<p>You do not provide proper reference to the source and there are several handbooks of macroeconomics so it is hard to discern what authors meant by just reading one sentence. However, stating that:</p>
<blockquote>
<p>physical capital contributes almost nothing to differences in GDP per worker across countries.</p>
</blockquote>
<p>Does not mean there is no relationship. It says the relationship is very weak.</p>
<p>In fact your data also show very weak relationship. You do not share the details of your regression, but just eyeballing the graph I would guess that the coefficient is very low and the regression explains only very little variation in the data. Moreover, when observations are clustered like that regression will be very sensitive to outliers.</p>
<p>Regarding your questions:</p>
<blockquote>
<p>When can we say X and Y have an association?</p>
</blockquote>
<p>For <strong>any</strong> association (as opposed to causality), it is enough to show there is significant correlation between two variables. Typically most textbooks would say that correlation coefficient (in absolute value) below 0.3 indicates weak (or none in case of zero or close to it) association, between 0.3 and 0.7 moderate and after 0.7 strong association. Positive (negative) coefficient indicates positive (negative) association.</p>
<blockquote>
<p>&quot;physical capital contributes almost nothing to differences in GDP per worker across countries&quot;? Is that his opinion?</p>
</blockquote>
<p>That depends on your definition of opinion. Any data analysis is in the end an opinion, but not all opinions are created equal. As my econometrics teacher used to say data never speak for themselves, statisticians speak for the data. Multiple skilled statisticians  can interpret exactly the same datasets differently. Statistical reasoning is inductive reasoning and from an epistemological perspective induction (as opposed to deduction) can't give you generally valid 'truth' and can be influenced by subjective perspectives (see discussion about it on <a href="https://www.wikilectures.eu/w/Statistical_Induction_Principle" rel="nofollow noreferrer">WikiLectures</a>).</p>
<p>This being said, rigorous  statistical methods are created in a way that can help us inductively analyze relationships in most objective way possible. Even if induction can't really give you general truth statistics allows researchers to create general conclusions from empirical data in the most objective inductive reasoning based on probability. However, in the end it is an opinion whether there is an relationship (e.g. if the coefficient would not be significant someone could claim there is too noise in the data, or if it is significant it could be just statistical artefact). If you read the handbook and if the author supports his statement by rigorous analysis I would say that comes as close to a fact as humanly possible, but using broad definition of opinion it would still qualify as an informed opinion.</p>
<blockquote>
<p>Or I should take it that the statement is only true in 2011?</p>
</blockquote>
<p>This can't be answered from that one sentence. English is high context language. Depending on the context he might have been talking just about 2011 or making more general claim.</p>
<blockquote>
<p>I was expecting the relationship between average <span class="math-container">$K_t/Y_t$</span> and average <span class="math-container">$Y_t/n_t$</span> to be insignificant, but it is the opposite. Can I say the author is wrong? And that capital or capital-output ratio matter for cross-country differences in  <span class="math-container">$Y_t/n_t$</span>?</p>
</blockquote>
<p>That depends on how rigorously you analyzed the data and what exactly you found.</p>
<p>First, the author does not say that there is no relationship, as I read it he says it can explain <strong>almost</strong> nothing, but not nothing.</p>
<p>Second, if you just regressed <span class="math-container">$Y_t/n_t$</span> on <span class="math-container">$K_t/Y_t$</span> I do not think you would have enough evidence to say that the author is wrong. Perhaps there are important covariates you omit and you found significant relationship just because of omitted variable bias.</p>
<p>If you run some state of the art empirical model and found significant and economically strong relationship you would have some solid grounds to claim the author is wrong. Otherwise, I would advise using some softer language, such as his result is inconsistent with yours or something like that.</p>

