Let $ Y_i = \alpha + \beta x_i + \epsilon_i $, where $ x_i $'s are fixed covariates, $ \alpha $ and $ \beta $ are unknown parameters, and $ \epsilon_i $'s are independent and identically distributed random variables with mean zero and finite variance. Let $ \hat{\alpha} $ and $ \hat{\beta} $ be the ordinary least squares estimators of $ \alpha $ and $ \beta $, respectively. Given the following observations: The value of $ \hat{\alpha} + \hat{\beta} $ equals _________ (round off to 2 decimal places).

Question

Let $ Y_i = \alpha + \beta x_i + \epsilon_i $, where $ x_i $'s are fixed covariates, $ \alpha $ and $ \beta $ are unknown parameters, and $ \epsilon_i $'s are independent and identically distributed random variables with mean zero and finite variance. Let $ \hat{\alpha} $ and $ \hat{\beta} $ be the ordinary least squares estimators of $ \alpha $ and $ \beta $, respectively. Given the following observations:   The value of $ \hat{\alpha} + \hat{\beta} $ equals _________ (round off to 2 decimal places).

cdquestions Admin · Accepted Answer

The ordinary least squares (OLS) estimators for $ \alpha $ and $ \beta $ are calculated using the following formulae: $$ \hat{\beta} = \frac{\sum_{i=1}^{n} (x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{n} (x_i - \bar{x})^2} $$ $$ \hat{\alpha} = \bar{y} - \hat{\beta} \bar{x} $$ Using the given data, we compute the estimates for $ \hat{\alpha} $ and $ \hat{\beta} $, and the sum $ \hat{\alpha} + \hat{\beta} = 6.33 $. Thus, the value of $ \hat{\alpha} + \hat{\beta} $ is $ 6.33 $.

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Correct Answer: 6.31

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