play a
bigger role and the primary goal might be conservation of wealth rather than profit. This, as can
be expected, drives the price of gold higher than might otherwise have been expected on the
basis of the independent variables.
Next, the ARIMA model will be discussed and why it was chosen. Later follows a description of
the data collected and an explanation of the methodology employed.
20
Chapter 3 – Data and Methods
This chapter will deal with the Auto-Regressive Integrated Moving Average (ARIMA) model and
its appropriateness to answer the research questions. It will also detail the data collected,
sources used and the research methodology employed.
The chapter will detail the extent to which gold prices since the end of Bretton Woods can be
explained by the independent variables inflation, real interest rates, silver prices, US dollar
money supply (M2), oil prices, the MSCI World index and the S&P 500. The gold price is a time
series: a series of data points. Analyses of time series are used to identify patterns over time.
They are also applied to forecast future patterns. A problem of the data set can be an
underlying trend or lingering effects, i.e. auto-correlated variables or error terms. Longer time
series also tend to show tendencies for periodical (and predictable) changes or patterns (named
seasonality). When data sets show qualities like these (violation of independence of errors and
patterns in the data), time series analysis is appropriate rather than standard multiple regression
(Tabachnik and Fidell 2010, chapter 18, pp. 5-6). As a test for the monthly gold price showed
that gold is positively auto-correlated, a time series model was deemed the most appropriate.
The most common model to research time series is the Auto-regressive, Integrated, Moving
Average (ARIMA) model (Sato 2013, p. 128), which was developed by George Box and Gwilym
Jenkins in 1976. ARIMA models can be used to fit time series data as they comprise a family of
models with the ability to deal with auto-correlation, non-stationary data and excessive volatility.
Basic univariate ARIMA models use present data from a time series to make a prediction of
future data within the time series. Sekular et al (2010, p. 194) summarise it like this:
“The ARIMA procedure analyses and forecasts equally spaced univariate time series data
[…] by using the autoregressive integrated moving-average (ARIMA) […]. An ARIMA
model predicts a value in a response time series as a linear combination of its own past
values, past errors (also called shocks or innovations), and current and past values of
other time series.”
A major advantage over univariate ARIMA models is that multivariate ARIMA models are
capable of accounting for the influence of independent time series on dependent time series
(Yanovitzky and van Lear, pp. 101-110). The multivariate ARIMA models (transfer function
models, sometimes called ARIMAX or MARIMA) are a generalisation of the univariate model
and include multiple independent time series that are both auto- and cross-correlated (Öller
1985, p. 143). Bagshaw 1987 (p. 5) compared the univariate ARIMA model, the multivariate
ARIMA model and the vector autoregression model (VAR) for their forecasting abilities and
concluded that multivariate ARIMA performed best.
21
3.1 The ARIMA model
The simplest form of the ARIMA model is similar to a linear regression model and is known as
the Auto-Regressive model (AR). The auto-regressive model is a time series that has no trend.
The Moving Average model (MA) is not a regression in the usual sense. It is a model that uses
past forecast errors, but instead of using all of the past observations it includes moving
averages and white noise. The combined ARMA model is for stationary series and thus
considers past values and shocks. Because the underlying data series show trends and cycles,
these must be removed to make the data stationary. This is done through differencing (Dixon
1992, p. 469-470).
In ARIMA (p, d, q), “p” stands for the number of auto-regressive terms, i.e. in a model with two
auto-regressive terms (p=2), an observation depends on two previous observations. “d”
represents the trend in the data and expresses the number of non-seasonal differences, i.e. the
terms needed to make a non-stationary time series stationary. A model with a d of 2 has to be
differenced twice to make it stationary. And “q” is the number of lagged forecast errors, i.e.
observations of a model with a q of 2 depend on two previous error terms or random shocks
respectively (Tabachnik and Fidell 2010, chapter 18, p. 4).
While the basic univariate ARIMA model is better known, the multivariate ARIMA model has
also been used in a number of studies (De Gooijer and Hyndman 2006, p. 447). Andreoni and
Postorino (2006), for example, investigated air transport demand in the Italian airport of Reggio
Calabria and compared the performances of a basic ARIMA model and a multivariate ARIMA
model using the independent variables income per capita and the yearly number of movements
in the airport (p. 9). In their comparison of the univariate with the multivariate ARIMA model, the
univariate model fared better as it provided the better fit. They acknowledge that both models
provided satisfactory results and while stating that one model would not necessarily have been
better than the other in this case, the univariate model was comparatively limited in its validity.
They viewed the main limitation in the multivariate model as its difficulty in identifying
independent variables wielding significant influence on the dependent variable (p. 13). The
subject matter of this dissertation is very different because five out of seven of the identified
seven independent variables correlate moderately to strongly with the gold price.
Hallquist et al. (1996) used a multivariate ARIMA model in an attempt to forecast WTI oil prices
because petroleum prices depended “not only on their past history but also on one or more
independent data series” (p. 662). In an iterative process they built a model they later used to
test the statistical significance of the independent variables (different types of crude oil and
22
gasoline) and to provide estimates of petroleum prices and publish them within two weeks of the
end of the reference month within one US cent accuracy. Their model to estimate the wholesale
gasoline price was able to provide a satisfactory estimate (p. 666).
Heuts and Bronckers (1988) investigated whether a multivariate ARIMA model was able to
improve the fitting and forecasting performances of a univariate ARIMA model. The research
dealt with the truck market’s performance in the Netherlands. In the multivariate ARIMA model
five independent variables were used, two truck sales’ series and three economic indicators (p.
57). They formed the conclusion that a multivariate ARIMA model provides a better fit to
historical data than a univariate ARIMA model as it substantially reduces residual variance.
However, they also said it remains unclear if a multivariate ARIMA model is also better suited
for forecasting (p. 78). These findings were confirmed by Tsitsika et al. (2007) who tested
univariate and multivariate ARIMA models to fit and forecast the monthly pelagic production of
fish species in the Mediterranean Sea from 1990–2005. They found that while the performance
of univariate and multivariate ARIMA models satisfactorily predicted total pelagic fish
production, the multivariate ARIMA performed better than the univariate ARIMA models in terms
of fitting accuracy.
3.2 Assumptions of an ARIMA model
To reliably fit an ARIMA model, a sufficiently large data set is required (Abdullah 2012, p. 154).
The random shocks/error terms of a good time-series model are considered to be independent,
normally and randomly distributed, to have a constant variance over time and a mean of zero.
(Tabachnik and Fidell 2010, chapter 18, p. 6). An upward trending mean can be corrected by
differencing once or twice; a changing variability may be made stationary by logarithmic
transformation. The number of times you have to difference defines the value of d. A d of zero
means the series is already stationary and shows no trend. Differencing once removes a linear
trend (Tabachnik and Fidell 2010, chapter 18, pp. 8-9).
The model should show no remaining autocorrelations or partial autocorrelations in neither the
Autocorrelation Function (ACF) nor the Partial Autocorrelation Function (PACF). If the model
shows no autocorrelation, the partial autocorrelation test PACF should be close to zero and the
Ljung-Box Q-statistic should be not significant (p > 0.05). Moving average processes have tops
in the first few lags (or just the first) of the ACF and an exponentially declining PACF. The
quantity of tops indicates the order of the moving average (SPSS Inc. 2007, p. 95).
23
Also, there shouldn’t be significant outliers. Existing outliers must be dealt with by deleting the
outlier and replacing it with an imputed value as they can distort the result significantly
(Tabachnik and Fidell 2010, chapter 18, p. 7). Values, p and q, are usually quite small. ACF is a
collection of correlation coefficients of the series and lags of the series over time. PACF,
however, is “the partial correlation coefficients between the series and lags of itself over time”
(Root 2011, p. 2).
Last but not least, the residuals must not correlate, follow a white noise process and show
random fluctuations (Alonso and Garcia-Martos 2012):
• The error series can be checked by computing the ACF and the PACF, which must not
be significantly different from zero. One or two high-order correlations, however, may
exceed the 95% confidence level (by chance). A large first- or second-order correlation
is a reliable indicator the model is misspecified (SPSS Inc. 1999, p. 59).
