Finding an intercept on percentage based data and using dummy variables

Question

How do I find an intercept on a percentage data? My data has percentage of grades ( I have converted to numbers where $A^*=8, A=7,B=6...U=0$) by ethnicity and other indicators which I want to test for using dummy variables. For example 90.3% of Chinese students got $A^*-C$ grade, Mixed race students got 87.3% etc. How do I interpret this to get an intercept? I have chosen the median 32.5 as the grades are 5 $A^*$ to $C$ (between $A^*(8\cdot5=40)$ and $C( 5\cdot5=25)$. Is the use of median in this case sensible?

My equation is going to be

$y =b_0 +b_1 +b_2+b_3+b_4+b_5+b_6+u$

where $y$ is the grade, $b_0$ is the median ( constant), $b_1$ is the free school meal, $b_2$ is Chinese, $b_3$ is Black, $b_4$ is Asian, $b_5$ is male, $b_6$ is female, and $u$ is the error term. White is the default.

Therefore if a Chinese male pupil does not get free school meals(proxy for poverty)it is $b_0 + b_2 + b_5$.

My question is as above, does my use of median make any sense and secondly since I already know that Chinese pupils perform better than the rest of the group do I need to use the percentage difference or use the dummy binary variables.

I want to simply find out the effect of poverty and race on the pupils expected grades. I do not have access to individual grades or panel data for income etc hence why I want to use the free school meal.

Thanks again for your answers.

Please see image below.

Answer 1

As Jamzy noted, run an OLS regression on grades against whatever variables you have.

$$\text{grades} = \beta_0 + \beta_1 x_1 + \cdots + \beta_i \ \text{race} + \cdots$$

$$\text{race} =\begin{array}{cc} \Bigg\{ & \begin{array}{cc} 0 & mixed \\ 1 & Chinese \\ \end{array} \end{array}$$

or vice versa with the 0 and 1. $\beta_0$ will be your intercept that you are looking for. If it ends up negative, try taking the log of both sides and see if a linear regression still has a good fit for you there.

Answer 2

This ends up being a bit of a mess. One should run a GLM with probit or logit link. The reason for this is that the regression is bounded, and we would not want your estimates to ignore the bounds and suggest grades of 130% or -20%. Such estimates can and likely will occur with OLS. See Stata Journal's post on this subject., or here.

Usually, these probit/logit regressions are used for binary data, consisting of 0 and 1. However, they will function well here, where the grade is the probability of getting any one question right.

The intercept has an analogue in these estimations, still. It is a constant term that will undergo a transformation depending on your choice of probit or logit. I believe there are other link functions available, but they are not particularly common in economics literature.

Given: $Y=\beta_0+x'\beta_{1..n}+\epsilon$ is your investigative target where

$0<Y<1$, $Y=grade,x=[gender,race,...]$

Assuming $\epsilon$ is logistically distributed, Logit: $\frac {1}{1+e^{-\beta_0}}$

Assuming $\epsilon$ is normally distributed, Probit: $\Phi(\beta_0)$

Of course it is possible your distribution is none of these, but these are considered standard.

It is also the case that OLS is unbiased in estimating an intercept for this type of data, but it may suggest impossible grades (for example, -0.2 or 1.3 as an intercept). The reason this is impossible is because one cannot get a -0.2 or a -1.3 as a percentile grade.