# fitted value converge in probability in OLS regression

Consider linear regression model as follows;

$$Y_i = X_i\beta+\epsilon_i, i=1,2,...,n$$.
$$\epsilon_i \sim N(0, \sigma_e^2)$$

We can obtain the OLS estimate of $$\beta$$, and also have the property that $$\hat\beta\xrightarrow{p} \beta$$ .

And I was wondering what about the fitted value $$\hat Y=X_i\hat \beta$$,

Suppose the possible value of $$X_i$$ is 0,1,and 2, and $$X_i \sim Multinomial(p_0, p_1, p_2)$$.

(1) Can we claim that $$\hat Y \xrightarrow{p} X_i \beta$$, if $$\hat\beta \xrightarrow{p}\beta$$ ?

(2) Is it correct that $$\frac{1}{n}\sum_{i=1}^{n}\hat Yv_i \xrightarrow{p} E(\hat Y v_i)$$, based on the large law of number, where $$v_i \sim N(0,\sigma_v^2)$$ ?

(3) If the (2) is true, then I was wondering if there are the other conditions (except $$v_i$$ is independent to $$\hat Y$$) to make $$\frac{1}{n}\sum_{i=1}^{n}\hat Yv_i \xrightarrow{p} E(\hat Y v_i)=0$$ ?

• What is the sense of the convergence here? Its meaning depends on how the explanatory variables $X_i$ are allowed to vary as $n$ grows.
– whuber
Oct 6, 2021 at 13:17
• Hmm, I only consider two scenarios (1) the value of $X_i$ is 0,1 or 2. ; (2) $X_i$ follows the normal distribution. In these two cases, is it possible to discuss the questions I posted? Oct 6, 2021 at 13:38
• The second scenario is sufficiently specific. The first is not, because it does not specify how the $X_i$ might vary as $n$ changes--and the result depends on that.
– whuber
Oct 6, 2021 at 14:10
• Maybe I misunderstand something. In the first scenario , the largest value of $X_i$ is 2, no matter how the sample size $n$ changes. Does the existence of $X_i$'s upper bound not sufficient enough to discuss the convergence? Oct 6, 2021 at 14:40
• Oh I see, I agreed this scenario (1) once $n>1, X_i=0$ won't have convergence. And I think what I really want is that $X_i$ follows a multinomial distribution ($p_0$,$p_1$,$p_2$). I will specify the distribution of $X_i$ and $v_i$ to make this question clearer. Thank you Oct 6, 2021 at 14:56