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

Year	Coded Year
1990	0
1991	1
1992	2
1993	3
1994	4

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

Year(t)	$x=t-\overline{t}$	$x=\frac{t-\overline{t}}{1/2}$
1990	-2.5	-5
1991	-1.5	-3
1992	-0.5	1
1993	0.5	1
1994	1.5	3
1995	2.5	5

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.