Question

Let \(g G D P_{t}\) denote the annual percentage change in gross domestic product and let int \(_{t}\) denote a short-term interest rate. Suppose that \(g G D P_{t}\) is related to interest rates by $$g G D P_{t}=\alpha_{0}+\delta_{0} i n t_{t}+\delta_{1} i n t_{t-1}+u_{t}$$ where \(u_{t}\) is uncorrelated with int, \(i n t_{t-1}\), and all other past values of interest rates. Suppose that the Federal Reserve follows the policy rule: $$i n t_{t}=\gamma_{0}+\gamma_{1}\left(g G D P_{t-1}-3\right)+v_{t}$$ where \(\gamma_{1}>0 .\) (When last year's GDP growth is above \(3 \%,\) the Fed increases interest rates to prevent an "overheated" economy.) If \(v_{t}\) is uncorrelated with all past values of int \(_{t}\) and \(u_{t},\) argue that \(i n t_{t}\) must be correlated with \(u_{t-1}\). (Hint: Lag the first equation for one time period and substitute for \(g G D P_{t-1}\) in the second equation.) Which Gauss-Markov assumption does this violate?

Short answer

The exogeneity assumption is violated as \(int_{t}\) correlates with \(u_{t-1}\).

Step-by-step solution

Identify the Equations

We have two equations: \( gGDP_{t} = \alpha_{0} + \delta_{0}int_{t} + \delta_{1}int_{t-1} + u_{t} \) and the Fed's policy rule: \( int_{t} = \gamma_{0} + \gamma_{1}(gGDP_{t-1} - 3) + v_{t} \). Our goal is to show that \( int_{t} \) is correlated with \( u_{t-1} \).

Lag the first equation

Lag the first equation by one period: \( gGDP_{t-1} = \alpha_{0} + \delta_{0}int_{t-1} + \delta_{1}int_{t-2} + u_{t-1} \). This represents the GDP growth from one period ago in terms of earlier interest rates and error term \( u_{t-1} \).

Substitute into the Fed's Policy Rule

Substitute \( gGDP_{t-1} \) from Step 2 into the Fed's rule: \( int_{t} = \gamma_{0} + \gamma_{1} ((\alpha_{0} + \delta_{0}int_{t-1} + \delta_{1}int_{t-2} + u_{t-1}) - 3) + v_{t} \).

Simplify the Fed's Policy Equation

Simplify the substitution from Step 3 to get \( int_{t} = \gamma_{0} + \gamma_{1}(\alpha_{0} - 3) + \gamma_{1}(\delta_{0}int_{t-1} + \delta_{1}int_{t-2}) + \gamma_{1}u_{t-1} + v_{t} \). Note that \( u_{t-1} \) now appears in the equation.

Analyze Correlation Relations

The term \( \gamma_{1}u_{t-1} \) is part of the \( int_{t} \) equation, which implies that \( int_{t} \) is influenced by \( u_{t-1} \). Since \( u_{t-1} \) is an error term from the GDP equation uncorrelated with past \( int \) values, using it to influence \( int_{t} \) constitutes new information affecting it, creating correlation.

Identify the Violated Gauss-Markov Assumption

The violation occurs in the assumption of exogeneity, which states that the error term \( u_{t} \) should not be correlated with the explanatory variables. Here, \( u_{t-1} \) impacts \( int_{t} \), signaling a violation of this assumption as \( int_{t} \) is now correlated with past errors.

Key concepts

GDP growth

Gross Domestic Product (GDP) growth is a key measure of economic performance. It indicates how much a country's economy is expanding over time. When GDP grows, it means production and services in the country are increasing, which usually signifies health in the economy.
Sometimes GDP may increase because of:

• Higher consumer spending
• Increased business investments
• More government spending
• Net exports rising

Each of these factors can contribute to higher GDP growth rates, leading to a perception of economic prosperity and strength.
Monitoring GDP growth is crucial for policymakers and investors as it affects employment levels, consumer confidence, and overall economic health.

interest rates

Interest rates refer to the cost of borrowing money or the reward for saving it. They are typically set by a country's central bank, like the Federal Reserve in the US. Low interest rates make borrowing cheaper, stimulating investment and spending but can also lead to inflation if kept too low.
Conversely, high rates can slow economic growth by making loans and mortgages expensive but help to control inflation. The impact of changing interest rates includes:

• Altered consumer spending habits
• Investment activity changes by businesses
• Influence on currency strength

Therefore, understanding interest rates helps in foreseeing economic trends and consumer behavior.

Gauss-Markov assumptions

The Gauss-Markov assumptions are important principles in statistics and econometrics, ensuring the reliability of regression models. These assumptions explain the conditions under which the Ordinary Least Squares (OLS) estimator is the Best Linear Unbiased Estimator (BLUE). For an estimator to be BLUE, it must follow criteria like:

• Linearity
• Unbiasedness
• Minimum variance

Some of the key Gauss-Markov conditions include:

• Linear relationship between dependent and independent variables
• No correlation between the error terms and the independent variables
• Constant variance in the error term
• No autocorrelation among the error terms

Violating any of these assumptions, such as through correlation between errors and explanatory variables, could result in biased estimates, misleading results, and poor predictions.

Federal Reserve policy

Federal Reserve policy incorporates tools and guidelines used by the Federal Reserve (the Fed) to manage the economy. Decision-makers at the Fed use instruments such as interest rates and open market operations to control inflation, stabilize prices, and ensure full employment. A key policy mechanism is the setting of interest rates. For instance:

• Raising rates to curb inflationary pressures
• Lowering rates to stimulate spending and investment

These policies significantly influence economic conditions, such as GDP growth and unemployment rates. The Fed's policies are data-driven and aim for sustainable economic growth, and a clear understanding of them helps anticipate economic trends.

exogeneity

Exogeneity is a concept in econometrics that refers to the independence of outside variables from the variables within a model. An exogenous variable is presumed unaffected by the system it is part of. It is crucial in determining how causal relationships can be validly interpreted from models.
A model's reliability is diminished when endogeneity exists, which is when variables are influenced by other variables in the system. Exogeneity is assumed in the Gauss-Markov theorem, which dictates that error terms must not correlate with explanatory variables. This ensures the resultant estimations are unbiased, providing a true picture of relationships between variables.
Understanding exogeneity is critical for constructing robust economic and statistical models, offering precise information about variable relationships.