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1 In the simple regression model under MLR. 1 through MLR.4, we argued that the slope estimator, $\hat{\beta}_{1}$, is consistent for $\beta_{1}$. Using $\hat{\beta}_{0}=\bar{y}-\hat{\beta}_{1} \mathrm{x}_{1}{ }_{1}$, show that plim $\hat{\beta}_{0}=\beta_{0}$. [You need to use the consistency of $\hat{\beta}_{1}$ and the law of large numbers, along with the fact that $\beta_{0}=$ $\left.\mathrm{E}(\mathrm{y})=\beta_{1} \mathrm{E}\left(\mathrm{x}_{1}\right).\right]$
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Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

We start with the given regression model and the consistency of the slope estimator $\hat{\beta}_{1}$

step 2

The slope estimator $\hat{\beta}_{1}$ is consistent for $\beta_{1}$, which means $\text{plim} \, \hat{\beta}_{1} = \beta_{1}$

step 3

Using the formula $\hat{\beta}_{0} = \bar{y} - \hat{\beta}_{1} \bar{x}_{1}$, we need to show that $\text{plim} \, \hat{\beta}_{0} = \beta_{0}$

step 4

By the law of large numbers, $\bar{y} \rightarrow \mathrm{E}(y)$ and $\bar{x}_{1} \rightarrow \mathrm{E}(x_{1})$ as the sample size increases

step 5

Substituting the limits, we get $\text{plim} \, \hat{\beta}_{0} = \text{plim} \, (\bar{y} - \hat{\beta}_{1} \bar{x}_{1}) = \mathrm{E}(y) - \beta_{1} \mathrm{E}(x_{1})$

step 6

Given that $\beta_{0} = \mathrm{E}(y) - \beta_{1} \mathrm{E}(x_{1})$, we conclude that $\text{plim} \, \hat{\beta}_{0} = \beta_{0}$

Answer

$\text{plim} \, \hat{\beta}_{0} = \beta_{0}$

Key Concept

Consistency of estimators

Explanation

The consistency of the slope estimator $\hat{\beta}_{1}$ and the law of large numbers ensure that the intercept estimator $\hat{\beta}_{0}$ converges in probability to the true intercept $\beta_{0}$.

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