Relationship between regimes in gold mining stocks and the gold price

University of Amsterdam

Faculty of Economics and Business

Master Thesis Financial Econometrics

Relationship between Regimes in Gold

Mining Stocks and the Gold Price

Author:

Rob Schuitemaker

Supervisor:

prof. dr. H.P. Boswijk

Second reader:

dr. S.A. Broda

July 11, 2017

A single-regime view is still a popular way of examining time series, but the estimation of multiple regimes is gaining ground, although the belief that markets can be partitioned into different regimes already originates from a century ago. There is a clear difference between regimes in equity markets, where the bull-bear classification implies a negative mean-variance relationship, and regimes in the gold price, for which a positive mean-variance relationship holds. However, whether regimes in gold mining stocks are more similar to regimes in equity markets or to regimes in the gold price is open for question. In this thesis we investigate the relationship between the gold price and a portfolio of gold mining stocks by employing the Markov Switching model. The estimated regimes in the portfolio clearly rule out a negative mean-variance relationship and behave similar to regimes in the gold price, but in a bivariate setting the relationship turns out to be more complicated. The presence of concurrent dependence and Granger causality shows a relationship, but the facts that the maximum-variance state of the portfolio does not correspond to maximum variance in the gold price and the lack of cointegration doubt this conclusion.

Contents

1 Introduction 1

2 Literature Review 3

2.1 History of determining regimes in the economy . . . 3

2.2 Rule-based methods . . . 5

2.3 Markov Switching model . . . 9

2.4 Diagnostic testing in Markov Switching models . . . 14

3 Methodology 16 3.1 Model . . . 16 3.2 Estimation . . . 17 3.3 Approach . . . 18 4 Data 19 4.1 Commodities . . . 19 4.2 Portfolios . . . 20 4.3 Predictor variables . . . 22 5 Results 24 5.1 Gold analysis . . . 24 5.2 Silver analysis . . . 38 5.3 Robustness checks . . . 47 6 Conclusion 50 References 52 Appendices 56 A Additional figures . . . 56

B Additional tables . . . 60

C Portfolio composition . . . 63

Statement of originality

This document is written by Rob Schuitemaker who declares to take full responsibility for the contents of this document.

I declare that the text and the work pre-sented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.

The Faculty of Economics and Business

is responsible solely for the supervision of completion of the work, not for the contents.

1

Introduction

The idea that financial markets can be partitioned into periods of rising and declining prices has been widely adopted in economics. When a market is in an upward trend, this state is often named a ‘bull market’, while ‘bear market’ is the term referring to worse economic times. Although the origin of these names is unclear, it probably refers to the way of attacking of these animals. Bears stand up and strike down with their claws, while bulls thrust their horns up. Knowing the state of the market is important for different levels in economics. From central banks, which are able to respond quicker to a market tendency, to a single investor, who can adjust his portfolio based on the market sentiment. However, the state of the market is latent for a number of reasons. First, there is a lag of information. Ideally one exactly knows the turning points of the market, which are peaks for switches from bull to bear and troughs for bear to bull. However, one can only identify these points after a certain time. For instance, Nyberg (2013) finds an average lag length of six months after applying the Bry and Boschan (1971) dating algorithm to the S&P 500 index. Second, it is not possible to formally say in which state a market is, since it is based on belief. For example, one investor might already have a pessimistic view after a small price drop, while another might stay optimistic believing the market will regain momemtum. Several attempts have been made to formalize investor’s belief into rules for detecting turning points.

Forecasting is regarded as an important goal of analyzing financial time series. There is a causal relationship in the sense that predictions will be better if the amount of overfitting in the underlying model reduces. For this reason a major focus in the literature has been to implement the existence of multiple states into models. Broadly speaking, the first attempt was made by Hamilton (1989), who set the standard by employing first-order Markov chains. It took a few years for this model to gain popularity due to computational constraints (Tsay, 2010, Chapter 12). Diebold, Lee, and Weinbach (1994) extend the so-called Markov Switching model to allow for time-varying transition probabilities, that are described by logistic functions of economic variables. Durland and McCurdy (1994) argue that duration could explain volatility clustering, mean-reversion, and non-linear cyclical patterns. They implement duration in the model by specifying a higher-order Markov system such that duration affects transition probabilities for multiple periods. Besides these extensions, different estimation techniques have been developed for the Markov Switching model. The EM-algorithm of Dempster, Laird, and Rubin (1977) is less demanding than ordinary maximum likelihood, while Albert and Chib (1993) describe how the Bayesian Gibbs sampling procedure of Geman and Geman (1984) can be applied to estimate Markov Switching models. Although it is not of interest for this thesis, it is worth noting that forecasting based on Markov Switching models has gained ground in the past years.

In this thesis the different states in commodity markets, especially gold, will be examined. This topic is underexposed in the literature regarding regime switching since the majority focuses on stock indices, followed by studies on macroeconomic variables and interest rates. Stock indices like S&P 500 have the undesirable feature of being in a bull market most of the time (Maheu, McCurdy, & Song, 2012, Figure 1). On top of that, indices quickly recover from crises which makes it hard to get accurate estimates for the bear markets. Also, peaks and troughs in commodity markets tend to follow each other rapidly, while the transition between regimes in stock markets can be more subtle. As an example, consider the gold price which rallied from $ 750 in 2009 to almost $ 1900 in 2011 but fell back to less than $ 1200 at the end of 2013. Note that it is questionable whether the theory on bull and bear markets is applicable to commodity markets. Researchers have argued for a positive relationship between return and volatility in the gold market based on the inventory effect and the safe haven property of gold. We will relate regimes in the gold price to regimes in a portfolio of companies operating in the gold market. More generally, the thesis aims to answer the question: “Are regimes in a portfolio of gold mining stocks comparable to regimes in the gold price and if so, how is it possible to quantify the relationship?”. At this point we follow the concept of Ntantamis and Zhou (2015), who relate prices of oil, gold, and other metals to prices of stocks corresponding to these commodities. They apply the rule-based dating algorithm of Lunde and Timmermann (2004) to indicate bull and bear markets. Our main focus, however, will be on the Markov Switching model; the rule-based algorithms will only serve as a starting point. Apart from that, Ntantamis and Zhou (2015) concentrate on the Canadian stock market, where mining and energy stocks play an important role, plus that they only consider five stocks per commodity. In our analysis we do not have a country of interest, though the stocks should be listed on the NYSE, and use only silver and, especially, gold as commodities.

Our research will be conducted in several steps. First, we implement rule-based methods to get a first impression of the bull and bear markets in each commodity and in each portfolio of stocks. These methods also serve as a benchmark for the results of Markov Switching models. Second, for each commodity and portfolio we select the optimal number of states and analyze the model of choice with both constant and time-varying transition probabilities. Third, we estimate a model for the commodity price and the portfolio of related stocks simultaneously in a multivariate Markov Switching model. These steps will be made for both gold and silver, where the latter will be used for comparison and validation. Finally, we test the robustness of our results.

The outline of this thesis is as follows: Section 2 will contain a literature review in which we concentrate on methods and their history. The methodology and data, which also contains a detailed description about the portfolio composition, will be outlined in Sections 3 and 4,

respectively. Section 5 will report and discuss the results and the conclusions and limitations follow in Section 6.

2

Literature Review

This section gives an overview of the literature concerning bull and bear markets. First, we shortly discuss the history of business cycles. There exists a vast amount of literature, conse-quently we focus on topics related to Markov Switching models, which are of special interest for our research. Identification methods are either based on rules or on models. Section 2.2 explains the ruled-based methods of Lunde and Timmermann (2004) and Bry and Boschan (1971), while Section 2.3 focuses on the Markov Switching model. Finally, in Section 2.4 we discuss methods for testing in Markov Switching models.

