Question

Consider the savings function $$\operatorname{sav}=\beta_{0}+\beta_{1} \operatorname{inc}+u, u=\sqrt{i n c} \cdot e$$ where \(e\) is a random variable with \(\mathrm{E}(e)=0\) and \(\operatorname{Var}(e)=\sigma_{e}^{2} .\) Assume that \(e\) is independent of inc. i. Show that \(\mathrm{E}(u | \text { inc })=0,\) so that the key zero conditional mean assumption (Assumption SLR.4) is satisfied. [Hint: If \(e \text { is independent of inc, then } \mathrm{E}(e | i n c)=\mathrm{E}(e) .]\) ii. Show that \(\operatorname{Var}(u | \text { inc })=\sigma_{c}^{2}\) inc, so that the homoskedasticity Assumption SLR.5 is violated. In particular, the variance of sav increases with inc. [Hint: Var(elinc) = Var(e) if \(e\) and inc are independent.] iii. Provide a discussion that supports the assumption that the variance of savings increases with family income.

Short answer

i. \(\mathrm{E}(u | \text{inc}) = 0\); ii. \(\operatorname{Var}(u | \text{inc}) = \sigma_e^2 \cdot \text{inc}\); iii. Higher income usually leads to greater savings variance.

Step-by-step solution

Calculate the Conditional Expectation of u

Start with \(u = \sqrt{\text{inc}} \cdot e\). Since \(e\) is independent of \(\text{inc}\), \(\mathrm{E}(e | \text{inc}) = \mathrm{E}(e)\). We know \(\mathrm{E}(e) = 0\). Thus, \(\mathrm{E}(u | \text{inc}) = \mathrm{E}(\sqrt{\text{inc}} \cdot e | \text{inc}) = \sqrt{\text{inc}} \cdot \mathrm{E}(e | \text{inc}) = \sqrt{\text{inc}} \cdot 0 = 0\). This satisfies Assumption SLR.4, the zero conditional mean assumption.

Calculate the Conditional Variance of u

To find \(\operatorname{Var}(u | \text{inc})\), start with \(u = \sqrt{\text{inc}} \cdot e\). Since \(e\) and \(\text{inc}\) are independent, \(\operatorname{Var}(e | \text{inc}) = \operatorname{Var}(e) = \sigma_e^2\). Therefore, \(\operatorname{Var}(u | \text{inc}) = \operatorname{Var}(\sqrt{\text{inc}} \cdot e | \text{inc}) = \sqrt{\text{inc}}^2 \cdot \operatorname{Var}(e | \text{inc}) = \text{inc} \cdot \sigma_e^2\). This indicates that the variance is proportional to \(\text{inc}\), violating Assumption SLR.5 of homoskedasticity.

Discuss Variance in Savings with Income

Higher income families often engage in diverse saving activities that are not linearly correlated, leading to greater variance in their savings. For instance, they might invest in volatile markets, which increases variances compared to lower-income households, where saving behaviors tend to be more consistent and predictable, resulting in less variance.

Key concepts

Homoskedasticity

Homoskedasticity in econometrics refers to the assumption that the variance of errors in a regression model is constant across all levels of an independent variable. When this assumption holds, it means that the variability of the dependent variable, in this case, savings (\( \operatorname{sav} \)), remains the same regardless of income levels (\( \operatorname{inc} \)). Homoskedasticity is important because it ensures that the model's predictions are equally reliable for all data points, regardless of their independent variable values.

In our exercise, however, the variance of the savings error term (\( u \)) was shown to increase with income, as calculated by the formula \( \operatorname{Var}(u | \text{inc}) = \text{inc} \cdot \sigma_e^2 \). This is a clear violation of the homoskedasticity assumption, known as heteroskedasticity. In practical terms, this means the reliability of predictions will vary depending on income levels, potentially biasing standard error estimates and hypothesis tests.

Zero Conditional Mean Assumption

The zero conditional mean assumption is a key requirement in linear regression analysis. It posits that the expected value of the error term (\( u \)) given any value of the independent variable (\( \operatorname{inc} \)) is zero, expressed mathematically as \( \mathrm{E}(u | \text{inc}) = 0 \).

This assumption ensures that the independent variable is not systematically related to the error term, allowing for unbiased and consistent estimates of the coefficients in a regression model. In the exercise, the assumption was fulfilled because \( e \), the random variable, is independent of \( \operatorname{inc} \), leading to \( \mathrm{E}(e | \text{inc}) = \mathrm{E}(e) = 0 \).

This is critical to ensure that the model captures the true relationship between income and savings, without being skewed by other unobserved factors that might also affect savings.

Conditional Variance

Conditional variance refers to the variance of a random variable given the value of another variable. In regression models, it describes how the spread of the dependent variable’s values changes depending on different levels of the independent variable. Calculating conditional variance helps identify whether variance remains constant or changes with the level of the independent variable.

For our problem, it was shown that the conditional variance of \( u \), the error term, upon given levels of \( \operatorname{inc} \) is \( \operatorname{Var}(u | \text{inc}) = \text{inc} \times \sigma_e^2 \). This indicates that as income (\( \operatorname{inc} \)) increases, the variance of savings also increases, due to its proportionality with income.

Understanding the concept of conditional variance helps us diagnose issues in regression analysis, such as heteroskedasticity, and reminds us to apply suitable transformation or robust methods to address these.

Income and Savings Relationship

The relationship between income and savings is a fundamental concept in economics and econometrics. It explores how variations in income affect the behavior and amount of savings by individuals or households.

This relationship is often non-linear; higher-income households might not only save more, but also engage in riskier investment options, enhancing the variability of their total savings. As per the discussion from the exercise, the variance in savings increases with income due to these varying saving activities. It's not just about saving more cash, but how that cash is utilized, which may include volatile investments.

In summary, understanding this relationship is crucial for creating economic models. It allows for more accurate forecasts and policy-making, especially in the contexts of wealth distribution and financial advising.