CC BY-SA 4.0
2022-07-09T00:26:16.080Z
Sorry...I have attached the regression results and the paper is "The Facts of Economic Growth" by C.I. Jones. So I guess your point is that the statement, "X explains almost none of the variations in Y" does not imply that the relationship between X and Y is insignificant. And I guess, similarly, the statement, "X explains almost all of the variations in Y" does not imply that the relationship between X and Y is significant. In both cases, we are omitting several variables by considering only X and Y. It seems you are attaching "almost none" to correlation coefficient, and not p-value, yeah?
Emmanuel Ameyaw
CC BY-SA 4.0
2022-07-09T00:32:10.223Z
If I understand your argument, when we want to make a statement about two variables, X and Y, the correlation coefficient between X and Y should be the important measure...not say, the p-value? Only use p-value when you have like state-of-the-art regression with many explanatory variables, right? For just two variables, use the correlation coefficient?
Emmanuel Ameyaw
CC BY-SA 4.0
2022-07-09T08:19:23.663Z
@EmmanuelAmeyaw I think you are little bit misunderstanding me "And I guess, similarly, the statement, "X explains almost all of the variations in Y"  does not imply that the relationship between X and Y is significant." That would be strong implication that the relationship is significant because otherwise the variable should not be able to explain a lot of variation. However, saying 'almost none' means that *some* variation can be explained. For example take your regression results. They show that if Y/K increases by 1 the Y/n would increase just by 0.25. That implies
1muflon1
CC BY-SA 4.0
2022-07-09T08:25:45.437Z
to increase Y/n just by 1 dollar (or I am not sure if you measured all dollar amounts in K or just did that for the graph, if they are all in k it would be 1k but thats still small) you would have to increase K/Y by 4. Based on your plot the biggest difference in K/Y between 2 countries is not larger than 15. According to your regression result increasing K/Y by 15 would increase the Y/n just by 3.75. In that case it would be valid to say that "physical capital contributes almost nothing to differences in GDP per worker across countries".
1muflon1
CC BY-SA 4.0
2022-07-09T08:30:04.533Z
Also, when I was talking about variation that was my guess about your regression, usually when coefficient is small the amount of variation explained by model is also small. In your case the model with that one variable can explain 17.4% which is not bad, but still the coefficient is so small that indeed you can say K/Y contributes almost nothing
1muflon1
CC BY-SA 4.0
2022-07-09T08:34:42.333Z
@EmmanuelAmeyaw also you should use p-value. But p value just tell you the probability that there is some relationship - p value tells you nothing about how strong the relationship is. If p>0.05 that just means the probability there is no relationship is higher than 5% which would be large enough probability in social sciences to say there is no relationship. You have p of close to zero, so probability there is some relationship is high (although maybe you have bias in your model so don't take the results too seriously), but having low p value tells you nothing about strength of the
1muflon1
CC BY-SA 4.0
2022-07-09T08:39:05.777Z
relationship and its economic meaningfulness. For example, if a medical research would show through some well designed study that coffee lowers your chance of getting cancer by 0.0000001% and this result would be significant with p value 0.0001 (meaning there almost certainly is some relationship), but the probability is so small that no serious doctor would recommended coffee as an anti-cancer food. With such small reduction in probability of getting cancer the coffee would be useless in preventing it even if there is true small relationship
1muflon1
CC BY-SA 4.0
2022-07-08T20:12:52.010Z
2022-07-08T20:12:52.010Z
1muflon1
<p>I was reading the first chapter of Handbook for macroeconomics. The authors show a table of capital-output ratios (<span class="math-container">$\frac{K_{2011}}{Y_{2011}}$</span>) (in 2011) for selected countries. Because Malawi and Kenya have similar capital-output ratios as the US and other developed economies, the author claims that &quot;physical capital contributes almost nothing to differences in GDP per worker across countries&quot;.</p>
<p>If I plot the mean capital-output ratios of all countries against the mean of GDP per worker from 1970 to 2019, I get the following figure (using Penn world tables dataset). If I run a regression for the two variables on the x-axis and y-axis, the regression coefficient is significant, so why the author can emphatically say that &quot;physical capital contributes almost nothing to differences in GDP per worker across countries&quot;? Is that his opinion? Or I should take it that the statement is only true in 2011? I was expecting the relationship between average <span class="math-container">$\frac{K_t}{Y_t}$</span> and average <span class="math-container">$\frac{Y_t}{n_t}$</span> to be insignificant, but it is the opposite. Can I say the author is wrong? And that capital or capital-output ratio matter for cross-country differences in <span class="math-container">$\frac{Y_t}{n_t}$</span>?</p>
<p><a href="https://i.sstatic.net/ZszG8.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/ZszG8.png" alt="enter image description here" /></a></p>
<p>Regression results. yn is Y/N and ky is K/Y.</p>
<p><a href="https://i.sstatic.net/gNNFC.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/gNNFC.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-07-08T21:42:26.873Z
I assume you mean that you did a linear regression, since for a nonlinear regression it is not easy to decide whether the difference is significant or not. In any case it could be that the author used a very small data set and in his case the regression coefficient would not have been significant (and he would not be lying, although he would have done a mediocre job).
Davius
<macroeconomics>
When can we say X and Y have an association?
Emmanuel Ameyaw
# When can we say X and Y have an association?