2.1

History of determining regimes in the economy

Nowadays a one-dimensional view on markets, in the sense that the market structure can be represented by a single regime, is still popular. In the past decades literature has paid much at-tention to linear models. ARMA models have been proposed to describe the linearity in returns, while ARCH models could deal with heteroskedasticity, such as volatility clustering and lever-age effects. Additionally, non-normality of stock returns and the effect of seasonality have been investigated among others. More recently, the existence of multiple regimes has been studied, though the partition in upward and downward movements has already been mentioned early in the twentieth century (Pagan & Sossounov, 2003). When the single-regime view is extended to a two-state classification, the phases are often called bull and bear markets. A bull market starts at a trough and finishes at a peak and is characterized by a positive mean return and usually a low volatility. The reverse is true for bear markets. In a Markov Switching setting, the existence of a negative mean-variance relationship between bull and bear markets in equity markets has been found by Turner, Startz, and Nelson (1989), Nyberg (2013), and Kole and Van Dijk (2017) among others.

The National Bureau of Economic Research (NBER) has played an important role in in-vestigating business cycles in the economy. The work of Burns and Mitchell (1946) is broadly considered to be a first attempt to describe and measure periods of expansions and contrac-tions, which are the terms referring to bull and bear phases in macroeconomic variables. In the years afterwards the NBER started dating business cycles in the United States. In 1978 the dating process has been formalized when the Business Cycle Dating Committee was formed.

The committee tries to determine turning points in the economy based on economic indicators as GDP, consumption, and unemployment. Bry and Boschan (1971) developed a set of rules by which the cycles determined by the committee could be replicated. A detailed description is provided in Section 2.2.

In the 1980s, modeling of cycles in the economy took a dive. Beveridge and Nelson (1981), Nelson and Plosser (1982) and Campbell and Mankiw (1987) used an autoregressive moving av-erage process that fluctuates around a deterministic trend. Others applied a linear unobserved-components model (Harvey, 1985; Watson, 1986; Clark, 1987), while King, Plosser, Stock, and Watson (1987) tried to fit a cointegration specification to business cycles. However, the common element is the assumption that the time series is integrated of order one. In other words, they assume the series follows a linear stationary process after taking first differences. Since then non-linear methods have been proposed, for example treshold autoregressive models, smooth transition autoregressive models, and duration models. Another non-linear procedure is known as the Markov Switching model and was initiated by Hamilton (1989). In Section 2.3 this model and its extensions will be thoroughly discussed.

By means of the Markov Switching model several types of time series have been investigated. The seminal work of Hamilton (1989) has an application to aggregate output. Additionally, in-terest rates and exchange rates have been topics in research. More relevant for our research, however, are applications to stock returns and commodity prices. Turner et al. (1989) dis-tinguish between a high- and low-variance state in stock indices. They find evidence for the existence of more than one regime since the higher variance is four times greater than the lower variance. Schaller and Van Norden (1997) further elaborate on their findings and also allow for switching in means. Their results support the general view that bull markets are characterized by high returns and low variance and vice versa for bear markets. Perez-Quiros and Timmer-mann (2000) estimate several size-sorted portfolios of stocks separately. Although significance is questionable, their results suggest that cyclical behaviour is stronger among small-sized firms and a similar conclusion has been derived by Guidolin and Timmermann (2006).

Applications of regime switching model to commodity markets are scarce, except for oil which is a rather popular topic. Fong and See (2001) propose a model that deals with regimes in volatility. They do not choose a specific commodity market but use the Goldman Sachs Com-modity Index, which consists of 22 commodities from five different sectors. They find evidence of regime shifts and using these shifts, GARCH effects are minimal. On top of that, forecasts improve considerably over single-regime models. Alizadeh, Nomikos, and Pouliasis (2008) inves-tigate the hedging effectiveness of energy commodities by estimating a regime switching VECM in which they use both spot and futures prices. The VECM enables them to examine whether the speed of adjustment of spot and futures markets to the long-run relationship changes across

different regimes. Choi and Hammoudeh (2010) use a univariate Markov Switching model to estimate high- and low-volatility regimes in several commodity markets, among which gold and silver. Although their sample size is fairly limited, it is interesting to see that the average du-ration of the high-volatility regime in gold is almost three times larger than in silver, while the duration of the low-volatility state is comparable. Besides, they find that gold and silver behave similarly, but the latter is more volatile. Baur and Lucey (2010) find that gold is a safe haven for US stocks, because the returns are negatively correlated. Chiarella, Kang, Nikitopoulos, and Tô (2016) argue that gold and silver are not popular in bull markets, since, in contrast with consumption commodities, gold and silver are especially bought for investment reasons and generate no cash flow. Due to the safe haven property, investors are more interested in investment commodities once there is more uncertainty in the market and hence this results in a positive return-volatility relationship. On top of that, the results of a stochastic volatility model on gold futures confirm the theoretical explanation.

In addition, multiple states have been taken into account. It is not hard to extend the Markov Switching model, but the classification and identification of states requires extra care. Ang and Bekaert (2002) investigate a joint specification of the interest rate data from the United States, Germany, and the United Kingdom. The interest rates are allowed to have two regimes such that there are eight states in total. Based on Granger causality they can impose some restrictions and hence estimation is feasible. Guidolin and Timmermann (2006), who con-sider both stock and bond returns, choose four states based on the Hannan-Quinn information criterion. After estimation they identify these regimes as crash, slow growth, bull and recovery. Maheu et al. (2012), however, first specify bull, bull correction, bear, and bear rally states and then impose restrictions on mean and variance such that the states are well defined. In other literature, for example Kim, Nelson, and Startz (1998) and Jiang and Fang (2015), the different states are not classified.

2.2

Rule-based methods

In the economic field there are many tools to analyze time series. One of the most popular indicators is the moving average (MA), where a number of past prices are used to remove noise and hence to find patterns. Especially for a sufficient number of lags the MA is a useful indicator of expansions and extractions. Based on the standard deviation of the MA process, Bollinger (1992) Bands can be added to provide confidence bounds. Other well-known indicators are the

Money Flow Index1, Relative Strength Index2, and Stochastic Oscillator3. For the identification of bull and bear regimes we concentrate on methods of Lunde and Timmermann (2004) and Pagan and Sossounov (2003), which are related to technical analysis in the sense that they simply apply a set of rules to a time series and do not involve any model estimation. The magnitude of price movements forms the basis of the algorithm of Lunde and Timmermann, while Pagan and Sossounov focus on the length of business cycles.

Lunde and Timmermann (2004, LT) specify two tresholds λ1 and λ2 that determine regime

switches. For price Pt at any time t ∈ {1, . . . , T } they follow:

• If the previous extreme value was a peak, Pmax_{, with corresponding time t}max_:

1. If Pt < (1 − λ2)Pmax, the maximum changes to a minimum, Pt = Pmin, at time

t = tmin and a bear phase has been established from tmax to tmin.

2. If (1 − λ2)Pmax≤ Pt≤ Pmax, no update takes place.

3. If Pt > Pmax, update the maximum price Pt= Pmax.

• If the previous extreme value was a trough, Pmin_{, with corresponding time t}min_:

1. If Pt < Pmin, update the minimum price, Pt= Pmin.

2. If Pmin _{≤ P}

t≤ (1 + λ1)Pmin, no update takes place.

3. If Pt > (1 + λ1)Pmin, the minimum changes to a maximum, Pt = Pmax, at time

t = tmax _{and a bull phase has been established from t}min _{to t}max_.

