1.5 Time Series

According to the efficient market hypothesis [1], a stock market reflects all the information available to the market participants at any give time. Given the vast information flow, the natural consequence is that financial markets are best described over time. More precisely, the definition of a time series is an ordered sequence of values of a variable at equally spaced time intervals [2].

The nature of time series data originated the applications generally known as Time Series Analysis and Time Series Forecasting. Both of which play an important role throughout this project.

Time series analysis

The analysis of data observed at different points in time leads to problems in statistical modelling and inference. More specifically, the correlation of adjacent points in time can restrict the applicability of conventional statistical methods which traditionally depend on the assumption that these adjacent observations are independent and identically distributed. A systematic approach by which one attempts to answer the mathematical and statistical questions posed by these time correlations is commonly referred to as time series analysis. Therefore, mathematical models are developed with the primary objective to provide plausible descriptions for sample data.

Random variables

A collection of random variables indexed according to the order they are obtained in time serves as a representation of the time series. For example, a time series with three data points ($x_1$, $x_2$, $x_3$) can be considered as a sequence of the random variables $x_1$, $x_2$ and $x_3$, where the random variable $x_1$ denotes the value of the first time period, the variable $x_2$ denotes the value for the second time period and $x_3$ denotes the value for the third time period. While graphically plotting the values of random variables, it is conventional to display the random variables on the vertical axis with the time scale as abscissa. [4]

One such example of a time series analysis, that is often times seen in the context of financial data, is the moving average. Moving average allows to smoothen the series by averaging the current value and its immediate neighbors in the past and future.

v_t = 1/3(x_{t-1} + x_t + x_{t+1})

Stationarity

Some of the time series behaviours, which will be presented within this body of work, may hint that a sort of regularity exist over time. We refer the notion of regularity using a concept called stationarity, introduced by [4].

A strictly stationary time series is one for which the proabilistic behavior of every collection of values

${x_t1, x_t2, ... ,x_tk}$

is identical to that of the time shifted set

${x_{t1+h}, x_{t2+h}, ... ,x_{tk+h}}$

for all time shifts $h = 0, ±1, ±2, ...$.

A weakly stationary time series, $x_t$, is a finite variance process such that

1. the mean value function, µt, defined in (1.9) is constant and does not depend on time t, and
2. the autocovariance function, $γ(s, t)$, defined in (1.10) depends on $s$ and $t$ only through their difference $|s − t|$.

Henceforth, we will use the term stationary to mean weakly stationary; if a process is stationary in the strict sense, we will use the term strictly stationary.

Time series forecasting

In statistics, prediction is a part of statistical inference. Providing a means of the transfer of knowledge about a sample of a population to the whole population, and to other related populations is one description of statistics. However, this is not necessarily equivalent to the process of predicting over time. This process, instead, is known as forecasting and describes the transfer of information across, often to very specific point in, time [3]. Hence the problem is defined as: forecasting future values $X_(t+h) where (h>0)$ of a weakly stationary process ${X_t}$ from the known values $X_s$ (s<=t) [5]. The integer $h$ is called lead time or forecasting horizon, whereas $h$ stands for horizon.

Forecasting methods can be classified, according to [6], into three types:

1. Judgemental forecasts: produce projections based on intuition, inside knowledge, and any other relevant information.
2. Univariate methods: forecast depends on present or past values of the time series on which the forecast is projected. Augmentation by a function of time is possible.
3. Multivariate methods: forecast depends on one or more additional time series variables or multivariate models.

Over the course of this work, we make use of univariate- and multivariate methods.

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[1] Malkiel, Burton G. "Efficient market hypothesis." Finance. Palgrave Macmillan, London, 1989. 127-134.

[2] https://www.itl.nist.gov/div898/handbook/pmc/section4/pmc4.htm

[3] https://www.wikiwand.com/en/Time_series

[4] Shumway, Robert H., and David S. Stoffer. "Time series analysis and its applications." Studies In Informatics And Control 9.4 (2000): 375-376.

[5] Itô, K. (Ed.). "Prediction Theory." §395D in Encyclopedic Dictionary of Mathematics, 2nd ed., Vol. 3. Cambridge, MA: MIT Press, pp. 1463-1465, 1987.

[6] Chatfield, Chris. Time-series forecasting. CRC Press, 2000.