• There should be white noise, i.e. the residuals should show no pattern. This can be
tested by using the Box-Ljung Q-test. The statistic should not be significant (SPSS Inc.
1999, p. 59).
3.3 Data collection and sources
The study required the collection of monthly data for the period from March 1973 until end of
2011 for all the independent variables and the dependent variable. The duration of the research
period is 38 years and 10 months, i.e. 466 months/observations per variable. The identified
independent variables are inflation, the real interest rate (nominal interest rate minus inflation),
silver prices, US dollar money supply (money stock M2), WTI oil prices, the MSCI World Index
and the S&P 500.
The data was collected from various sources (see Table 1). The data pool of the Federal
Reserve Bank of St. Louis (US) proved particularly valuable because it provided access to a
wide range of data required for the analysis. The following table shows dependent and
independent variables and the data sources.
24
Variable
Data Source
URL
Monthly gold price in USD (fixed at
last labour day of each month)
USA Gold
http://www.usagold.com/reference/pri
ces/history.html
US Inflation rate
Coin News Media Group
http://www.usinflationcalculator.com/i
nflation/historical-inflation-rates/
Real interest rate (= Nominal
Interest Rate - Inflation (Expected
or Actual))
Federal Reserve
Nominal interest rate (10 year
Treasury-bond yield):
http://www.federalreserve.gov/releas
es/h15/data.htm
Inflation rate:
http://www.usinflationcalculator.com/i
nflation/historical-inflation-rates/
Coin News Media Group
Silver price
The London Bullion Market
Association
http://www.lbma.org.uk/pages/index.c
fm?page_id=54&title=silver_fixings
and
Goldmasters USA
http://goldmastersusa.com/silver_hist
orical_prices.asp
Money supply (M2, money stock in
billions of dollars, balance on first
day of each month)
Federal Reserve Bank of
St.Louis
http://research.stlouisfed.org/fred2/da
ta/M2NS.txt
WTI crude oil price (per barrel) in
USD, monthly average
Signal Trend Inc.
http://www.forecast-chart.com/chartcrude-oil.html
MSCI World Standard (Large and
mid caps)
MSCI
http://www.msci.com/products/indice
s/performance.html
S&P 500 Index
Federal Reserve Bank of
St.Louis
http://research.stlouisfed.org/fred2/se
ries/SP500/downloaddata
Table 1: Data sources
3.4 Defining an ARIMA model to fit the gold price
To address the efficacy of the multivariate ARIMA model using independent variables to gauge
the gold price over time, the study will answer the following questions already mentioned in
Chapter 1.4:
A. How effective is the model including these independent variables in explaining gold price
variations in times of so-called ‘normality’ and in times of crises?
B. What is the explanation for the differences in variability described by the model during times
of so-called ‘normality’ and times of crises?
25
Until a few years ago developing an ARIMA model (uni- or multivariate) meant going through an
extensive process of trial and error until a satisfactory model could be identified. Today, SPSS’
Expert Modeler command automatically finds the significant independent variables and
identifies the best fitting ARIMA model. However, the researcher has to choose the appropriate
criteria and must also be able to interpret the output (“fit measures”). Since SPSS developed
and made the Expert Modeler available, it has become widely used by researchers engaged in
time series analyses that try to fit data and/or forecast asset prices or phenomena such as
malaria transmission or road accidents. Loha and Lindtjorn (2010) used the Expert Modeler to
investigate data from southern Ethiopia in an attempt to predict a certain type of malaria for the
period between 1998- 2007. In their multivariate ARIMA analysis they worked with
meteorological variables such as monthly rainfall, temperature and relative humidity. They found
that past data on the illness was a better predictor of its future dispersion than meteorology.
Jakasa et al. (2013) analysed daily data of German electricity prices from 2000 to 2011 to build
a univariate ARIMA forecasting model. They used two thirds of the observations to build the
model and the remaining third to test it. They found the Expert Modeler suited their needs as it
allowed them to model the electricity price “adequately”. Their model showed a mean absolute
percentage error (MAPE) of 3.55%.
Many other researchers have also worked with SPSS’ Expert Modeler when trying to build an
ARIMA model – with seemingly satisfactory results. For example, Hota and Sahu (2012)
compared whether a univariate ARIMA model or exponential smoothing fared better in
predicting the share value of the State Bank of India (SBI). They worked with data from January
2003 to May 2011 and concluded that the Expert Modeler performed better in predicting the
share price when benchmarking for the MAPE and the stationary R-Squared. Sarani et al.
(2012) used a univariate ARIMA model (and the Expert Modeler) in their attempt to predict
Malaysian road fatalities for 2020. They found ARIMA performed well and was able to explain
almost 98% or the variation in the data. It also performed better than the other two models
considered, the Poisson model and Negative Binomial model. These results provide a solid
basis on which to select ARIMA as a model as well as for the suitability of the SPSS Expert
Modeler.
3.5 Evaluation of the ARIMA model
The goal of this study is to find an ARIMA model capable of fitting the gold price. As defined in
the research question, the “best” model chosen shall be the one with the best combination of R26
squared and the mean absolute percentage error (MAPE) and BIC, which are among the most
common indicators for the fit of the ARIMA model (Schendera 2008, p.401). R-squared, the
coefficient of determination, is the proportion of the explained variation. BIC scores the addition
of parameters and penalises extraneous parameters; the lower the BIC value, the better
(Chatfield 1996, 499). SPSS’ Expert Modeler is used to find the “best-fit” ARIMA model; the
programme is able to detect outliers automatically, which is why they don’t pose a problem. To
analyse the data and answer the research questions, the following steps were followed.
Question A: How effective is the model including these independent variables in explaining gold
price variations in times of so-called ‘normality’ and in times of crises?
1. For the whole sample, descriptive statistics will be generated and presented for each
variable, dependent and independent. Correlation coefficients between independent
variables as well as between independent and the dependent variable will be calculated.
2. The model suggested by SPSS Expert Modeler will be checked for stationarity of the data.
In order to do so, autocorrelation and partial autocorrelation functions of the data (ACF and
PACF) will be used. Residuals should show no patterns. This can be controlled for by using
the Box-Ljung Q-test.
3. The model will be evaluated for its goodness of fit, based on the criteria R-squared, mean
absolute percentage error (MAPE) and BIC. The individual independent variables
contributing significantly to the fit of the model will also be discussed.
4. For the samples times of crises (January 1978 to January 1981 and August 2007 to
December 2011) and “normal” times (all other data points) ARIMA Expert Modeler will be
used to find the best fitting models and independent variables for each period.
5. The goodness of fit of the models in each period will be compared as well as the factors
contributing to the quality of the model.
Question B: What is the explanation for the differences in variability described by the model
during times of so-called ‘normality’ and times of crises?
Based on the analysis that was undertaken to answer question A and in particular its results
(steps 4 and 5), the study will endeavour to explain the differences in the goodness of fit of the
model. As it was assumed that markets are not efficient and investors influenced by
psychological biases, behavioural explanations such as herding (group think), excessive
extrapolation and/or the safe value bias might play an important role in helping to explain
27
potential differences. If the model explains a much higher percentage of the variability during
“normal” times than during crises, this might be an indication for irrational investor behaviour.
3.6 Conclusion
This chapter provided an explanation of how the research questions outlined in the introductory
chapter would be addressed. It provides answers to the questions of why and how a particular
model was chosen. While the basic ARIMA model is more common than the multivariate ARIMA
model to fit time series, several independent time series are strongly linked to the gold price.
This is why it is expected that a multivariate ARIMA model is able to explain a higher
percentage of gold price movements than a basic, univariate ARIMA model.
The chapter also outlined a five-step manual to explain the disparity in performance of the
model in “normal” times to the model in times of crises. Investor psychology may explain the
potentially distinct model performance between the different periods. Phenomena in the field of
behavioural science were discussed in Chapter 2 and may provide the key to making sense of
the differences in the model fit in different times.
