Time Series - 1 Method of Least Squares - Fitting of Linear Trend - Odd number of years
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Definitions
“A time series may be defined as a sequence of values of same variable corresponding to successive points in time.” – W. Z. Hersch
“A time series may be defined as a sequence of repeated measurement of a variable made periodically through time.” – Cecil H. Mayers
Analysis of Time Series
“The main object of analyzing time series is to understand, interpret and evaluate changes in economic phenomena in the hope of more correctly anticipating the course of future events.” – Hersch
A time series is a dynamic distribution, which reveals a good deal of variations over time. Statistical methods are, therefore, required to analyze various types of movements in a time series. There may be cyclical variations in general business activity and there may be short duration seasonal variations. There are also some accidental and random variables. The primary purpose of the analysis of time series is to discover and measure all such types of variations, which characterize a time series.
Time series analysis means analyzing the historical patterns of the variable that have occurred in past as a means of predicting the future value of the variable. It helps to identify and explain the following:
(i) Any regular or systematic variation in the series of data which is due to seasonality- the ‘seasonal’
(ii) Cyclical patterns.
(iii) Trends in the data.
(iv) Growth rates of these trends.
This method can be useful when no major environmental changes are expected and it does highlight seasonal variations in sales and consumer demand. However, time series analysis is limited when organizations face volatile environments.
Components of Time series –
The time series are classified into four basic types of variations which are analyzed below:
T = Trend
S = Seasonal variations
C = Cyclic variations
I = Irregular fluctuations.
This composite series is symbolized by the following general terms: O = T x S x C x I
Where
O = Original data
T = Trend
S = Seasonal variations
C = Cyclic variations
I = Irregular components.
This Multiplicative model is to be used when S, C, and I are given in percentages. If, however, their true (absolute) values are known the model takes the additive form i.e., O=T+C+S+I.
Algebraic Method For Finding Trend (Method of curve fitting by the principle of Least Squares)
Fitting of Linear Trend
Let the straight line trend between the given time series values (y) and time (x) be given by the standard equation: y = a + bx
Then for any given time ‘x’ the estimated value of ye as given by the equation is ye = a + bx
The following two normal equations are used for estimating 'a' and 'b'.
Σy = na + bΣx
Σxy = aΣx + bΣx^2
When Odd No. of Years, [X = (Year – Origin) / Interval]
Case
Given below are the figures of sales (in '000 units) of a certain shop. Fit a straight line by the method of least square and show the estimate for the year 2017:
Year: 2010 2011 2012 2013 2014 2015 2016
Sales: 125 128 133 135 140 141 143
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