ECON 266: Multivariate OLS

In a multivariate ordinary least squares (OLS) regression, under what specific condition would Stata automatically remove variables from the model?

Within an econometric model, consider the scenario where an independent variable, $X_i$, is measured with error such that $X_i = X_i^* + \nu_i$, where $X_i^$ is the true, unobserved value, and $\nu_i$ is a random error term with mean zero and uncorrelated with both $X_i^$ and the error term $\epsilon_i$. If one estimates a regression model, what is the consequence of this measurement error?

If the true model is given by $Y_i = \beta_0 + \beta_1X_{1i} + \beta_2X_{2i} + \epsilon_i$, what is the interpretation of $R_j^2$ within the context of assessing multicollinearity?

Consider an OLS regression where one of the independent variables, $X_1$, is a perfect linear combination of another independent variable, $X_2$ (i.e., $X_1 = a + bX_2$). What implication does this have for the estimation procedure and the properties of the OLS estimators?

In the context of econometric modeling, what is the most accurate interpretation of the term 'attenuation bias'?

Suppose the true model is $Y_i = \beta_1 X_i^* + \epsilon_i$, but we observe $X_i = X_i^* + v_i$. Given that $plim(\hat{\beta_1}) = \frac{cov(X, Y)}{var(X)}$, derive an expression for $plim(\hat{\beta_1})$ in terms of $var(X^*)$, $var(v)$, and $\beta_1$.

In the context of estimating a multiple regression model, what is the effect of increasing the $R^2$ from the auxiliary regression of an independent variable $X_j$ on all other independent variables in the model on the variance of the estimated coefficient, $var(b_j)$?

What methodological approach can be used to address multicollinearity? (Select all that apply)

Consider $Y_i = \beta_0 + \beta_1X_{1i} + \beta_2X_{2i} + \epsilon_i$, where $X_{1i}$ and $X_{2i}$ are highly correlated. What are the implications for hypothesis testing regarding $\beta_1$?

What are the consequences of using a biased model versus having a low $R^2$?

If you are considering adding more independent variables, is this necessary a good thing? Explain.

Assume the true variance of estimated coefficient $b_1$ is $Var(b_1) = \frac{\sigma^2}{N var(X_i)}$, where $\sigma^2$ is the error variance, $N$ is the sample size, and $var(X_i)$ is the variance of the independent variable. What does this suggest when the model is multivariate?

What does a precise estimated coefficient mean?

The variance of sums of random variables can influence the coefficients in the regression. Evaluate:

In the context of regression analysis, how could the rules about the covariance or variance of random variables be applied to deal with measurement errors?

In an econometric study analyzing the determinants of educational attainment, researchers include years of schooling as one of the explanatory variables. However, due to limitations in data collection, they use household expenditure on education as a proxy for years of schooling. Discuss the potential consequences and econometric issues that may arise from using this proxy variable.

Describe the implications of perfect multicollinearity and provide measures through the use of auxiliary equations.

In a multiple regression model, the adjusted R-squared is often used to assess the goodness of fit. How does the adjusted $R^2$ specifically address the limitations of the regular $R^2$ in model selection, and what are the implications of using the adjusted $R^2$ for comparing models with different numbers of predictors?

In the context of multivariate regression analysis, how does the variance inflation factor (VIF) quantify the severity of multicollinearity, and what strategies can be employed to mitigate its impact on coefficient estimation and statistical inference?