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
19900
19911
19922
19933
19944

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
1992-0.51
19930.51
19941.53
19952.55

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.

Muhammad Imdad Ullah

Currently working as Assistant Professor of Statistics in Ghazi University, Dera Ghazi Khan. Completed my Ph.D. in Statistics from the Department of Statistics, Bahauddin Zakariya University, Multan, Pakistan. l like Applied Statistics, Mathematics, and Statistical Computing. Statistical and Mathematical software used is SAS, STATA, GRETL, EVIEWS, R, SPSS, VBA in MS-Excel. Like to use type-setting LaTeX for composing Articles, thesis, etc.

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