Category: Introduction

Seasonal Variations: Estimation

We have to find out a way of isolating and measuring the seasonal variations. There are two reasons for isolating and measuring the effect of seasonal variation.

  • To study the changes brought by seasons in the values of the given variable in a time series
  • To remove it from the time series to determine the value of the variable

Summing the values of a particular season for a number of years, the irregular variations will cancel each other, due to independent random disturbances. If we also eliminate the effect of trend and cyclical variations, the seasonal variations will be left out which are expressed as a percentage of their average.

A study of seasonal variation leads to more realistic planning of production and purchases etc.

Seasonal Index

When the effect of the trend has been eliminated, we can calculate a measure of seasonal variation known as the seasonal index. A seasonal index is simply an average of the monthly or quarterly value of different years expressed as a percentage of averages of all the monthly or quarterly values of the year.

The following methods are used to estimate seasonal variations.

  • Average percentage method (simple average method)
  • Link relative method
  • Ratio to the trend of short time values
  • Ratio to the trend of long time averages projected to short times
  • Ratio to moving average

The Simple Average Method

Assume the series is expressed as


Considering the long time averages as trend values and eliminate the trend element by expressing a short time observed value as a percentage of the corresponding long time average. In the multiplicative model, we obtain

\frac{\text{short time observed value} }{\text{long time average}}\times &= \frac{TSCI}{T}\times 100\\
&=SCI\times 100

This percentage of long time average represents the seasonal (S), the cyclical (C) and the irregular (I) component.

Once $SCI$ obtained, we try to remove $CI$ as much as possible from $SCI$. This is done by arranging these percentages season-wise for all the long times (say years) and taking the modified arithmetic mean for each season by ignoring both the smallest and the largest percentages. These would be seasonal indices.

If the average of these indices is not 100, then the adjustment can be made, by expressing these seasonal indices as the percentage of their arithmetic mean. The adjustment factor would be

\frac{100}{\text{Mean of Seasonal Indiex}} \rightarrow \frac{400}{\text{sums of quarterly index}} \,\, \text{ or } \frac{1200}{\text{sums of monthly indices}}

Question: The following data is about number of automobile sold.

YearQuarter 1Quarter 2Quarter 3Quarter 4

Calculate the seasonal indices by the average percentage method.


First, we obtain the yearly (long term) averages

Year Total11311098127213381353
Yearly Average1131/4=282.75274.50318.00334.50338.25

Next, we divide each quarterly value by the corresponding yearly average and express the results as percentages. That is,

YearQuarter 1Quarter 2Quarter 3Quarter 4 
Total (modfied)
Mean (modified)

* on values represents smallest and largest values in a quarter that are not included in the total.

Read about Component of Time Series

Coding Time Variable

Coding Time Variable by Taking Origin at Beginning

Suppose we have time-series data for the years 1990, 1991, 1992, and 1994.

We can take the origin of a time series at the beginning and assign $x = 0$ to the first period and $1, 2, 3, …$ to other periods. The code for the year will be

YearCoded Year

Coding Time Variable by Taking Middle Years as Zero

To simplify the trend calculations, the time variable t (year variable) is coded by taking deviations $t-\overline{t}$, where $\overline{t}$ is average number computed as $\overline{t}=\frac{First\, Period + Last\, Period}{2}$. Taking $x=t-\overline{t}$ we get
$$\sum x = 0 = \sum x^3 = \sum x^5 = \cdots$$

There for two cases:

  • When there are odd number of Years:
    For odd number of years (as in period 1990 to 1994) the $\overline{t}$ as the middle poin4t. The $\overline{t}$ is $\overline{t} = (1990+1994)/2=1992$ the code for the year $t$ is $x=t-\overline{t}$. For t=1990, we have $x=1990-1992 =0$. Thus the coded year is zero at $\overline{t}$. Now after taking x=0 at the middle of an odd number of years, we assign $-1, -2, …$ for the years before the middle of year and $1,2,…$ for the years after the middle year.
    Year (t) $x=t-\overline{t}$
    1990 -2
    1991 -1
    1992 0
    1993 1
    1994 2
  • When there are even numbers of years
    Suppose we have time-series data for the years 1990, 1991, 1992, 1993, 1994, 1995.The value of middle point is $\overline{t} = (1990+1995)/2 = 1992.5$. So $x=0$ halfway between the years 1992 and 1993 (in the middle of 1992 and 1993). For $t=1992$, we have $x=t-\overline{t}=1992-1992.5=-0.5$. Thus coding the middle of an even number of years as $x=0$, we assign $-0.5, -1.5, -2.5, …$ for the years before the middle year and $0.5, 1.5, 2.5, …$ for the years after the middle year as shown below

