# Simple regression: not including the intercept term

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1. Sep 29, 2014

### 939

1. The problem statement, all variables and given/known data

The simple regression model is y = α + βx + u, where u is the error term. If you don't include α, when is β unbiased?

2. Relevant equations
y = α + βx + u

3. The attempt at a solution

Not including α doesn't affect whether β is unbiased because α is a constant.

2. Sep 29, 2014

### Ray Vickson

If the true model is $y = \beta x + \epsilon$, you get an unbiased estimate of $\beta$ by using the least-squares method on the model $\hat{y} = a + b x$---including the intercept! The point is that $a, b$ are both unbiased for the true model $y = \alpha+\beta x + \epsilon$, and this is true even if it happens that $\alpha = 0$. Therefore, my guess would be that the estimated obtained from the no-intercept fit $\hat{y} = bx$ would be biased. After all, the two estimates of $\beta$ would be given by different formulas in the $(x_i, y_i)$ data points, and one of the formulas gives an unbiased result.

3. Sep 30, 2014

If $\alpha \ne 0$ but you fit the "no intercept" model then the estimate of the slope will be biased. To see this begin with
$$E(b) = E[\left( X'X \right)^{-1}X' y] = E[\left( X'X \right)^{-1}X' \left(\alpha + X \beta + \epsilon \right)]$$
Regression without the intercept is rarely a good idea, for this reason AND for the fact that it means the traditional $R^2$ statistic is rendered useless (there are other issues as well).