# Does the linear probability model require the regressand to be zero/one-valued?

Typically, the dependent variable in a linear probability model (LPM) is a 0/1-valued binary variable. What if the dependent variable $y_i$ is still binary but take on general values $a$ and $b$ rather than 0 and 1? Technically, the resulting predictor still retains its nice properties, e.g., it is the minimum mean squared error (MMSE) linear predictor. Again, we may also transform $y_i$ into a 0/1 variable; but sometimes we want to keep the original values, say, for interpretation.

My question is: Can we still call the estimator the LPM estimator? If not, what should we call it?

## 2 Answers

The "LPM" label refers to the structure of the equation, not to the estimator. LPM models can be estimated not only by least-squares methods but also by maximum-likelihood for example.

As regards the nature of the dependent variable, we are talking about an affine transformation here. Let a model be with a binary dependent variable and a single regressor for simplicity,

$$y_i = \beta_0 +\beta_1x_i +e_i$$

with $y_i \in \{0,1\}$. We are then told that the original variable was

$$z_i = a+(b-a)y_i$$

so $y_i=0\implies z_i =a$ and $y_i=1 \implies z_i =b$.

Then the model for $z_i$ is

$$z_i = a+(b-a)[\beta_0 +\beta_1x_i +e_i]$$

or

$$z_i = \gamma_0 +\gamma_1x_i + u_i$$

$$\gamma_0 = a+(b-a)\beta_0,\;\;\; \gamma_1 = (b-a)\beta_1,\;\;\; u_i = (b-a)e_i$$

The structure of the model remains the same, an equation linear in parameters, so it is an LPM model still.

Since $(a,b)$ are presumably known values, after estimating the $z$-model, and obtaining estimates for the gammas, we can recover the estimates for the betas also, if they are of interest.

An LPM is a model in which the probability of the binary dependent variable having a particular value is linear in parameters. For example, if $y$ is a 0/1-valued variable and $P(y=1)=x\beta$, it (the equation for the probability) is called a linear probability model. Likewise, if $y \in \{ -1, 1 \}$ and $P(y=-1) = x\beta$, it’s a linear probability model. Or if $y$ is an apple/orange-valued variable and you believe that the probability of $y$ being an apple is $x\beta$, your belief is said to be a linear probability model. Zero/one mean nothing real here; they are only labels. You can label them as man/woman instead if you want. You lose no information by this relabeling as long as you remember the new labels. Also, as Papadopoulus said, it is a model, not an estimator, though we just understand an “LPM estimator” casually as an estimator of parameters in an LPM.