28
Chapter 4 – Analysis and Results
This chapter will present the results of the analysis. Descriptive statistics and scatter plots will
give an overview of the data before the model is checked for stationarity and evaluated for its
goodness of fit. Then, the performance of the model during different periods (“normal” times and
crises) is discussed and the potential reasons for different model performance during different
times.
4.1 Data description
As the graph on Page 9 shows, the average monthly gold price was trending upwards from
1973 until 2011 with temporary, albeit small setbacks. The price ranged from 89.25 to 1,813.5
US dollars an ounce. Looking at the independent variables, silver fluctuated between 2.18 and
42.8 US dollars an ounce and was more volatile than gold (higher standard deviation). Inflation
ranged from -2.1% to 14.8% and the real interest rate from -4.87% to +9.36%. The M2 US dollar
money supply was expanded year after year and in many cases month after month. It seldom
showed any monthly falls. The M2 money supply ranged from 816 billion US dollars in March
1973 to 9,691 billion US dollars in December 2011. Oil, which is especially prone to political
crises and wars, cost on average 3.6 US dollars a barrel in early 1973 and reached its peak in
June 2008 during the latest financial crisis. From then on and until the end of 2011 it declined
steeply before rising in price again; the average price in December 2011 was 98.5 US dollars a
barrel. The minimum and maximum values of the MSCI World Standard index and the S&P 500
index as well as standard deviations were very close (74.45 vs. 72.56, 1682.35 vs. 1539.66 and
457.75 vs. 482.6).
Although the fear indicator VIX (it measures implied volatility) was not included in the ARIMA
model due to lack of sufficient data points (VIX is only available since 1990), it is still worth
mentioning that from 1990 to 2011 it ranged between 10.42 in the very quiet times when
investors felt very safe to almost 60 in October 2008, when investors expected the financial
system to collapse. VIX showed relatively weak but significant correlation of 0.223 with the gold
price during the period for which it was available (from 1990 until the end of 2011). During the
latest financial crisis - since August 2007 - correlation between the VIX and the gold price was
not significant because of the relatively small number of observations (53).
29
N
End of the month gold
price in USD an ounce
Silver price an ounce in
cents
Inflation rate at time 0
M2 Money Stock in
billions of USD
WTI oil prices in USD a
barrel of oil
World Standard (Large
and mid cap) in USD
S&P 500 Stock Price
Index
Real interest rate
Minimum
Maximum
Mean
Std. Deviation
466
89.25
1813.50
436.2299
291.461
466
217.96
4279.79
818.5881
645.760
466
-2.1
14.8
4.458
3.10
466
815.60
9691.20
3856.5867
2330.084
466
3.60
133.90
31.8487
23.735
466
74.45
1682.35
640.5781
457.752
466
72.56
1539.66
604.1542
482.615
466
-4.87
9.36
2.6840
2.708
Valid N (listwise)
466
Table 2: Descriptive statistics of dependent and independent variables: minimum, maximum, mean and standard
deviation.
The silver price was the independent variable that showed the highest correlation with the gold
price (see Table 3). The Pearson correlation coefficient of gold and silver prices was 0.876. The
correlation between the oil price and the gold price (Pearson correlation of 0.847) was almost as
high. The gold price also correlated strongly with the M2 money stock (Pearson correlation of
0.757). The MSCI showed a significant but moderate correlation with the gold price (Pearson
correlation of 0.508); the S&P 500 and the gold price correlated slightly weaker with a Pearson
correlation of 0.465. The correlations of the gold price with the remaining independent variables
– the inflation rate and real interest rate – were rather small and negative with Pearson
correlations of -0.256 (inflation rate at time 0) and -0.124 (real interest rate).
Some of the correlations between the independent variables were also very high.
Unsurprisingly, the highest correlation exists between the MSCI World Standard Index and the
S&P500 Index (Pearson correlation coefficient of 0.985). This is why it was expected that only
one – if any – of these two factors would contribute significantly to the best ARIMA model fitting
the gold price because of the problem of multicollinearity (Martz 2013). Other independent
variables also correlated strongly. The M2 money stock in particular showed strong correlations
with several of the independent variables, such as the inflation rate (-0.626), the oil price (0.789)
and the two stock indices (0.898 and 0.876 respectively). The oil price showed very high
correlations with the M2 money stock (0.789), the silver price (0.752), the MSCI and the S&P
500 (both over 0.6). Besides correlating very strongly with each other, they both showed very
30
high correlations with the M2 money stock. This is a contemporary phenomenon as the Federal
Reserve expands M2 and stock markets keep rising – for now.
31
End of
the
month
gold
price in
USD an
ounce
Inflation
rate at
time 0
Real
interest
rate
Pearson
Correlation
Sig. (2tailed)
N
Pearson
Correlation
Sig. (2tailed)
N
Pearson
Correlation
Sig. (2tailed)
N
Pearson
Correlation
Silver
price an
ounce in
cents
Sig. (2tailed)
N
M2
Pearson
Money
Correlation
Stock in
Billions
Sig. (2of USD
tailed)
N
WTI oil
Pearson
price in
Correlation
USD a
barrel of Sig. (2WTI oil
tailed)
N
World
Pearson
Standard Correlation
(Large
and mid Sig. (2cap) in
tailed)
USD
N
S&P 500 Pearson
Stock
Correlation
Price
Index
Sig. (2tailed)
N
End of
the
month
gold
price in Inflation
rate at
USD an
ounce
time 0
**
1
-.256
Real
interest
rate
**
-.124
.000
.007
.000
.000
.000
.000
.000
466
466
466
466
466
466
466
1
**
.070
**
**
**
.000
.131
.000
.000
.000
.000
466
1
466
**
-.255
466
-.084
466
**
-.204
466
-.087
466
**
-.120
.000
.069
.000
.059
.010
466
1
466
**
.503
466
**
.752
466
**
.296
466
**
.283
.000
.000
.000
.000
466
1
466
**
.789
466
**
.892
466
**
.876
.000
.000
.000
466
1
466
**
.631
466
**
.608
.000
.000
466
1
466
**
.985
466
**
-.256
M2
Silver
Money
price an Stock in
ounce in Billions
cents
of USD
**
**
.876
.757
-.544
.000
-.626
MSCI
World
Standard
Oil price
(Large
in USD a and mid
barrel of cap) in
WTI oil
USD
**
**
.847
.508
-.199
-.615
466
**
-.124
466
**
-.544
.007
.000
466
**
.876
466
.070
466
**
-.255
.000
.131
.000
466
**
.757
466
**
-.626
466
-.084
466
**
.503
.000
.000
.069
.000
466
**
.847
466
**
-.199
466
**
-.204
466
**
.752
466
**
.789
.000
.000
.000
.000
.000
466
**
.508
466
**
-.615
466
-.087
466
**
.296
466
**
.892
466
**
.631
.000
.000
.059
.000
.000
.000
466
**
.465
466
**
-.588
466
**
-.120
466
**
.283
466
**
.876
466
**
.608
466
**
.985
.000
.000
.010
.000
.000
.000
0.000
466
466
466
466
466
466
466
**. Correlation is significant at the 0.01 level (2-tailed).
Table 3: Correlation between the gold price and the independent variables
32
S&P
500
Stock
Price
Index
**
.465
**
-.588
0.000
466
1
466
4.2 The best fitting ARIMA model
The fit measures chosen for SPSS’ Expert Modeler were stationary R-squared, mean absolute
percentage error (MAPE) and normalised BIC. SPSS suggested the best fitting ARIMA model
(see table 4) to be ARIMA (0,1,0)(0,0,0), which is an ARIMA model with one order of
differencing and no seasonal differencing. Figure 2 shows the stationary gold price after the
removal of the trend by differencing once. It is obvious that the volatility was particularly high
during the identified crises, from 1978 to 1981 and from 2007 to 2011.
