Introduction to Time Series Lecture 01

Introduction to Time Series 

Lecture 01 

Definition

A sequence of well-defined observations measured on a specific variable in a chronological manner. The observations for a specific event or phenomenon in a time series are denoted by Y1, Y2, ..., Yt and are typically at evenly spaced points in time (days, months, quarters, years).

Mathematically it is defined as

Yt = f(t) + et    t = 0, 1, 2, ⋯⋯

Where:

Yt is the value of a certain variable at time "t" following a defined sequence "f(t)" and a systematic pattern of variation called a signal. The value of Yt under f(t) in the real world is not unique; a term et, often known as the error or white noise term, is attached to f(t). The term "noise," sometimes known as "random shock," refers to an unpredictable component of time series that includes factors that are irregular and ad hoc in their approach to the phenomenon.

Assume that our goal is to compile data on Pakistan's per-acre wheat production during the past 20 years (1990 to 2020). The annual production of wheat per acre in Pakistan is then our f(t), and et is designed to account for any irregular or unintended variations in that production.

Examples:

i. Daily average temperature in the month of June.

ii. Weekly petrol price in the international market.

iii. Monthly earnings of a firm.

iv. Annual industrial production of Pakistan. 

Noise Term "et"

The noise, sometimes called the white noise term, is used to represent random fluctuation or unexplained variation in a phenomenon. The noise term arises due to various sources, like measurement error, outliers, missing observations, and random fluctuations.

Assumptions of Noise Term

i. The mean of the noise term is zero.

E (et) = 0

ii. The variance of the noise term is constant.

Var (et) = σ²

iii. The correlation between lags of the noise term is zero.

ret, et-1 = 0

Types of Time Series

1.      Continuous Time Series

2.      Discrete Time Series

Continuous Time Series

In continuous time series, the observations are measured on a variable or an activity at an irregular instance of time and represented by Yt; t ϵ R. 

e.g., temperature readings, flow of rivers, concentration of a chemical process, etc. can be recorded as a continuous time series.

 

Discrete Time Series

The observations are measured on a variable or an activity at discrete points in time. Usually in a discrete time series the consecutive observations are recorded at equally spaced time intervals, such as hourly, daily, weekly, monthly, or yearly time separations.

e.g., the population of a particular city, the production of a company, and exchange rates between two different currencies may represent discrete time series.

Objectives of Time Series Analysis

Following are the main objectives of time series analysis:

i. To understand the past pattern of time series data and forecast the future pattern.

ii. To study and analyse the causes or sources of variation in time series data.

iii. To develop techniques to minimise the causes or sources of variation in the field.

vi. To develop smooth field operations in planning and administration. 

v. To sketch out the entire phenomenon or activity under study.

• Read More: Components of Time Series