(Asked by: username_0 on 2022-07-08)

I was reading the first chapter of Handbook for macroeconomics. The authors show a table of capital-output ratios ($\frac{K\_{2011}}{Y\_{2011}}$) (in 2011) for selected countries. Because Malawi and Kenya have similar capital-output ratios as the US and other developed economies, the author claims that "physical capital contributes almost nothing to differences in GDP per worker across countries".

If I plot the mean capital-output ratios of all countries against the mean of GDP per worker from 1970 to 2019, I get the following figure (using Penn world tables dataset). If I run a regression for the two variables on the x-axis and y-axis, the regression coefficient is significant, so why the author can emphatically say that "physical capital contributes almost nothing to differences in GDP per worker across countries"? Is that his opinion? Or I should take it that the statement is only true in 2011? I was expecting the relationship between average $\frac{K\_t}{Y\_t}$ and average $\frac{Y\_t}{n\_t}$ to be insignificant, but it is the opposite. Can I say the author is wrong? And that capital or capital-output ratio matter for cross-country differences in $\frac{Y\_t}{n\_t}$?

[![enter image description here](https://i.sstatic.net/ZszG8.png)](https://i.sstatic.net/ZszG8.png)

Regression results. yn is Y/N and ky is K/Y.

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

**Tags:** macroeconomics

> **Comments:**
> **username_1**: I assume you mean that you did a linear regression, since for a nonlinear regression it is not easy to decide whether the difference is significant or not. In any case it could be that the author used a very small data set and in his case the regression coefficient would not have been significant (and he would not be lying, although he would have done a mediocre job).  

---

## 1 answers

### Answer by username_2 on 2022-07-08. Score: 0 | Accepted answer

You do not provide proper reference to the source and there are several handbooks of macroeconomics so it is hard to discern what authors meant by just reading one sentence. However, stating that:

> physical capital contributes almost nothing to differences in GDP per worker across countries.

Does not mean there is no relationship. It says the relationship is very weak.

In fact your data also show very weak relationship. You do not share the details of your regression, but just eyeballing the graph I would guess that the coefficient is very low and the regression explains only very little variation in the data. Moreover, when observations are clustered like that regression will be very sensitive to outliers.

Regarding your questions:

> When can we say X and Y have an association?

For **any** association (as opposed to causality), it is enough to show there is significant correlation between two variables. Typically most textbooks would say that correlation coefficient (in absolute value) below 0.3 indicates weak (or none in case of zero or close to it) association, between 0.3 and 0.7 moderate and after 0.7 strong association. Positive (negative) coefficient indicates positive (negative) association.

> "physical capital contributes almost nothing to differences in GDP per worker across countries"? Is that his opinion?

That depends on your definition of opinion. Any data analysis is in the end an opinion, but not all opinions are created equal. As my econometrics teacher used to say data never speak for themselves, statisticians speak for the data. Multiple skilled statisticians can interpret exactly the same datasets differently. Statistical reasoning is inductive reasoning and from an epistemological perspective induction (as opposed to deduction) can't give you generally valid 'truth' and can be influenced by subjective perspectives (see discussion about it on [WikiLectures](https://www.wikilectures.eu/w/Statistical_Induction_Principle)).

