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


1 Answer 1


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.

Usually, people are interested in three-month moving average because the value for each month is highly volatile and can't be used with much precision.


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