I'm reading a post about the consistency of coefficients of SLR models:

Consistency of estimators in simple linear regression

Now I'm wondering whether there will be some similar conditions for SLR models through the origin? That's to say, consider a set of given data $\{(X_{i},Y_{i})\}_{i=1}^{n}$ and a SLR model through the origin: $$Y = \beta X + \epsilon$$ The OLS estimate of $\beta$ is given by: $$\hat{\beta} = \frac{\sum_{i=1}^{n}X_{i}Y_{i}}{\sum_{i=1}^{n}X_{i}^2}$$ Similar to the post above, can we develop some condition that ensures $\hat{\beta}$ is a consistent estimator of $\beta$?


Yes, dropping the intercept term (regression through the origin) will still be consistent. Consider,

$$\hat\beta = \frac{\sum_i X_iY_i }{\sum_i X_i^2}=\frac{n^{-1}\sum_i X_i(X_i\beta + \epsilon) }{n^{-1}\sum_i X_i^2}=\frac{n^{-1}\beta\sum_i X_i^2 }{n^{-1}\sum_i X_i^2} + \frac{n^{-1}\sum_i X_i\epsilon_i}{n^{-1}\sum_i X_i^2}=\beta + \frac{n^{-1}\sum_i X_i\epsilon_i}{n^{-1}\sum_i X_i^2}$$

By the WLLN we have,

$$n^{-1}\sum_i X_i^2\overset{p}{\to}\mathbb{E}[X_iX_i]$$ $$n^{-1}\sum_i X_i\epsilon_i\overset{p}{\to}\mathbb{E}[X_i\epsilon_i]=0$$

Under the assumption that $Cov(X,\epsilon)=0$. Then by Slutsky's theorem and the continuous mapping theorem we have,

$$\hat\beta \to \beta + \mathbb{E}[x_ix_i]^{-1}\cdot 0 = \beta$$


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