<p>Formally, mathematically, random variable is not actually a variable but a function that assigns possible values from the probability space to the measurable space.</p>
<p>Or in short variable <span class="math-container">$Y$</span> is a random variable if:</p>
<p><span class="math-container">$$Y: \Omega \to M$$</span></p>
<p>Where <span class="math-container">$\Omega$</span> is the probability space, and <span class="math-container">$M$</span> measurable space.</p>
<p>In applied economics, i.e. statistically speaking, GDP is clearly random variable, as the observed GDP is simply the mapping from the probability space of all GDP values that could exist to some measurable outcome that we measure.</p>
<p>By the way this has nothing to do with social sciences, statistically speaking, even a measured mass of proton is a random variable, since when you try to estimate the mass of proton there is a probability space of various possible masses, and every taken measurement (which will necessarily have some measurement error) can be thought of as a result of mapping from the probability space to measurement space.</p>

CC BY-SA 4.0
2024-03-05T08:57:14.437Z
Hi! Sorry, I don't get your argument. You give a formal definition of random variables and then argue that the randomness of GDP in time series models does actually come only from measurement?
Giskard
CC BY-SA 4.0
2024-03-05T10:56:04.607Z
@Giskard my argument, if its not clear from the answer is as follows; 1. Random variables have certain definition, 2. Recorded GDP satisfy the definition, this is among other things because of measurement error. For example if it helps, Output (which is equivalent to GDP since the output is given by sum of all consumption, investment, government spending and net exports) in undergraduate IS-LM is not a random variable
1muflon1
CC BY-SA 4.0
2024-03-05T11:30:51.330Z
Okay, but 1. why is a mathematical definition necessary here? You never use the notions again. Are you padding the answer like one would an academic submission to make it look more sciency? 2. Yeah, undergrad IS-LM model treats both demand and supply as deterministic, so no surprise there.
Giskard
CC BY-SA 4.0
2024-03-05T11:31:06.737Z
I just don't see how your answer answers the original question I guess? It is "randomness can also follow from measurement errors", just very very long? Do you think this is the most relevant source of randomness in time series models?
Giskard
CC BY-SA 4.0
2024-03-05T18:04:36.087Z
@Giskard what do you mean why is it necessary? You don’t think definition of category is necessary when discussing whether something belongs to that category? How do you decide if something is mammal when you don’t know what mammal means (and OP was clearly not aware of definition otherwise there wouldn’t have been any confusion) I don’t think it’s necessary the most relevant but it’s the easier one to understand that’s just an example
1muflon1
CC BY-SA 4.0
2024-03-05T18:09:52.217Z
. OP clearly doesn’t understand what *random variable* means and that is ok because the term random variable is very misleading since ironically random variable is not actually variable in standard mathematical sense but function. Any empirically measured quantity will be such function, even physical constants such as speed of light are random variables in applied work, due to measurement error but there are also other reasons, but one just need to mention one example to show it’s a random variable
1muflon1
CC BY-SA 4.0
2024-03-05T21:01:48.447Z
Wow. It was only a few days ago that you as a [mod read the comment thread](https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95847_57815) under the other question where the answerer has claimed to have full insight into what OP meant. Then it turned out they didn't. (And refused to acknowledge it!)
Giskard
CC BY-SA 4.0
2024-03-05T21:43:53.697Z
@Giskard what are you talking about? I usually move comments to chat when there is a lot of back and forth, even if some of it is useful. That is what I was told by other older moderators even outside E.SE that is the right thing to do, even when some comments are useful. I was told that multiple times by senior mods in teachers lounge. Some mods automatically do that for any case where there is automatic flag from a bot. I am also happy to respond to community so I brought your comments back even though a regular mod would probably just kept them in chat anyway.
1muflon1
CC BY-SA 4.0
2024-03-05T21:46:10.973Z
Also what am I supposed to acknowledge here? The comments got flagged by bot, there was a lot of bickering, so I moved them to chat, and then you made a reasonable request so I brought some back... also if you have issues with this raise them on meta, what does this have to do with my answer here?
1muflon1
CC BY-SA 4.0
2024-03-06T06:45:37.047Z
I was referring to this part of your comment: "*OP clearly doesn’t understand what random variable means*" Other guy I was "bickering" with was just as arrogant.
Giskard
CC BY-SA 4.0
2024-03-06T09:44:39.320Z
@Giskard I am not trying to be arrogant in this thread, but this is in my opinion clear from the question
1muflon1
CC BY-SA 4.0
2024-03-05T07:39:58.987Z
2024-03-05T07:39:58.987Z
1muflon1
<p>By the same argument, a die roll is also not random,<sup>1</sup> because when I measure the outcome, i.e., look at it, that is much more reliably deterministic than the measuring of GDP.</p>
<p>The models probably do not treat the measurement as random, but the future value of the variable itself. In which case GDP, a component of which is strongly weather dependent agricultural output, is very clearly random from our point of view. (Without magic supercomputers + apriori.)</p>
<hr />
<p><sup>1</sup>Yes yes, a physics super computer might be able to predict all die rolls depending on the exact nature of the universe. This is beside the point.</p>