To avoid decimal in the coded year, we can take the unit of measurement as $\frac{1}{2}$ year. Therefore, after coding $x=0$ in the middle of an even number of years, we assign $-1,-3, -5,…$ for the year before the middle year and $1,3,5,…$ for the years after the middle year as shown above.

Multiplicative and Additive Model

Here we will discuss about multiplicative and additive model.

The analysis of a time series is the decomposition of a time series into its different components for their separate study. The process of analyzing a time series is to isolate and measure its various components. We try to answer the following questions when we analyze a time series.

  1. What would have been the value of the variable at different points of time if it were influenced only be long time movements?
  2. What changes occur in the value of the variable due to seasonal variations?
  3. To what extent and in what direction has the variable been effected by cyclical fluctuations?
  4. What has been the effect of irregular variations?

The study of a time series is mainly required for estimation and forecasting. An ideal forecast should base on forecasts of the various types of fluctuations. Separate forecasts should be made of the trend, seasonal and cyclical variations. These forecasts become doubtful for a forecast of irregular movements. Therefore, it is necessary to separate and measure various types of fluctuations present in a time series.

A value of a time series variable considered as the resultant of the combined impact of its components. The components of a time series follow either the multiplicative or the additive model.

Let $Y$= original observation, $T$= trend component, $S$=seasonal component, $C$=cyclical component, and $I$=irregular component.

Multiplicative Model

It is assumed that the value $Y$ of a composite series is the product of the four components. That is

$$Y = T \times S \times C \times I,$$

where $T$ is given in original units of $Y$, but $S$, $C$, and $I$ are expressed as percentage unit-less index numbers.

Additive Model

It is assumed that the value of $Y$ of a composite series is the sum of the four components. That is

$$Y = T + S + C + I,$$

where $T$, $S$, $C$, and $I$ all are given in the original units of $Y$.

Time series analysis is the analysis of a series of data-points over time, allowing one to answer a question such as what is the causal effect on a variable $Y$ of a change in variable $X$ over time? An important difference between time series and cross-section data is that the ordering of cases does matter in time series.

Rather than dealing with individuals as units, the unit of interest is time: the value of $Y$ at time $t$ is $Y_t$. The unit of time can be anything from days to election years. The value of $Y_t$ in the previous period is called the first lag value: $Y_{t-1}$. The jth lag is denoted: $Y_{t-j}$. Similarly $Y_{t+1}$ is the value of $Y_t$ in the next period. So a simple bivariate regression equation for time series data looks like: \[Y_t = \beta_0 + \beta X_t + u_t\]

$Y_t$ is treated as random variable. If $Y_t$ is generated by some model (Regression model for time series i.e. $Y_t=x_t\beta +\varepsilon_t$, $E(\varepsilon_t|x_t)=0$, then ordinary least square (OLS) provides a consistent estimates of $\beta$.

Read about Introduction to Time Series Data


Here we will discuss the graphical representation of time series data, called historigram.

As we have discussed in the introduction to Time Series Data, 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 an independent variable) and the observed value ($Y_t$) on Y-axis (as the dependent variable: as a function of time). Such a graph will show various types of fluctuations and other points of interest.

A historigram is a graphical representation of a time series that reveals the changes that occurred at different time periods. A first step in the prediction (or forecast) of a time series involves an examination of the set of past observations. In this case, historigram may be a useful tool. The construction of a historigram involves the following steps described below:

  • Use an appropriate scale and take time $t$ along $x$-axis as an independent variable.
  • Use an appropriate scale, plot the observed values of variable $Y$ as a dependent variable against the given points of time.
  • Join the plotted points by line segments to get the required graphical representation.

Example: Draw a historigram to show the population of Pakistan in various census years.

Census Year195119611972198119982017
Population (Million33.4442.8865.3183.78130.58200.17
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