Shifting of origin is a technique used in trend analysis to adjust the trend line to start from a different point in time. This is useful when we want to compare trends over different time periods. Shifting of origin involves subtracting a constant value from the time variable (usually the x-axis) so that the new trend line starts from the desired point in time.
For example, suppose we have annual data from 2010 to 2020 and we want to shift the origin to start from 2015. We can subtract 2015 from each year, so that the new time variable ranges from -5 to 5 (instead of 2010 to 2020), and then fit a new trend line to the shifted data.
Conversion of Annual Linear Trend Equation to Quarterly/Monthly Basis:
When working with time series data, we often want to analyze the data on a different time scale than the original data. For example, we may want to analyze quarterly or monthly data, even if the original data is annual.
To convert an annual linear trend equation to a quarterly or monthly basis, we need to adjust the slope of the trend line to account for the shorter time period. Suppose we have an annual trend equation of the form:
y = a + bx
where y is the dependent variable (e.g., sales), x is the time variable (in years), a is the intercept, and b is the slope. To convert this equation to a quarterly basis, we can use the following formula:
y = a + (b/4)x
where the slope coefficient b is divided by 4 to account for the fact that there are 4 quarters in a year. To convert the equation to a monthly basis, we can use the following formula:
y = a + (b/12)x
where the slope coefficient b is divided by 12 to account for the fact that there are 12 months in a year.
Vice-Versa: Conversion of Quarterly/Monthly Linear Trend Equation to Annual Basis:
Conversely, we may also want to convert a quarterly or monthly trend equation to an annual basis. To do this, we need to adjust the slope of the trend line to account for the longer time period. Suppose we have a quarterly or monthly trend equation of the form:
y = a + bx
where y is the dependent variable, x is the time variable (in quarters or months), a is the intercept, and b is the slope. To convert this equation to an annual basis, we can use the following formula:
y = a + 4b*x
for quarterly data or
y = a + 12b*x
for monthly data.