**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 *yt* , the consumption *ct*, the investment

*The Global Business Cycle within the New Commodities and the Financial Cycle… DOI: http://dx.doi.org/10.5772/intechopen.111482*

*It*, and the inflation *Inft* . We use as commodities energy such as the crude oil price *pt* and the gold such as the price of gold per ounce *gt* , and the financial cycle through the price of three global Assets *At*, which is a set of three global assets such as the FTSE, the SP and the MSCI. The macroeconomic variables, *zt* ∈ *yt* ,*ct*,*It*,*Inft* � �, are with an independent permanent component, *znb*, for oil price, gold price and financial asset price respectively, with a permanent component determined by *β<sup>z</sup> pt* , *αzgt* , and *σzAt*, oil price, gold price, and financial assets respectively; an independent transitory component ~*znb* of the oil price, the gold price and financial asset respectively; and transitory component ~*zb* determined by oil price, gold price and financial assets respectively. The parameter *β<sup>z</sup>* , *αz*, and *σ<sup>z</sup>* are the long-term elasticity of the macroeconomic variables *zt* with respect to the oil price, the gold price and the financial assets price respectively.

$$\mathbf{y}\_t = \overline{\mathbf{y}}^{no} + \beta^p \mathbf{p}\_t + \tilde{\mathbf{y}}^{no} + \tilde{\mathbf{y}}^{\rho} + \tilde{\mathbf{y}}^{\mathbf{g}} + \alpha^p \mathbf{g}\_t + \tilde{\mathbf{y}}^{\mathbf{g}} + \tilde{\mathbf{y}}^{\mathbf{g}} + \sigma^\mathbf{y} \mathbf{A}\_t + \tilde{\mathbf{y}}^{\mathbf{n}A} + \tilde{\mathbf{y}}^A + \tilde{\mathbf{y}}^{\mathbf{n}A} \tag{9}$$

$$\varepsilon\_{t} = \overline{\boldsymbol{c}}^{\prime \prime \prime} + \boldsymbol{\beta}^{\varepsilon} \boldsymbol{p}\_{t} + \overline{\boldsymbol{c}}^{\prime \prime \prime} + \overline{\boldsymbol{c}}^{\prime} + \overline{\boldsymbol{c}}^{\prime \prime} + \boldsymbol{a}^{\varepsilon} \mathbf{g}\_{t} + \overline{\boldsymbol{c}}^{\prime \prime \mathbf{g}} + \overline{\boldsymbol{c}}^{\mathbb{g}} + \boldsymbol{\sigma}^{\prime} \mathbf{A}\_{t} + \overline{\boldsymbol{c}}^{\prime \rm NA} + \overline{\boldsymbol{c}}^{\prime A} + \overline{\boldsymbol{c}}^{\prime A} \tag{10} \boldsymbol{\epsilon}$$

$$I\_t = \overline{I}^{no} + \beta^l p\_t + \tilde{I}^{no} + \tilde{I}^{\rho} + \tilde{I}^{\mathbf{g}} + a^l \mathbf{g}\_t + \tilde{I}^{\mathbf{g}} + \tilde{I}^{\mathbf{g}} + \sigma^l A\_l + \tilde{I}^{NA} + \tilde{I}^A + \tilde{I}^{nA} \tag{11}$$

$$\begin{split} \text{Inf}\_{t} &= \overline{\text{Inf}}^{\text{no}} + \beta^{\text{lnf}} p\_{t} + \bar{\text{Inf}}^{\text{no}}\_{t} + \bar{\text{Inf}}^{\text{o}} + \overline{\text{Inf}}^{\text{ng}} + \alpha^{\text{lnf}} \mathbf{g}\_{t} + \bar{\text{Inf}}^{\text{ng}} + \bar{\text{Inf}}^{\text{g}} + \sigma^{\text{lnf}} A\_{t} \\ &+ \bar{\text{Inf}}^{\text{NA}} + \bar{\text{Inf}}^{\text{A}} + \overline{\text{Inf}}^{\text{nA}} \end{split} \tag{12}$$

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:

$$
\begin{pmatrix} \overline{\mathcal{Y}}\_{t}^{ub} \\ \overline{c}\_{t}^{ub} \\ \overline{I}\_{t}^{ub} \\ \overline{\mathcal{I}\eta}\_{t}^{ub} \end{pmatrix} = \begin{pmatrix} 1 \\ \lambda b^{c} \\ \lambda b^{l} \\ \lambda b^{l\eta^{l}} \end{pmatrix} \mu\_{t}^{b} + \begin{pmatrix} \overline{\mathcal{Y}}\_{t-1}^{ub} \\ \overline{c}\_{t-1}^{ub} \\ \overline{I}\_{t-1}^{ub} \\ \overline{\mathcal{I}\eta}\_{t-1}^{ub} \end{pmatrix}, \mathbf{b} = \mathbf{o}, \mathbf{g}, \text{and } \mathbf{A} \tag{13}
$$