Whether P0 is supposed to be a minimum or maximum does not really matter, since one

usually cuts off the first phase due to uncertainty about the start date of the phase. Figure 1

illustrates two examples of the LT-algorithm. In the left panel the bear market ends at tmin

and Pmax changes to Pmin. Also note the change of treshold value from λ2 to λ1 at tmin. The

right panel shows how a peak gets updated. Crucial in the LT-algorithm algorithm is the choice

of treshold values λ1 and λ2. The magnitude of these filters differs substantially from 10% to

25%. Lunde and Timmermann (2004) apply (20, 15), (20, 10), (15, 15) and (15, 10) as tresholds

for (λ1, λ2), but they do not find strong evidence in favor of one treshold combination. On the

other hand, Pagan and Sossounov (2003) argue that a rule of thumb in the financial press is to qualify states as either bull or bear after a rise or fall of either 20% or 25%. The usual approach

is to set λ1 > λ2, which results in less bear markets and hence deals with the upward drift

in stock indices. However, in our case of commodity prices this restriction is not so obvious

1_{For details, see http://www.investopedia.com/terms/m/mfi.asp} 2_{For details, see http://www.investopedia.com/terms/r/rsi.asp}

Figure 1: Examples of LT-algorithm Pmax (1 − λ2)Pmax λ2 Pmin (1 + λ1)Pmin λ1 Price Time

a) Trough after peak tmin tmax Pmax Pmax new λ2 Price Time b) New peak tmax

as for stock indices. Ntantamis and Zhou (2015) choose λ1 = 25% and λ2 = 20% based on

observed minimum and maximum values in monthly returns. This seems reasonable for their gold and stock returns, but is questionable for oil prices that do not show a broad upward movement. The price of one barrel of crude oil is 40 dollars at both the start and end of their sample in 1982 and 2011 respectively. It is true that oil rose since its 10 dollar low in 1999, but then the authors disregard the decline in the early years. Chiang, Lin, and Yu (2009) do not choose certain percentages in advance, but let the data determine the filters. They study (σ, σ), (0.5σ, 0.5σ), (σ, 0.5σ) and (0.5σ, σ), where σ is the standard deviation of the underlying time series. Unfortunately, they do not argue which filter works best.

It is important to note that Lunde and Timmermann (2004) do not take duration into account. They mention it adds extra complexity since it requires additional filters besides the tresholds. Duration, however is the leading determinant in the algorithm of Bry and Boschan (1971, BB). They developed a quantitative approach to replicate the qualitative turning points that are determined by the Business Cycle Dating Committee of the NBER. Pagan and Sos-sounov (2003) argue that the nature of macroeconomic variables that are used in the recognition of business cycles is different from stock prices. Nevertheless, the basic idea is just finding pat-terns in time series and therefore the BB-algorithm is still useful after some modifications. First, the original algorithm smooths the time series due to the macroeconomic character. However, in the identification of bull and bear markets there is special interest in extrema, which will ‘soften’ by smoothing the data. Second, Pagan and Sossounov add an exception regarding the phase of the market. The BB-algorithm states a period from peak to trough or trough to peak

in a single month the price rises or falls with more than 20%.4

The revised BB-algorithm can be applied by taking the following steps:

1. Find Pmin

t and Ptmax for each time t ∈ {0, . . . , T } in the time interval [t − τwindow, t +

τwindow]. The larger τwindow, the more minima and maxima will overlap and hence less unique extrema will be found.

2. Create an alternating series of peaks and troughs by selecting the highest value among subsequent maxima and the lowest among minima.

3. Censor the first and last τcensor periods. In these regions no peak or trough is allowed

since we do not know what happened in the periods before the start of our series and the same argument holds for the period after time T . If the first extremum in the censored series is a peak (trough), the initial state will be regarded as bull (bear), while a peak (trough) as final extremum would indicate a bear (bull) market at the end of the series.

4. If a cycle lasts shorter than τcycle periods, the cycle is eliminated. In case of a cycle from

peak to peak, the trough and the lowest peak will be removed and vice versa for trough-to-trough-cycles. This is necessary to keep the alternation of peaks and troughs in the series.

5. If a phase lasts shorter than τphase periods, the phase will be eliminated. If the absolute

price change in a single month exceeds 20% the minimum-duration rule does not apply. 6. If a peak (trough) next to an endpoint is lower (higher) than this endpoint, the peak

(trough) is eliminated.

An example of the BB-algorithm is given in Figure 2. The red dots and blue dots denote the unique minima and maxima, respectively, that follow from moving the window (green bracket) along the time axis. The selected peaks and troughs have been marked by arrows.

Finally, we discuss the choice of the duration parameters for there are some striking differ-ences. Bry and Boschan (1971) stay close to the duration of business cycles which are longer than 15 months. Pagan and Sossounov (2003) and Kole and Van Dijk (2017) vary a little by

setting τcycle = 16 months and τcycle = 70 weeks (or approximately 16.2 months) respectively.

Bry and Boschan use τphase = 5 months for the minimum phase duration. Pagan and Sossounov

mention that early literature regarding bull and bear markets points at a minimum duration of

4_{Pagan and Sossounov (2003, Appendix B) do not mention that the ‘20%-rule’ only holds for single months} instead of a complete phase. It is worth noting that Kole and Van Dijk (2017) follow the modifications of the BB-algorithm but apply the 20%-rule to a complete phase.

Figure 2: Example of BB-algorithm

Time Price

τcensor [t − τwindow, t + τwindow] τcensor cycle if duration ≥ τcycle

phase if duration ≥ τphase,

unless rise or fall of more than γ%

three months and choose a middle ground by specifying τphase = 4 months, which is followed by

Kole and Van Dijk. Regarding the choice of τwindow, Pagan and Sossounov substantially differ

from the 4 months of Bry and Boschan. They argue a longer period is appropriate in order to compensate for the lack of smoothing, but they are in the dark what the influence is of taking stock prices instead of real quantities. The settlement of 8 months is followed by Kole and Van Dijk, but they choose a much smaller censor region. Bry and Boschan and Pagan and Sossounov both choose a length of six months or 26 weeks. Kole and Van Dijk believe that investors know the state of the market earlier than after 26 weeks and propose a censor period of 13 weeks.

2.3

Markov Switching model

Although it is easier to investigate bull and bear markets separately using the LT- and re-vised BB-algorithm, model-based methods have already been applied years before. Hamilton (1989) developed the Markov Switching model that deals with non-linearities in time series. An important advantage of non-linearity over linearity involves stationarity. Markov Switching models require that linear stationarity only holds within a market regime instead of over the full time series. Elaborating on the Markov switching regression of Goldfeld and Quandt (1973),

Hamilton divides the observed time series ˜yt in two parts

˜

where both nt and ˜zt are not observed. Typically, ˜yt = log Pt in case of a price series. The

Markov trend nt= α0+ α1St+ nt−1 depends on the unobserved state St∈ {0, 1}, for which the

transition is governed by a first-order Markov chain with probabilities P[St= 0|St−1= 0] = p0,0

and P[St= 1|St−1 = 1] = p1,1. The state-independent part ˜ztis assumed to follow an ARIMA(r,

1, 0) process. By first-differencing the following system of equations are obtained:

yt= α0+ α1St+ zt

zt= φ1zt−1+ φ2zt−2+ · · · + φrzt−r+ εt,

(2)

where yt= ˜yt−˜yt−1and zt = ˜zt−˜zt−1. Hamilton adds two assumptions regarding the innovations

εt, namely εt ∼ i.i.d. N (0, σ2) and independence from nt−j for all j. Subsequently, he uses a

filter operation to obtain the joint conditional probability and conditional likelihood from the joint conditional probability of one period ago. The most challenging part lies in the first input of the algorithm. Due to computational reasons Hamilton does not start with the conditional probability

P[Sr = sr, . . . , S1 = s1|yr, . . . , y1],

but with the joint unconditional probability, which follows from setting P[S−r+1 = 1] = π and

P[S−r+1 = 0] = 1 − π, where π is the limiting distribution of the first-order Markov chain such

that π = (1 − p0,0)/(1 − p1,1+ 1 − p0,0). Finally, the conditional log likelihood

log f (yT, yT −1, . . . , yr+1|yr, . . . , y1) =

T X

t=1

log f (yt|yt−1, yt−2, . . . , yt−r), (3)

where f (·) denotes the density, is maximized over θ = (α0, α1, p0,0, p1,1, σ2, φ1, . . . , φr) with the

restriction α1 > 0 such that state 1 represents the bull market.

