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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 $\frac{1}{n}\sum_{i=1}^{n}\hat Yv_i \xrightarrow{p} E(\hat Y v_i)=0$ ?

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  • $\begingroup$ 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. $\endgroup$
    – whuber
    Oct 6, 2021 at 13:17
  • $\begingroup$ 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? $\endgroup$
    – Amy Chang
    Oct 6, 2021 at 13:38
  • $\begingroup$ 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. $\endgroup$
    – whuber
    Oct 6, 2021 at 14:10
  • $\begingroup$ 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? $\endgroup$
    – Amy Chang
    Oct 6, 2021 at 14:40
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    $\begingroup$ 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 $\endgroup$
    – Amy Chang
    Oct 6, 2021 at 14:56

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