*λbc* , *λb<sup>I</sup>* , *λbInf* are loading parameters for the common growth rate components. It is assumed that the long-run growth *μ<sup>b</sup> <sup>t</sup>* the random walk of oil price, gold prices and financial assets price are as follow:

$$
\mu\_t^b = \mu\_{t-1}^b + u\_t^b, \ u\_t^b \sim \mathcal{N}\left(\mathbf{0}, \sigma\_{u^b}^2\right) \tag{14}
$$

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

$$
\begin{pmatrix} p\_t \\ g\_t \\ A\_t \end{pmatrix} = \begin{pmatrix} p\_{t-1} \\ g\_{t-1} \\ A\_{t-1} \end{pmatrix} + \begin{pmatrix} \eta\_t^b \\ \eta\_t^b \\ \eta\_t^b \end{pmatrix}, \ \eta\_t^b \sim \mathcal{N}\left(\mathbf{0}, \sigma\_{\eta^b}^2\right) \tag{15}
$$

The independent transitory of the variables ~*znb* shares a common transitory component *q<sup>b</sup> <sup>t</sup>* described by the AR(2) as follow:

$$q\_t^b = \rho\_1 q\_{t-1}^b + \rho\_2 q\_{t-2}^b + \varepsilon\_t^b, \varepsilon\_t^b \sim \mathcal{N}\left(0, \sigma\_{\varepsilon^b}^2\right) \tag{16}$$

*New Topics in Emerging Markets*

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The model is based on stochastic disturbances in Eq. (5). *γ<sup>c</sup>* , *γ<sup>I</sup>* , and *γInf* are loading parameters for the transitory components. ~*ynbi <sup>t</sup>* , ~*c nbi <sup>t</sup>* , <sup>~</sup>*<sup>I</sup> nbi <sup>t</sup>* , and <sup>~</sup>*Inf nbi <sup>t</sup>* are idiosyncratic transitory components for the GDP, the consumption, the investment, and the inflation respectively.

$$
\begin{pmatrix} \tilde{\boldsymbol{\mathcal{y}}}\_{t}^{no} \\ \tilde{\boldsymbol{\mathcal{y}}}\_{t}^{ng} \\ \tilde{\boldsymbol{\mathcal{y}}}\_{t}^{nA} \end{pmatrix} = \begin{pmatrix} \boldsymbol{q}\_{t}^{o} \\ \boldsymbol{q}\_{t}^{g} \\ \boldsymbol{q}\_{t}^{A} \end{pmatrix} + \begin{pmatrix} \tilde{\boldsymbol{\mathcal{y}}}\_{t}^{noi} \\ \tilde{\boldsymbol{\mathcal{y}}}\_{t}^{ngi} \\ \tilde{\boldsymbol{\mathcal{y}}}\_{t}^{nAi} \end{pmatrix} \tag{17}
$$

$$
\tilde{\boldsymbol{c}}\_t^{nb} = \boldsymbol{\gamma}^c \boldsymbol{q}\_t^b + \tilde{\boldsymbol{c}}\_t^{nbi} \tag{18}
$$

$$
\hat{I}\_t^{nb} = \chi^l q\_t^b + \hat{I}\_t^{nbi} \tag{19}
$$

$$\left\|\tilde{\boldsymbol{\Im}}\right\|\_{t}^{nb} = \boldsymbol{\gamma}^{\text{I}\text{pf}}\boldsymbol{q}\_{t}^{b} + \boldsymbol{\Gamma}\tilde{\boldsymbol{\eta}}\_{t}^{\text{nbi}}\tag{20}$$

