Question 1

Given the regression results Unemployment Rate_it = α + 0.8 (0.3) × Alabama_it - 0.4 (0.2) × South Carolina_it + 0.2 (0.3) × North Carolina_it + 0.4 (0.12)MinimumWage_it - 0.1 (0.1)Year_t, where the coefficients are reported with their standard errors in parenthesis, how should we interpret the coefficient for minimum wage? A) An increase in minimum wage will increase the unemployment rate. B) An increase in minimum wage will increase the unemployment rate, holding fixed (time constant) differences across states. C) An increase in minimum wage will increase the unemployment rate, holding fixed the downward trend in unemployment rates occurring across all of these three states. D) An increase in minimum wage will increase the unemployment rate, holding fixed (time constant) differences across states, and the downward trend in unemployment rates occurring across all of these three states.

Accepted Answer

The coefficient for minimum wage is positive, indicating an increase in minimum wage is associated with an increase in the unemployment rate. This interpretation holds fixed (time constant) differences across states due to the inclusion of state-specific variables (Alabama, South Carolina, North Carolina) and the downward trend in unemployment rates across these states, indicated by the inclusion of the Year variable.

Question 2

A potential downside of using within estimation is:
A) r-squared is less meaningful.
B) the coefficients will be biased.
C) the estimator will not yield estimates of the fixed effects themselves.
D) the standard errors will be larger.

Accepted Answer

Within estimation only provides estimates of the time-varying effects and not the fixed effects. It assumes that the fixed effects are constant over time and therefore cannot estimate them.

Question 3

Generally speaking, what are the methods available to the econometrician who wants to estimate a linear model with a fixed effects model design (i.e., dummy variable for individual units observed over multiple periods)?
A) Within Estimator and Fixed Effects regression
B) Diff-in-diff and Two-stage least squares
C) Two-stage least squares and Fixed Effects regression
D) Within estimator and instrumental variable regression

Accepted Answer

The methods available to the econometrician who wants to estimate a linear model with a fixed effects model design are Within Estimator and Fixed Effects regression. Within estimator, also known as the "residual" method, subtracts the time-invariant individual effects from the data and then estimates the parameters on the transformed data. Fixed Effects regression includes a dummy variable for each of the individual units observed over multiple periods, and then estimates the parameters on the transformed data. While other methods, such as Diff-in-diff, Two-stage least squares, and instrumental variable regression can be useful in other contexts, they are not directly applicable to a linear model with a fixed effects model design.

Question 4

Interpreting the coefficients on fixed effects will always be based on what?
A) If the regression was run using STATA or Excel.
B) The sign of the coefficient on the instrumental variable in the first stage regression.
C) The base group omitted from the regression.
D) The standard error on the treatment variable.

Accepted Answer

The coefficients on fixed effects reflect the difference between each of the treatment groups and the base group. Therefore, interpreting these coefficients correctly requires knowledge of which group was chosen as the base group and what characteristics were held constant across groups.

Question 5

Given the role of cross sectional fixed effects in the empirical strategy for identifying causal effects, instead of conducting hypothesis tests of any one coefficient on a fixed effect being zero, it is typical to conduct what sort of hypothesis test?
A) A hypothesis test for a fixed effect coefficient being equal to 1 instead of zero.
B) A hypothesis test for a fixed effect coefficient being equal to -1 instead of zero.
C) A joint hypothesis test for all of the fixed effect coefficients being equal to zero.
D) A joint hypothesis test that the fixed effect coefficients are all equal to the treatment effect.

Accepted Answer

Conducting a joint hypothesis test for all of the fixed effect coefficients being equal to zero is typical in the empirical strategy for identifying causal effects. This tests whether the fixed effects are jointly significant in explaining the outcome variable. Hypothesis tests for individual fixed effect coefficients being equal to specific values (such as zero, one, or negative one) are not typically conducted because the interpretation of these coefficients can be complex and not directly related to the research question at hand. Hypothesis tests that the fixed effect coefficients are all equal to the treatment effect are also not typical, as the fixed effects are not usually interpreted as measures of treatment effects.

