<p>My best guess is that they detrended by computing the discounted value. (Although I'm not sure if this is really what they've done.)</p>
<p>Let <span class="math-container">$r = 0.015$</span> be the growth rate and let <span class="math-container">$t$</span> be the time period and <span class="math-container">$t_0$</span> the base year, (e.g. <span class="math-container">$t = 2000$</span> and <span class="math-container">$t_0 = 1982$</span>).</p>
<p>If <span class="math-container">$Y_t$</span> is GDP in period <span class="math-container">$t$</span> then the detrended GDP is given by:
<span class="math-container">$$
\frac{Y_t}{(1+r)^{t-t_0}}.
$$</span></p>
<p>If you work in quarters, <span class="math-container">$t$</span> should also be transformed accordingly. For example, if the period is <span class="math-container">$2010Q1$</span>, you should use <span class="math-container">$t = 2010.25$</span>.</p>

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2021-10-07T14:38:56.817Z
2021-10-07T14:38:56.817Z
tdm
<p>What does it mean to detrend a time series at a given rate, say, 1.5%?</p>
<p><a href="https://i.sstatic.net/qNGll.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/qNGll.png" alt="enter image description here" /></a></p>
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<p>What is the formula here? <span class="math-container">$X_t - trend$</span>? where trend = average growth rate? Say <span class="math-container">$X_t - 0.015$</span>?</p>

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2021-10-07T09:13:36.683Z
Your understanding of the formula is correct, if I am not mistaken. Probably more relevantly however is the process of estimating the trend such as through the residuals.
EB3112
<macroeconomics><business-cycles>
What does it mean to detrend a time series at a given rate?
Emmanuel Ameyaw
# What does it mean to detrend a time series at a given rate?

(Asked by: username_0 on 2021-10-07)

What does it mean to detrend a time series at a given rate, say, 1.5%?

[![enter image description here](https://i.sstatic.net/qNGll.png)](https://i.sstatic.net/qNGll.png)

[![enter image description here](https://i.sstatic.net/RkRSI.png)](https://i.sstatic.net/RkRSI.png)

What is the formula here? $X\_t - trend$? where trend = average growth rate? Say $X\_t - 0.015$?

**Tags:** macroeconomics, business-cycles

> **Comments:**
> **username_1**: Your understanding of the formula is correct, if I am not mistaken. Probably more relevantly however is the process of estimating the trend such as through the residuals.  

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## 1 answers

### Answer by username_2 on 2021-10-07. Score: 1

My best guess is that they detrended by computing the discounted value. (Although I'm not sure if this is really what they've done.)

Let $r = 0.015$ be the growth rate and let $t$ be the time period and $t\_0$ the base year, (e.g. $t = 2000$ and $t\_0 = 1982$).

If $Y\_t$ is GDP in period $t$ then the detrended GDP is given by:
$$
\frac{Y\_t}{(1+r)^{t-t\_0}}.
$$

If you work in quarters, $t$ should also be transformed accordingly. For example, if the period is $2010Q1$, you should use $t = 2010.25$.


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