Model Type
Model ID
End of the month gold
price in USD per ounce
Table 4: ARIMA model description
Model_1
ARIMA(0,1,0)(0,0,0)
Figure 2: Gold time-series made stationary by differencing once
The independent variables: Inflation rate, real interest rate, silver price, oil price and MSCI
World Standard Index were found to contribute significantly to the predictive power of the
model. The independent variables included in the model were these that could be expected to
33
have a high correlation with the gold price – with the exception of M2 money stock that highly
correlated with all of the independent variables of the model except the real interest rate and
was therefore not contributing significantly because of the problem of multicollinearity.
As is desirable for a model that shows no autocorrelation, Figure 3 shows partial autocorrelation
(PACF) was very close to zero with no significant outliers. The PACF only crosses the 95%
confidence interval once – which can be attributed to chance. The graph of the autocorrelation
function (ACF) looks equally satisfactory. It is safe to conclude that the model complies with the
assumptions of an ARIMA model regarding autocorrelations or partial autocorrelations, the lack
of significant autocorrelation respectively.
Figure 3: Partial autocorrelation function (PACF) and autocorrelation function (ACF) of the model
As Table 5 shows, the model was found to be a very good estimator of the price of gold, giving
a stationary R-squared of 0.602 compared to a stationary r-squared of only 0.334 for an ARIMA
model without the benefit of contributions from the independent time series. The mean average
percentage error (MAPE) was 2.861% (MAPE of ARIMA model only using the gold price:
3.712%). Normalised BIC, the parameter that penalises an increased number of parameters in
34
the model, was 6.340 compared to 6.892 for the model using only the gold price. The
insignificance of Ljung-Box statistics (p>0.05) is consistent with the lack of autocorrelation,
which was to be avoided. This confirms the residuals show no pattern.
Model
Number of
Predictors
Model Fit statistics
Stationary
MAPE
Ljung-Box Q(18)
Normalize
R-squared
Statistics
DF
Number
Sig.
of
Outliers
d BIC
End of the
month gold
price in USD
5
.602
2.861
6.340
22.008
18
.232
9
an ounceModel_1
Table 5: ARIMA model statistics
The independent variables could contribute to the estimation of the gold price through a variety
of transformations that made an analytical closed form for the overall time series difficult to
express. The table below describes the contributions of the independent variables: inflation rate,
real interest rate, silver price, oil price and MSCI World Standard index to the ARIMA model,
along with the significance of the contributions, in terms of the numerator and denominator
effects. The M2 money stock and the S&P 500 did not contribute to the quality of the model. As
the S&P 500 has an almost perfect correlation with the MSCI World Standard Index it does not
come as a surprise that only one of these independent variables contributed significantly to the
quality of the model because of their collinearity.
35
End of the month gold
price in USD an ounce
Natural Log
Difference
Inflation rate at time 0
No
Transformation
Delay
Numerator
Estimate
1
No
Transformation
Silver price an ounce in Natural Log
cents
Oil price in USD a
barrel of WTI oil
Natural Log
World Standard (Large
and mid cap) in USD
Natural Log
t
Sig.
2
Lag 0
Difference
Real interest rate
SE
-.021
.007 -3.207
.001
.126 -5.192
.004 -5.658
.000
.000
1
Denominator Lag 1
Numerator
Lag 0
-.656
-.021
Lag 2
.021
.005
3.993
.000
Lag 0
1
.389
.023 17.185
.000
Difference
Numerator
Difference
1
Denominator Lag 1
-.264
.056 -4.698
.000
Lag 2
.127
10
.059
2.161
.031
Lag 0
.080
.021
3.878
.000
.016 -2.401
.017
.062
.000
Delay
Numerator
Difference
Delay
Numerator
1
1
Lag 0
Difference
-.038
1
Denominator Lag 1
-1.588
Lag 2
-.875
25.752
.067
13.135
.000
Table 6: ARIMA model parameters contributing to the fit of the monthly price of gold
Figure 4 below illustrates the fit of the ARIMA model to the actual price of gold over the time
span under consideration – almost four decades. The model relied on data lagged by no more
than two time steps. As the mean absolute percentage error (MAPE) was very small (2.861%),
the model fits the data almost perfectly.
Figure 4: Observed gold price and gold price model fit (MAPE for the model: 2.861%)
36
4.3 ARIMA model fit during normal times and crises
The times of crisis during the period under consideration were identified as January 1978 to
January 1981 and August 2007 until the end of the sample (December 2011). Non-crises, i.e.
“normal” times, were identified as lasting from March 1973 to December 1977 and from
February 1981 to July 2007. The three main findings were:
(1) An ARIMA model using the gold price and the independent variables explained a much
higher percentage of the variability during normal times than during times of crisis (see
tables 8 and 9 in the appendix: R-squared: 0.651 vs. 0.493; MAPE: 2.836 vs. 4.943).
Normalised BIC also performed better for the ARIMA model for the data during normal
times (Normalized BIC: 5.206 vs. 8.288) than during crises.
(2) The model performed better during normal times than it did for the entire time frame
regarding R-squared, while normalised BIC was comparable (R-squared: 0.651 vs.
0.602; MAPE: 2.836 vs. 2.861; normalized BIC: 5.206 vs. 6.340).
(3) Some of the independent variables lost their value for the fit of the gold price during
crises: For the months defined as “normal times” the inflation rate, real interest rate, silver
price an ounce, M2 money stock and the oil price had a significant fitting value for the
gold price, while for months during crises only the silver price and M2 money stock had
significant influence. The inflation rate, real interest rate and the oil price lost their
significance for the model during crises (see tables 10 and 11 in the appendix). Over the
entire period, the variables inflation rate, real interest rate, silver prices, oil prices and the
MSCI World Standard Index contributed to the quality of an ARIMA model fitting the gold
price. S&P 500 was not considered in any model due lack of significance. Table 7 below
summarises these findings.
Complete period
Normal times
Inflation Rate
X
X
Real Interest Rate
X
X
Silver Price
X
X
Oil Price
X
X
MSCI World Standard
X
X
M2 Money Stock
Crises
X
X
S&P 500
Table 7: Independent variables contributing to the quality of an ARIMA model during the complete period (March
1973 to December 2011), during normal times and during crises (January 1978 to January 1981)
In line with the assumption of Dieupart et al. (2013) that there are three groups of investors –
rational informed agents, irrational informed agents and non-informed agents – it can be
expected that during crises the irrational informed agents and the non-informed agents drive the
37
price of gold as their actions are influenced by emotions (such as fear) and cognitive biases as
well as from observing the market and/or the gold chart. Non-informed agents are most likely to
buy when others buy, too, which contributes to irrational exuberance.
4.4 Explaining divergences of the model fit during normal times and crises
As we have seen during normal times, almost two thirds of the variability of the gold price can
be explained by an ARIMA model of the gold price and five (out of seven) independent
variables. When testing the same seven independent variables for the combined periods of
crises (1978 to 1981 and 2007 to 2011) only two independent variables significantly contribute
to the fit of the model (the silver price and M2 Money Stock) and the model “only” explains
roughly 50% of the variability. Also, volatility of the gold price was much higher during crises
which indicates investors’ nervous trading activities.
The performance difference of 16 absolute percentage points is in line with the theory presented
in Chapter 2.1 that the gold market is not efficient as understood by the efficient market
hypothesis. In an efficient (gold) market according to the efficient market hypothesis, all market
participants are perfectly informed, have rational expectations and possess the same level of
information. Independent variables explained two thirds of the variability of the gold price during
normal times but only 50 percent during crises: This means that investors treat the same
information differently during specific periods of time, i.e. during crises. This phenomenon can
be seen as an indicator of the irrationality of investors because psychological factors – such as
fear – become more important while others become less so. It indicates Fama’s EHM is not a
realistic picture of how the (gold) market works. This supports Solt and Swanson’s (1981)
findings and Kindleberger and Aliber’s (2005) claim that markets are not (always) rational and
that irrational exuberance occurs fairly often.