Question 6

When one uses within-group differences in variables to estimate parameters in the data generating process, you are using what approach?
A) Two-stage least squares
B) Nonlinear least squares
C) Within estimation
D) Time trends

Accepted Answer

The approach described in the question is known as within estimation or fixed effects regression. It involves estimating parameters based on differences between observations within groups (such as individuals or countries) rather than between groups. This approach can help control for unobserved heterogeneity across groups, but it assumes that all time-invariant characteristics of individuals are captured by the within-group differences.

Question 7

All of the following coefficients/statistics will be the same across the dummy variable and within estimator approaches for estimating a fixed effects model except:
A) coefficient on treatment variable.
B) coefficient on time trend.
C) coefficient on additional control variables.
D) R-squared

Accepted Answer

In both approaches, the coefficients on the treatment variable, time trend, and additional control variables will be the same. However, the R-squared may differ between the two approaches due to differences in the assumptions made about the error term.

Question 8

Given the regression results Unemployment Rate_it = α + 0.8 (0.3) × Alabama_it - 0.4 (0.2) × South Carolina_it + 0.2 (0.3) × North Carolina_it + 0.4 (0.12)MinimumWage_it - 0.1 (0.1)Year_t, where the coefficients are reported with their standard errors in parenthesis, what might be a fact that would make you concerned about the interpretation of the coefficient on minimum wage as being causal? A) North Carolina has consistently had a higher share of college educated in the population relative to other states in the analysis. B) National policies have led to stricter conditions to apply for unemployment insurance. C) The minimum wage rate is different across many states in the southern region of the U.S. D) In most instances, when a state raises the minimum wage it is also accompanied by state legislation that changes fiscal policy towards economic activity.

Accepted Answer

The fact that state legislation that changes fiscal policy towards economic activity often accompanies a minimum wage increase suggests that there may be confounding factors that are affecting the relationship between minimum wage and unemployment rate. This makes it difficult to interpret the coefficient on minimum wage as a causal effect.

Question 9

In estimating a fixed effects model using panel data, which of the following variables will not be effective controls if you use a full set of (cross sectional) fixed effects for individuals?
A) Year to year changes in income for individuals
B) Yearly (cumulative) education attainment for individuals
C) Income taxes paid
D) Birthplace of the individual

Accepted Answer

If you are using a full set of fixed effects for individuals, you are essentially controlling for all individual-specific factors, including birthplace. Therefore, birthplace would not be an effective control variable in this case as it is already accounted for in the fixed effects. Variables such as year to year changes in income, cumulative education attainment, and income taxes paid would still be useful control variables as they vary within individuals over time and are not fully accounted for by the fixed effects.

Question 10

A potential upside of using within estimation besides the reduction in the number of parameters to be estimated is:
A) r-squared is more meaningful.
B) the coefficients will be biased.
C) the estimator will yield estimates of the fixed effects themselves.
D) the standard errors will be larger.

Accepted Answer

Within estimation, by focusing on variation within entities (e.g., individuals, firms, countries) over time, effectively controls for all unobserved time-invariant characteristics. This leads to a more meaningful R-squared because it reflects the proportion of the variance explained by the variables that change over time within those entities, providing a clearer picture of the model's explanatory power regarding the within-entity dynamics.

Quiz 8: Advanced Methods for Establishing Causal Inference

A potential downside of using within estimation is:

Generally speaking, what are the methods available to the econometrician who wants to estimate a linear model with a fixed effects model design (i.e., dummy variable for individual units observed over multiple periods) ?

Interpreting the coefficients on fixed effects will always be based on what?

Given the role of cross sectional fixed effects in the empirical strategy for identifying causal effects, instead of conducting hypothesis tests of any one coefficient on a fixed effect being zero, it is typical to conduct what sort of hypothesis test?

When one uses within-group differences in variables to estimate parameters in the data generating process, you are using what approach?

All of the following coefficients/statistics will be the same across the dummy variable and within estimator approaches for estimating a fixed effects model except:

In estimating a fixed effects model using panel data, which of the following variables will not be effective controls if you use a full set of (cross sectional) fixed effects for individuals?

A potential upside of using within estimation besides the reduction in the number of parameters to be estimated is: