ECON 266: Hypothesis Testing and Statistical Power

In the context of hypothesis testing, what is the most critical implication of failing to minimize both Type I and Type II errors?

Assuming a fixed Type I error rate, how does an increase in the standard error of an estimated coefficient, $b_1$, affect the statistical power of a hypothesis test?

Considering the variance of an estimated coefficient in Ordinary Least Squares (OLS), $var(b_1) = \frac{\sigma^2}{N \times var(X)}$, what strategic adjustments can be made to the sample data to minimize this variance and thereby enhance the statistical power of the test, assuming $\sigma^2$ is irreducible?

When is the arsenal of hypothesis testing deemed essentially impotent in econometric analysis?

Given that a $t$-test result provides information predominantly about statistical significance, what critical aspect of the estimated effect does it fail to directly communicate, potentially leading to misinterpretations in policy or theory?

In scenarios with exceedingly large samples, what challenge arises regarding the interpretation of $t$-statistics, particularly in distinguishing between statistical and substantive significance?

Conversely, how might a small sample size misleadingly impact the assessment of a coefficient's significance, especially when the true underlying effect is, in fact, substantively meaningful?

Elaborate on the fundamental definition of a confidence interval in the context of econometric estimation and its essential role in statistical inference:

Within the framework of confidence intervals, under what precise condition should the null hypothesis $H_0 : \beta_1 = 0$ be rejected, providing a clear decision rule for hypothesis testing?

Consider the implications of employing Ordinary Least Squares (OLS) on observational data, particularly regarding the potential for endogeneity. Why is observational data inherently 'lousy' with endogeneity, and what fundamental assumption of OLS does this violate?

To make legitimate causal inferences from observational data, what essential methodological shift is required to transcend the limitations of bivariate Ordinary Least Squares (OLS)?

Multivariate OLS is often touted as a method to mitigate endogeneity. Explain the theoretical basis for this claim.

In the context of multivariate OLS, consider the model $Y_i = \beta_0 + \beta_1X_{1i} + \beta_2X_{2i} + ... + \beta_mX_{mi} + \epsilon_i$. What is the interpretation of the coefficient $\beta_1$?

Regarding the relationship between bivariate and multivariate OLS, what is a core tenet?

Given the multivariate OLS model $Y_i = b_0 + b_1X_{1i} + b_2X_{2i} + ... + b_mX_{mi}$, explain the nuances in interpreting the estimated coefficients compared to the bivariate OLS context.

What is the most effective way to address endogeneity in a time series model?

The regression results for car prices are shown in bivariate and multivariate form. The bivariate equation of price on mpg resulted in a coefficient of -238.9. When controlling for weight and foreign, the coefficient on mpg changes to 21.85. What is a possible interpretation?

In the context of omitted variable bias (OVB), how does multivariate OLS aim to 'fight' endogeneity arising from the omission of relevant variables?

Consider a scenario where the 'true' model is given by $Y_i = \beta_0 + \beta_1X_{1i} + \beta_2X_{2i} + v_i$, but instead, one estimates $Y_i = \beta_0^{OX_2} + \beta_1^{OX_2}X_{1i} + \epsilon_i$. Further, assume the relationship between $X_{1i}$ and $X_{2i}$ is given by $X_{2i} = \delta_0 + \delta_1X_{1i} + \tau_i$, where $\tau_i$ is uncorrelated with $v_i$ and $X_1$. What is the formula for the omitted variable bias?

In the context of omitted variable bias, what key factors amplify the magnitude of the resulting bias in the coefficient estimate of an included variable?

Consider a scenario where you suspect omitted variable bias in your regression model. You have prior knowledge about the likely sign of the effect of the omitted variable on the dependent variable ($\beta_2$) and the sign of the correlation between the included and omitted variables ($\delta_1$). If you know that $\beta_2 > 0$ and $\delta_1 < 0$, how would the omission of the variable likely affect the coefficient estimate of the included variable?

An investigator seeks to estimate the effect of education ($X_1$) on income ($Y$). However, individual ability ($X_2$) is an unobserved confounder: more able people tend to pursue more education and earn higher incomes. Assuming education and ability are positively correlated, and that ability has a positive direct effect on income, how would omitting ability from the regression model likely affect the estimated effect of education on income?

In a regression model examining the impact of smoking ($X_1$) on health outcomes ($Y$), socioeconomic status (SES) ($X_2$) is a potential confounder. Poorer individuals are more likely to smoke and experience worse health outcomes. Assuming that lower SES leads to both increased smoking and poorer health, how would omitting SES from the regression model likely affect the estimated impact of smoking on health?

To clarify, assuming the 'true' model is given by $Y_i = \beta_0 + \beta_1X_{1i} + \beta_2X_{2i} + v_i$, instead modeling the equation $Y_i = \beta_0 + \beta_1X_{1i} + \epsilon_i$ leads to omitted variable bias. If we know that $X_{1}$ and $X_{2}$ are substitutes, and that $X_{2}$ has a positive impact on $Y$, would the OLS estimate of $\beta_1$ be higher or lower than the true parameter value?