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

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 [84–88] 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:

$$\mathcal{Y}\_t = \mathfrak{r}\_t + \mathfrak{c}\_t + \mathfrak{e}\_t \tag{5}$$

$$
\pi\_t = \pi\_{t-1} + d\_{t-1} + w\_t, w\_t \sim i.i.d.N(0, \sigma\_w^2) \tag{6}
$$

$$d\_t = d\_{t-1} + u\_t, u\_t \sim i.i.d.N \left(0, \sigma\_u^2\right) \tag{7}$$

$$\mathbf{c}\_{t} = \phi\_{1}\mathbf{c}\_{t-1} + \phi\_{2}\mathbf{c}\_{t-2} + \boldsymbol{\nu}\_{t}, \boldsymbol{\upsilon}\_{t} \sim i, i, d.N \left(\mathbf{0}, \sigma\_{\boldsymbol{\upsilon}}^{2}\right) \tag{8}$$

where *τt*, *ct*, and *ε<sup>t</sup>* 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 ð Þ *dt* . The trend and seasonal components are modeled by linear dynamic stochastic processes, which depend on disturbances. The drift ð Þ *dt* 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. *ε<sup>t</sup>* 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 *τ<sup>t</sup>* is modeled as a random walk process, *N* 0, *σ*<sup>2</sup> *w* � � refers to normally independently distributed series with mean zero and variance *σ*2. The disturbance series is serially independent and mutually independent of all other disturbance series related to *yt* . *wt*, *ut*, and *vt* are innovation to trend, trend growth rate, and cycle respectively. The model is rewritten as follows:

$$\boldsymbol{\gamma}\_{t} = H\boldsymbol{\xi}\_{t} + \boldsymbol{\varepsilon}\_{t}$$

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

$$H = \begin{pmatrix} \mathbf{1} & \mathbf{1} & \mathbf{0} & \mathbf{0} \end{pmatrix} \boldsymbol{\xi}\_{t} = \begin{pmatrix} \boldsymbol{\varepsilon}\_{t} & \boldsymbol{c}\_{t} & \boldsymbol{c}\_{t-1} & \boldsymbol{d}\_{t} \end{pmatrix}$$

$$F = \begin{pmatrix} \mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{1} \\ \mathbf{0} & \boldsymbol{\phi}\_{1} & \boldsymbol{\phi}\_{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{pmatrix} \boldsymbol{Q} = \begin{pmatrix} \boldsymbol{\sigma}\_{w}^{2} & \boldsymbol{\sigma}\_{wu} & \boldsymbol{\sigma}\_{uv} & \mathbf{0} \\ \boldsymbol{\sigma}\_{wu} & \boldsymbol{\sigma}\_{u}^{2} & \boldsymbol{\sigma}\_{uv} & \mathbf{0} \\ \boldsymbol{\sigma}\_{uv} & \boldsymbol{\sigma}\_{uv} & \boldsymbol{\sigma}\_{v}^{2} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \end{pmatrix}$$

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].
