Paired or Unpaired t test

[Sidra Ilyas]

Hi Dear,

If there is economic growth data about two countries over the same thirty years, should it be considered as paired or unpaired data for two sample t test?

Thank you.

[Nick Cox]

It's paired by year, but

if your pairs are say

Freedonia in 1989 vs Sylvania in 1989
Freedonia in 1990 vs Sylvania in 1990
,,,
Freedonia in 2018 vs Sylvania in 2018

then the usual t test will be void if there is serial dependence.

Conversely, I am not (any longer) an economist and can't advise on the state of the art on this macroeconomic question.

[Sidra Ilyas]

Thanks Nick,

Is there a way to check for that serial dependence?

Also, in case if that exists, any solution?

Thanks again.

[Nick Cox]

Naturally, through an autocorrelation calculation, or a plot of difference versus previous difference, The last sentence of #2 still applies,

[Sidra Ilyas]

Thanks Nick,

Actually I understand the basic concept of autocorrelation, and know that there are autocorrelation tests after running regressions. But, when we have data like the case we are discussing here, are there any Stata commands to test autocorrelation of that data? I have already tried to google and search for the answers, but did not find anything myself.

Thanks again.

[Nick Cox]

In your case you need a wide layout with time defining observations and different countries defining different variables. Then you can tsset the data and apply ac or corrgram. You need this layout any way for any paired t tests. For paired t tests the focus is on how the differences behave.

The bigger issue here is whether this is a good way to compare countries, which I doubt without (#2 again) know what people do. There must be a literature on this....

[Sidra Ilyas]

Thanks Nick,

Do you mean to say that for paired t tests the focus is on how the differences "in individual" pairs behave? If this is the case, can you please tell me what is the focus of unpaired t tests?

Also, I have a question about panel data. If we have to run t tests of two groups of countries. Both groups of countries consist of different number of countries. For example, group one has 6 countries and group two has 3 countries. Data for each country is for same 30 years. The t tests have to be conducted at the group level. Do you think unpaired t test can work for such type of panel data set?

Many thanks again.

[Nick Cox]

For unpaired tests imagine 30 men and 30 women completely unrelated and compare their heights. You are comparing the mean male height and mean female height and the context is the variability of male heights and of female heights. For paired tests imagine 30 heterosexual married couples and compare height for each husband with each wife.

Your panel example is still paired because there is pairing by years. I will keep repeating my disclaimer: you are asking me about panel data econometrics, which I don't practise and don't pretend to understand. But I would say that applying such a test at least with the usual machinery would probably be regarded as very naive. You need a model that recognises a structure of countries within groups and accounts for serial dependence. The descriptive statistics would often make sense: on average Asian countries grew more in a certain period than European countries, or whatever it is, but getting a defensible P-value out of that comparison is harder (for me at least).

I suggest that you start a new thread flagging interest in panel data comparisons. This thread is not attracting panel data economists or econometricians who are the people you want to be reading your questions.

That's it from me. Hope it helped a bit.

[Sidra Ilyas]

Thanks Nick,

I have written a new post. And I think I am getting the basic concept of paired and unpaired data from your kind explanation.

But I still have a question here. Have you ever seen two sample unpaired t test being conducted for two variables with different number of observations?

Thanks again.

[Nick Cox]

Often!!! Indeed usually. Don't they teach this any more?

sysuse auto, clear
ttest mpg, by(foreign)

is an easy example. It is much of the point that without pairing you aren't constrained to have equal subsample sizes -- which is not to say that unequal subsample sizes are a good idea. But they often arise.

[Sidra Ilyas]

Thanks Nick,

Unfortunately the education system in every part of the world is not the same.

Anyways, there are further questions. In the document explaining the t test commands in Stata, I found that there are options for the assumptions of equal and unequal variances. Can you please clarify if they apply to both paired and unpaired t test? I am not being able to assure it from the document.

Also, what I understand is that the assumption of equal and unequal variances is that "whether we assume underlying population variances to be equal or not". Am I getting it right?

Moreover, how do we choose between the two? Mostly, populations and therefore their variances are unknown.

Thanks again.

[Carlo Lazzaro]

Sidra:
the option -equal-/-unequal- variances applies to unpaired -ttest- only.
Your interpretation about the assumption is correct.
When the option is available, it's in general safer to assume .unequal- variances.

[Sidra Ilyas]

Thanks Carlo,

The Stata document says "welch specifies that the approximate degrees of freedom for the test be obtained from Welch’s formula (1947) rather than from Satterthwaite’s approximation formula (1946), which is the default when unequal is specified. Specifying welch implies unequal".

Can you please tell me how to decide between welch or Satterthwaite’s specifiction?

Thanks again.

[Carlo Lazzaro]

Sidra:
I would go

Satterthwaite’s approximation formula (1946), which is the default when unequal is specified. Specifying welch implies unequal

[Sidra Ilyas]

Thanks Carlo,

But I am sorry I am unable to get your response.

Can you please rephrase it in a simple and easy way?

Many thanks.