In the multiple regression model, [math: yi=β1+β2xi+β3zi+ui]y_i=\beta _1+\beta _2x_i +\beta _3z_i+u_i, which of the following leads to improved precision on the estimates [math: β^2]\hatβ_2, (that is, smaller [math: se(β^2|x,z)]se(\hat{\beta}_2|x,\,z)?[Fill in the blank]

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To evaluate what improves the precision of the estimated slope on x, we should consider factors that affect the standard error of hatβ2 in a multiple regression.
Option a: Smaller sample size. In general, reducing the sample size increases the standard errors of all coefficients, making estimates less precise. So this would worsen, not improve, precision. 
Option b: Larger correlation between the explanatory variables x and z. High multicollinearity (strong correlation between regressors) inflates the variance of the coefficient estimates, particularly hatβ2, reducing precision. Therefore this would harm precision rather than improve it.
Option c: Larger var(u|x,z). If the conditional error variance increases, the standard errors of the coefficients rise, which again lowers precision. Hence this option would not improve precision.
Option d: More variation in the explanatory variable x around its mean. Greater spread of x provides more information about the relationship between x and y, reducing the variance of hatβ2 (all else equal). This improves precision because the slope estimate is estimated with more distinct changes in y for changes in x. 
Thus, the option that leads to improved precision is: More variation in the explanatory variable x around its mean.

In the multiple regression model, [math: yi=β1+β2xi+β3zi+ui]y_i=\beta _1+\beta _2x_i +\beta _3z_i+u_i, which of the following leads to improved precision on the estimates [math: β^2]\hatβ_2, (that is, smaller [math: se(β^2|x,z)]se(\hat{\beta}_2|x,\,z)?[Fill in the blank]

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