However, if rationality is understood by the Merriam-Webster dictionary’s definition as “a rational
opinion, belief or practice” an investor’s decision to invest in gold because he/she feels relatively
more urgency to preserve his/her wealth and gives more weight to personal feelings (“fear”)
during a crisis then their behaviour might be considered perfectly rational. Investors seek gold
because they fear other assets might lose their value (such as during the crisis of 2007) or
because they speculate the price of gold might rise (for example during political crises/wars).
The independent variables that contributed to the fit of the gold price during normal times
suddenly lost their value for an ARIMA model during crises. This is an indicator that during
crises, investors buy gold because they fear holding cash and they shift money from stocks into
38
the gold market. Psychological factors such as fear become more important in an attempt to
explain this phenomenon.
When we look at the gold price chart during the almost four decades under observation, we see
that during the first crisis of that period – 1978 to 1981 – gold rose to a temporary peak of over
850 US dollars an ounce in 1980 (monthly average peak of over 660 US dollars an ounce in
September 1980), which was its highest value until the next crisis began in 2007. In this case,
again we saw gold prices rising - this time to levels that were never seen before (in absolute
terms and not adjusted for inflation). While in both periods of crises the safe value bias is likely
to have played a role, by looking at the gold chart it is probably the case that herding (group
think) and especially excessive extrapolation played key roles, particularly during the period
between 2007-2011 as was claimed by Utkus (2011), who identified this as one of the drivers of
the continuous, inexorable price rises since 2000. During that time, gold also moved from an
elitist to a mainstream investment vehicle (Barkhordar 2009, p. 2). During the 1978 to 1981
crisis it was different as the monthly average gold price didn’t rise for years, but rose and fell
within a rather short time period compared to the 2007 crisis .
It is also worth noting in comparing the two crises of 1978 to 1981 and 2007 to 2011 that the
first of these was mainly a political crisis, characterised by the Iranian Revolution and the
subsequent oil crisis (as described in Chapter 1.1.). Whereas the crisis of 2007 was a global
financial one in which central banks around the world intervened. Firstly, these provided the
markets – and financial institutions in particular – with cheap liquidity. This led investors to
expect higher inflation and all that normally goes with it. Consequently, they searched for a
traditional safe haven: gold.
4.5 Conclusion
This chapter discussed the results of the analysis. It was found that an ARIMA model fits the
gold price better providing it includes independent variables. Additionally, it was found that an
ARIMA model encompassing the gold price and various independent variables performed
differently during periods defined as “normal” and “crises” – and significantly better during
normal times than during crises. This confirms the theory that independent variables strongly
related to gold contribute to an ARIMA model attempting to fit the gold price. The analysis
shows a distinct performance of the model during crises when the investing landscape is
insecure – as opposed to “normal” times. During crises only silver and the money supply (M2
money stock) significantly contributed to the fit of the model. This can be seen as an indication
of the crucial influence played by psychology as fearful investors bought gold during crises and
39
independent variables lost their significance to fit the gold price. The data and the chart, in
particular, indicate further that during the most recent crisis herding and excessive extrapolation
have been key drivers.
The following discussion chapter will contain a summary of the results of the study, as well as
thoughts on its limitations and suggestions for those researchers who wish to research this topic
further.
40
Chapter 5 – Discussion and Conclusions
In this concluding chapter, the results of the study will be summarised and discussed in an
attempt to explain the results. Furthermore, the limitations of the study will be considered as
well as the direction for future research.
5.1 Summary
Since 1973 gold has been allowed to fluctuate freely. It is strongly related to a number of
independent variables. Inflation, the real interest rate, the silver price, the US dollar money
supply (M2), the oil price, the MSCI World index and the S&P 500 have been identified (Table 3,
page 32). A multivariate ARIMA (auto-regressive, integrated, moving average) model was used
to test variability and to ascertain how the model performed during normal times and crises. An
ARIMA model is capable of correcting for autocorrelations, non-stationary data and excessive
volatility and is the most frequently used model to investigate time series data. The periods from
1978 to the end of January 1981 (during the oil crisis and the Iranian revolution) and from the
start of the latest financial crisis of 2007 to the end of 2011 (which was not the end of the crisis
but the end of the period under observation) were Identified as crises with major impacts on oil
and gold prices as well as on stock markets.
The study investigated the extent to which a multivariate ARIMA model and the independent
variables identified (evaluation criteria: R-squared, mean absolute percentage error (MAPE) and
BIC) could explain gold price movements since the end of Bretton Woods. The study answered
the following two questions:
• How effective is the model including these independent variables in explaining gold price
variations in times of so-called ‘normality’ and in times of crises?
• What is the explanation for the differences in variability described by the model during
times of so-called ‘normality’ and times of crises?
As so far no comprehensive theory, neither for gold valuation nor for precious metals prices,
exists, market efficiency of precious metals markets and theories on gold price movements
during crises and investor psychology were discussed to provide a theoretical framework for this
dissertation. Most researchers now agree markets are not rational and that Eugene Fama’s
Efficient Market Hypothesis (EMH) does not reflect reality. The widely acknowledged view is
that markets are at least sometimes irrational and that anomalies and irrational investors are at
the root of mania and panics. EMH fails to recognise that psychology plays an important part in
investment decisions. Herding (group think) is important when seeking to explain gold market
41
anomalies. Also, positive price performance cannot always be explained by a change in
fundamentals. The longer the price trend continues the more investors hop on the train, which
further drives the price up.
A third crucial bias is the safe haven allure of gold. This at least partially explains investors’
actions during crises when demand for gold is particularly strong. Financial instability and
decisions of central banks as well as political tensions, wars or distrust in the policies and
prospects of nations can all trigger demand for gold as investors look for ways to protect their
assets. Such events engender insecurity and investors purchase gold as a defensive asset and
as a substitute investment.
The performances of the ARIMA model during normal times and during crises were evaluated
by using SPSS’ Expert modeler, which is capable of finding the significantly contributing
independent variables and identifying the best fitting ARIMA model based on a user’s criteria.
The suggested ARIMA model was tested for stationarity of the data. In order to do so,
autocorrelation and partial autocorrelation functions of the data (ACF and PACF) were used. To
test for undesired patterns in the residuals a Box-Ljung Q-test was undertaken. The model was
further evaluated for its goodness of fit (criteria: R-squared, mean absolute percentage error
(MAPE) and BIC). Later, the whole sample was split into two based on criteria guiding “normal”
times and crises and the same ARIMA model was used to evaluate the performance of the
model for each time frame. Based on the results of the empirical evidence garnered, the
differences in the goodness of fit of the model were discussed, with a focus on behavioural
science approaches such as herding (group think), excessive extrapolation and the safe value
bias – all of which provide logical explanations for the phenomena observed.
The empirical analysis threw up some remarkable results. For the period from March 1973 to
the end of 2011, the overall performance of a multivariate ARIMA model using independent
variables in addition to the gold price was significantly superior to a univariate ARIMA model
using only the gold price. The ARIMA model using independent variables explained almost
twice as much of the variability of the gold price (r-squared of 0.602 vs. r-squared of 0.334), the
mean absolute percentage error (MAPE) was lower (2.861% vs. 3.712%) and the normalised
BIC (6.340 vs. 6.892) meant a better fit of the model - including independent variables.
The analysis also showed that the model explained a higher percentage of the variability (rsquared of 0.651) during normal times than for the whole period (normal times and crises) and
especially compared to during times of crises alone (R-squared of 0.493, MAPE: 4.943,
42
normalised BIC: 8.288). Remarkably, during crises the number of independent variables
significantly contributing to the fit of the model decreased from five over the complete period
and during normal times to only two during crises. For the complete time period, the
independent variables inflation rate, real interest rate, silver price, oil price and the MSCI World
Standard Index contributed significantly to the fitting quality of the model, while M2 Money Stock
and S&P 500 did not. During normal times the MSCI lost its significance for the model but the
M2 Money Stock was included. During the crises’ periods only the silver price and M2 Money
Stock contributed significantly to the fit of the model.