This being said, rigorous statistical methods are created in a way that can help us inductively analyze relationships in most objective way possible. Even if induction can't really give you general truth statistics allows researchers to create general conclusions from empirical data in the most objective inductive reasoning based on probability. However, in the end it is an opinion whether there is an relationship (e.g. if the coefficient would not be significant someone could claim there is too noise in the data, or if it is significant it could be just statistical artefact). If you read the handbook and if the author supports his statement by rigorous analysis I would say that comes as close to a fact as humanly possible, but using broad definition of opinion it would still qualify as an informed opinion.

> Or I should take it that the statement is only true in 2011?

This can't be answered from that one sentence. English is high context language. Depending on the context he might have been talking just about 2011 or making more general claim.

> I was expecting the relationship between average $K\_t/Y\_t$ and average $Y\_t/n\_t$ to be insignificant, but it is the opposite. Can I say the author is wrong? And that capital or capital-output ratio matter for cross-country differences in $Y\_t/n\_t$?

That depends on how rigorously you analyzed the data and what exactly you found.

First, the author does not say that there is no relationship, as I read it he says it can explain **almost** nothing, but not nothing.

Second, if you just regressed $Y\_t/n\_t$ on $K\_t/Y\_t$ I do not think you would have enough evidence to say that the author is wrong. Perhaps there are important covariates you omit and you found significant relationship just because of omitted variable bias.

If you run some state of the art empirical model and found significant and economically strong relationship you would have some solid grounds to claim the author is wrong. Otherwise, I would advise using some softer language, such as his result is inconsistent with yours or something like that.

> **Comments:**
> **username_0**: Sorry...I have attached the regression results and the paper is "The Facts of Economic Growth" by C.I. Jones. So I guess your point is that the statement, "X explains almost none of the variations in Y" does not imply that the relationship between X and Y is insignificant. And I guess, similarly, the statement, "X explains almost all of the variations in Y" does not imply that the relationship between X and Y is significant. In both cases, we are omitting several variables by considering only X and Y. It seems you are attaching "almost none" to correlation coefficient, and not p-value, yeah?  
> **username_0**: If I understand your argument, when we want to make a statement about two variables, X and Y, the correlation coefficient between X and Y should be the important measure...not say, the p-value? Only use p-value when you have like state-of-the-art regression with many explanatory variables, right? For just two variables, use the correlation coefficient?  
> **username_2**: @username_0 I think you are little bit misunderstanding me "And I guess, similarly, the statement, "X explains almost all of the variations in Y"  does not imply that the relationship between X and Y is significant." That would be strong implication that the relationship is significant because otherwise the variable should not be able to explain a lot of variation. However, saying 'almost none' means that *some* variation can be explained. For example take your regression results. They show that if Y/K increases by 1 the Y/n would increase just by 0.25. That implies  
> **username_2**: to increase Y/n just by 1 dollar (or I am not sure if you measured all dollar amounts in K or just did that for the graph, if they are all in k it would be 1k but thats still small) you would have to increase K/Y by 4. Based on your plot the biggest difference in K/Y between 2 countries is not larger than 15. According to your regression result increasing K/Y by 15 would increase the Y/n just by 3.75. In that case it would be valid to say that "physical capital contributes almost nothing to differences in GDP per worker across countries".  
> **username_2**: Also, when I was talking about variation that was my guess about your regression, usually when coefficient is small the amount of variation explained by model is also small. In your case the model with that one variable can explain 17.4% which is not bad, but still the coefficient is so small that indeed you can say K/Y contributes almost nothing  
> **username_2**: @username_0 also you should use p-value. But p value just tell you the probability that there is some relationship - p value tells you nothing about how strong the relationship is. If p>0.05 that just means the probability there is no relationship is higher than 5% which would be large enough probability in social sciences to say there is no relationship. You have p of close to zero, so probability there is some relationship is high (although maybe you have bias in your model so don't take the results too seriously), but having low p value tells you nothing about strength of the  
> **username_2**: relationship and its economic meaningfulness. For example, if a medical research would show through some well designed study that coffee lowers your chance of getting cancer by 0.0000001% and this result would be significant with p value 0.0001 (meaning there almost certainly is some relationship), but the probability is so small that no serious doctor would recommended coffee as an anti-cancer food. With such small reduction in probability of getting cancer the coffee would be useless in preventing it even if there is true small relationship  

---

