The Global Business Cycle within the New Commodities and the Financial Cycle: An Empirical Evidence Based on a Multivariate Unobserved Components Model (UCM)

Abstract

The recent analysis focused on gold and energy as hedge fund and channel for the cyclicity of the economic system. Within the occurrence of the prominent crisis, the financial cycle became a nexus relation with the real sphere. The aim of this chapter is to investigate the impact of the commodities and the stock market on the global business cycle within filtering based on HP and Band Pass filter for the period Q1–1984 to Q4–2020. We review the literature on the business cycle, commodities, and financial cycle research. We find which HP trend is close to the estimated trend of global GDP based on a multivariate UC model. We provide a Band-Pass filter to examine the upswing and downswing commodities cycle as well as the financial cycle. We construct a multivariate unobserved component model that includes the gold price, the crude oil price, and the financial cycle as the independent component within the model. The estimation of the model found that oil has a lag impact on the fluctuation of the business cycle, it reacts as an impulsion for the crisis, compared to the gold and the global financial cycle which reacts simultaneously with the trend of the global business cycle.

1. Introduction

The vulnerability of financial stability toward any risk might arise shed light on the linkage between global demand and financial stability as a key factor that influences the economic system and the trend of the business cycle. Studying the business cycle is the top macroeconomic topic which is interlinked for any economic and non-economic issue that might arise. The concept of the business cycle is a macroeconomic concept based on different definitions with reference to the context referred to, such as the macroeconomic variables investment, inflation, consumption, GDP, and production. Within this study, we will refer to the business cycle within the GDP. Schumpeter [1] is initiating the concept of the business cycle through the innovation cycle in the economy as “innovation clusters”. The long cycle in the economy is complemented by Mensch [2], Haustein and Neuwirth [3], Van Duijn [4], and Kleinknecht [5] by further empirical studies.

Clarke et al. [6] pointed out an increase in the number of critical innovations during the recovery of the long cycles. Singer [7] has complemented the theory of the long cycle by finding that the trade gaps between standardized and innovative products follow a deteriorating trend. The application of these studies in the commodities economy has brought important literature with different approaches. The fluctuation of the long cycle and the commodities have lay an important role in the variation of the intensity of use of productive inputs, where the commodities such as precious metals and energy have played a relevant role. For the expansion phase, the expanding demand for precious metals drags prices upwards. Respectively in the new period of stagnation, the intensity of demand drops as well as the prices of precious metals. It is by analogy applying to the energy sector as well as the financial market. Studying the long cycle of commodities and the financial cycle and their connection to the global economic dynamics is an area to be more and more explored, as these sectors are open for more interconnected development.

Within this context, this chapter is novel as it might contribute to an area of limited research, which is less studied by combining the commodities market and the financial market. The macroeconomic impact of the commodities on the business cycle has been studied in isolation or within the monetary policy but not in conjunction with the standards of the business cycle. Studying the interlink of the commodities and the business cycle is a big literature with different empirical approaches, studies, and theoretical focusing on detecting the breaks of the time series components.

Therefore, decomposing and forecasting the business cycle within the commodities and the stocks frame can be efficient for policy decisions and the synchronization of the interference and the trade-off between the concerned targeted policies. The business cycle is considered as the main framework for the evaluation of the economic system in general and the policymakers. We attempt to interlink the financial sector and commodities with the business cycle to give a more-clear overview about the economic system in general. The interdependence between the business cycle and the financial cycle was with a prominent literature taking the credit gap and house prices as a reference for the financial sector [8, 9, 10, 11].

Within our study, we are referring to the stock prices for the financial cycle. Several literatures treat the stock market and the commodities apart for the interlink with the business cycle of the economy. The study of the volatility of the business cycle allows for more vulnerability toward any economic and extra-economic shocks such as the buoyant of the stock market and the pro-cycle of the commodities.

Within this chapter, we aim to investigate the impact of the energy commodity e.g., the crude oil, and the non-energy commodity e.g., the gold and the financial cycle on the global business cycle. It is a univariate unobserved component model followed by an extension to the multivariate model using the commodities and the financial cycle to decompose the macroeconomic aggregates. The eventual connection between the long cycle of the commodities and the financial cycle with the economic cycle considers the further relationship between the business cycle and the demand for the essential commodities selected as well as the proxies of the financial cycle. In economic theory, the long cycle classified with different groups basic on different factors to study the cyclicity of the economy for different horizons: the business cycle (2–8 years), the Kitchin cycle (3–5 years), the Juglar cycle (7–11 years), the Kuznet swings (15–25 years), the medium term business cycle, and long economic cycle wave or the Kondratiev wave (K-wave) currently recognized with the period of 45–60 years [12]. Within this study, the chosen period is 36 years, quarterly data, dated from 1984-Q1 to 2020-Q4 studying the global business cycle.

The chapter is structured as follows: In the first section, we decompose the trend of the commodities cycle using the filtering approach. The second section is dedicated to analyzing the financial cycle for the selected global assets. The third section is devoted to the modeling using the univariate and multivariate approach for the Unobserved Components Model. The fourth section presents the estimation of the model and reports our numerical results. The last section contains a discussion of further policy recommendations and concluding remarks.

2. Related literature

Our study constructs on an extended literature, which is focus on the business cycle, the financial cycle, the energy commodities such as oil, and the non-energy commodities such as gold.

The interlink between the financial sphere and the real sphere within the economic activity, shed the light to any macroeconomic or monetary policy is conducted. Several documented studies highlight the interaction of the financial cycle and the business cycle within the occurrence of the multi-type of crisis. Helbling et al. [13] provide an analysis of the interlink between the credit market and the global business cycle naming that credit shocks play an important role in shaping the US business cycle. Drehmann et al. [14] provide a justification on the finding of the prior studies that financial cycles are longer and more ample than business cycles. Claessens et al. [10] report that there are strong links between different phases of business cycles and financial cycles. Recessions associated with financial disruption tend to be longer and deeper than other recessions and recoveries associated with rapid growth in credit and house prices are often stronger. Gerdrup et al. [15] and Detken et al. [16] argue that the average length of the financial cycle is around four times that of the business cycle.

In the same perspective Borio [8] argues that the financial cycle and the business cycle are find out within three properties. The financial cycle is most parsimoniously described in terms of credit and property prices, it has a much lower frequency than the traditional business cycle (2–8 years), its peaks are closely associated with the financial crisis, and it helps to detect financial distress risk in real-time. Stremmel [17], and Stremmel and Zsamboki [18] argue that the financial cycle is less synchronized in tranquil periods and more synchronized in a period of common financial stress. Gorton and Ordonez [19] show that not all credit booms are followed by financial crises. Schüler et al. [20] find that financial cycles exhibit higher amplitude and persistence than business cycles. Rünstler and Vlekke [21] find out that the financial cycle is heterogeneous across European countries having a much longer and more ample financial cycle. The link between the business cycle, the movement in dividends, stock prices have been studied extensively in the macroeconomic and asset pricing literature such as in the works of Lucas [22] and Blanchard [23].

Rangvid [24] finds that the stock price-output ratio is a predictor of expected US stock returns. Cooper and Priestley [25] argue that the output gap has in-ample and out-of-ample predictive power for stock returns in the US and other G7 countries. From the same perspective, Vivian and Wohar [26] analyze whether the US output gap predicts the return of portfolio formed on size and value. Gold is a traded asset globally as an alternative investment class to the ordinary portfolio comprising stocks and bonds. Baur and McDermott [27] argue that gold is a stabilizing factor for the financial system since it minimizes losses for market participants and portfolio managers in the event of negative market shocks. Beaudry et al. [28] argue that the metaphor of profit-driven by fluctuation called gold rushes provides a period of economic boom associated with expenditure aimed at securing a claim near a new found vein of gold.

Pierdzioch et al. [29] found that the international business cycle has out-of-ample predictive power for gold price fluctuations. Apergis and Eleftheriou [30] found that the business cycle asymmetrically affects gold returns, while these returns respond stronger during the recessionary than booming phases of the cycle. Within the same perspective applying to several types of precious metals such as gold, silver and platinum, Kucher and McCoskey [31] found that the co-integrating relationships between precious metal prices are not stable over time with significant shifts in the price relation around business cycle peaks and during recessions. Within the last decade, after the oil shocks and petroleum crisis, the constraint of adjustment within the capital market has probably less influenced by the size and level of stock trading which for some countries targeted their policies for the optimum arrangements for their portfolios. For the policy economic makers, they have to take their decisions on the volume and pricing of oil with the desired level of oil revenues and their use (consumer goods, investment goods, and financial investments). The interlink between oil as a main key for the financial sphere and the real sphere has been documented in several studies on the impact of the volatility of pricing oil on the business cycle. The oil is acting as the hedge fund for several investments, the interlink with the cyclicity of the economic system as a source of the business cycle is a topic for a prominent literature review.

Hamilton [32] is shedding light on oil as a source of the business cycle in the case of the US, by finding out that an increase in oil is leading to a recession in the US. The volatility of oil prices such as an increase has a positive effect on the output of the exporting countries [33]. The prominent works by Estrella and Hardouvelis [34], Estrella and Mishkin [35], Chauvet and Potter [36], Nyberg [37], and Ng [38] studying the impact of oil price shocks on business cycle fluctuations, by modeling the probability of the recession and the Probit and Logit models as binary dependent variable models. The main result of these models is identifying the term spread and stock market returns as useful predictors of US recessions. Michael [39] found that shocks of the oil price explain the reduced fraction of the real GNP growth and inflation variance in US and Japan. Sadorsky [40] found that oil price volatility shocks have asymmetric effects on the market activity. From a different empirical approach Elder and Serletis [41] argued that oil price volatility has a negative and statistically significant impact on several measures of investment, durable consumption, and aggregate output.

Moreover, using a GARCH-in mean empirical method, Elder and Serletis [41] found that the volatility in oil price shocks has a negative and statistically significant effect on different measures of investment, durable consumption, and aggregate output. Jo [42] showed the negative effect of oil price uncertainty shock on world industrial production using a quarterly vector autoregressive model with stochastic volatility in mean. In the same perspective, [43] use the oil supply and demand shocks to estimate the US dividend yields components. Using a measure oil market uncertainty Yin and Feng [44] studied the dynamic relationship between oil market uncertainty and international business cycles, the authors have found that oil market uncertainty has a linear leading effect on international business cycles. Pönkä and Zheng [45] studied the role of oil prices in forecasting the Russian recession period using a Probit model. The author suggests that fluctuations in nominal oil prices are useful predictors of the Russian business cycle, with the term spread turning out to be the most powerful predictor of future recessions.

Our contribution to the literature by summing up the three sources of international business cycles such as the financial cycle, the gold and the oil on the trend of the international business cycle, is the use of the Unobserved Component Model Univariate and Multivariate based on the significant studies of Polbin [46], Grant and Chan [47], and Yoon [48].

3. The business cycle and the commodity

The Schumpeter theory Schumpeter [1] brings an overview of the business cycle, the trend of an economic system. According to Schumpeter [1], the business cycle is a sum of perpetual economic cycles or an overlapping cycle. His main theory is focusing on the business cycle within the process of creative destruction, for which the introduction of innovation boosts investment opportunities and creates economic growth and at the same time decay in the obsolete sector of production. This process is containing an expansion phase and a recession phase where the economy assimilates the innovation across sectors. The commodity prices are involved in the same perspective regarding the demand. Within the expansion phase, the competition for commodities product such as gold and energy tend to increase their prices compared to manufacturing goods, the introduction of innovation as imitation reduces the opportunities for investment to obtain economic rent, decreasing the demand for commodities. Schumpeter is among the economists who reject the frame that the decline in prices might be a result of a slow-down in terms of output and growth, as he explains that within the great recession (1878–1896) for the case of the falling of price due to a decline in the production of gold which result in profit squeeze and the decrease of the investment. From the same perspective as Schumpeter, for the economic phase of the commodities and the manufactured good on the business cycle, we find Prebisch [49], Singer [50], as well as Ocampo [51] and Ocampo and Parra [52] as prominent literature studying the commodities prices and the business cycles.

According to IMF five years ahead forecast for 2017, the expected long-term growth boosted by the boom of the commodity price has been revised down from 4 to 3%. The boom in the commodities prices has an increasing effect in the short run on the real GDP by raising the value and production and lifting the demand for ancillary goods and services. An increase in the investment in the resource sector, such as metal or energy, may raise the potential output, which in turn boosts the financial resource for the investment in the other sector. However, in the long term, commodities boosting growth is a controversial question. According to Corden [53], the positive term trade and income shock associated with the commodity boom shift production out of non-commodity tradable and into the non-tradable service sectors with lower productivity. The global economic crisis starting in 2012 was the result of the boom in commodity prices characterized by unprecedented magnitude and duration, as the price reached the highest level in history, this phase was characterized as a phase of mineral boom. In fact, within the depression of the global economy after the subprime crisis, which slow down the demand for the commodity price, however, the recover for the price was surprisingly fast and the world economy experienced a boom in commodity prices which might be seen as a continuation of 2004–2008. The upswing demand in commodity lifts the resilience of the growth performance of major developing countries and producers’ countries. Within the several literatures, it is tempting to believe that there are causality links between the business cycle in terms of output and commodities.

Among many econometric approaches such as the SVAR, UC, and VECM, prove the biased data with low-frequency movement. Fernald [54] noted that low-frequency movement in an hour per capita may bias the VAR model with long-run restriction. Differencing removes the low-frequency movement from the data [55]. In opposite Hamilton [56] affirms that differencing a bounded series may involve misspecification issues by suggesting a filtering approach prior to the estimation model [57]. However, Gospodinov et al. [58] found that filtering the data prior to the estimation removes necessary information to identify these stocks using long-run restrictions.

Within our study, we aim to estimate the model of the Univariate and Multivariate Unobserved Component Model using the HP filtering data and the Band-Pass filtering data. Filtering data, through the selected filters, with a filter window such that cycles are generated at such frequencies. The decomposition method isolates major fluctuations in the deviation of a macroeconomic variable, such as the GDP, around its trend through a combination of detrending procedures and smoothing techniques [59]. Within the literature, determining the long wave is also based on the filtering approach, such as in Baxter and King [60] and Christiano and Fitzgerald [61], the time series is considered as a summation of different frequencies and the filtering approach consists to determine the filter coefficients so as to isolate specific frequencies and to show the course of the pre-specified frequency component in the time domain. The choice of both filters is mentioned in the next section.

4. The HP filter

The Hodrick-Prescott filter has been the common use of filtering approach in many documented works studying the cyclicity of the economic system. Our first part of the study will study the real GDP and the trend through the HP filter. The trend of the data series $\left\{y_t\right\}$, $t=1,\dots ,T$, is the solution $\left\{x_t\right\}$ of the following minimization:

$$\sum _{t=1}^T{\left(y_t-x_t\right)}^2+\lambda \sum _{t=3}^T{\left({∆}^2{x}_t\right)}^2$$E1

T is the sample size and $∆$ denote the difference operator. $y_t$ is the log of real GDP and * is the potential or unobserved level

$$∆y_t^{\ast }=∆y_{t-1}^{\ast }+{\epsilon }_{0,t},{\epsilon }_{0,t}\sim N\left(0,{\sigma }_0^2\right)$$E2

$$y_t=y_t^{\ast }+{\epsilon }_{1,t},{\epsilon }_{1,t}\sim N\left(0,{\sigma }_1^2\right)$$E3

$$∆x_t=x_t-x_{t-1}\text{and}{∆}^2{x}_t=∆x_t-∆x_{t-1}=x_t-2x_{t-1}+x_{t-2}$$E4

The first component in Eq. (1) measures the error $y_t-x_t$, the second component measures the smoothness of the trend. ${∆}^2{x}_t,\lambda \ge 0$ is a regularization or smoothing parameter (or a turning parameter) that controls the trade-off between the size of the error and the smoothness of the trends. Eq. (1) is penalized the least square problem, penalizing the smoothness of its solution. The solution of the Eq. (1) is denoted as $x^{\mathit{hp}}\left(\lambda \right)\in R^T$. It is common to calibrate the variance error term so that their ratio is equal $\lambda =1600$, which correspond to a business cycle of 8 year, for quarterly data is not always appropriate.

Within our study we refer to the minimization of Eq. (1) to find the value of $\lambda$. Figure 1 denotes the real global GDP, shown in red and the trend shown in blue. At a further step later in the model we ask which HP trend is close to the estimated trend of global GDP from the multivariate UC model. The search for $\lambda$ in Eq. (1) is limited to positive integers. We find that the HP trend with $\lambda =\mathrm{540,000}$ minimizes Eq. (1).

Figure 1.

Global real GDP and the estimated smoothed trend, 1984:Q1–2020:Q4. The red line shows the global real GDP, and the blue line is the estimated smoothed trend. Source: Own study.

Figure 2 shows the global GDP from 1984 to the fourth quarter of 2020, which is our sample data in the study and the trend using the HP filter. Based on that filtering we have 6 downswings, 1984–1986, 1993–1996, 2001–2004, 2008–2012, 2015–2018, and 2020 and 4 upswings, 1987–1992, 1997–2000, 2005–2007, 2013–2014. The analyzed swings are short as the data sampled cycle is for 36 years, we have an average of 4 years of upswings and 3 years of downswings. We can understand from this figure, considering the second petroleum shock and the Asiatic, the Mexican, the Russian crisis and the subprime crisis, that we have an average of 4 years of upswing and an average downswing of about 3 years, for a total average cycle length of 7 years. The presence of several crises and monetary shocks impulsion as well as the petroleum shock are bringing down this average.

Figure 2.

The estimated smoothed trend and HP trend with $\lambda =54000$ of global GDP. Source: Own study.

In order to reframe the trend of the global GDP with a close trend to the smoothed estimated trend in Figure 2, we calculate the value of $\lambda$ based on the minimization of the Eq. (1). The blue line shows the HP trend with a smoothing parameter equal to 400,000. Making a comparison between Figures 1 and 2 for the trends we can understand that both trends are not identical, which we can affirm that a standard choice of $\lambda =16000$ for quarterly data is not appropriate to show the real trend.

5. The financial cycle

Within the literature, the financial cycle proxies are with several definitions such as the house-pricing asset, the interest rate, the stock market, and the exchange rate. The literature review opts for several approaches to measure the financial cycle within the connectedness with the business cycle. Drehmann et al. [14] propose two approaches to measure the financial cycle such as the turning point analysis and the Band-Pass filter using five financial variables such as the credit, the credit to GDP ratio, property price, equity price, and an aggregate asset price index, the study distinguishes a short term financial cycle and a medium-term financial cycle. Borio [8] recommended the use of credit and property price as the interlink between credit and saving and investment to measure the volatility and the cyclicity of the financial cycle.

Gorton and He [62] used real estate as collateral for credit and property price fluctuations affecting the credit and induce pro-cyclicality of credit and real estate price. The credit to GDP is considered as leverage measure in the macroeconomic context and an indirect indicator of the absorptive capacity of the financial system [63]. Regarding the stock market price, there is a controversial debate about a measure for the financial cycle. Drehmann et al. [14] argued that share prices do not fit as a proxy component of the financial cycle because they exhibit comparatively higher volatility at short-term frequencies and co-move far less with the other series. However, Schüler et al. [64] find that share price creates important common cyclicality with credit and residential price, while Tölö et al. [65] claimed that stock returns are considered as an early indicator warning of crisis.

In the same perspective of a combined measure Drehmann et al. [14], and Stremmel [17], use various financial measures such as the credit-to-GDP ratio, credit growth, and the ratio of house price to income. Drehmann et al. [14], Stremmel [17], Aglietta and Brand [66], Merler [67], Galati et al. [68], Schüler et al. [20] stress that despite there is no consensus on the best measure of the financial cycle that a composite financial cycle exploiting the co-movement of credit growth and house price is the best indicator of the systemic banking crisis.

The main goal of this chapter is to investigate the interconnectedness of the financial cycle and the global business cycle and not to provide a more developed measure of the financial cycle. Therefore, we opt on the chapter for using one indicator and not the composite variables such as the asset price indicator of three global assets such as the MSCI, S&P, and the FTSE. To measure the financial cycle, we construct an index of the combined three global assets based on the variable proposed by the literature such as the aggregate asset price index, all the series are normalized to Q1 1984, and in logs. As a first step, we use the filter band pass to filter the data and remove the low-frequency movement. We build the combined index of the financial cycle indicator by taking the average of the three filtered time series of the weighted stock prices. Figure 3 plots the cycle of each global asset apart obtained by the Band-Pass filter.

Figure 3.

The cycle of each global asset apart obtained by band-pass filter. Source: Own study.

The blue line in Figure 3 shows the FTSE world cycle, the green line shows the S&P world cycle, and the red dotted line shows the MSCI world cycle. For the MSCI and the FTSE, they follow the same trend with the same peaks, we can discern 4 peaks in 1990, 2000, 2008, and 2015 implying a peak-to-peak average cycle of 5 years. After the Asiatic and the Russian crisis, there are three peaks, 2000, 2008, and 2015, implying two cycles of 8 years from peak to peak. Regarding the S&P, they keep the same trend as the MSCI and FTSE with a lag of two years. There are short periods of cyclical movement within these cycles. Figure 4 shows the financial cycle for the global stock market based on the selection of three global assets with a Band Pass filter.

Figure 4.

The financial cycle for the global stock market. Source: Own study.

Figure 3 shows the financial cycle for the global economy based on the variable of the aggregate stock market returns for the main global asset. Based on that filtering approach we have four upswings, 1990, 2000, 2007, and 2014, and four downswings, 1991, 2003, 2009, 2016. The dipper downswing is in 2009, corresponding to the subprime crisis, as it is the biggest crash for the stock market compared to the other financial crisis.

6. The commodities trend: gold and energy

Gold and energy are interlinked to the financial system as a payment system and investment. The integration of the volatility of both commodities influences the business cycle within the demand and the investment. Using the band pass filter, we are aiming at removing the low frequencies of data. Figure 5 shows the data for real non-oil commodity prices such as the gold price and the oil commodity price such as crude oil.

Figure 5.

The cycle of the gold price and the crude oil price with band pass filter. Source: Own study.

The blue line shows the cycle of the crude oil, and the dashed red line shows the cycle of the gold price.

For the energy crude oil, according to Figure 5, we can discern eight peaks, 1986, 1990, 1997, 2001, 2006, 2008, 2012, 2014, implying a peak-to-peak average cycle length of around 5 years. However, related to the gold we realize that the number of peaks is different to the crude oil. We can discern four peaks, 1984, 1994, 2004, 2014, implying a peak-to-peak average cycle length of around 10 years.

7. The model

The Basic Unobserved Components Models GDP and Inflation.

The UC model is based on studying or capturing the prior trend about each component based on the approach of Structural Time Series (STs) developed by Engle [69], Gersch and Kitagawa [70], Harvey and Todd [71], and Harvey [72], or through the ARIMA model based on Box et al. [73], Burman [74], Hillmer and Tiao [75], Bell and Hillmer [76], Maravall and Pierce [77]. Most of the model-based approaches use a linear assumption with different statistical features specified for each model. In Clark [78], all shocks to the component of the UC model are assumed to be orthogonal. Morley & Nelson [79] found that the orthogonality assumption results in the over identification for UC models, the authors pointed out that in the UC model within constant growth rate and a cyclical component followed the AR(2) process, the correlation between the trend and cycle is estimable and the resulting negative estimate of the correlation leads to a substantially more volatile trend estimate and less volatile cycle estimate. Oh and Zivot [80] find out that based on the given AR(2) specification for the cyclical component suggest two feasible cases of the exact identification, the first, the trend shock and the cycle shock are allowed to be correlated but the trend growth rate shock is independent of the other two shocks and they are refereeing to this case as the trend-cycle; the second, the trend growth rate shock and the cycle shock are allowed to be correlated but the trend shock is independent from the other two shocks and they are referring to this case as the drift-cycle case.

The literature review studying the trend of the GDP within the UC model is diversified and based on different models within different statistical approaches. Morley & Nelson [79] found that within the feature data of the US, the dataset does not contain a sufficient amount of variation in the long-run growth rate. Ma and Wohar [81] estimate a multivariate UC model of output, consumption and investment with common trends and common cycles. Yoon [48] found in the case of US real GDP a sequence of mostly negative shocks, rather than a few extraordinarily large ones, are responsible for the change in the US real GDP trend. Morley & Nelson [79] showed that the difference between widely used trend-cycle decomposition such as in Beveridge and Nelson’s [82] decomposition and Watson [83] Unobserved Component Model is entirely due to one restriction imposed in the UC model, the correlation between the innovations to the trend and the cycle is assumed to be zero.

Without this restriction, Morley & Nelson [79] found that the two trend-cycle decomposition are identical, and both approaches yield to output gap estimated are noisy and small in amplitude. The filtering approach before the estimation is a well-recognized step in any econometric model dedicated to decomposing the data. The well-used filter HP [84, 85] for decomposition is often criticized as it might bring a biased result for the decomposition of the data, among others [84, 85, 86, 87, 88] while others affirm that the use of HP filter by removing the low-frequency movement in the data might lead to poor model fit and forecasts. Morley & Nelson [79] find out that the estimating of the US output gap using the correlated UC model is close to zero, however, by using the HP filter the corresponding estimate is as large as 3%.

To overcome the black box character of filtering and the lack of a proper statistical model for filters limit the importance of the filtering approach in terms of detecting the cases in which the filter is not appropriate for the series at hand, which brings as the main statistical overview that there is no systematic procedure to overcome the filter inadequacies. Filtering yields an estimator of the unobserved component. Within this part, we will focus on the Univariate Component model. The Univariate UC model [78], makes a distinction between “the smooth trend” and “the irregular trend” models. The developed setup of the Clark [78] model is as follows:

$$y_t={\tau }_t+c_t+{\epsilon }_t$$E5

$${\tau }_t={\tau }_{t-1}+d_{t-1}+w_t,w_t\sim i.i.d.N\left(0,{\sigma }_w^2\right)$$E6

$$d_t=d_{t-1}+u_t,u_t\sim i.i.d.N\left(0,{\sigma }_u^2\right)$$E7

$$c_t={\varphi }_1{c}_{t-1}+{\varphi }_2{c}_{t-2}+v_t,v_t\sim i,i,d.N\left(0,{\sigma }_v^2\right)$$E8

where ${\tau }_t$, $c_t$, and ${\epsilon }_t$ represent trend, cyclical, and seasonal components respectively. In Eq. (6) the trend component captures the productivity shocks that tend to have a permanent effect on the GDP. The trend component follows a random walk with a drift $\left(d_t\right)$. The trend and seasonal components are modeled by linear dynamic stochastic processes, which depend on disturbances. The drift $\left(d_t\right)$ follows another random walk as the trend and the cycle. The components are not deterministic, they are formulated to be allowed to change over time. The disturbances driving the components are independent of each other. The cyclical component captures the business cycle features of a time series and corresponds to deviations of the actual output from its long run or potential level. Within that model, we are referred to the stationary AR(2) process to model the cyclical component. There is a variety of stochastic specifications of the cycle component that can be considered. ${\epsilon }_t$ are the seasonal component of the model and they represent the seasonal effect at time t that is associated with season S. The trend component ${\tau }_t$ is modeled as a random walk process, $N\left(0,{\sigma }_w^2\right)$ refers to normally independently distributed series with mean zero and variance ${\sigma }^2$. The disturbance series is serially independent and mutually independent of all other disturbance series related to $y_t$. $w_t$, $u_t$, and $v_t$ are innovation to trend, trend growth rate, and cycle respectively. The model is rewritten as follows:

$$y_t=H{\xi }_t+{\epsilon }_t$$

$${\xi }_t=F{\xi }_{t-1}+{\eta }_t,{\eta }_t\sim i.i.d.N\left(0,Q\right)$$

$$H=\left(1100\right){\xi }_t=\left(\begin{array}{cccc}{\tau }_t& c_t& c_{t-1}& d_{t.}\end{array}\right)$$

$$F=\left(\begin{array}{cccc}1& 0& 0& 1\\ 0& {\varphi }_1& {\varphi }_2& 0\\ 0& 1& 0& 0\\ 0& 0& 0& 1\end{array}\right)Q=\left(\begin{array}{cccc}{\sigma }_w^2& {\sigma }_{\mathit{wu}}& {\sigma }_{\mathit{wv}}& 0\\ {\sigma }_{\mathit{wu}}& {\sigma }_u^2& {\sigma }_{\mathit{uv}}& 0\\ {\sigma }_{\mathit{uv}}& {\sigma }_{\mathit{uv}}& {\sigma }_v^2& 0\\ 0& 0& 0& 0\end{array}\right)$$

The generalized UC model univariate estimated by a variant of the Kalman Filter. The UC models are developed based on parametric models which are very close to AR, MA, or ARIMA model popularized by Box and Jenkin [89].

8. The extension: the multivariate unobserved component model

The unobserved component model is among the feature econometric models to detrend the cyclicity of economic activities. We are referring to our modeling of the Multivariate Unobserved Components Model to the model of Polbin [46] as he developed a model of Multivariate Unobserved Components for the growth, consumption and investment with a common growth and cyclical components, the oil price. Within this model, we are referring to the same methodology as in the proposed model. We use macroeconomic variables such as the GDP $y_t$, the consumption $c_t$, the investment $I_t$, and the inflation ${\mathit{Inf}}_t$. We use as commodities energy such as the crude oil price $p_t$ and the gold such as the price of gold per ounce $g_t$, and the financial cycle through the price of three global Assets $A_t$, which is a set of three global assets such as the FTSE, the SP and the MSCI. The macroeconomic variables, $z_t\in \left\{y_t,c_t,I_t,{\mathit{Inf}}_t\right\}$, are with an independent permanent component, ${\overline{z}}^{\mathit{nb}}$, for oil price, gold price and financial asset price respectively, with a permanent component determined by ${\beta }^z{p}_t$, ${\alpha }^z{g}_t$, and ${\sigma }^z{A}_t$, oil price, gold price, and financial assets respectively; an independent transitory component ${\stackrel{\sim }{z}}^{\mathit{nb}}$ of the oil price, the gold price and financial asset respectively; and transitory component ${\stackrel{\sim }{z}}^b$ determined by oil price, gold price and financial assets respectively. The parameter ${\beta }^z$, ${\alpha }^z$, and ${\sigma }^z$ are the long-term elasticity of the macroeconomic variables $z_t$ with respect to the oil price, the gold price and the financial assets price respectively.

$$y_t={\overline{y}}^{\mathit{no}}+{\beta }^y{p}_t+{\stackrel{\sim }{y}}^{\mathit{no}}+{\stackrel{\sim }{y}}^o+{\overline{y}}^{\mathit{ng}}+{\alpha }^y{g}_t+{\stackrel{\sim }{y}}^{\mathit{ng}}+{\stackrel{\sim }{y}}^g+{\sigma }^y{A}_t+{\stackrel{\sim }{y}}^{\mathit{nA}}+{\stackrel{\sim }{y}}^A+{\overline{y}}^{\mathit{nA}}$$E9

$$c_t={\overline{c}}^{\mathit{no}}+{\beta }^c{p}_t+{\stackrel{\sim }{c}}^{\mathit{no}}+{\stackrel{\sim }{c}}^o+{\overline{c}}^{\mathit{ng}}+{\alpha }^c{g}_t+{\stackrel{\sim }{c}}^{\mathit{ng}}+{\stackrel{\sim }{c}}^g+{\sigma }^c{A}_t+{\stackrel{\sim }{c}}^{\mathit{NA}}+{\stackrel{\sim }{c}}^A+{\overline{c}}^{\mathit{nA}}$$E10

$$I_t={\overline{I}}^{\mathit{no}}+{\beta }^I{p}_t+{\stackrel{\sim }{I}}^{\mathit{no}}+{\stackrel{\sim }{I}}^o+{\overline{I}}^{\mathit{ng}}+{\alpha }^I{g}_t+{\stackrel{\sim }{I}}^{\mathit{ng}}+{\stackrel{\sim }{I}}^g+{\sigma }^I{A}_t+{\stackrel{\sim }{I}}^{\mathit{NA}}+{\stackrel{\sim }{I}}^A+{\overline{I}}^{\mathit{nA}}$$E11

$${\mathit{Inf}}_t=\begin{array}{l}{\overline{\mathit{Inf}}}^{\mathit{no}}+{\beta }^{\mathit{Inf}}{p}_t+{\stackrel{\sim }{\mathit{Inf}}}^{\mathit{no}}+{\stackrel{\sim }{\mathit{Inf}}}^o+{\overline{\mathit{Inf}}}^{\mathit{ng}}+{\alpha }^{\mathit{Inf}}{g}_t+{\stackrel{\sim }{\mathit{Inf}}}^{\mathit{ng}}+{\stackrel{\sim }{\mathit{Inf}}}^g+{\sigma }^{\mathit{Inf}}{A}_t\\ +{\stackrel{\sim }{\mathit{Inf}}}^{\mathit{NA}}+{\stackrel{\sim }{\mathit{Inf}}}^A+{\overline{\mathit{Inf}}}^{\mathit{nA}}\end{array}$$E12

It is assumed that the growth rates of the permanent components such as the oil price the gold and the financial asset share the common path as follow:

$$\left(\begin{array}{c}{\overline{y}}_t^{\mathit{nb}}\\ {\overline{c}}_t^{\mathit{nb}}\\ {\overline{I}}_t^{\mathit{nb}}\\ {\overline{\mathit{Inf}}}_t^{\mathit{nb}}\end{array}\right)=\left(\begin{array}{c}1\\ {\mathit{\lambda b}}^c\\ {\mathit{\lambda b}}^I\\ {\mathit{\lambda b}}^{\mathit{Inf}}\end{array}\right){\mu }_t^b+\left(\begin{array}{c}{\overline{y}}_{t-1}^{\mathit{nb}}\\ {\overline{c}}_{t-1}^{\mathit{nb}}\\ {\overline{I}}_{t-1}^{\mathit{nb}}\\ {\overline{\mathit{Inf}}}_{t-1}^{\mathit{nb}}\end{array}\right),\mathrm{b}=\mathrm{o},\mathrm{g},\text{and}\mathrm{A}$$E13

${\mathit{\lambda b}}^c$, ${\mathit{\lambda b}}^I$, ${\mathit{\lambda b}}^{\mathit{Inf}}$ are loading parameters for the common growth rate components. It is assumed that the long-run growth ${\mu }_t^b$ the random walk of oil price, gold prices and financial assets price are as follow:

$${\mu }_t^b={\mu }_{t-1}^b+u_t^b,u_t^b\sim \mathcal{N}\left(0,{\sigma }_{u^b}^2\right)$$E14

The oil price, the gold price, and the financial asset prices are described by a random walk process.

$$\left(\begin{array}{c}{p}_t\\ g_t\\ A_t\end{array}\right)=\left(\begin{array}{c}{p}_{t-1}\\ g_{t-1}\\ A_{t-1}\end{array}\right)+\left(\begin{array}{c}{\eta }_t^b\\ {\eta }_t^b\\ {\eta }_t^b\end{array}\right),{\eta }_t^b\sim \mathcal{N}\left(0,{\sigma }_{{\eta }^b}^2\right)$$E15

The independent transitory of the variables ${\stackrel{\sim }{z}}^{\mathit{nb}}$ shares a common transitory component $q_t^b$ described by the AR(2) as follow:

$$q_t^b={\rho }_1{q}_{t-1}^b+{\rho }_2{q}_{t-2}^b+{\epsilon }_t^b,{\epsilon }_t^b\sim \mathcal{N}\left(0,{\sigma }_{{\epsilon }^b}^2\right)$$E16

The model is based on stochastic disturbances in Eq. (5). ${\gamma }^c$, ${\gamma }^I$, and ${\gamma }^{\mathit{Inf}}$ are loading parameters for the transitory components. ${\stackrel{\sim }{y}}_t^{\mathit{nbi}}$, ${\stackrel{\sim }{c}}_t^{\mathit{nbi}}$, ${\stackrel{\sim }{I}}_t^{\mathit{nbi}}$, and ${\stackrel{\sim }{\mathit{Inf}}}_t^{\mathit{nbi}}$ are idiosyncratic transitory components for the GDP, the consumption, the investment, and the inflation respectively.

$$\left(\begin{array}{c}{\stackrel{\sim }{y}}_t^{\mathit{no}}\\ {\stackrel{\sim }{y}}_t^{\mathit{ng}}\\ {\stackrel{\sim }{y}}_t^{\mathit{nA}}\end{array}\right)=\left(\begin{array}{c}{q}_t^o\\ q_t^g\\ q_t^A\end{array}\right)+\left(\begin{array}{c}{\stackrel{\sim }{y}}_t^{\mathit{noi}}\\ {\stackrel{\sim }{y}}_t^{\mathit{ngi}}\\ {\stackrel{\sim }{y}}_t^{\mathit{nAi}}\end{array}\right)$$E17

$${\stackrel{\sim }{c}}_t^{\mathit{nb}}={\gamma }^c{q}_t^b+{\stackrel{\sim }{c}}_t^{\mathit{nbi}}$$E18

$${\stackrel{\sim }{I}}_t^{\mathit{nb}}={\gamma }^I{q}_t^b+{\stackrel{\sim }{I}}_t^{\mathit{nbi}}$$E19

$${\stackrel{\sim }{\mathit{Inf}}}_t^{\mathit{nb}}={\gamma }^{\mathit{Inf}}{q}_t^b+{\stackrel{\sim }{\mathit{Inf}}}_t^{\mathit{nbi}}$$E20

The idiosyncratic transitory component for the macroeconomic variables is described by AR(1) process as follow:

$$\left\{\begin{array}{c}{\stackrel{\sim }{y}}_t^{\mathit{nbi}}={\zeta }_b^y{\stackrel{\sim }{y}}_{t-1}^{\mathit{nbi}}+{\xi }_t^y,{\xi }_t^y\sim \mathcal{N}\left(0,{\sigma }_{{\xi }^y}^2\right)\\ {\stackrel{\sim }{c}}_t^{\mathit{nbi}}={\zeta }_b^c{\stackrel{\sim }{c}}_{t-1}^{\mathit{nbi}}+{\xi }_t^c,{\xi }_t^c\sim \mathcal{N}\left(0,{\sigma }_{{\xi }^c}^2\right)\\ {\stackrel{\sim }{I}}_t^{\mathit{nbi}}={\zeta }_b^I{\stackrel{\sim }{I}}_{t-1}^{\mathit{nbi}}+{\xi }_t^I,{\xi }_t^I\sim \mathcal{N}\left(0,{\sigma }_{{\xi }^I}^2\right)\\ {\stackrel{\sim }{\mathit{Inf}}}_t^{\mathit{nbi}}={\zeta }_b^{\mathit{Inf}}{\stackrel{\sim }{\mathit{Inf}}}_{t-1}^{\mathit{nbi}}+{\xi }_t^{\mathit{Inf}},{\xi }_t^c\sim \mathcal{N}\left(0,{\sigma }_{{\xi }^{\mathit{Inf}}}^2\right)\end{array}\right.$$E21-24

The dynamic of the transitory component of the macroeconomic variables ${\stackrel{\sim }{z}}^{\mathit{nb}}$ determined by the oil price, the gold price and the financial assets are described as follows:

$${\stackrel{\sim }{y}}_t^b={\psi }_b^y{\stackrel{\sim }{y}}_{t-1}^b+{\theta }_b^y{\eta }_t$$E25

$${\stackrel{\sim }{c}}_t^b={\psi }_b^c{\stackrel{\sim }{c}}_{t-1}^b+{\theta }_b^c{\eta }_t$$E26

$${\stackrel{\sim }{I}}_t^b={\psi }_b^I{\stackrel{\sim }{I}}_{t-1}^b+{\theta }_b^I{\eta }_t$$E27

$${\stackrel{\sim }{\mathit{Inf}}}_t^b={\psi }_b^{\mathit{Inf}}{\stackrel{\sim }{\mathit{Inf}}}_{t-1}^b+{\theta }_b^{\mathit{Inf}}{\eta }_t$$E28

${\theta }^y$, ${\theta }^c$, ${\theta }^I$, and ${\theta }^{\mathit{Inf}}$ are the shocks sensitivity parameters for GDP, consumption, investment, and inflation respectively. These shocks parameters are negative, as the actual macroeconomic variables take time to adapt to their permanent level, and are instantaneously changed by the oil price shock, the gold price shock, and the financial assets shock.

${\beta }^y+{\theta }_b^y$, ${\alpha }^y+{\theta }_b^y$, and ${\sigma }^y+{\theta }_b^y$ are the short-run elasticities of the GDP to the oil price, the gold price, and the financial assets price.

${\beta }^c+{\theta }_b^c$, ${\alpha }^c+{\theta }_b^c$, and ${\sigma }^c+{\theta }_b^c$ are the short-run elasticities of the consumption to the oil price, the gold price, and the financial assets price.

${\beta }^I+{\theta }_b^I$, ${\alpha }^I+{\theta }_b^I$, and ${\sigma }^I+{\theta }_b^I$ are the short-run elasticities of the investment to the oil price, the gold price, and the financial asset price.

${\beta }^{\mathit{Inf}}+{\theta }_b^{\mathit{Inf}}$, ${\alpha }^{\mathit{Inf}}+{\theta }_b^{\mathit{Inf}}$, and ${\sigma }^{\mathit{Inf}}+{\theta }_b^{\mathit{Inf}}$ are the short-run elasticities of the investment to the oil price, the gold price, and the financial assets price.

9. Estimation results

Within this section, we report the cycle estimates and other parameters of interest under the multivariate unobserved component model by the maximum likelihood method using the Kalman filter. In a second step, we proceed to calculate the c and Beta parameters for the trend, cycle, and seasonal parameters using LLT and State Space model using as instruments the gold, energy, and the financial cycle.

The parameters’ estimates for the likelihood estimator are shown in Table 1. All the parameters are statistically significant. Global consumption and global inflation are significantly negatively determined by the oil, the gold and the financial cycle, which is confirming the expected result compatible with the theory, as a higher price of commodities have a negative impact on the consumption and enhances the inflation which leads to affecting the global economy negatively. In opposite global investment is significantly positively determined by the oil, the gold, and the financial cycle as the oil, the gold are used as hedge funds by the investors as well as the trading of the stock market which promotes the global economy and affects the cyclicity of the global business cycle within the peak. The long-run global investment elasticity of the global output is the highest equal to 0.119. This is confirming that the three determinants are used as a hedge fund for the investors and the payment system and are consistent with the capital channel of the global system, which means that the commodities are distributed mainly for investment other than consumption. Figure 6a–d shows the trend of the global business cycle within the Multivariate Unobserved Components model. The time lag of one period for the global GDP has not pronounced an impact on the cyclicity of the global business cycle, which might be explained as the deterministic commodities influence the upswing and downswing of the business cycle. However, basic on the filtering data, it is confirmed that oil has a lag impact on the fluctuation of the business cycle, it reacts as an impulsion for the crisis, compared to the gold and the global financial cycle which react simultaneously with the trend of the global business cycle.

Dependent Variable: GLOBALGDP
Method: LIML / K-Class
Sample (adjusted): 1984Q2 2020Q4
Included observations: 147 after adjustments
Covariance type: K-Class
Constant added to instrument list
Variable	Coefficient	Std. Error	t-Statistic	Prob.
GLOBALGDP(−1)	0.959091	0.055487	17.28497	0
GLOBALCONSUMPTION	−0.06819	0.088511	−0.77039	0.9423
GLOBAL_INVESTMENT	0.119578	0.149275	0.801058	0.9244
GLOBALCPI	−0.00811	0.006946	−1.16793	0.9448
R-squared	0.998838	Mean dependent var		16.11314
Adjusted R-squared	0.998813	S.D. dependent var		0.57896
S.E. of regression	0.019945	Sum squared resid		0.056886
Durbin-Watson stat	0.793426	LIML min. Eigenvalue		1

Table 1.

Result maximum likelihood estimator with Kalman filter for multivariate unobserved model of the global GDP.

The estimate of the generalized UC model for the global GDP. The estimation result of the generalized UC model is reported. The sample period is 1984:Q 1–2020:Q4. Within the estimation, we are limiting to the standard format of the multivariate unobserved model using the maximum likelihood estimator with the Kalman filter. Source: Own study.

Figure 6

a. the residual, actual, and fitted estimation of global GDP with maximum likelihood estimator. b. the residual, actual, and fitted estimation of global consumption with maximum likelihood estimator. c. the residual, actual and fitted estimation of global investment with maximum likelihood estimator. d. the residual, actual and fitted estimation of global inflation with maximum likelihood estimator. Source: Own study.

Related to the investment, the inflation, and the consumption the related graph will show the univariate unobserved component model for each variable determined by the oil, the gold, and the financial cycle. According to the main result, we can consider that the determinant oil, gold, and financial cycle was a passive influencer for the global cycle of investment, consumption, and inflation. It is in opposite to the theory; however, it can be explained economically that the macroeconomic aggregate is an endogenous aggregate of the fluctuation of the commodities and not the opposite. However, related to the global business cycle, the commodities are reacting as an intermediary between the global business cycle and the investment, consumption, and inflation. Our result confirms that the long-run relationship between commodities and the business cycle is influenced by the economic condition. The relationship between gold and the financial cycle is stable with significant shifts during peak and recession.

10. Conclusion

This chapter uses a Multivariate Unobserved Component Model to determine the impact of the commodities and the financial cycle on the global business cycle. It shed light on the cyclicity of the unobserved components as a determinant of the business cycle using the filtering approach Band-Pass filter. One distinctive feature of our research is that it combines the financial and non-financial markets as the realm of economic systems to examine the cyclicity of the business cycle.

Our model produces several implications, which are discussed in the chapter. We use the maximum likelihood estimator with the Kalman filter to estimate the model. The main results show that oil has a lag impact on the fluctuation of the business cycle, it reacts as an impulsion for the crisis, compared to the gold and the global financial cycle which reacts simultaneously with the trend of the global business cycle. Our results are in accordance with the prominent literature treating the energy as an impulsion for any upswing or downswing within the cyclicity of the business cycle. Within the recent reframing of the economic system and the interconnectedness of the real and the financial spheres, the gold and the stock market are acting as hedge funds and as investment reacts simultaneously with the trend of the global business cycle. Our study might bring new features that the energy is acting as a hedge fund rather than just a consumption creating a combined business cycle with prominent expectations of their fluctuations.