Question

Suppose that, for a given state in the United States, you wish to use annual time series data to estimate the effect of the state-level minimum wage on the employment of those 18 to 25 years old \((E M P) .\) A simple model is $$ g E M P_{t}=\beta_{0}+\beta_{1} g M I N_{t}+\beta_{2} g P O P_{t}+\beta_{3} g G S P_{t}+\beta_{4} g G D P_{t}+u_{t} $$ where \(M I N_{t}\) is the minimum wage, in real dollars; \(P O P_{t}\) is the population from 18 to 25 years old; \(G S P_{t}\) is gross state product; and \(G D P_{t}\) is U.S. gross domestic product. The \(g\) prefix indicates the growth rate from year \(t-1\) to year \(t,\) which would typically be approximated by the difference in the logs. i. If we are worried that the state chooses its minimum wage partly based on unobserved (to us) factors that affect youth employment, what is the problem with OLS estimation? ii. Let USMIN_t be the U.S. minimum wage, which is also measured in real terms. Do you think \(g U S M I N_{t}\) is uncorrelated with \(u_{t} ?\) iii. By law, any state's minimum wage must be at least as large as the U.S. minimum. Explain why this makes \(g U S M I N_{t}\) a potential IV candidate for \(g M I N_{t}\).

Short answer

OLS estimation can be biased due to endogeneity. USMIN_t is likely uncorrelated with state-specific errors, making it a good IV for MIN_t.

Step-by-step solution

Identify the Problem with OLS Estimation

Ordinary Least Squares (OLS) estimation requires that the independent variables are uncorrelated with the error term, denoted as \(u_t\). If the state sets its minimum wage \(MIN_t\) based on unobserved factors that affect youth employment (\(EMP_t\)), it implies that \(MIN_t\) could be correlated with \(u_t\), which leads to endogeneity. This correlation would mean that one of the key OLS assumptions (exogeneity) is violated, resulting in biased and inconsistent OLS estimators.

Analyze the Correlation of gUSMIN_t and u_t

To determine if \(g USMIN_t\) (growth rate of the U.S. minimum wage) is uncorrelated with \(u_t\), consider that \(u_t\) represents unobservable factors at the state level. \(g USMIN_t\) is determined at the federal level, separate from individual state economic conditions. Thus, it's reasonable to assume that \(g USMIN_t\) is uncorrelated with state-specific unobservable factors (\(u_t\)), ensuring it doesn't share the same source of endogeneity as \(g MIN_t\).

Evaluate gUSMIN_t as an Instrumental Variable for gMIN_t

An Instrumental Variable (IV) can be used if it is correlated with the endogenous explanatory variable \((gMIN_t)\) but uncorrelated with the error term \((u_t)\). Because the U.S. minimum wage sets a floor above which state minimum wages are set, changes in \(g USMIN_t\) are likely correlated with \(g MIN_t\) even if states set their wages independently. Additionally, since \(g USMIN_t\) is federally determined, it is argued to be uncorrelated with \(u_t\), satisfying the criteria for an IV: relevance and exogeneity.

Key concepts

Instrumental Variables

An Instrumental Variable (IV) is a powerful tool used in econometrics when dealing with endogeneity problems. In the given exercise, we encounter such a situation where the state's minimum wage setting ( MIN_t) might be based on factors unobservable to the researchers but affecting youth employment. This potential correlation creates a challenge for estimation.

• IVs help isolate causal relationships by providing a source of variation in the independent variable that is not related to the error term.
• An IV must be correlated with the endogenous explanatory variable (here, state minimum wage growth, gMIN_t) but uncorrelated with the error term ( u_t).

In this context, the U.S. minimum wage growth ( g USMIN_t) stands as a potential IV. Since states are legally required to set their minimum wage at or above the U.S. minimum, there is a natural connection between g USMIN_t and g MIN_t, ensuring relevance. Furthermore, because g USMIN_t is driven by federal policy, it is presumably uncorrelated with the state-specific error term, thus fulfilling the condition of exogeneity. These characteristics make it a robust candidate for addressing endogeneity in the model.

Ordinary Least Squares

Ordinary Least Squares (OLS) is one of the most commonly used methods for estimating the parameters of a linear regression model. This approach minimizes the sum of the squared differences between observed values and the values predicted by the model, achieving what is known as the 'best linear unbiased estimator' when specific conditions are met.

• OLS requires certain key assumptions, notably the independence of the explanatory variables and the error term ( u).
• A violation of any OLS assumptions leads to biased or inconsistent estimates, which is problematic especially in policy-related studies like the impact of minimum wage on employment.

In our exercise, the potential endogeneity, where the state minimum wage ( MIN_t) might correlate with unobserved determinants of youth employment, threatens the OLS assumption of exogeneity. Such correlation can alter the results, leading to misinterpretion of the relationship strength and direction between minimum wage changes and employment, a crucial focus for policy makers.

Endogeneity

Endogeneity arises when an explanatory variable is correlated with the error term in a regression model. This is problematic, as it violates the OLS assumption of exogeneity, causing bias in the estimated coefficients and undermining the model's reliability.

• Endogeneity can emerge from various sources, such as omitted variable bias, measurement error, or simultaneity.
• In policy analysis, understanding and addressing endogeneity is critical to obtaining accurate and credible estimates.

In the exercise, the concern is whether the minimum wage set by a state ( MIN_t) is determined with respect to unobserved factors that simultaneously affect youth employment. This situation indicates a simultaneity problem, where cause and effect may mutually influence each other, thus obscuring the true causal effect. Employing Instrumental Variables can help disentangle these effects by providing valid causal inference when direct estimation via OLS fails.

Time Series Analysis

Time Series Analysis involves statistical techniques used to model and interpret data points collected or indexed over time. This is highly pertinent in econometrics as it helps in observing trends, seasonality, and other temporal variables that might influence the dependent variable.

• Time series data are particularly useful for observing the dynamic effects of policies, like changes in the minimum wage over a given period.
• They allow for the exploration of the relationship between past values and current or future values.

In the provided model, the growth rates of variables such as employment ( g EMP), minimum wage ( g MIN), and economic indicators like GSP and GDP are analyzed over time. The focus is on identifying the impact of these changes across different years. This approach helps in accounting for serial correlation and can provide insights into whether the effects of changes persist over time or are transient. Understanding time series dynamics is essential for policymakers and researchers aiming to forecast economic trends and the potential impact of policy shifts on a demographic group.