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

$$\begin{split} \tilde{\jmath}\_{t}^{nbi} &= \tilde{\varsigma}\_{b}^{\mathcal{V}} \tilde{\jmath}\_{t-1}^{nbi} + \tilde{\varsigma}\_{t}^{\mathcal{V}}, \tilde{\varsigma}\_{t}^{\mathcal{V}} \sim \mathcal{N}\left(\mathbf{0}, \sigma\_{\tilde{\varsigma}^{\mathcal{V}}}^{2}\right) \\ \tilde{c}\_{t}^{nbi} &= \tilde{\varsigma}\_{b}^{c} \tilde{\varsigma}\_{t-1}^{nbi} + \tilde{\varsigma}\_{t}^{c}, \tilde{\varsigma}\_{t}^{c} \sim \mathcal{N}\left(\mathbf{0}, \sigma\_{\tilde{\varsigma}^{\mathcal{V}}}^{2}\right) \\ \tilde{I}\_{t}^{nbi} &= \tilde{\varsigma}\_{b}^{I} \tilde{I}\_{t-1}^{nbi} + \tilde{\varsigma}\_{t}^{I}, \tilde{\varsigma}\_{t}^{I} \sim \mathcal{N}\left(\mathbf{0}, \sigma\_{\tilde{\varsigma}^{\mathcal{V}}}^{2}\right) \\ \tilde{\ln} \tilde{\jmath}\_{t}^{nbi} &= \tilde{\varsigma}\_{b}^{\mathcal{I}n\tilde{\jmath}} \ln \tilde{\jmath}\_{t-1}^{nbi} + \tilde{\varsigma}\_{t}^{\mathcal{I}n\tilde{\jmath}}, \tilde{\varsigma}\_{t}^{c} \sim \mathcal{N}\left(\mathbf{0}, \sigma\_{\tilde{\varsigma}^{\mathcal{V}}}^{2}\right) \end{split} \tag{21.24}$$

The dynamic of the transitory component of the macroeconomic variables ~*znb* determined by the oil price, the gold price and the financial assets are described as follows:

$$
\bar{\mathbf{y}}\_t^b = \boldsymbol{\nu}\_b^\mathcal{V} \bar{\mathbf{y}}\_{t-1}^b + \boldsymbol{\theta}\_b^\mathcal{V} \boldsymbol{\eta}\_t \tag{25}
$$

$$
\tilde{\boldsymbol{\varepsilon}}\_{t}^{b} = \boldsymbol{\mu}\_{b}^{\varepsilon} \tilde{\boldsymbol{\varepsilon}}\_{t-1}^{b} + \boldsymbol{\theta}\_{b}^{\varepsilon} \boldsymbol{\eta}\_{t} \tag{26}
$$

$$
\tilde{I}\_t^b = \psi\_b^I \tilde{I}\_{t-1}^b + \theta\_b^I \eta\_t \tag{27}
$$

$$\left\| \tilde{\boldsymbol{\text{n}}} \right\|\_{t}^{b} = \boldsymbol{\psi}\_{b}^{\text{lvf}} \boldsymbol{\tilde{\text{n}}} \boldsymbol{\tilde{\text{f}}}\_{t-1}^{b} + \boldsymbol{\theta}\_{b}^{\text{lpf}} \boldsymbol{\eta}\_{t} \tag{28}$$

*θy* , *θ<sup>c</sup>* , *θ<sup>I</sup>* , and *θInf* 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.

*<sup>β</sup><sup>y</sup>* <sup>þ</sup> *<sup>θ</sup> y <sup>b</sup>*, *<sup>α</sup><sup>y</sup>* <sup>þ</sup> *<sup>θ</sup> y <sup>b</sup>*, and *<sup>σ</sup><sup>y</sup>* <sup>þ</sup> *<sup>θ</sup> y <sup>b</sup>* are the short-run elasticities of the GDP to the oil price, the gold price, and the financial assets price.

*<sup>β</sup><sup>c</sup>* <sup>þ</sup> *<sup>θ</sup><sup>c</sup> <sup>b</sup>*, *<sup>α</sup><sup>c</sup>* <sup>þ</sup> *<sup>θ</sup><sup>c</sup> <sup>b</sup>*, and *<sup>σ</sup><sup>c</sup>* <sup>þ</sup> *<sup>θ</sup><sup>c</sup> <sup>b</sup>* are the short-run elasticities of the consumption to the oil price, the gold price, and the financial assets price.

*<sup>β</sup><sup>I</sup>* <sup>þ</sup> *<sup>θ</sup><sup>I</sup> <sup>b</sup>*, *<sup>α</sup><sup>I</sup>* <sup>þ</sup> *<sup>θ</sup><sup>I</sup> <sup>b</sup>*, and *<sup>σ</sup><sup>I</sup>* <sup>þ</sup> *<sup>θ</sup><sup>I</sup> <sup>b</sup>* are the short-run elasticities of the investment to the oil price, the gold price, and the financial asset price.

*<sup>β</sup>Inf* <sup>þ</sup> *<sup>θ</sup> Inf <sup>b</sup>* , *<sup>α</sup>Inf* <sup>þ</sup> *<sup>θ</sup> Inf <sup>b</sup>* , and *<sup>σ</sup>Inf* <sup>þ</sup> *<sup>θ</sup> Inf <sup>b</sup>* are the short-run elasticities of the investment to the oil price, the gold price, and the financial assets price.

*The Global Business Cycle within the New Commodities and the Financial Cycle… DOI: http://dx.doi.org/10.5772/intechopen.111482*
