# Time series analysis

A time series is a series of data points indexed (or listed or graphed) in time order. Most commonly, a time series is a sequence taken at successive equally spaced points in time. Thus it is a sequence of discrete-time data. Examples of time series are heights of ocean tides, counts of sunspots, and the daily closing value of the Dow Jones Industrial Average. Time series are very frequently plotted via line charts .

## Discrete Time Series^{[1]}

**A time series is said to be discrete when observations are taken at specific time, usually equally spaced. The term discrete is used for series of this type even when the measured variable is continuous variable.**

Most of Macroeconomic and finance data comes in form of time series. GNP or Stock Return is example fo time series data.

We can write a series as {x1,x2,x3,⋯,xT}{x1,x2,x3,⋯,xT} or {xt}{xt}, where t=1,2,3,⋯,Tt=1,2,3,⋯,T. xtxt is treated as random variable.

Time series analysis refers to the branch of statistics where observations are collected sequentially in time, usually but not necessarily at equal spaced time points. The arcane difference between time series and other variable is use of subscript.

Time series analysis comprises methods for analyzing time series data in order to extract some useful (meaningful) statistics and other characteristics of the data, while Time series forecasting is the use of a model to predict future values based on previously observed values.

Given an observed time series, the first step in analyzing a time series is to plot the given series on a graph taking time intervals (t) along X-axis (as independent variable) and the observed value (YtYt) on Y-axis (as dependent variable). Such a graph will show various types of fluctuations and other point of interest.

Note

YtYt is treated as random variable. If YtYt is generated by some model (Regression model for time series i.e. Yt=xtβ+εtYt=xtβ+εt, E(εt|xt)=0E(εt|xt)=0, then ordinary least square (OLS) provides a consistent estimates of ββ. Time series interchangeably used for sample {xt}{xt} and probability model. A possible probability model for the joint distribution of a time series {xt}{xt} is xt=εtxt=εt, εt∼iidN(0,σ2ε)εt∼iidN(0,σε2)

Time series are typically not iid (Independent Identically Distributed) e.g. If GNP today is unusually high, GNP tomorrow will also likely to be unusually high.