This heading is divided into two sub-headings. The first presents the fundamental concepts related to time series, including their characteristics, components and classifications, and establishes their relationship with the analysis of time series and forecasting, which is the objective of this report. The second sub-heading delves into the characteristics of share prices, starting with a general description and moving towards more specific aspects, as well as the importance of the structure in which the data related to these are found.
2.1.1 Time series and their characteristics
Time series are a type of stochastic process that is characterized by ordering random variables according to time. This means that each moment is associated with a value of the variable that depends on chance and that can change over time. According to Ruiz (2011), a stochastic process is “a collection or family of random variables, ordered according to a subscript that is usually time” (p.01). The analysis of the time series can have different purposes, such as describing the behaviour of the variables or predicting or forecasting their future values, which is especially relevant for financial series.
Time series analysis is a statistical tool that allows studying the behaviour of a variable over time. However, there is no single consensus on the components that should be considered in this type of analysis. Some authors, such as Kocenda and Cerný (2017) and Anderson et al. (2017), propose that time series can be decomposed into three components: trend, seasonality, and noise. Other authors, such as Dodge (2008) and Espallargas and Solís (2012), suggest that a fourth component should be added: the cycle. Finally, there are authors who suggest that time series can have up to five components, these are the cases of IBM (2021) and Chirinos (2018).
Trend: It is the long-term pattern of change that is observed in a series of data. It can be defined as the general and persistent direction of the variations of the series over time. It can be classified as positive (Figure 1), negative (Figure 2) or null (Figure 3), depending on whether the series increases, decreases or remains constant in the long term. The trend can be identified by graphical analysis or by statistical methods. This component is important to understand the historical behaviour and project the future of a series of data, it is common in the different criteria mentioned.
Seasonality: Also called regular cyclical variation: It refers to the variation corresponding to the movements of the series that occur every certain period of time, Figure 4. This component is, like the trend, common in the aforementioned criteria. Differentiating in that those authors who expose four and five components call seasonality the periodic variations corresponding to periods less than or equal to one year (such as daily, weekly, monthly, or annual periodicity), while the periodic variations corresponding to longer periods. They contemplate a component called cyclical variations. Therefore, to determine the seasonality of a time series, it is necessary to analyse them in a period of no less than two years.
One component that cannot be explained by the other elements of the time series is the irregular variation or error. This component is also known as random variation, noise, or residual, and is shown in Figure 4. Irregular variation is common in all three criteria mentioned above. Some authors distinguish between irregular variation, which is occasional and random, and atypical variation, which is caused by isolated events that alter the behaviour of the series. Atypical variation can be classified into several types: additive, innovation, level change, transient, additive seasonality, and local trend.
One way to categorize the time series is according to the degree of variability that they present over time. According to what was exposed in Villagarcía (2006), it is possible to distinguish between homoscedastic and heteroscedastic series. The homoscedastic series are those that maintain a constant range of variation, as shown in Figure 3. On the contrary, heteroscedastic series are those that change the range of variation, increasing or decreasing its amplitude, as illustrated in Figure 1 and Figure 2.
A key concept in time series analysis is that of stationarity. A time series is stationary when its statistical properties, such as the mean, variance, and covariance, do not change with time. This implies that the series does not present a trend, cycles or seasonality. As Castillo and Varela (2010), Villavicencio (2010) and Ruiz (2011) point out, stationarity is a necessary condition to be able to predict the future behaviour of a time series using statistical techniques. An example of a stationary time series is shown in Figure 3.
Financial time series present heteroscedasticity, that is, variances that change over time. This implies that they are not stationary and that their behaviour depends on external factors. To verify the stationarity of a time series, different methods can be used, such as the correlogram, which shows the autocorrelation and partial autocorrelation functions of the series, or unit root tests, such as Dickey Fuller’s or Phillips Perron’s. , which test the null hypothesis that the series has a unit root. These methods are explained in more detail in Castillo and Varela (2010), Villavicencio (2010) and Ruiz (2011). The Figure 5 illustrates an example correlogram for a financial time series.
2.1.2 Pricing Features
Investing in stocks or any other asset listed on the stock market is a complex and challenging task, requiring a thorough understanding of market trends and fluctuations. At the core of this understanding is the ability to analyse and interpret stock market price data, providing key insights into the behaviour of market participants and the factors that drive market movements. The purpose of this sub-section is to provide a comprehensive overview of the stock price environment and how they are commonly represented, pointing out the most important aspects for the application of the techniques that will be explored in the following sections.
