<p>Let's say the raw series is <span class="math-container">$x_1, x_2, \ldots$</span>. The three-month moving average is given by <span class="math-container">$y_n = \frac 1 3 \cdot (x_{n-2}+x_{n-1}+x_n)$</span>. What you have, though is <span class="math-container">$z_n = y_n - y_{n-1} = \frac 1 3 \cdot(x_n - x_{n-3})$</span>.</p>

<p>Without knowing any of the <span class="math-container">$x_n$</span>'s, you can't back out <span class="math-container">$x_n$</span>'s from <span class="math-container">$z_n$</span>'s. Knowing only <span class="math-container">$x_1$</span>, however, would give you <span class="math-container">$x_1, x_4, x_7, \ldots$</span>. It's likely that you would know this, since the series we're interested in is an index, which usually gives the base year.</p>

<p>In all, you need to know at least <span class="math-container">$x_1, x_2, x_3$</span> (or, to be more precise, at least one <span class="math-container">$x_i$</span> where <span class="math-container">$\mod(i, 3) = j$</span> for <span class="math-container">$j \in \{1, 2, 3\}$</span> to be able to back out the entire series of <span class="math-container">$x_n$</span>'s.</p>

<p>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.</p>

CC BY-SA 4.0
2019-11-19T10:00:33.957Z
2019-11-19T10:00:33.957Z
Art
<p>I am trying to understand the meaning of an economic index which is measured in the following way:</p>

<p>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).</p>

<p>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.</p>

<p>Thanks, Tom</p>

<macroeconomics><united-kingdom>
Meaning of UK Employment Change Index - 3 Months Average
tom schneorson
# Meaning of UK Employment Change Index - 3 Months Average

(Asked by: username_0 on 2019-11-14)

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

**Tags:** macroeconomics, united-kingdom

---

## 1 answers

### Answer by username_1 on 2019-11-19. Score: 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.


---

