Coding Time Variable by Taking Origin at the 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
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 the 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 are two cases when coding a Time Variable (when taking zero in the Middle):
- When there are an odd number of Years:
For an odd number of years (as in the period 1990 to 1994) the $\overline{t}$ is the middle point. 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 the 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, and 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 decimals 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.
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