During crises such as significant wars, oil crises or collapses in a monetary system, demand for
gold is usually high because investors tend to flee to assets that are considered to conserve
value. The ability of the model to explain a higher percentage of the variability during normal
times is in line with behavioural science’s view that during crises markets (and investors) are
less rational. Phenomena such as “safe value” bias, herding (group-think) and excessive
extrapolation all play an important role in explaining the weaker performance of the ARIMA
model during crises. During the financial crisis since 2007 in particular, the chart pattern
suggests herding and investors’ extrapolation of the gold price into the future also played a role,
as well as fear of future inflation as the central banks around the world loosened their monetary
policies. Unlike the findings of Andreoni and Postorino (2006), in this study several of the
independent variables used in the multivariate ARIMA model are strongly correlated with the
gold price, with the result that a multivariate ARIMA model fared far better than a univariate
ARIMA model confined solely to the price of gold.
An empirical analysis of why the model performed differently during crises and normal times
was outwith the scope of this dissertation.
5.2 Implications
Several studies have employed multivariate ARIMA or multivariate ARMA models in various
research projects, among them Tsitsika et al. (2007), Andreoni and Postorino (2006), Hallquist
et al. (1996) and Heuts and Bronckers (1988). The comprehensive application of such
modelling for mapping gold price movements does not yet exist. However, the results of this
study show a multivariate ARIMA model that also uses independent variables far outperformed
the univariate ARIMA model fitting the gold price during the observed time frame. It also
indicates that a multivariate ARIMA model using high correlative independent variables can
explain almost two thirds of the variance of the gold price. The fit was almost perfect with a
mean absolute percentage error of below three percent. This recommends a multivariate
43
ARIMA model for evaluating different methods to fit a time series, especially where a number of
independent variables can be identified as being highly correlated with the dependent time
series.
The results also provide further evidence of the shortcomings of the ‘efficient market’
hypothesis. During times of crises compared with “normal” times, the ARIMA model was able to
explain significantly less variability. Also, during crises the independent variables – the inflation
rate, real interest rate and oil prices – lost their significance for the model. During the two crises
from 1978 to 1981 and especially since 2007 behavioural explanations of the gold price
movements gained importance. Herding, excessive extrapolation and safe haven purchases
provide the most compelling explanations for the differences in the ARIMA model’s performance
during normal times and crises. For investors this means they should be especially careful when
the price of an asset keeps rising without fundamentals changing in line.
5.3 Limitations
This dissertation has several limitations. The main shortcomings are:
(1) Some of the independent variables came from different sources;
(2) there was no data available for other independent variables with a close relationship
with gold price movements;
(3) the psychological factors (behavioural explanations of the gold price’s movements)
were not researched empirically.
The data for the gold price and for the independent variables was collected from different
sources and it is highly probable the various sources used somewhat different methodologies
when collecting it. Unfortunately, this limitation was unavoidable as there is no single source
available to provide all the data for the dependent and independent variables. The data was
mined from nine separate sources. Two sources were used to determine real interest rates and
silver prices. The real interest rate had to be calculated (therefore data for the nominal interest
rate and for inflation was needed) and the silver price data had to be collected from two different
sources as no source provided the necessary silver price data for the complete time frame
(1973 to 2011). As the independent variables were selected for their strong relationship with
gold, it can be expected that if this limitation had any influence on the study, it lowered the
correlation between the gold price and the independent variables. ‘Better’ data might have
increased the efficacy of the ARIMA model. Future research might be able to overcome this
limitation if more and better data becomes available from fewer sources.
44
Several factors were identified to having a potentially high influence on the gold price and could
have improved the quality of the ARIMA model, such as political risk, official sector activity and
central bank gold reserve sales. However, such variables are practically impossible to quantify
because even if the data does exist, it and/or its timing is likely to be politically sensitive and so
kept secret. The volatility index, which is known as the “fear gauge” (Prousis 2013), is known to
be highest when investors fear for their assets. The study considered its inclusion as an
indicator to perceived political risk but VIX is only available from 1990 and when the available
data was included in the analysis it was found to be insignificant when fitting the gold price. In
an attempt to improve the fitting properties of an ARIMA model for the gold price, researchers
may be able to identify alternative independent variables capable of representing investors’
actions/emotions when confronted with (expected) political risk. Also, in the future the data for
central bank gold in-/divestments might become available on a monthly basis at least.
In this dissertation empirical data was used to build an ARIMA model for the gold price and it
was evaluated for its fitting qualities, most notably the percentage of explained variance. One
part of the remaining variance can be explained by psychology but there is every likelihood that
some of it cannot be explained at all. The study discussed which psychological and behavioural
biases provide the most coherent explanations for gold price movements that remain
unexplained by the ARIMA model. However, these were not researched empirically. Future
research might consider investigating empirically the magnitude of psychological factors on
investors’ actions – especially during crises. Although the hypothesis that gold is considered the
perennial safe haven can be investigated empirically by controlling for gold’s correlation during
a market crash (Baur 2010, p. 217) to know exactly why any individual or group of investors
acts in a certain way at any point in time requires asking them. This would entail the compilation
of a survey with a large enough sample to being able to draw meaningful conclusions. But even
then, some intangibles may never be accounted for as it is also not sure whether investors
would be honest and/or transparent about their motives.
5.4 Direction for Future Research
This dissertation has shown that a multivariate ARIMA model can serve well when attempting to
fit a time series in cases where independent variables are strongly correlated with the
dependent time series. A better theoretical framework that provides a foundation for how
precious metal prices are formed might help in interpreting the study results. Also, a better
database for the gold price and for the independent variables would improve the results further,
e.g. weekly data from a single data source. It would also be interesting to benchmark the results
of an ARIMA model of the gold price with an alternative approach that is also able to fit time45
series data. Another future study might focus on the percentage of the gold price an ARIMA
model cannot explain, i.e. the behavioural science aspects affecting gold price movements.
5.5 Reflections
The dissertation sought to ascertain how well gold price movements since the end of Bretton
Woods were explicable by a multivariate ARIMA model that uses independent variables highly
correlative with the gold price. It also investigated if such an ARIMA model fared better (or
worse) during crises than during periods considered “normal”. The lack of a solid theoretical
foundation – as no comprehensive theory of gold valuation exists – was a challenge, especially
when interpreting the study results. Newer behavioural science approaches provide coherent
explanations for the phenomena observed.
This study indicates that a number of independent variables that correlate strongly with the gold
price explain almost two thirds of gold price’s variance during normal times. During crises,
however, investors become fearful and supposedly more emotional and the percentage of the
variance that can be explained drops to 50%. Future research might seek to measure the extent
of that emotion in an attempt to assess its impact.
46
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53
Appendices
Tables:
Model Fit statistics
Ljung-Box Q(18)
Number
Stationary
Normalized
Number of
of
Model
Predictors R-squared MAPE
BIC
Statistics DF
Sig.
Outliers
End of the month
2
.493 4.943
8.288
17.254
18
.506
2
gold price in
USD an ounceModel_1
Table 8: ARIMA model statistics during crises (January 1978 to January 1981 and August 2007 to December
2011).
Model Fit statistics
Ljung-Box Q(18)
Number
Stationary
Normalized
of
Predictors R-squared MAPE
BIC
Statistics DF
Sig.
5
.651 2.836
5.206
18.514
18
.422
Number of
Model
Outliers
End of the
8
month gold price
in USD an
ounce-Model_1
Table 9: ARIMA model statistics during normal times (March 1973 to December 1977 and from February 1981 to
July 2007).
End of the month gold
price in USD an ounce
No Transformation Difference
Silver price an ounce in
cents
No Transformation Numerator Lag 0
M2 Money Stock in
Billions of Dollars
No Transformation Numerator Lag 0
Difference
Difference
Estimate
1
.105
SE
t
Sig.
.019
5.450
.000
.010
3.602
.001
1
.035
1
Table 10: ARIMA model parameters contributing to the fit of the monthly price of gold during crises (January 1978
to January 1981 and August 2007 to December 2011).
