<p>For negative values alone you can define relative change as:</p>
<p><span class="math-container">$$\frac{X_t-X_{t-1}}{|X_{t-1}|}$$</span></p>
<p>This is quite common way to deal with rates of change when you have negative numbers. However, when the denominator is zero then the growth rate is not defined. This can solve any issue when zero <strong>is not</strong> a base.</p>
<p>When zero is the base unfortunately there is no way how you can calculate growth rate while keeping the zero there. This is because if you start from zero any change represents <span class="math-container">$\infty$</span> increase. Going from 0 to 1 is <span class="math-container">$\infty$</span> growth rate as well as going from 0 to 10 or 0 to 1000. You have to either get rid of a zero in some way or another or simply do not calculate growth rate but some other measure of change (e.g. simple change <span class="math-container">$X_t -X_{t-1}$</span>).</p>
<blockquote>
<p>I have already thought of adding an arbitrary but reasonable positive number to data points, to bring everything into a positive domain. But this is a very crude and not an elegant way to do this.</p>
</blockquote>
<p>Yes, this is very crude way of doing it but perhaps there are more elegant alternatives. Using the temperature as an example a solution would be to convert degrees Celsius to Kelvin (with <span class="math-container">$T_{(K)} = T_{(°C)} + 273.15$</span>) which would solve the problem.</p>
<p>Perhaps there is an another way to measure the quantity you want. Otherwise, adding an arbitrary constant would be a solution. It is hard to recommend any specific way without knowing all the details of the research.</p>

CC BY-SA 4.0
2021-01-09T11:44:39.997Z
Sir, Thank you for your answer. The definition you gave for the relative change still has a problem of dealing with zeros. Value of zero has a substantive meaning and I cannot remove the data points with a value of zero from my sample.
DrStrangeLove
CC BY-SA 4.0
2021-01-09T11:46:44.890Z
The temperature was just an example to explain the problem clearly so that my issue is understandable to a large audience. Nightlights are measured on a relative scale by an unit called Digital Number (DN).
DrStrangeLove
CC BY-SA 4.0
2021-01-09T11:49:09.350Z
@DrStrangeLove oh ok, I though you actually had a problem with temperature since you been vague about it. Well, if you have zero as a base growth rates are simply not defined, there is no direct way how to apply it. As a matter of fact if your base is zero any increase from zero represents $\infty$ growth. You either need to find some alternative transformation like in the example with temperature (or worst case scenario just adding any constant), or what is the reason why you want to calculate changes of that variable? There might be some alternative solution for that reason
1muflon1
CC BY-SA 4.0
2021-01-09T11:51:23.410Z
@DrStrangeLove Maybe you should improve your question then. In your question 80% of what you write are about temperature. Sure, it is only an example but when you offer no real description of what kind of data on nightlights you are considering, it is impossible to help you in any other way than dealing with the example. If the example does not match the data you have in some way, maybe you have chosen a bad example.
Jesper Hybel
CC BY-SA 4.0
2021-01-09T12:01:12.333Z
@Jesper Sure I will do that. But the problem I am raising is more general and not specific to me. How do you calculate growth rates when you have negative values and zeroes in your dataset?
DrStrangeLove
CC BY-SA 4.0
2021-01-09T12:02:36.453Z
@DrStrangeLove Fair enough.
Jesper Hybel
CC BY-SA 4.0
2021-01-09T12:03:42.157Z
@DrStrangeLove those are 2 separate questions. As mentioned in my answer A) for negative values you can use the formula above. B) for zero base the growth rate simply is not defined. That is the answer it’s like asking how do you square a circle. Only solution is either not to calculate growth and use different transformation to measure change (e.g. simple difference) or transform the series to remove zero.
1muflon1
CC BY-SA 4.0
2021-01-09T12:07:09.970Z
I am studying convergence, so not calculating growth rates or using a simple difference is out of question. If you could suggest an elegant transformation to remove zeroes so that the distribution of growth is not affected
DrStrangeLove
CC BY-SA 4.0
2021-01-09T12:14:24.123Z
@DrStrangeLove well given that for base 0 growth rate does not exist any transformation will affect the distribution of growth rates (as when zero is there as a base those observations wont enter the distribution). However, adding a constant there could be a solution.
1muflon1
CC BY-SA 4.0
2021-01-09T10:53:19.177Z
2021-01-09T12:12:29.203Z
2021-01-09T12:12:29.203Z
1muflon1
<p>I am studying nightlights data and trying to model cross-sectional convergence in nightlights in India. However data has some stray light corrections, which has made some of the measurements below zero. Another point being, that the value of 0 has a representative meaning.</p>
<p>Hence any conventional way of measuring growth rates in literature is not applicable here.</p>
<p>In simple words: Nightlights are measured on a relative scale from 0 to 63. The corrections for cloud cover and ephemeral lights makes some of the data points negative. So:</p>
<p>It is analogous to ask what is the percentage growth in nightlights if it increases from - 1 DN to 0 DN</p>
<p>Even more problematic question is what is the percentage growth in nightlights if it increases from 0 DN to 2 DN.</p>
<p>I have already thought of adding an arbitrary but reasonable positive number to data points, to bring everything into a positive domain. But this is a very crude and not an elegant way to do this.</p>