As explained in CNMV (n.d.b), stock exchanges are organized markets where shares and other securities are traded, such as fixed income, warrants, certificates and exchange-traded funds. In BME (n.d.) it is stated that, in Spain, there are four traditional stock exchanges (Madrid, Barcelona, Bilbao and Valencia) that are part of the holding Stocks and Spanish Market (hereinafter, BME, for its acronym in Spanish, Bolsas y Mercados Españoles), which also integrates other segments and trading, clearing and settlement systems values. Being, as explained in CNMV (n.d.c), the Spanish Stock Market Interconnection System (hereinafter, SIBE, for its acronym in Spanish, Sistema de Interconexión Bursátil Español) is the platform that allows continuous and electronic trading of all securities admitted to trading on the four Spanish stock
As CNMV (n.d.a) exposes, shares are transferable securities that represent a proportional part of the share capital of a public limited company, and their holders are proprietary partners of the same. Shares may be traded on stock exchanges or other authorized secondary markets.
From what was stated in Mitchell (2020), Pinset (2021) and C. Team (2023) it can be concluded that, to explain the price of a company’s shares, the following factors can be considered:
The supply and demand of shares in the market: if there are more buyers than sellers, the price will rise and vice versa. This depends on the expectations and confidence of investors in the future of the company.
Changes in the management or production of the company: if the company improves its efficiency, its profitability or its innovation, the price of its shares may increase. On the contrary, if the company has internal problems, loses competitiveness or is affected by external crises, the price may fall.
The company’s reputation: If the company has a good public image, is associated with successes or achievements, or receives good ratings from analysts, its share price may rise. Conversely, if the company is embroiled in scandals, lawsuits, or controversies, or receives poor ratings from analysts, the price may drop.
In the texts Pinset (2021), T. I. Team (2022) and C. Team (2023) also point out the importance of differentiating the price of a company or its share from its intrinsic value. Being able to summarize considering what is indicated in these texts and what has been previously stated that the price of a company or action is what buyers and sellers are willing to pay for it at a given moment, while the intrinsic value of a company or action depends largely on the methodology used to value the companies and the objectives of the evaluator.
Once the environment in which share prices are found has been contextualized in a general way and some of the factors that may affect them have been explained, the structure in which these data usually appear is explained below. Generally, the prices of the shares are registered periodically (daily, weekly, monthly, annually, etc.). registering for each period the opening price, the highest price, the lowest, the closing price, the volume and the adjusted closing price, see Table 1.
From what was exposed in Barone (2022), Chen (2022), Downey (2022), Hayes (2021) and Ganti (2020) it can be understood that:
The opening price is the first price at which a financial asset trades in a trading session. This price may be different from the closing price of the previous session, as there may be changes in supply and demand during the period when the market is closed. The opening price usually indicates the tone or trend of the market for that day.
The highest price is the highest price at which a financial asset trades in a trading session. This price reflects the highest level of buyer interest for that asset on that day. The higher price can be an indicator of an asset’s strength or weakness, as well as its volatility.
The lowest price is the lowest price at which a financial asset trades in a trading session. This price reflects the minimum level of interest of sellers for that asset on that day. The lower price can be an indicator of an asset’s pressure or resistance, as well as its volatility.
The closing price is the last price at which a financial asset is traded in a trading session. This price is the one used to calculate the market value of that asset at the end of the day. The closing price is usually the most important for investors, as it summarizes the result of the day’s operations and shows the direction of the market.
Volume is the number of units of a financial asset traded in a trading session. Volume shows the level of activity or liquidity of a market or an asset. Volume often accompanies price movements, as it indicates the degree of consensus or divergence among market participants.
Adjusted closing price is the closing price of a financial asset that is changed to consider events such as dividends, splits, mergers or acquisitions that affect the value of the asset. The adjusted closing price allows you to compare the historical performance of an asset with greater precision and consistency.
Based on what was stated in Hayes (2021) and Ganti (2020), it is understood that the difference between the closing price and the adjusted closing price is of great importance, since the former can give a distorted image of the performance of a share throughout the year. while the second reflects the actual value of the stock after adjusting for the factors that alter it.
For example, a company’s board of directors may decide to divide the company’s shares 3 by 1. Thus, the company’s outstanding shares increase by a multiple of three, while its share price is divided by three. Let’s say a stock closed at $300 the day before your stock split. In this case, the closing price is adjusted to $100 ($300 divided by 3) per share to maintain a consistent standard of comparison. Similarly, all other previous closing prices for that company would be divided by three to get the adjusted closing prices. Ganti (2020)
Due to this, the adjusted closing price is better for the application of time series analysis techniques, since it allows comparing the behaviour of a stock over time without the distortions caused by corporate events. The time series most commonly used in market price analysis studies is that made up of returns calculated from the adjusted closing price.