54
End of the month gold
price in USD an ounce
No Transformation Difference
Inflation rate at time 0
No Transformation Delay
Estimate
1
3
2.320
3.358
.001
Lag 1
5.314
1.691
3.142
.002
1
No Transformation Delay
2
Numerator Lag 0
-5.070
1.267
-4.002
.000
Lag 1
-6.218
1.765
-3.524
.000
1
No Transformation Numerator Lag 0
.167
.014
12.312
.000
Lag 2
-.047
.012
-3.758
.000
Difference
M2 Money Stock in
Billions of Dollars
1
No Transformation Numerator Lag 0
.091
.025
3.662
.000
Lag 4
.084
.024
3.461
.001
Difference
Oil price in USD an
barrel of oil
Sig.
7.789
Difference
Silver price an ounce in
cents
t
Numerator Lag 0
Difference
Real interest rate
SE
1
No Transformation Numerator Lag 0
1.047
.303
3.456
.001
Lag 2
.782
.300
2.605
.010
Difference
1
Table 11: ARIMA model parameters contributing to the fit of the monthly price of gold during normal times (March
1973 to December 1977 and from February 1981 to July 2007).
55
Project Proposal
Name: Stefan Frank Heini
Student ID number: 049004610
Programme title: MSc Finance
Module title: Dissertation (Proposal)
Dissertation tutor: Prof. Chin Bun Tse
Determinants of the Gold Price since the End of Bretton Woods:
A Multiple Regression Model Analysis
Project Proposal
August 2012
56
Abstract
Gold is better known as a financial asset than for its industrial use. But first of all, it is used by
many as a hedge against inflation. It played an important part in monetary policy, too, the last
time during the era of Bretton Woods (1944 until 1971), when the currencies of 43 nations were
tied to the dollar and the dollar to the gold price. From 1973 on, these currencies were floating
freely again. Fixed exchange rates were replaced by flexible rates.
Since then, the gold price and factors that help explaining it have caught the attention of
academics and practitioners alike. Besides factors such as political risk, official sector activity
and central bank gold reserve sales which are all very difficult to grasp and – especially –
quantify their likely effect the gold price, other factors have been identified, too. These are –
among others – currency exchange rates, inflation, the crude oil price or the gold supply. This
study wants to build and optimize a multiple regression model that explains the gold price since
the end of the Bretton Woods era until today by testing the independent factors researchers
claim to have a significant influence on the gold price.
57
Table of Contents
1.
Introduction ...............................................................................................................59
2.
Relation to previous research .................................................................................60
3.
Proposed methods ...................................................................................................62
3.1 Methodology ......................................................................................................62
3.2 Data collection...................................................................................................63
4.
Reflections ................................................................................................................64
4.1 Potential and practical empirical obstacles ....................................................64
4.2 Conceptual and theoretical problems and difficulties ...................................65
4.3 Ethics .................................................................................................................65
4.4 Political field and reflection..............................................................................66
5.
Conclusion ................................................................................................................66
6.
Timetable ...................................................................................................................68
7.
References ................................................................................................................69
58
1.
Introduction
Gold has a special place in the economy and economic history. It is a precious metal that is
used in the industry but it is more widely used (and known) as a financial asset or for its use in
the jewellery sector. Investors use it as a hedge against geopolitical and/or economic risk or
against inflation. In 1944 the Bretton Woods system of pegged exchange rates was introduced.
Between 1944 and 1971 the currencies of 43 nations were tied to the dollar and the dollar to the
gold price. The International Monetary Fund (IMF), however, was given the authority to
intervene in case of an imbalance of payments.
The purpose of this study is to develop a multi factor model explaining the price of gold (in USD,
GBP and YEN) since the reintroduction of free floating currencies in 1973 (March) until the end
of 2011. The aim is to find the determinants (independent factors) that have significant
predictive value for the gold price.
The central research question is:
-
How well could the gold price since the end of Bretton Woods be explained by a multiple
regression model?
Sub questions that will be discussed are:
-
Which factors had the highest predictive quality for the gold price between the end of
Bretton Woods and today?
-
How big is the proportion of variability the model accounts for?
-
What could explain the proportion of variability that cannot be explained by the model?”
-
How could the model be improved?”
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2.
Relation to previous research
Models that were used to estimate the relationship between these independent factors and the
gold price are multiple regression models using time series data (i.e. GARCH-models that
model the residuals of a time series regression) and others that use cross-sectional data. This
study will use a multiple regression model using cross-sectional data.
Many scientists (and practitioners) have tried to estimate and/or explain the gold price.
However, they have used different independent variables, time periods and methods. Factors
that were found to have influence on the gold price are inflation, the oil price, the silver and
copper prices, gold supply and US dollar money supply (M1).
a.
Inflation
According to Pukthuanthong and Roll (2011, p.2070) the dollar price of gold can be “associated
with currency depreciation in every country”. Sjaastad and Scacciavillani (1996, p. 893) claim
that currency depreciation is associated with a rising gold price. Other researchers, too, claim
that inflation correlates with the gold price (i.e. Capie, Mills and Woods 2004, Ruggiero 2002),
and gold was a reliable hedge against inflation. However, a study of Credit Suisse and LBS
(2012) found that gold “failed to serve as an inflation hedge”.
b.
Oil price
An analysis of Hammes and Wills (2005) showed that in the decade between 1970 and 1980
gold and oil prices – both in US dollars - were highly correlated (they did not look at the
relationship after that period). Shafiee and Topal (2010), too, found that from 1968 until 2008
gold and crude oil prices were positively correlated.
c.
Silver and copper prices
60
Vural (2003) as cited in Toraman, Basarir and Bayramoglu (2011) found a positive correlation of
silver and copper prices with the gold price. Sari, Hammoudeh and Ewing (2007) found a
positive correlation between silver and gold prices and a hardly existing correlation between the
price of copper and the gold price.
d.
Gold supply
The gold price is set by the supply and demand for gold. It can therefore be expected that a
change in supply/production of the commodity influences its price. Shafiee (2010), for example,
found that a decrease in gold production (due to increased mining costs) contributed to the rise
of the gold price since 1997.
e.
US dollar money supply (M1)
Ismail, Yahya and Shabri (2009) used a multiple linear regression model to identify factors that
influence the gold price and found that the US dollar money supply (M1) was positively
correlated with the gold price.
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3.
Proposed methods
The purpose of the planned study is to develop a multiple regression model that serves best to
explain the gold price since the end of the Bretton Woods system until 2011. Such a model
takes the following form (Newton and Spurrell 1967, Tranmer and Elliott 2008, Agresti and
Finlay 2007):
Y = a + β1 * x1 + β2 * x2 + β3 * x3 + βp * xp + E
Y: Dependent variable (gold price)
a: Constant
β 1 : The (relative) importance of predictor x1 in predicting Y
x 1 : factor 1
β 2 : The (relative) importance of predictor x2 in predicting Y
x 2 : independent variable 2
β 3 : The (relative) importance of predictor x3 in predicting Y
x 3 : independent variable 3
E: Residual
3.1 Methodology
To get a comprehensive data overview and to evaluate how good the gold price can be
estimated by different independent factors that have significant influence, several steps are
undertaken. All factors identified during the literature research 1 are being tested by using the
Statistical Package for Social Sciences (SPSS). The independent variables tested are inflation,
the oil price, the silver and copper prices, gold supply, and (US dollar) money supply (M1).
Those without significant influence will be eliminated (backward elimination). The specific steps
that will be undertaken are the following:
1
The most important and most frequently tested factors researchers have found to have a significant predictive
value for the gold price.
62
•
First of all, descriptive statistics are computed. To get a graphic representation to check
for a linear relationship, scatter plots are produced.
•
To see how the different variables correlate and to check for the significance of their
correlation, a correlation analysis is undertaken. The linear correlation coefficient r and
the coefficient of determination r square (r2) are computed (by undertaking a multiple
regression analysis) to see how strongly the different factors correlate (“r”) and to
compute how much of the variation of firm’s returns can be explained by the different
models (“r2”). The independent variables with significant influence are used to complete
the regression model and explain the relationship between the gold price and the
independent variables.
