# Meaning of UK Employment Change Index - 3 Months Average

I am trying to understand the meaning of an economic index which is measured in the following way:

The value for month n is determined by calculating the average of the last 3 months (n, n-1, n-2) and subtracting the average of the previous (non-overlapping) 3 months (n-3, n-4, n-5).

I wonder if anyone has an idea as to how to isolate a certain month (that data is not published) or how to better "understand" the meaning of the value of a specific month - the economic meaning (difference between one month and another) is not that clear to me.

Thanks, Tom

Let's say the raw series is $$x_1, x_2, \ldots$$. The three-month moving average is given by $$y_n = \frac 1 3 \cdot (x_{n-2}+x_{n-1}+x_n)$$. What you have, though is $$z_n = y_n - y_{n-1} = \frac 1 3 \cdot(x_n - x_{n-3})$$.
Without knowing any of the $$x_n$$'s, you can't back out $$x_n$$'s from $$z_n$$'s. Knowing only $$x_1$$, however, would give you $$x_1, x_4, x_7, \ldots$$. It's likely that you would know this, since the series we're interested in is an index, which usually gives the base year.
In all, you need to know at least $$x_1, x_2, x_3$$ (or, to be more precise, at least one $$x_i$$ where $$\mod(i, 3) = j$$ for $$j \in \{1, 2, 3\}$$ to be able to back out the entire series of $$x_n$$'s.