Basic Statistics and Data Analysis

Statistics Lecture Notes, Online MCQs

Category: Analysing the Secular Trend

Analysing the Secular Trend / Time Series Analysis and Forecasting

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The Method of Free Hand Curve

The secular trend is measured by the method of the free-hand curve in the following steps:

  • Take the time periods along $x$-axis by taking appropriate scaling
  • Plot the points for observed values of the $Y$ variable as the dependent variable against the given time periods
  • Join these plotted points by line segments to get a historigram
  • Draw a free hand smooth curve (or a straight line) through the histogram

In this method we draw the given times series data on graph paper, then we draw a free hand trend line through the plotted graph according to the trend of the graph. Then we read trend values from this free-hand trend line.

It is generally preferred to use a curve instead of a straight line to show the secular trend.

Merits:

  • The free-hand curve method is a simple, easy, and quick method for measuring secular trends.
  • A well-fitted trend line (or curve) gives a close approximation to the trend based on a mathematical model.

Demerits:

  • It is a rough and crude method.
  • It is greatly affected by personal bias as different persons may fit different trends to the same data.
    The estimates are not reliable due to personal bias.

Question: The following time series shows the number of road accidents in Punjab for the year 1972 to 1978.

Year1972197319741975197619771978
No. of Accidents2493263826993038374540794688
  • Obtain the historigram showing the number of road accidents and a free hand trend line by drawing a straight line
  • Find the trend values for this time series

Solution:

Method of Free Hand Curve
YearValueTotalMeanTrend Value
19722493  2200
19732638  2550
19742699  2950
1975303823338033403340
19763745  3650
19774079  4050
19784688  4499

Analysing the Secular Trend / Time Series Analysis and Forecasting

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The Secular Trend

For the estimation of the secular trend of a time series, the most commonly used method is to fit a straight line $\hat{y} = a+bx$, an exponential curve $\hat{y}=ab^x$, and a second-degree parabola $\hat{y}=a +bx+ cx^2$, etc, where $y$ is the value of a time series variable, $x$ representing the time and all others are constants (the intercept $a$, and the slope $b$). The method of least squares is a widely used method to determine the values of the constants appearing in such an equation.

It is used

  • For the purpose of prediction (or projection) into the future
  • The detrending process (removal of trend) in a time series for studying other non-trend fluctuations.
  • It is used for historical description

The secular trend can be represented either by a straight line or by some type of smooth curve. It is measured by the following methods:

Least Squares Method (secular trend)

The secular trend may be used whether in determining how a time series has grown in the past or in making a forecast. The trend line is used to adjust a series to eliminate the effect of the secular trend in order to isolate non-trend fluctuations.

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