CC BY-SA 4.0
2024-03-05T18:12:13.550Z
“The models probably do not treat the measurement as random” - this is false in applied economics, you can have look at any empirical work. Even past observations of GDP are considered to be random variable in applied work. You can literally open any work using let’s say regression analysis and GDP as dependent or even independent variable and it will be treated as random variable during the whole process
1muflon1
CC BY-SA 4.0
2024-03-05T21:05:13.907Z
Disclaimer: I am 100% sure you have more insight into this area than I do, I was a game theorist. However, I am not sure we disagree here. I meant that they do not consider the act of measuring random. In any point of time the variables ahead of that point of time will be treated as random variables.
Giskard
CC BY-SA 4.0
2024-03-05T21:39:27.300Z
what do you mean by act of measuring? Act of measuring does not need to be random, but any empirically observed quantity is a random variable prima facie because it is generated by some unknown data generating process (DGP), What has this to do with game theory? In a theoretical game theory models you probably worked with there arent any DGPs
1muflon1
CC BY-SA 4.0
2024-03-06T06:42:26.200Z
Did you read my comment or just skim it? "*I am 100% sure you have more insight into this area than I do, I was a game theorist.*"
Giskard
CC BY-SA 4.0
2024-03-06T06:43:14.573Z
"*Act of measuring does not need to be random, but any empirically observed quantity is a random variable prima facie because it is generated by some unknown data generating process (DGP)*" I mean, we can juggle with what we mean by the word "act", but I think we are saying pretty much the same thing.
Giskard
CC BY-SA 4.0
2024-03-06T06:44:06.560Z
Sidenote: I would probably not use the abbrevation DGP in a context where we are talking about GDP.
Giskard
CC BY-SA 4.0
2024-03-06T09:54:30.417Z
1. Sorry I misread your comment as saying I have more insight my bad. 2. I am actually a bit confused because when I think of the act of measuring I imagine statistician simply recording values for example from tax returns to find the figure for total consumption, or in physics case some lab assistant recording mass of proton as reported by machine scale. That act itself is not random. But the measurement value on screen is by definition random variable. Ex ante there is a distribution of possible masses of a proton. Each measurement is a draw from this distribution.
1muflon1
CC BY-SA 4.0
2024-03-05T06:49:49.013Z
2024-03-05T06:49:49.013Z
Giskard
<p>I have seen in time series models that GDP is considered a random variable. At first, I found this troubling because I could not see any random process underlying the measurement. However, the argument for this claim is both sound and odd at the same time: in theory, GDP can take any value depending on political, natural and economic conditions (among other factors that affect this measure). Therefore, every measure of GDP is just a realization of something that could have occurred in a counterfactual world.</p>
<p>If this is so, then almost every measurable thing in social sciences, no matter if the source is a survey or a census, must be a random variable too as those things that affect GDP and make it follow an stochastic process also affect every aspect of human life. Is this correct?</p>

CC BY-SA 4.0
2024-03-05T03:27:46.200Z
Yeah, that's pretty much correct. 
Galen
CC BY-SA 4.0
2024-03-05T06:46:54.793Z
"Is this correct?" type questions are a poor fit for the SE format.
Giskard
<time-series><econometrics>
Why is GDP a random variable?
Santiago Valdivieso
# Why is GDP a random variable?

(Asked by: username_0 on 2024-03-05)

I have seen in time series models that GDP is considered a random variable. At first, I found this troubling because I could not see any random process underlying the measurement. However, the argument for this claim is both sound and odd at the same time: in theory, GDP can take any value depending on political, natural and economic conditions (among other factors that affect this measure). Therefore, every measure of GDP is just a realization of something that could have occurred in a counterfactual world.

If this is so, then almost every measurable thing in social sciences, no matter if the source is a survey or a census, must be a random variable too as those things that affect GDP and make it follow an stochastic process also affect every aspect of human life. Is this correct?

**Tags:** time-series, econometrics

> **Comments:**
> **username_1**: Yeah, that's pretty much correct.   
> **username_2**: "Is this correct?" type questions are a poor fit for the SE format.  

---

## 2 answers

### Answer by username_2 on 2024-03-05. Score: 2 | Accepted answer

By the same argument, a die roll is also not random,1 because when I measure the outcome, i.e., look at it, that is much more reliably deterministic than the measuring of GDP.

The models probably do not treat the measurement as random, but the future value of the variable itself. In which case GDP, a component of which is strongly weather dependent agricultural output, is very clearly random from our point of view. (Without magic supercomputers + apriori.)

---

1Yes yes, a physics super computer might be able to predict all die rolls depending on the exact nature of the universe. This is beside the point.

