# Interpreting coefficient, marginal effect from Linear Probability Model

I am regressing part time as a binary dependent variable (0 who dont work part time and 1 people work part time) with different parameter listed below

partime – variable=1 if employee works part time, 0 otherwise age=age of respondent in years ethbg=categorical variable (1-5) indicating the ethnic background of an individual female –variable if female, 0 otherwise hqual=categorical variable (0-4) indicating the highest educational qualification achieved marstat=categorical variable (1-3) indicating the ‘marital’ status of the individual reg=categorical variable (1-11) indicating the region of residence ind=categorical variable (1-9) showing industry of employment

Now I have added age and age squared into my model and drop one category in each variable in order not to enter into dummy variable trap which I get the result

1.) Now the problem is, how to interpret the marginal effect? I know it is just the coefficient of age. So would it a unit change in age , on average the probability of people work in part time job fall by 2.49%? 2.) On the age-squared variable, how do i interpret the coefficient? As age rise, people works in part time job increase at an increasing rate at 0.03% rate? This doesn't make sense at all if we combined with question 1... 3.) How do I interpret the constant term? 4.) If the p-value on the coefficient is signficant, is it saying that this coefficient is explaining the model. E.g. 0.0000<5% sig level. 5.) I understand that in LPM we cannot use R-squared as a measure of goodness of fit... because binary variable takes on 0 and 1. What else we can I do to show the goodness of fit?

Many thanks!

• You should not be using least squares for this regression: a binary response needs a generalized linear model (GLM) instead.
– whuber
Jun 12 '14 at 19:42
• I must also admit curiosity as to why you'd choose the linear probability model. Is there a reason you need an identity link? Jan 13 '15 at 10:00
• In addition to the above excellent comments, it is not possible to have marginal effects from an improperly linear probability model because they will fail to recognize the constraints that probabilities must be in $[0,1]$, i.e., they will ignore strange interactions that must be added to the model to make it mathematically legitimate. Jun 3 '16 at 11:36

Before I answer your questions, I will give some thoughts on using the linear probability model (LPM).

Using the LPM ones has to live with the following three drawbacks:

1. The effect $\Delta P(y = 1 \mid X = x_0 + \Delta x)$ is always constant

2. The error term is by definition heteroscedastic

3. OLS does not bound the predicted probability in the unit interval

1. For example if y is a work force participation indicator, and the x variable under study is the number of children, then the effect of 1 additional child always have the same predicted effect. Literally this means the effect on going from 0 to 1 children is the same as going from 10 to 11 children – this is clearly a strict and unrealistic assumption. You could try to loosen it, by adding interactions to your model. But it is seldom clear exactly how you should group things.

2. This is “easy” to get free from, use robust standard errors. In fact, you know that the variance is $p(1-p)$, so technically you could use WLS to get a more efficient estimate, but this of course requires the assumptions that you can estimate $p$ with consistency (cf. above).

3. This follows directly from that fact that OLS does not impose “range”-conditions on the response, if you increase (or decrease) the x-variables enough, then at some point you will cross the border, and get seemingly meaning less predictions (this actually happens quite often, when you use the LPM – and makes the use WLS difficult).

This is why people, in the comments, suggests that you instead use a GLM, covering these models is beyond the scope here but try searching on Logit and Probit, there are excellent questions (and answers) on this site and google. Suffice it to say that GLM models directly eliminate the problems outlined above – on the other hand, they do require that you make arbitrary distributional assumptions (which you don’t have to do with OLS).

That said, the LPM has some advantages in ease of interpretation, and can work quite well around the means of the independent variables, when all you want is an on average partial effect. For classification the LPM is, in my experience, extremely horrible.

1. Remember that the partial effect, in any model, is given by the derivative $\frac{\partial y}{\partial age}$, you cannot change age and hold $age^2$ fixed – it just doesn’t make sense.
3. The same as always, it is the value of $y$ when all variables $(x)$ are set to 0 – you use a lot of dummy variables, so it will take so figuring out.
5. I don’t know who told you that, you can use $R^2$ the same always – it means exactly the same thing. As an alternative you could use the percentage correctly predicted measure. You form predictions for all individuals in the sample, if $p < 0.5$ you predict $y=0$ and$1$ otherwise. Then you calculate how many individuals you predicted $y$ correctly for. But as I said above, the LPM often work horrible for classification.