# Regression with percentage points

Suppose I want to find out how growth in average income affects the growth in income of the poor. For this I have 60 countries and for each country the income of the poor and average income for 6 years. I am unsure on how to perform a regression of this.

I have been calculating the average annual growth in percentage for average income and income for the poor. From what I understand this gives me percentage points change instead of percentage change, thus it would be a simple linear regression.

Yet I have seen some tutorials that log the percentage points, creating a log-log regression. Which of these two methods are correct?

EDIT: Example values:

Country A:

• Average income for the poor for 4 years: 1990:150, 1991: 260, 1992:300, 1993: 400
• Average income for 4 years: 1990:340, 1991:600, 1992:710, 1993:1000

Compound annual growth poor: 38.67%
Compound annual growth all: 43.28%

Say I do this for 60 countries, so i will get for average annual growth of income of the poor (in percent):

21.10 , 16.23 , 12.34 , 23.25 , 11.12 , 13.55 etc

and for average annual growth if income of all:

43.13 , 35.31 , 56.31 etc

and I do a regression on them and get say y = 2.433 + 0.6 x . So if income of all increases one unit, the income of the poor increases 0.6 units.

Is it correct to say than that a 1 percent change increase in average income of all will result in 0.6 percent change average income of the poor? Or is it percentage points change?

Say I do this for 60 countries, so i will get for average annual growth of income of the poor (in percent):

21.10 , 16.23 , 12.34 , 23.25 , 11.12 , 13.55 etc

and for average annual growth if income of all:

43.13 , 35.31 , 56.31 etc

as I said:

I consider you put in the values as 21.10 and not as 0.2110?

So if income of all increases one unit, the income of the poor increases 0.6 units.

Wrong, remeber your variables, these are not income variables but annual growth rates!

So if the average annual growth rate of the income of all increases by one unit, the annual average growth rate of the income of the poor will increase by 0.6 units on average.

Is it correct to say than that a 1 percent change increase in average income of all will result in 0.6 percent change average income of the poor? Or is it percentage points change?

I suppose you put in the values as 21.10 and so on. The regression output is

y = 2.433 + 0.6 x


where x is the average annual growth rate of the income of all and y is the average annual growth rate of the income of the poor. You put in the values, as e.g. 21.10. If x (average annual growth rate of the income of all) is increased by one unit (from e.g. 21.10 to 22.20, so from 0.2110 percent to 0.2220 percent, this is a change by 0.01 percentage point) the y (average annual growth rate of the income of the poor) will increase by 0.6. So from

$y = 2.433 + 0.6 * 21.10=15.093$

to

$y= 2.433 + 0.6*22.10= 15.693$

so from 15.693 to 15.093, which is from 0.15093 percent to 0.15693 percent, which is a change by 0.006 percentage point. Or an increase by $(0.15693-0.15093)/0.15093= 0.03975353$. So an increase by about 4 percent.

• I noticed, that the value of 21.10 in your statement above belongs to y and not x. But this does not matter, since it is just an example, so imagine it is a value for the x variable. Commented Apr 7, 2013 at 18:53
• I edited my answer, since I had a typo, when calculating the values with R. Commented Apr 7, 2013 at 19:07

I am not really sure, if I got your right......

Just to have a specification, you have:

income of the poor, given in "normal" (not percentage) values for 6 years

average income, given in "normal" (not percentage) values for 6 years

so with this, you have calculated:

y = average annual growth of income of the poor

x = average annual growth of income

So y and x are univariate? Like

y             x
value 1      value 1
value 2      value 2


Did you use the geometric or arithmetic mean?

I would suggest using the geometric, look at a simple example for this (consider this to be e.g. income values):

40000, 42000, 45000 and 47000

This gives the percentage values: 0.05, 0.07143, 0.0444

the geometric mean is: $(1.05 * 1.07142857 * 1.04444)^{(1/3)}= 1.0552257$

This is a factor, whereas 0.0552257 is percent

i.e. each year, the income increases by 5.52257 percent on average:

40000*1.0552257=42209.028
42209.028*1.0552257=44540.05112
44540.05112*1.0552257=46999.8 (difference due to unprecisely rounding)


So this is not percentage point? Or what did you mean? Percentage point is: If the annual growth rate of 5% increased to 7%, this is an increase of 2 percentage point or an increase by 40 percent!

So you have now the geometric annual growth rate of the income in percent (let's say you use the percent value and not the factor value) and the geometric annual growth rate of the income of the poor, right?

If you use the values as following:

y     x
5.5   4.5
6.5   5.5
and so on


and not

y    x
0.055 0.045
and so on


Now, you do the regression of y,x and you have a regression coeff of e.g. 0.7.

This means, if you increase x by one unit the variable y will increase by 0.7 on average.

So if the annual growth rate of the income is increased by one unit, e.g. from 5 to 6, the y variable will increase by 0.7.

So if the annual growth rate of the income is increased from 5% to 6%, the y variable will increase by 0.7% (on average).

Did you mean this?

Depending on log on one side or taking log-log on both side, the interpretation changes. This basically depends on the dependence of your data. If you take the x as log values, this gives not a linear dependence anymore, but a logarithmic dependence. You have to do model identification to see, what would be appropriate. Plot y and x in a scatter plot, is it a linear dependency or logarithmic?

In case of the interpretation look at this thread: In linear regression, when is it appropriate to use the log of an independent variable instead of the actual values? , especially the second answer with 17 thumps up.

• maybe also this paper will help you: www-stat.wharton.upenn.edu/~stine/stat621/handouts/… Commented Apr 7, 2013 at 17:36
• @Crazzzy12 or specify the problem more detailed, give example values or originial values Commented Apr 7, 2013 at 17:47
• I have used the geometrical average and calculated as you did. But is it then defined as percentage change or percentage point change in the end? If average annual growth rate income increases 1 unit and average annual growth of income of the poor increases 0.7 units, it is a change in percentage points right? I've plotted y and x in a scatter and it seems to be a linear dependency. Commented Apr 7, 2013 at 17:57
• I will give example values in the text to make it clearer! Commented Apr 7, 2013 at 18:00
• @Crazzzy12 I added an answer. Commented Apr 7, 2013 at 18:51