The literature usually refers to the work of Hamilton (1989) but an almost similar approach has already been demonstrated by Sclove (1983) a few years earlier. One of the differences is that

Sclove maximizes with respect to both θ and states (s1, . . . , sT)0, while Hamilton first obtains

ˆ

θ and uses these estimates to draw inferences on (s1, . . . , sT)0. On the other hand, Hamilton

includes r lagged terms in the Gaussian part zt, but Sclove assumes no autocorrelation.

The Markov Switching model can be extended by adding a dimension and states, allowing transition probabilities to vary over time, or using higher-order Markov chains. An increased number of states is computationally no different, but be aware that the number of parameters grows fast and extra identification restrictions are often necessary. Markov Switching Vector

Autoregressive Models (VAR) can be estimated by extending the dimension, where yt becomes

a vector and {φj}

r

j=1 and σ

2 _{matrices. How the Markov Switching model can allow for}

time-varying transition probabilities has been shown by Diebold et al. (1994) and Filardo (1994). Both authors use a logistic specification of the form

pi,i,t = P[St= i|St−1 = i, xt−1] =

exp(αi+ x0t−1βi)

1 + exp(αi+ x0t−1βi)

where xt−1contains economic indicator variables and may include further lags. The subscript t−

1 is just to make sure xt−1does not consist of variables in current time. To test for time variation

an F -test with null hypothesis βi = 0 for both i = 0 and i = 1 is sufficient. If β0 and β1 are zero,

the probabilities p0,0,t and p1,1,t are constant over time and the model simplifies to the Markov

Switching model discussed above. Diebold et al. (1994) track the performance of the model with time-varying transition probabilities relative to constant probabilities by employing a Monte Carlo analysis. They find an improvement over the specification with constant transition probabilities considering that the MSE is more than twice as small and the log likelihood increases, such that a LR test rejects in favor of the specification with time-varying transition probabilities. Filardo (1994) has an application to the industrial production in the United States. He finds support for time-varying over fixed transition probabilities by the statistical significance of the parameters and the improved regime classification as shown by the smoothed transition probabilities. The logistic functional form for time-varying transition probabilities has been used by, for instance, Whitelaw (2000), Ang and Bekaert (2002) and Kole and Van Dijk (2017).

Different specifications have been proposed by Gray (1996) and Bazzi, Blasques, Koopman, and Lucas (2017). The former bases the time-varying transition probabilities on the standard normal distribution with cdf Φ(·),

pi,i,t = Φ(αi+ x0t−1βi), i = 0, 1. (5)

A disadvantage with respect to the logistic distribution is, however, that it requires an additional transformation to ensure the probabilities sum to one. The standard normal approach has been used by Schaller and Van Norden (1997) and Perez-Quiros and Timmermann (2000) among others. Nevertheless, the differences between a logistic and normal specification are small. Basically, it is just a way of forcing the fitted values in the (0, 1)-interval. The dynamic

transition probabilities of Bazzi et al. (2017) are driven by an updating equation for ft in the

following way:

pi,i,t = δi,i+ (1 − 2δi,i)

exp(−fi,i,t) 1 + exp(−fi,i,t) , i = 0, 1, ft+1= (f0,0,t+1, f1,1,t+1)0 = ω + Ast+ Bft, st = St∇t, ∇t = ∂ ∂ft log P(yt|ft, ψ∗, It−1), (6)

where ω is a vector of constants, A and B are coefficient matrices, It−1 denotes the

informa-tion up to time t − 1 and ψ∗ = (σ2_{, ω, A, B).}5 _{A feature of these transition probabilities is}

5

For details on the conditional observation density P(yt|ft, ψ∗, It−1) for which the score is computed, we refer to Bazzi et al. (2017).

the parameter δi,i which allows the researcher to influence the variation in the probabilities. If

δi,i = 0, then pi,i,t can take all values in (0, 1), while pi,i,t is constant for δi,i = 0.5.

Addition-ally, Bazzi et al. (2017) show how this dynamic procedure can be applied to more than two states. In an application to the industrial production of the United States they show that the dynamic transition probabilities fit the data better than both the constant and time-varying specifications, although the improvement over logistic time-varying probabilities is relatively small. The results of a Monte Carlo analysis are, however, more convincing in the sense that the score-driven model always has the best performance after the correctly specified model.

Another way to influence the transition probabilities and hence the identification of states is by using higher-order Markov chains. Durland and McCurdy (1994) argue that duration de-pendence could play a role elaborating on the idea that a regime switch becomes less likely as a state grows older. Additionally, duration dependence could explain typical features of time series like volatility clustering, mean-reversion, and non-linear cyclical patterns. They allow duration to affect the transition probabilities for τ periods by employing a higher-order Markov chain. Similar to Diebold et al. (1994) and Filardo (1994) a logistic form is used to influence the transition probabilities,

pi,i = P[St = i|St−1= i, Dt−1= d] =          exp(αi+ βid) 1 + exp(αi+ βid) , if d ≤ τ exp(αi+ βiτ ) 1 + exp(αi+ βiτ ) , if d > τ , i = 0, 1, (7)

where d, d = 1, 2, 3, . . ., is a variable that counts the time spent in a state. If βi = 0 for all

i, then the specification collapses to the constant transition probabilities of Hamilton (1989). In their application to U.S. GNP growth, they set τ = 9 quarters based on a grid search. The

estimate of β0 is significantly different from zero meaning that duration dependence influences

the probability of a transition out of the low-growth state into the high-growth state. However,

it does not work both ways since ˆβ1 is not significant. More importantly, they find support for

positive duration dependence in the sense that transition probabilities increase with the time of being in a certain state. Based on a LR test their model performs better than a baseline AR(4) model, but the latter is not rejected in favor of the Markov Switching model with constant transition probabilities. There is limited literature on duration dependence in Markov Switching models. Maheu and McCurdy (2000) test different specifications of the mean equation in a duration-dependent Markov Switching setting using S&P 500 data. Shibata (2012) and Bejaoui and Karaa (2016) apply the different models of Maheu et al. (2012) to the Japanese and Tunisian stock market, respectively. Ghysels (1993), however, provides a different view on duration dependence that does not involve higher-order Markov chains. He basically allows the transition parameters to differ between seasons. For example, in case of quarterly data the probability of a switch from state i to j consists of four values, each corresponding to a quarter.

Besides extending Markov Switching models, there are various ways to estimate them. Turner et al. (1989) apply the EM-algorithm of Dempster et al. (1977) to their specification with high- and low-variance states. Maximization of the log likelihood is complicated due to

the recursive structure with many local optima (Kole, 2010). In a two-state model St is either

0 or 1, which can be seen as a draw from a Bernoulli distribution. Although we need to add an extra parameter ζ to account for the lack of history, the log likelihood can be simplified,

`(YT, ST; θ) = T X t=1  (1 − St) log f (yt) + Stlog f (yt)  + T X t=2  (1 − St)(1 − St−1) log p0,0+ (1 − St)St−1log(1 − p1,1) + St(1 − St−1) log(1 − p0,0) + StSt−1log p1,1  + (1 − S1) log ζ + S1log(1 − ζ), (8)

where Yt= {y1, . . . , yt} and St = {S1, . . . , St}. Then, the EM-algorithm can be applied:

1. Choose set of starting values θ(k−1) _{= θ}(0)_.

2. Calculate θ(k)_{= arg max}

θ E[`(YT, ST; θ)|YT; θ

(k−1)_].