> **Comments:**
> **username_3**: “The models probably do not treat the measurement as random” - this is false in applied economics, you can have look at any empirical work. Even past observations of GDP are considered to be random variable in applied work. You can literally open any work using let’s say regression analysis and GDP as dependent or even independent variable and it will be treated as random variable during the whole process  
> **username_2**: Disclaimer: I am 100% sure you have more insight into this area than I do, I was a game theorist. However, I am not sure we disagree here. I meant that they do not consider the act of measuring random. In any point of time the variables ahead of that point of time will be treated as random variables.  
> **username_3**: what do you mean by act of measuring? Act of measuring does not need to be random, but any empirically observed quantity is a random variable prima facie because it is generated by some unknown data generating process (DGP), What has this to do with game theory? In a theoretical game theory models you probably worked with there arent any DGPs  
> **username_2**: Did you read my comment or just skim it? "*I am 100% sure you have more insight into this area than I do, I was a game theorist.*"  
> **username_2**: "*Act of measuring does not need to be random, but any empirically observed quantity is a random variable prima facie because it is generated by some unknown data generating process (DGP)*" I mean, we can juggle with what we mean by the word "act", but I think we are saying pretty much the same thing.  
> **username_2**: Sidenote: I would probably not use the abbrevation DGP in a context where we are talking about GDP.  
> **username_3**: 1. Sorry I misread your comment as saying I have more insight my bad. 2. I am actually a bit confused because when I think of the act of measuring I imagine statistician simply recording values for example from tax returns to find the figure for total consumption, or in physics case some lab assistant recording mass of proton as reported by machine scale. That act itself is not random. But the measurement value on screen is by definition random variable. Ex ante there is a distribution of possible masses of a proton. Each measurement is a draw from this distribution.  

---

### Answer by username_3 on 2024-03-05. Score: 1

Formally, mathematically, random variable is not actually a variable but a function that assigns possible values from the probability space to the measurable space.

Or in short variable $Y$ is a random variable if:

$$Y: \Omega \to M$$

Where $\Omega$ is the probability space, and $M$ measurable space.

In applied economics, i.e. statistically speaking, GDP is clearly random variable, as the observed GDP is simply the mapping from the probability space of all GDP values that could exist to some measurable outcome that we measure.

By the way this has nothing to do with social sciences, statistically speaking, even a measured mass of proton is a random variable, since when you try to estimate the mass of proton there is a probability space of various possible masses, and every taken measurement (which will necessarily have some measurement error) can be thought of as a result of mapping from the probability space to measurement space.

> **Comments:**
> **username_2**: Hi! Sorry, I don't get your argument. You give a formal definition of random variables and then argue that the randomness of GDP in time series models does actually come only from measurement?  
> **username_3**: @username_2 my argument, if its not clear from the answer is as follows; 1. Random variables have certain definition, 2. Recorded GDP satisfy the definition, this is among other things because of measurement error. For example if it helps, Output (which is equivalent to GDP since the output is given by sum of all consumption, investment, government spending and net exports) in undergraduate IS-LM is not a random variable  
> **username_2**: Okay, but 1. why is a mathematical definition necessary here? You never use the notions again. Are you padding the answer like one would an academic submission to make it look more sciency? 2. Yeah, undergrad IS-LM model treats both demand and supply as deterministic, so no surprise there.  
> **username_2**: I just don't see how your answer answers the original question I guess? It is "randomness can also follow from measurement errors", just very very long? Do you think this is the most relevant source of randomness in time series models?  
> **username_3**: @username_2 what do you mean why is it necessary? You don’t think definition of category is necessary when discussing whether something belongs to that category? How do you decide if something is mammal when you don’t know what mammal means (and OP was clearly not aware of definition otherwise there wouldn’t have been any confusion) I don’t think it’s necessary the most relevant but it’s the easier one to understand that’s just an example  
> **username_3**: . OP clearly doesn’t understand what *random variable* means and that is ok because the term random variable is very misleading since ironically random variable is not actually variable in standard mathematical sense but function. Any empirically measured quantity will be such function, even physical constants such as speed of light are random variables in applied work, due to measurement error but there are also other reasons, but one just need to mention one example to show it’s a random variable  
> **username_2**: Wow. It was only a few days ago that you as a [mod read the comment thread](https://economics.stackexchange.com/questions/57806/proving-the-global-stability-in-the-solow-swan-model/57815?noredirect=1#comment95847_57815) under the other question where the answerer has claimed to have full insight into what OP meant. Then it turned out they didn't. (And refused to acknowledge it!)  
> **username_3**: @username_2 what are you talking about? I usually move comments to chat when there is a lot of back and forth, even if some of it is useful. That is what I was told by other older moderators even outside E.SE that is the right thing to do, even when some comments are useful. I was told that multiple times by senior mods in teachers lounge. Some mods automatically do that for any case where there is automatic flag from a bot. I am also happy to respond to community so I brought your comments back even though a regular mod would probably just kept them in chat anyway.  
> **username_3**: Also what am I supposed to acknowledge here? The comments got flagged by bot, there was a lot of bickering, so I moved them to chat, and then you made a reasonable request so I brought some back... also if you have issues with this raise them on meta, what does this have to do with my answer here?  
> **username_2**: I was referring to this part of your comment: "*OP clearly doesn’t understand what random variable means*" Other guy I was "bickering" with was just as arrogant.  
> **username_3**: @username_2 I am not trying to be arrogant in this thread, but this is in my opinion clear from the question  

---