<economic-growth><economic-measurement><growth-accounting>
Measuring growth rate when values are negative
DrStrangeLove
# Measuring growth rate when values are negative

(Asked by: username_0 on 2021-01-09)

I am studying nightlights data and trying to model cross-sectional convergence in nightlights in India. However data has some stray light corrections, which has made some of the measurements below zero. Another point being, that the value of 0 has a representative meaning.

Hence any conventional way of measuring growth rates in literature is not applicable here.

In simple words: Nightlights are measured on a relative scale from 0 to 63. The corrections for cloud cover and ephemeral lights makes some of the data points negative. So:

It is analogous to ask what is the percentage growth in nightlights if it increases from - 1 DN to 0 DN

Even more problematic question is what is the percentage growth in nightlights if it increases from 0 DN to 2 DN.

I have already thought of adding an arbitrary but reasonable positive number to data points, to bring everything into a positive domain. But this is a very crude and not an elegant way to do this.

**Tags:** economic-growth, economic-measurement, growth-accounting


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## 1 answers

### Answer by username_1 on 2021-01-09. Score: 2

For negative values alone you can define relative change as:

$$\frac{X\_t-X\_{t-1}}{|X\_{t-1}|}$$

This is quite common way to deal with rates of change when you have negative numbers. However, when the denominator is zero then the growth rate is not defined. This can solve any issue when zero **is not** a base.

When zero is the base unfortunately there is no way how you can calculate growth rate while keeping the zero there. This is because if you start from zero any change represents $\infty$ increase. Going from 0 to 1 is $\infty$ growth rate as well as going from 0 to 10 or 0 to 1000. You have to either get rid of a zero in some way or another or simply do not calculate growth rate but some other measure of change (e.g. simple change $X\_t -X\_{t-1}$).

> I have already thought of adding an arbitrary but reasonable positive number to data points, to bring everything into a positive domain. But this is a very crude and not an elegant way to do this.

Yes, this is very crude way of doing it but perhaps there are more elegant alternatives. Using the temperature as an example a solution would be to convert degrees Celsius to Kelvin (with $T\_{(K)} = T\_{(°C)} + 273.15$) which would solve the problem.

Perhaps there is an another way to measure the quantity you want. Otherwise, adding an arbitrary constant would be a solution. It is hard to recommend any specific way without knowing all the details of the research.

> **Comments:**
> **username_0**: Sir, Thank you for your answer. The definition you gave for the relative change still has a problem of dealing with zeros. Value of zero has a substantive meaning and I cannot remove the data points with a value of zero from my sample.  
> **username_0**: The temperature was just an example to explain the problem clearly so that my issue is understandable to a large audience. Nightlights are measured on a relative scale by an unit called Digital Number (DN).  
> **username_1**: @username_0 oh ok, I though you actually had a problem with temperature since you been vague about it. Well, if you have zero as a base growth rates are simply not defined, there is no direct way how to apply it. As a matter of fact if your base is zero any increase from zero represents $\infty$ growth. You either need to find some alternative transformation like in the example with temperature (or worst case scenario just adding any constant), or what is the reason why you want to calculate changes of that variable? There might be some alternative solution for that reason  
> **username_2**: @username_0 Maybe you should improve your question then. In your question 80% of what you write are about temperature. Sure, it is only an example but when you offer no real description of what kind of data on nightlights you are considering, it is impossible to help you in any other way than dealing with the example. If the example does not match the data you have in some way, maybe you have chosen a bad example.  
> **username_0**: @username_2 Sure I will do that. But the problem I am raising is more general and not specific to me. How do you calculate growth rates when you have negative values and zeroes in your dataset?  
> **username_2**: @username_0 Fair enough.  
> **username_1**: @username_0 those are 2 separate questions. As mentioned in my answer A) for negative values you can use the formula above. B) for zero base the growth rate simply is not defined. That is the answer it’s like asking how do you square a circle. Only solution is either not to calculate growth and use different transformation to measure change (e.g. simple difference) or transform the series to remove zero.  
> **username_0**: I am studying convergence, so not calculating growth rates or using a simple difference is out of question. If you could suggest an elegant transformation to remove zeroes so that the distribution of growth is not affected  
> **username_1**: @username_0 well given that for base 0 growth rate does not exist any transformation will affect the distribution of growth rates (as when zero is there as a base those observations wont enter the distribution). However, adding a constant there could be a solution.  

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