3. Plug in θ(k) for θ(k−1) in step 2 and repeat until convergence.

According to Dempster et al. (1977) the likelihood is maximized after convergence, although one should be careful with the starting values and is advised to check the optima for different

θ(0)_.

Another field in econometrics is concerned with Bayesian estimation. General maximum likelihood estimation (MLE) results in a point estimate for θ but the Bayesian approach treats θ as a random variable. It requires specifying prior distributions which in combination with the likelihood results in a posterior distribution for θ (Cameron & Trivedi, 2005). Albert and Chib (1993) argue that direct Bayesian estimation is not an attractive option due to a complicated

log likelihood. Alternatively, they treat {St}Tt=1 as additional unknown parameters and analyze

them jointly with θ using a Monte Carlo method known as Gibbs sampling (Geman & Geman, 1984).

Consider again model (1) of Hamilton (1989). In this case the set of parameters is θ =

(α0, α1, φ, σ2, P), where φ = (φ1, . . . , φr) and P is the transition matrix. In order to perform

Gibbs sampling, the full conditional distributions of both ST and θ are necessary. The

dis-tribution of St follows from using the lack of memory in Markov chains and applying Bayes’

a normal distribution for α0, α1 and φ, an inverse Gamma distribution for σ2, and a Dirichlet distribution for each row of P. Then the Gibbs sampling procedure is:

(a) Specify initial values S(0) _{and θ}(0) _{and set i = 1.}

(b) Draw from state distributions:

1. Draw S₁(i) from the conditional distribution f (s1|YT, S2(i−1), . . . , S

(i−1) T , θ

(i−1)_{), where s} 1

is a realization from S1.

2. Draw S₂(i) from f (s2|YT, S

(i) 1 , S (i−1) 3 , . . . , S (i−1) T , θ (i−1)_). .. .

T . Draw S_T(i) from f (sT|YT, S

(i)

1 , . . . , S

(i) T −1, θ

(i−1)_).

(c) Draw (α(i)₀ , α(i)₁ ) from f (α0, α1|YT, S

(i)

1 , . . . , S

(i)

T , σ2(i−1), φ(i−1), P(i−1)).

(d) Draw φ(i) _{from f (φ|Y}

T, S (i) 1 , . . . , S (i) T , α (i) 0 , α (i) 1 , σ2(i−1), P(i−1)).

(e) Draw σ2(i) from f (φ|YT, S

(i) 1 , . . . , S (i) T , α (i) 0 , α (i) 1 , φ(i), P(i−1)).

(f) Draw for each row pj of P with size k × k, resulting in P(i),

from f (pj1, . . . , pjk|YT, S (i) 1 , . . . , S (i) T , α (i) 0 , α (i) 1 , σ2(i), φ(i)).

(g) Set i = i + 1 and repeat (b)-(f) until i = T .

Albert and Chib (1993) mention that θ(T ) → θ and S(T ) _{→ S as T → ∞. After dropping the}

first M estimates to remove initial values, one can use the remaining θ(i)_{, i = M + 1, ..., M + N ,}

simulations as an approximate sample for the marginal conditional distribution of θ and use it to do inference.

2.4

Diagnostic testing in Markov Switching models

Diagnostic testing of Markov Switching models is more complicated than in linear models. Linear specification tests are not directly applicable since the residuals are unobserved due to the latency of the state variable (Bazzi et al., 2017). Hamilton (1996) shows how the Lagrange Multiplier (LM) statistic can be used for specification testing. Consider a Markov Switching model with likelihood

P(yt|Zt, St; θ), (9)

where Zt = (yt−1, . . . , yt−r, wt, . . . , w1) with (wt, . . . , w1) observed exogenous variables and r

θ includes the model parameters to estimate. Let P = (p1,1, . . . , p1,k−1, . . . , pk,1, . . . , pk,k−1)0 and

note that the redundant transition probabilities pi,k for i = 1, . . . , k are omitted from P. Based

on (9) and P, the likelihood `t(λ) can be calculated, which is parameterized by λ = (θ0, P0)0.

The key element of the LM statistic, the score ht(λ), can be obtained by differentiating with

respect to every element in θ and P and then stacking the derivatives.

Suppose the null hypothesis H0 : q = 0, consisting of m restrictions, is tested. First, the

score for the new parameter vector λ∗ = (λ0, q0) with constrained ML estimate ˜λ∗ = (ˆλ0, 0)0

needs to be computed. Under the null hypothesis ht(˜λ∗) is relatively easy obtained since the

additional elements introduced by q can just be stacked below ht(ˆλ). Then, the LM statistic

can be calculated using the outer product:

LM = 1 T T X t=1 ht(˜λ∗) !0" 1 T T X t=1 ht(˜λ∗)ht(˜λ∗)0 #−1 _T X t=1 ht(˜λ∗) ! d −→ χ2_{(m). (10)}

Depending on the underlying model the LM statistic can be used for different purposes. Hamilton lists tests for autocorrelation, ARCH effects, and omitted variables in mean and variance. In a Monte Carlo analysis he tests the small sample properties of the LM statistic. For sample sizes of T = 50 and T = 100 the LM statistic performs fairly well in terms of critical values and size.

Gray (1996) tests his model by computing in-sample and out-of-sample forecasts for differ-ent estimation samples. He compares his specification with a constant-variance and GARCH

model by means of the RMSE, MAE and an adapted R2_{, where}

R2 = 1 − PT t=1(vt− v e t)2 PT t=1v2t ,

where vt = e2t is the actual volatility and vet = E[e2t|Ft−1] the forecast volatility, with et the

generalized residual and Ft−1 all information available at time t − 1.

Ang and Bekaert (2002) propose the Regime Classification Measure (RCM) which is based

on smoothed transition probabilities ps

t = P(St = 1|FT) in a two-state model. Ideally, the

probabilities are close to 0 or 1 and this can be tested as follows:

RCM = 400 1 T T X t=1 ps_t(1 − ps_t).

A value of 0 indicates perfect regime classification, while the smoothed transition probabilities are constantly 0.5 for RCM = 100. In the general case of k states the statistic becomes

RCM = 100k2 1 T T X t=1 k Y i=1 ps_i,t ! .

The statistic is not useful for comparing estimations with a different number of regimes, which would in general lead to a lower value for estimations with a higher number of states. In the end, the RCM statistic is just an easy way of getting insight in the quality of your estimations.

3

Methodology

This section contains the empirical approach. First, we set out the model and explain the estimation procedure with special focus on the filter algorithm of Hamilton (1989). Hereafter, we describe our approach step by step.

3.1

Model

We consider both univariate and bivariate Markov Switching models for our analysis. The general setup is   y1,t y2,t  =   a1,St a2,St  +   φ1,1,1,St φ1,1,2,St φ1,2,1,St φ1,2,2,St     y1,t−1 y2,t−1  + · · · +   φr,1,1,St φr,1,2,St φr,2,1,St φr,2,2,St     y1,t−r y2,t−r  + εt, εt ∼ N  0,   σ2_1,St σ1,2,St σ2,1,St σ 2 2,St    , (11)

where yt denotes the log returns and St, St= 1, . . . , k, the state variable. Note that we do allow

for regime switching in the autoregressive terms and do not add additional regressors for both feasibility and simplicity. Moreover, if it is not necessary to correct for serial correlation, it is better to exclude the autoregressive lags from the specification. In addition, leaving them out facilitates interpretation as the constant reflects the mean. The switching process is governed by a first-order Markov chain with either constant transition probabilities,

pi,j = P[St = j|St−1 = i, . . . , St−r, Ft−1] = P[St = j|St−1 = i], i, j = 1, . . . , k, (12)

or time-varying transition probabilities, pi,j,t =

exp(αi,j+ x0t−1βi,j)

Pk

q=1exp(αi,q+ x0t−1βi,q)

, i, j = 1, . . . , k. (13)

In the constant case we have to impose the restrictionPk

q=1pi,q,t = 1 and hence there are k(k−1)

probabilities to estimate. The time-varying probabilities are driven by economic variables xtin a

multinomial logit form, which keeps the probabilities between 0 and 1. Besides, the multinomial logit specification has one redundant state, such that we can simplify expression (14),

pi,j,t=         

exp(αi,j + x0t−1βi,j)

1 +Pk−1

q=1exp(αi,q+ x0t−1βi,q)

, for j ∈ {1, 2, . . . , k − 1}

1

1 +Pk−1

q=1exp(αi,q+ x0t−1βi,q)

, for j = k

, i = 1, . . . , k. (14)

Given that there are l variables in xt, we have to estimate k(k − 1)(l + 1) parameters in case of

are only subtle, but for more than two states the former is preferred, since the probabilities resulting from the normal distribution do not sum to one by construction and as a result another transformation is necessary.

