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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∻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 $ ext{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 $ ext{plim} \, \hat{\beta}_{0} = \beta_{0}$. ‖ ‖ step 4 ⋮ By the law of large numbers, $\bar{y} ightarrow \mathrm{E}(y)$ and $\bar{x}_{1} ightarrow \mathrm{E}(x_{1})$ as the sample size increases. ‖ ‖ step 5 ⋮ Substituting the limits, we get $ ext{plim} \, \hat{\beta}_{0} = ext{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 $ ext{plim} \, \hat{\beta}_{0} = \beta_{0}$. ‖ ∻Answer∻ ⚹ $ ext{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 $\beta_{0}$.⚹◊What is the relationship between the consistency of $\hat{\beta}_{1}$ and proving that $plim \hat{\beta}_{0} = \beta_{0}$ using the law of large numbers?⍭ Generate me a similar question◊

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