**3. Methods**

### **3.1 Decoupling hypothesis**

Prior to the 2008 crisis, there were many studies showing that emerging markets had sustained economic growth, had built up sufficient economic and technological capacity, and that their economic cycles had become more independent of those of developed countries—the decoupling hypothesis [23]. Since the 2008 crisis, however, developed countries have changed their monetary policy substantially and have become more focused on stimulating their productive forces. The trade wars between the United States and China, which began in 2018, allow us to talk about direct restrictions in the exchange of technology between the United States and China.

At the same time, China and other major developing countries are facing a slowdown in economic growth. As a result of these processes, emerging markets are facing capital outflows and devaluation of national currencies.

It is also important to distinguish between the issue of the synchronicity of economic cycles and the fragmentation of the economies of the global economy. From our perspective, the existence of the synchronicity of economic cycles is objective in nature. The synchronicity of economic cycles confirms the interaction of the economic systems of national economies and confirms the decoupling hypothesis. In addition, the fragmentation of economies can be caused by various geopolitical events, such as sanctions and various prohibitive measures that impede global trade and technology exchange. Now we see a trade war between the United States and China, anti-Russian sanctions, and an almost complete termination of free trade

between Russia and the United States, Russia, and Europe. It is obviously a powerful geoeconomic fragmentation of the LEE. Nevertheless, whether their business cycles are synchronized remains a big question.

### **3.2 Hodriсk-Preskott filtering method**

Hodriсk-Preskott filtering is a simple and straightforward method to extract the trend and the cyclic component of a time series. We considered the GDP data as a time series and, due to isolation of the cyclical component, determined the difference between actual and potential GDP or output gap, in other words.

By "trend," we mean a certain steady, systematic change over a long period. However, no matter how long the series is, we can never be sure that the trend is not just a part of a slow fluctuation. After having separated the trend from seasonal fluctuations, the remainder of the series is a function of cyclical fluctuations. Seasonal fluctuations are the easiest to detect, isolate, and study.

When defining a trend, we understand that any movement observed over a sufficiently long period can be smoothed. It means that, at least locally, the component corresponding to the trend can be expressed by a polynomial of time.

Thus, in our case, GDP is a trend and fluctuations around this trend [24, 25]:

$$
\mathcal{Y}\_t = \mathcal{Y}\_t^\mathcal{Y} + \mathcal{Y}\_t^\mathcal{C}, \tag{1}
$$

where *y g <sup>t</sup>* is a trend or structural component of the time series and *y<sup>с</sup> <sup>t</sup>* is the cyclical component of the time series.

We imposed a minimization condition on the cyclic component to obtain a smoothed series:

$$\sum\_{t=0}^{\infty} \left( \mathcal{y}\_t^\varepsilon \right)^2 + \lambda \sum\_{t=0}^{\infty} \left[ \left( \mathcal{y}\_{t+1}^\mathcal{g} - \mathcal{y}\_t^\mathcal{g} \right) - \left( \mathcal{y}\_t^\mathcal{g} - \mathcal{y}\_{t-1}^\mathcal{g} \right) \right]^2 \to \min \tag{2}$$

where *λ* is a Lagrange multiplier. For annual data, *λ* ¼ 100.

After elimination of the trend, we investigated the reminder of the series by spectral analysis methods.

### **3.3 Spectral analysis**

To investigate the reminders of the time series, we applied spectral analysis techniques to check for correlations between time series members and to determine the period of major fluctuations in the reminders of the time series [26].

Suppose there is autocorrelation for any pair of values:

$$\rho\_j = \frac{cov\left(u\_t, u\_{t-j}\right)}{\sigma^2},\tag{3}$$

where *ρ<sup>j</sup>* is the correlation between members of the time series after filtering by *j*,*cov ut*, *ut*�*<sup>j</sup>* � � is the covariance between members of the time series after filtering by *j*,*σ*<sup>2</sup> is the dispersion of the rest of the time series.

The total sum of coefficients *ρ*0, *ρ*1, *ρ*2, … is called the correlogram of the series. By determining the correlation of the other series gradually, without the components of

### *Hodrick-Prescott Filtering of Large Emerging Economies and Decoupling Hypothesis DOI: http://dx.doi.org/10.5772/intechopen.112176*

the main trend, we can build a correlogram that allows us to graphically trace the interdependence between the members of the time series.

Then during studying of various kinds of periodic processes (we mean processes repeated over a certain period, including economic processes), it is best to decompose periodic functions describing these processes into trigonometric series. The simplest periodic functions are trigonometric functions sin *x* and cos *x*. The period *T* of these functions is 2*π*.

The simplest periodic process is a simple harmonic oscillation described by a function:

$$y = A \* \sin\left(at + \varphi\_0\right) \tag{4}$$

$$t \ge 0 \tag{5}$$

where *A* is the amplitude of the oscillation,

*ω* is the frequency,

*φ*<sup>0</sup> is the phase offset.

This kind of function (and its graph) is called a simple harmonic. The fundamental period of the function is *<sup>T</sup>* <sup>¼</sup> <sup>2</sup>*<sup>π</sup> <sup>ω</sup>* , which means that one full oscillation occurs in the span of time <sup>2</sup>*<sup>π</sup> <sup>ω</sup>* . The denominator *ω* shows how many oscillations will occur within the time unit 2*π*.

Complex harmonic oscillation, which occurs as a result of applying a finite (or infinite) number of simple harmonics, is also described by the functions *sinx* and *cosx*. This way, a constant periodic function can be expressed with this system of Eqs. (6)–(8):

$$A = \frac{2}{\pi} \sum\_{t=1}^{n} u\_t, \cos \frac{2\pi t}{\lambda}, \lambda = \frac{2\pi}{a} \tag{6}$$

$$B = \frac{2}{\pi} \sum\_{t=1}^{n} u\_t, \sin \frac{2\pi t}{\lambda} \tag{7}$$

$$S^2 = A^2 + B^2 = \frac{4}{n}\sigma^2 \varpi(\lambda)\tag{8}$$

where *ut* is a member of the time series after removing the trend:

*λ* is the wavelength;

*S*<sup>2</sup> is the intensity of the oscillations.

The graph of *S*<sup>2</sup> is dependence on the wavelength *λ* is called a periodogram.