•
The residuals are computed: the difference between the actual values of the dependent
variable and the values will be predicted by the regression equations. With the saved
residuals, residual plots are drawn. Like this the data can be controlled for
heteroscedasticity.
•
Simple transformations of the regression equation are undertaken to maximize the
goodness of fit of the model (r2)
3.2
Data collection
It is intended to collect data of the monthly gold prices from 1978 until today from the website of
the World Gold Council (www.gold.org). The data for monthly gold prices in USD from 1973 until
1978 is obtained from usagold.com, the same data for prices in Yen and British Pound could not
be found. This is why for the years from 1973 to 1978 the model can only be tested for USD.
However, the accuracy of the model can be expected to be the same for whatever currency it is
tested.
The data for the independent variables will be collected from these sources:
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4.
Inflation rate (US dollar buying
power)
http://www.usinflationcalculator.com/inflation/historic
al-inflation-rates/
Crude oil price (per barrel) in
USD
http://www.forecast-chart.com/chart-crude-oil.html
Silver price
http://www.inbullion.com/
Copper price (only annual
data)
http://minerals.usgs.gov/minerals/pubs/commodity/c
opper/240798.pdf
Gold supply (world production)
http://minerals.usgs.gov/ds/2005/140/gold.pdf
Money supply (M1)
http://research.stlouisfed.org/fred2/data/M1.txt
Reflections
Factors that might or almost certainly have an influence on the gold price but cannot be tested
by using a factor model analysis (due to missing data or data collection problems) are political
risk, official sector activity and central bank gold reserve sales. These factors might be possible
reasons that could help to explain the percentage of the gold price a factor model cannot
explain.
4.1 Potential and practical empirical obstacles
Because for the development of this proposal the likely sources that will be used to gather data
for the regression model were already identified, no major difficulties regarding data are
expected. However, it is possible that the model using the factors identified is not very accurate
in estimating the gold price. If this was the result of the analysis, the conclusion and reflection
what other reasons might have influenced the gold price in general or at a certain point in time
become even more important.
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4.2 Conceptual and theoretical problems and difficulties
One possible issue is that the factors identified do not serve to estimate the gold price or that
the residuals (“real” data compared to data estimated by the model) show patterns, which they
are not supposed to do (“heteroscedasticity” is when residuals variance is not constant, i.e.
residuals are not normally distributed).
The standard deviations ought to be constant and should not depend on the independent
variable (Downs and Rocke 1979, p.816). By modifying, i.e. optimizing, the model, the problem
of heteroscedasticity can sometimes be solved.
Regarding theory and previous studies in the field/on the topic, there is some uncertainty
because the relevant studies found use many different approaches, come to different
conclusions, for different time periods and test for a vast amount of (different) factors. For this
study it is planned to build a multiple regression model, to test for the possible independent
variables that were identified by other researchers and eliminate those that fail to significantly
contribute to the predictive quality of the model.
4.3 Ethics
“Ethics is a philosophical discipline that deals with rules of "correct", "good",
"moral" human behaviour” (Bruckstein 2005, p.3). Ethics are ubiquitous, but describing ethics
(or morals) can cause problems because it does not mean the same for everyone. Even to find
a common description is not easy. One could say it is about right and wrong and – especially –
about being able to distinguish between what is right or wrong (Resnik 2012). That there are
many discussions about ethical disputes indicates that not everyone means the same when he
or she talks about ethics or that – at least – not everyone acts or tries to act in an ethical way. In
an academic context, ethical behaviour means being honest and fair in the sense that the
65
researcher/author/academic only take credit for his or her own work and never maliciously
discredit other people’s work (Bruckstein 2005, p.16)
4.4 Political field and reflection
The author of this study does not work in the financial industry or any other field with an interest
in manipulating assumptions of outcome of the study. Also, pre-conceived ideas and/or political
viewpoints on the topic do not exist. From the author’s perspective, there is very little risk for a
biased interpretation or analysis. However, one must be cautious when selecting (the sources
of) raw data.
5.
Conclusion
The project’s aim is to build a multiple regression model with several independent factors that
explains the gold price. It is crucial to identify the factors with significant influence and to have
access to the data necessary. Important when interpreting the results are historical events and
factors such as political risk and official sector activity that cannot be included in the factor
analysis (or at least not within an acceptable monetary and temporal limit). This requires getting
an overview of all the events – historical und political – that had or possible had influenced the
gold price in a significant way. Where actual and predicted values differ most, one of these
events might be the explanation.
The importance of the project benefits from the fact that – to the knowledge of the author –
there is no other study available that included data from 1974 until now (or recently).
Upon approval of the proposal by the University of Leicester, the methodology will be adapted
based on the reviewers’ feedback. Then, further literature research and analysis will be
66
undertaken to have a more complete overview of existing studies, data and possible
independent factors. When the researcher is fairly confident that the data collected is complete
and the approach sound, first tests with SPSS will be undertaken.
67
6.
Timetable
Task
Further literature research and
review
References and appendices
collection
Amendments to the
project/methodology based on
examiner's feedback
Research methodology
elaboration
Write theoretical part of the
dissertation
Data collection and entering in
SPSS
Data analysis and
interpretation
Summarize findings
Review of study and findings
Conclusion
Proofreading, editing, binding
1
2
3
4
Project Week
5 6 7 8
68
9
10
11
12
13
7.
References
AGRESTI, A. and FINLAY, B. 2007. Statistical Methods for the Social Sciences. 4th ed. New
Jersey: Prentice Hall.
BRUCKSTEIN, A.M. 2005. Ethics in Academia: Principles for Ethical Behavior in Advanced
Studies and Research. [pdf] Available at:
http://www.graduate.technion.ac.il/Heb/Principles%20of%20Ethics3.pdf [Accessed 12th of
August 2012].
CAPIE, F., MILLS, T.C. and WOOD, G. 2005. Gold as a hedge against the dollar. Journal of
International Financial Markets, Institutions and Money, 15, pp.343-352.
CREDIT SUISSE, 2012. Credit Suisse Global Investment Returns Yearbook 2012 [pdf]
Available at: https://www.creditsuisse.com/investment_banking/doc/cs_global_investment_returns_yearbook.pdf [Accessed
13rd of August 2012].
DIMSON, E., MARSH, P. and STAUNTON, M. 2012. The real value of money. Credit Suisse
Global Investment Returns Yearbook 2012, pp.5-15.
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Political Science, 23 (4), pp.816-828.
GARBER, P.M. 1993. The Collapse of the Bretton Woods Fixed Exchange Rate System. In:
Bordo, M.D. and Eichengreen, B., ed. 1993. A Retrospective on the Bretton Woods System:
Lessons for International Monetary Reform. Chicago: University of Chicago Press, pp.461-494.
HAMMES, D. and WILLS, D. 2005. Black Gold. The End of Bretton Woods and the Oil-Price
Shocks of 1970s. The Independent Review, 9(4), pp. 501-511.
ISMAIL, Z., YAHYA, A. and SHABRI, A. 2009. Forecasting Gold Prices Using Multiple Linear
Regression Method. American Journal of Applied Sciences, 6(8), pp.1509-1514.
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34(6), p.30.
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Analysis of Routine Data. Journal of the Royal Statistical Society, 16(1), pp.51-64.
PUKTHUANTHONG, K. and ROLL, R. 2011. Gold and the Dollar (and the Euro, Pound and
Yen). Journal of Banking and Finance, 35(8), pp.2070-2083.
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p.48-51.
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commodity futures prices. Geopolitics of Energy, 29, pp. 1–12.
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SJAASTAD, L. and SCACCIAVILLANI, F. 1996. The price of gold and the exchange rate.
Journal of International Money and Finance, 15(6), pp. 879-897.
SHAFIEE, S. and TOPAL, E. 2010. An overview of global gold market and gold price
forecasting. Resources Policy, 35, pp. 178-189.
TORAMAN, C., BASARIR, C. and BAYRAMOGLU, M.F. 2011. Determination of Factors
Affecting the Price of Gold: A Study of MGARCH Model. Business and Economics Research
Journal 2(4), pp.37-50.
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