3.2

Estimation

Now that the model has been set up, it can be estimated. An important part of the estimation

procedure is to obtain the joint conditional probabilities of {St−m}rm=0, for which an iterative

procedure known as the Hamilton (1989) filter is used. The joint probabilities can be used to compute the smoothed joint probabilities, but more importantly it leads to the elements of the log likelihood.

Following Hamilton (1989), we carry out the algorithm for a specification with r autore-gressive lags. The key is to obtain the joint conditional probability at time t,

P[St = st, . . . , St−r+1 = st−r+1|Ft], (15)

from the joint conditional probability at t−1. In the first step we use the independence property of a first-order Markov chain to compute the predicted conditional probability conditional on

Ft−1,

P[St= st, . . . , St−r+1 = st−r+1|Ft−1] = P[St= st|St−1 = st−1]

× P[St−1 = st−1, . . . , St−r+1= st−r+1|Ft−1].

(16)

The conditional likelihood f (yt|St = st, . . . , St−r = st−r, Ft−1) is calculated by plugging in

the conditional mean and variance per state in the assumed distribution. The result of the

previous step and the conditional likelihood lead to the joint conditional likelihood of yt and

(St, . . . , St−r),

f (yt, St= st, . . . , St−r = st−r|Ft−1) = f (yt|St= st, . . . , St−r = st−r, Ft−1)

× P[St= st, . . . , St−r = st−r|Ft−1].

The third step is to calculate the conditional likelihood of yt by summing over the states,

f (yt|Ft−1) = k X st=1 · · · k X st−r=1 f (yt, St= st, . . . , St−r = st−r|Ft−1).

In the fourth step we calculate,

P[St = st, . . . , St−r = st−r|Ft] =

f (yt, St= st, . . . , St−r = st−r|Ft−1)

f (yt|Ft−1)

, and finally this leads to,

P[St= st, . . . , St−r+1 = st−r+1|Ft] =

k X

st−r=1

Note that the filter operation is substantially simplified when no autoregressive lags are in-cluded.

There are several ways to set the initial regime probabilities. As initial transition

probabil-ities we take pi,j = 1/k for i, j = 1, . . . , k. In the case of constant transition probabilities, the

steady-state values are used as initial regime probabilities, while in case of time variation we use the transition probabilities associated with the first observation. We optimize the parameters θ by ML over `(θ) = log(yT, . . . , yr+1|yr, . . . , y1) = T X t=1 log f (yt|Ft−1),

where we do not impose any restrictions on the mean and variance of the states.

After optimization we calculate smoothed marginal probabilities conditional on all infor-mation by recursion. First, we obtain the predicted joint probabilities,

P[St= st, . . . , St−r = st−r|Ft−1] = P[St−1 = st−1, . . . , St−r = st−r|Ft−1]

× P[St= st|St−1 = st−1],

and by using the predicted probabilities from (16) we find the smoothed joint probabilities,

P[St= st, . . . , St−r = st−r|FT] =

P[St = st, . . . , St−r+1 = st−r+1|FT] × P[St = st, . . . , St−r = st−r|Ft−1]

P[St= st, . . . , St−r+1 = st−r+1|Ft−1]

. As a result, the smoothed marginal transition probabilities follow by integrating out lagged state variables, P[St = st|FT] = k X st−1=1 · · · k X st−r=1 P[St = st, . . . , St−r = st−r|FT].

3.3

Approach

In our analysis we consider both gold and silver, but our main interest will lie in gold, because we expect the best results for this commodity due to the data availability and the fact that only a few stocks primarily focus on silver mining. The analysis of silver, however, will be used to validate the results of the gold analysis. Except for some macroeconomic variables, we have daily data available, but due to a capricious portfolio price and computational burden we use a weekly frequency. In the first instance we apply the LT- and BB-algorithms to the commodity price and the portfolio price of stocks operating in that particular commodity. This approach is fairly limited, but it is a useful way to get a first impression of the bull and bear regimes in the series.

Subsequently, the Markov Switching models come into view. First, we concentrate on uni-variate models with constant transition probabilities. Based on the maximization of the log likelihood and minimization of several selection criteria we choose a number of regimes and analyze the corresponding model. Then, we estimate the model again using time-varying tran-sition probabilities, which will be driven by economic variables that are described in Section 4.3. Next, we move towards multivariate models. This allows us to investigate whether similar regimes are present in both series and by using a VAR-specification we can test for Granger causality among other things.

As mentioned, we apply the same but less extensive analysis to the silver price and the corresponding portfolio of stocks. We will note the key similarities and differences between the results. Besides, we perform multiple robustness checks on the composition of the portfolio, such as by removing the weight constraints or by using equal weights. Also, we test the robustness of our results by selecting a different sample and by using a daily instead of a weekly frequency.

4

Data

This section describes the data. It is explained how the individual stock prices are converted into a portfolio price. Besides commodity and stock prices we use several economic variables in our analysis. These variables are usually available at a lower frequency and hence we have to make assumptions. Furthermore, we transform some variables to ensure stationarity.

4.1

Commodities

Due to limited stock data we set the sample period from April 1, 1980 up to and including February 28, 2017. The daily closing price of gold has been obtained from the World Gold

Council6 _{and is reported in US dollars. The World Gold Council is a market development}

organization whose purpose is to stimulate and sustain demand for gold. The largest gold mining companies are members of this organization. USAGOLD, which is a consultancy company for gold and silver investors, is our source for the silver price, which is reported in US dollars. However, we encountered some missing data for which we had to find a solution. First, we have used eye-balling to fill in the first three months of 1995, whereby we use a historical chart

from TradingCharts.7 _{Second, we have taken the COMEX settlement price for the missing}

months January and February 2012 and January to May 2013.8 The gold and silver prices are

6_{Source: gold.org}

7_{Source: http://futures.tradingcharts.com/historical/SV/1995/0/continuous.html} 8_{Source: http://www.pyromet999.com/daily-silver-price.html}

spot prices, such that we do not have sudden jumps of rolling from one futures contract into

another. In our analysis we compute the returns, say rt, as rt = 100 log(Pt/Pt−1), where Pt is

the price of the commodity or portfolio. For easy referencing, we simply refer to them as returns instead of log returns.

4.2

Portfolios

For each commodity we create a portfolio of stocks that are linked to the commodity. All stocks have to be listed on the New York Stock Exchange (NYSE) for several reasons. First, the NYSE lists the largest amount of stocks and therefore allows us to obtain longer time series for more stocks. Second, all commodity prices are reported in US dollars and hence we do not have to deal with differences in valuta. Third, all stocks are traded in the same time frame and have almost analogous non-trading days. Due to data limitations we cannot construct a balanced portfolio. The earliest stock data go back to 1980, but the number is limited. When we start the sample some years later we throw away useful information and, more importantly, have less data available to estimate the characteristics of market regimes.

The portfolio weights are based on the fraction of the total market capitalization, which is the sum of the individual adjusted market capitalizations. Not every company is exclusively operating in a particular commodity market and hence we cannot use the overall market cap-italization. For instance, stock A has twice more market capitalization than stock B, but only thirty percent of his operations are related to that commodity while B is fully operational in that market, then A gets too much weight in our portfolio. We deal with this issue by multiply-ing the market capitalization by an earnmultiply-ings ratio, which is computed by dividmultiply-ing the earnmultiply-ings corresponding to the commodity by the total earnings. We derive this ratio from the most recent annual report of the company and assume that it is constant over the full time span. Due to time and data constraints we have to make this strong assumption, but its influence is rather limited. The market capitalization is calculated as the price level multiplied by the number of shares outstanding, which are obtained from YCharts. For some stocks the number of outstanding shares is missing in the beginning of the sample. In order to solve this issue we extrapolate the series by fitting a polynomial through the data. It is not necessary to drop the corresponding prices for it involves only a couple years for a few stocks and, besides, the number of shares is constant most of the time.

Gold

The portfolio for gold consists of 23 stocks from the end of 2010 until the end of February 2017. However, there are less stocks available in the early years of our sample. When we compute the

Figure 3: Commodity price and stock portfolio price (a) Gold

1985 1990 1995 2000 2005 2010 2015

0

500

1000

1500

$ 2000

0

5

10

15

$ 20

Commodity

Stocks

1985

1990

1995

2000

2005

2010

2015

0

10

20

30

40

$ 50

0

5

10

$ 15

Commodity

Stocks

portfolio weights, our portfolio is dominated by a few stocks in the twentieth century. Even in recent years the portfolio weights of a single stock could be higher than twenty percent. This is regarded as an unattractive feature and hence we impose an upper bound on portfolio weights. Nevertheless, the maximum weights should vary over time because a barrier of, say, twenty percent would not make sense in the years that the portfolio only consists of five stocks. By

trial and error we set the restricted weight wres

t at time t according to the following formula:

wres_t = 0.1 − 1

We cannot use the restricted portfolio weights immediately since the portfolio weights do not sum to one anymore. Therefore the revised portfolio weights are computed by dividing by the new total portfolio weight, otherwise it would not have made any sense to use a higher max-imum when the number of stocks in the portfolio increases. After imposing the restriction, Newmont Mining and Gold Fields Limited only determine approximately sixty percent of the portfolio price in the early years of the sample instead of over ninety percent. In recent years

there are five heavyweights9 in the gold market, that influence over eighty percent of the stock

price. By imposing the restriction their combined share falls back to sixty percent. We believe the advantages of this approach surpass the drawbacks. The portfolio is not composed in the usual way, but the portfolio price is to a lesser extent driven by the performances of individual stocks such that it is a better representation of the market of stocks operating in gold mining. Figure 3a shows the resulting portfolio price compared to the gold price over the time span. The descriptive statistics of the individual stocks are reported in Table C.1 in the Appendix. Ad-ditionaly, Figure C.1 plots the revised portfolio weights over time. As one can see, the portfolio becomes more diversified when time passes. As a result we have more confidence that the port-folio price of the last ten years maps the market portport-folio for gold stocks than the first ten years.

Silver

The portfolio of silver mining stocks consists of three stocks in the first fourteen years, evolves to nine stocks in 2010, and then stays constant till the end of our sample. Again we encounter extreme portfolio weights and in order to prevent this we restrict the portfolio weights according to formula (17). Figure C.2 plots the portfolio composition. Up to 2005 the portfolio price is driven by only four companies, but in recent years the portfolio is more diversified. Figure 3b shows how the portfolio price relates to the silver price. We observe erratic behaviour in the first years which is probably a result of the small number of stocks in the early years. Descriptive statistics of the included stocks are reported in Table C.2.

4.3

Predictor variables

If we want to allow for time variation in the transition probabilities, then we need variables that drive the probabilities. One way to proceed is to use lagged versions of the dependent variable; another is to use macroeconomic variables. The first is the easier approach but the second seems more valid. We basically follow Kole and Van Dijk (2017) for the selection of the predictor variables. Except the dividend-to-price (D/P) ratio, all data series are obtained from the Federal Reserve Bank of St. Louis (FRED) database. For each variable we test for the

Table 1: Descriptive statistics

Variable Freq. Obs. Mean St. Dev. Min 25% Median 75% Max Skew Kurt.

Commodity: Gold W 1925 0.049 2.405 −13.790 −1.195 0.073 1.287 14.694 −0.020 4.001 Silver W 1925 0.011 4.279 −26.716 −2.008 0.000 2.142 24.009 −0.289 4.781 Portfolio: Gold W 1925 0.068 5.535 −35.451 −2.996 0.007 3.183 28.601 0.006 2.603 Silver W 1925 -0.030 6.959 −50.920 −3.644 -0.207 3.735 41.401 0.029 4.159 Predictors: 3-month T-Bill W 1926 4.368 3.558 0.000 0.948 4.640 6.080 16.760 0.757 0.491 Inflation M 443 0.253 0.285 −1.770 0.129 0.233 0.371 1.377 −0.574 7.487 Credit spread W 1926 1.102 0.478 0.000 0.780 0.960 1.290 3.470 1.808 4.112 Term spread D 9631 1.634 1.873 −16.305 0.890 1.960 2.780 5.410 −3.173 17.077 Unemployment M 443 6.372 1.627 3.800 5.150 5.900 7.400 10.800 0.695 −0.319 Industrial growth M 431 2.024 4.136 −15.429 0.472 2.543 4.275 12.184 −1.240 3.444 D/P ratio M 442 0.330 0.144 0.140 0.222 0.271 0.412 0.789 1.008 0.326 Dollar index W 1926 93.004 15.162 68.147 82.690 90.582 99.412 146.410 0.928 0.733 Notes: The second column denotes the frequency of the data series, i.e. D for daily, W for weekly and M for monthly data. The descriptive statistics for gold and silver are based on returns.

presence of a unit root by applying an Augmented Dickey-Fuller (ADF) test. The number of included autoregressive terms is based on the Akaike Information Criterion and the inclusion of a constant, a trend or none of them is based on the shape of the series.

The daily 3-month Treasury Bill rate is used as a proxy for the risk-free rate. The null hypothesis of the ADF statistic is the presence of a unit root and hence non-stationarity is rejected for the 3-month T-Bill with a p-value of 0.078. Following Kole and Van Dijk (2017), the series are transformed by subtracting a 40-week moving average and for the resulting series the ADF test rejects. The growth in industrial production is constructed by taking the yearly change of a monthly index for industrial production. We find no evidence for the presence of a unit root, but we still have to transform the data to a weekly frequency. Instead of assuming the growth rate to be constant within a month, we interpolate linearly between months for it makes the series smoother. Regarding the unemployment rate, the presence of a unit root is rejected for a 5% significance level. Although it does no harm to transform the data, we decide to leave it untouched. As a measure for the inflation rate we use the month-to-month change in the Consumer Price Index (CPI), which is stationary for a 1% significance level. We compute the term spread as the difference between the long- and short-run risk-free yield, or more specifically, the difference between the 10-year bond yield and the 3-month T-Bill rate. The ADF test shows that the presence of a unit root is not a concern. The credit spread is computed as the difference between the yields of Baa and Aaa corporate bonds and is stationary for the p-value of the corresponding ADF statistic is 0.002. The D/P ratio is also relevant for

the gold price, because one of the striking differences between stocks and commodities is the payment of dividends. Gold generates no cash flow and hence a higher D/P ratio would make stocks more attractive for investors. The D/P ratio is computed as

D/P_t=

P12

j=1Dt−j

Pt

,

where Dt and Pt are the dividends and price level of the S&P 500 index in month t, which are

obtained from Shiller10. After taking first-differences the series is stationary for a 1% significance

level. Again we transform the monthly data to a weekly frequency by linear interpolation. Finally, we add the US Dollar Index to our list of predictor variables. The index measures the value of the US dollar relative to a basket of major currencies. This measure is a relevant predictor for precious metals, especially gold. Academic studies have argued that there is a negative relationship between gold and the dollar. For gold is mostly traded in US dollars, a weaker dollar makes gold cheaper for other countries and hence the gold price increases as a result of higher demand. In order to create a stationary series, we take the percentage change with four weeks ago. Table 1 presents the descriptive statistics of the data, where the predictor variables are untransformed.

5

Results

In this section we present and analyze the results. As explained in Section 3.3, our focus will be on the analysis of gold. The rule-based methods provide a first impression for the next step in which we estimate various Markov Switching models. Hereafter, we carry out the same but less extensive analysis for silver. Finally, we investigate the robustness of our results by making some alterations.

5.1

Gold analysis

In order to get a feeling of the bull and bear markets in the gold market, we employ the LT- and

BB-algorithm, which are discussed in Section 2.2. Regarding the LT-algorithm we set λ1 = 25%

and λ2 = 20% for the commodity price. The reason for an asymmetric filter is the upward trend

in the gold price, but this does not seem to be necessary for the portfolio price. Due to the erratic behaviour of the portfolio price we have to use higher λ’s. Given that the LT-algorithm is based on price changes, it is hard to improve the regime classification of the portfolio price. A higher filter will surely remove some switches, but is harder to justify since 30% is already

10

Figure 4: Gold analysis – Regime classification by LT- and BB-algorithm

(a) Commodity – LT (b) Portfolio – LT

(c) Commodity – BB (d) Portfolio – BB

Notes: The red-shaded regions indicate a bull market. The plots on the left side (a, c) show the bull and bear classification for the gold price and on the right (b, d) are the plots for the portfolio price of gold mining stocks. The LT-algorithm uses a (25,20)-filter for the commodity price and a (30,30)-filter for the portfolio price. The BB-algorithm uses τwindow = 32, τcensor = 13, τcycle= 70, and τphase= 16 weeks for both prices.

an extreme price change. The BB-algorithm, however, is a better tool to indicate bull and bear regimes, because the compution is based on duration instead of price changes. Nevertheless,

a drawback is that there is a certain delay based on the choice of window length τwindow. For

instance, if τwindow is three months, then the BB-algorithm can only tell three months after a

certain time t whether t was part of a bull or a bear phase. The LT-algorithm is less sophisti-cated and more realistic, because, when the price crosses a certain level, there is a regime switch.

For the BB-algorithm, we follow Kole and Van Dijk (2017) by setting τwindow = 32, τcensor = 13,

τcycle = 70, and τphase = 16 weeks. We do not use different parameters for the portfolio price, because the algorithm does a good job in filtering out irrelevant price movements. The regime classification of the gold price and portfolio price using the LT-algorithm are given in Figure 4a and 4b, respectively. Similarly, the plots of the BB-algorithm are shown in Figure 4c and 4d.

We can learn a number of things from the graphs of Figure 4. A comparison between the up-per and lower charts clearly shows the differences between both approaches. The BB-algorithm accurately indicates regime switches at local extrema, while the LT-algorithm needs some time to know a regime switch has taken place. The latter is more realistic in practice, but the for-mer might be more relevant for our analysis. Another point is the smaller amount of regime switching in the BB-algorithm compared to the LT-algorithm. The BB-algorithm deals better with the directionless behaviour of the portfolio price in the first years of our sample. From Figure 3a we notice that the commodity and portfolio price behave largely in the same way. It is no suprise that the gold price is less volatile and less affected by crises, but the big picture is much alike. Besides, this follows from graphs were the portfolio price exhibits more and hence shorter regimes.

Table 2 shows the average statistics of the bull and bear markets for each series and algo-rithm. The average number of weeks spent in any regime is clearly higher in the case of the gold price. Especially for the BB-algorithm, there is a difference between the average weekly returns of bull and bear markets, however the standard deviation of bull and bear markets is almost the same. Generally speaking, in stock markets bull states have lower volatility than bear states, but gold has a positive return-volatility relationship. The similar standard devia-tions could indicate that the classic properties of bull and bear markets do indeed not hold in the gold market. On the other hand, the statistics do not confirm a positive return-volatility relationship.

The next step is to involve Markov Switching models. As explained, we do not want to include AR terms if it is not necessary. Therefore, we start with the univariate version of model (11), in which we only include an intercept in the mean equation. The number of states in the gold market is unknown and hence we use several tools to choose this number, namely the log likelihood and the Akaike (AIC), Bayesian (BIC), and Hannan-Quinn (HQIC) information criteria. Note that a higher value of the log likelihood is preferred, while we want to minimize the information criteria. In Table 3a we present the outcomes of these measures for a different number of regimes in gold returns. We do not allow more than six regimes given computational burden and the lack of a theoretical explanation. For the single regime model we estimate an AR(1) model, where the lag order is based on the minimization of information criteria. The results clearly reject the existence of one or two regimes. The information criteria choose three states, followed by four. The log likelihood is barely different after the third regime, but it still improves towards the optimum of six regimes. We choose to consider four states as it is a middle ground between log likelihood and information criteria.

A similar strategy is followed for the gold portfolio, for which the results are shown in Table 3b. The choice is less obvious than in the previous case. We can firmly ignore the linear

Table 2: Gold analysis – Regime statistics of rule-based methods

Commodity Portfolio

LT BB LT BB

Bull Bear Bull Bear Bull Bear Bull Bear

Duration 99.11 129.13 85.30 115.33 57.61 26.55 64.64 76.23 Mean 0.130 −0.039 0.928 −0.356 0.070 −0.811 1.191 −0.894 St. dev. 2.541 2.861 2.842 2.408 5.636 6.877 5.368 5.311 Min −7.190 −7.314 −5.433 −7.647 −11.664 −14.113 −13.758 −14.888 25% −1.255 −1.790 −0.751 −1.790 −3.119 −4.028 −2.229 −3.901 Median 0.221 0.093 0.584 −0.223 −0.317 −0.025 0.816 −0.631 75% 1.723 1.586 2.513 1.137 2.935 3.467 4.570 2.091 Max 6.944 8.589 8.263 6.147 14.436 10.591 15.029 13.876 Skewness −0.067 0.315 0.320 −0.227 0.351 −0.228 0.004 0.017 Kurtosis 1.486 2.841 1.182 1.679 1.006 0.925 0.990 1.127

Notes: The statistics are averages of the individual phase statistics for both the bull and bear regime. Duration is measured in weeks.

model, but the other regimes all have some advantages and disadvantages. If we prefer minimal information criteria, a two-state specification would be the choice. However, the log likelihood value in this case is one of the worst and is maximized with six regimes. On the other hand, the bad ranking of the log likelihood for three states is rather misleading, since the improvement is only small. Again we would rather be neutral and hence choose three states.

At this point we want to look more closely to the results of the univariate Markov Switching models for the chosen regimes, but we encounter a problem. Regimes are expected to switch from time to time and the transition probabilities should at least be higher than 0.5 for some times in the sample period, otherwise it would be wise to reduce the number of regimes. A graph of the smoothed transition probabilities of gold returns with four states is shown in Figure A.1. An undesired pattern occurs for state 2 and 3 in the sense that the transition probabilities are never close to 1 and both states seem to be present at the same time. It is probably better to consider three states such that state 2 and 3 can be combined. Figure 5a plots the case of three regimes. State 1 and 3 correspond to state 1 and 4 in Figure A.1, such that state 2 is a combination of state 2 and 3 in Figure A.1. The pattern of the smoothed transition probabilities changes considerably and supports our decision to consider three regimes.

A similar problem is detected in the smoothed transition probabilities of the gold portfolio returns with three states, which are shown in Figure A.2. Most of the time the ‘true’ state seems to be a combination state 1 and 2. When we move to a two-state specification, we obtain