Question

Use APPLE to verify some of the claims made in Section \(6-3\) . (i) Run the regression ecolbs on ecoprc, regprc and report the results in the usual form, including the \(R\) -squared and adjusted \(R\) -squared. Interpret the coefficients on the price variables and comment on their signs and magnitudes. (ii) Are the price variables statistically significant? Report the \(p\) -values for the individual tests. (iii) What is the range of fitted values for ecolbs? What fraction of the sample reports ecolbs = 0? Comment. (iv) Do you think the price variables together do a good job of explaining variation in ecolbs? Explain. (v) Add the variables faminc, hhsize (household size), educ, and age to the regression from part (i). Find the \(p\) -value for their joint significance. What do you conclude? (vi) Run separate simple regressions of ecolbs on ecoprc and then ecolbs on regpre. How do the simple regression coefficients compare with the multiple regression from part (i)? Find the correlation coefficient between ecoprc and regprc to help explain your findings. (vii) Consider a model that adds family income and the quantity demanded for regular apples: \(e c o l b s=\beta_{0}+\beta_{1} e c o p r c+\beta_{2} r e g p r c+\beta_{3}\) faminc \(+\beta_{4} r e g l b s+u\) From basic economic theory, which explanatory variable does not belong to the equation? When you drop the variables one at a time time, do the sizes of the adjusted \(R\) -squareds affect your answer?

Short answer

Run a regression with `ecoprc` and `regprc`, report coefficients, check \(p\)-values for significance, compare simple vs. multiple regressions, and consider model extension with more variables for improved fit.

Step-by-step solution

Set Up the Regression Model

Define the regression model to be utilized. You will set up a multiple linear regression where `ecolbs` is the dependent variable, and `ecoprc` and `regprc` are the independent variables. This can be expressed as: \[ ecolbs = \beta_0 + \beta_1 \cdot ecoprc + \beta_2 \cdot regprc + \epsilon \] where \(\epsilon\) represents the error term.

Perform Regression and Report Results

Run the regression using statistical software, ensuring to report the coefficient estimates for \(\beta_0, \beta_1,\) and \(\beta_2\), along with standard errors, \(t\)-values, and \(p\)-values. Also, compute and report the \(R\)-squared and adjusted \(R\)-squared values which indicate the model's explanatory power.

Interpret Coefficients

Analyze the sign and magnitude of the coefficients for `ecoprc` and `regprc`. Positive coefficients suggest an increase in `ecolbs` with an increase in the price of eco apples or regular apples, while negative coefficients suggest an inverse relationship. Consider economic intuition to interpret these results.

Evaluate Statistical Significance

Examine the \(p\)-values associated with the coefficients of `ecoprc` and `regprc`. A \(p\)-value less than 0.05 typically indicates statistical significance, suggesting a meaningful relationship between the price variable and `ecolbs`.

Analyze Range and Zeros in Fitted Values

Calculate the range of fitted values from the regression to understand the predicted values' spread. Determine what fraction of the sample has `ecolbs` equal to zero and contemplate why this occurs.

Assess Explanatory Power of Price Variables

Consider whether the `ecoprc` and `regprc` together provide a comprehensive explanation of variation in `ecolbs`, basing your assessment on the significance of the coefficients and the value of adjusted \(R\)-squared.

Extend Model with Additional Variables

Enhance the regression model by adding `faminc`, `hhsize`, `educ`, and `age`. Check for joint significance of these new variables using an F-test and examine the p-value to decide if they significantly enhance model explanation.

Compare with Simple Regressions

Run two additional regressions: one with `ecolbs` on `ecoprc` alone and one with `ecolbs` on `regprc` alone. Compare coefficients with the ones obtained in the multiple regression to interpret any notable differences.

Correlation and Model Considerations

Calculate the correlation coefficient between `ecoprc` and `regprc` which may reveal multicollinearity. This information helps explain the differences observed in coefficient estimates.

Consider Alternative Model

For a model adding `faminc` and `reglbs`, assess whether each explanatory variable is theoretically justified. Drop variables one by one and analyze how changes in adjusted \(R\)-squared affect the model, comparing it to economic theory.

Key concepts

Econometric Analysis

Econometric analysis is a powerful tool that helps researchers estimate relationships between variables in a dataset. It uses statistical methods to test hypotheses, predict future outcomes, or explore complex data structures. In the given exercise, a multiple linear regression is used to understand how the prices of eco-friendly (ecoprc) and regular apples (regprc) affect their consumption levels (ecolbs). By defining a model and analyzing the results, you can draw conclusions on how tightly these variables correlate within a specific set of observations. Econometric analysis aims to bridge the gap between theory and real-world data, providing meaningful insights.

Coefficient Interpretation

After running a regression, interpreting the coefficients becomes crucial. In this exercise, the coefficients for `ecoprc` and `regprc` reveal how changes in the prices of eco and regular apples affect the quantity consumed (ecolbs). A positive coefficient suggests that as the price increases, consumption also tends to increase, which can seem counterintuitive unless other factors are at play. Conversely, a negative coefficient implies that as prices increase, consumption decreases. The magnitude of these coefficients tells us the strength of this relationship. It's important to connect these insights with economic reasoning to ensure a grounded interpretation.

Statistical Significance

Statistical significance helps determine if the relationships observed in a regression analysis are likely to be genuine or if they occurred by chance. In this context, examining the p-values for the coefficients of `ecoprc` and `regprc` is essential. A p-value lower than a common threshold, like 0.05, indicates that you can be reasonably confident the relationship is genuine. This gives more weight to the regression results, allowing you to assert that the price of eco-friendly and regular apples meaningfully impacts their consumption. The significance test adds credibility to the regression analysis by confirming the reliability of the observed relationships.

Income Effect

The income effect refers to how changes in a person's income level affect their consumption choices. By including `faminc` in the regression model, the influence of family income on apple consumption is analyzed. If the coefficient of `faminc` is statistically significant, it suggests that income variations meaningfully alter apple consumption levels. This is crucial to understand consumer behavior across different income brackets, highlighting how essential variables like income can modify purchasing decisions beyond price changes. Contextualizing the role of income provides a more comprehensive understanding of the patterns that drive apple consumption.

Model Specification

Model specification is about selecting the correct variables to include in a regression model. It's a delicate balance between simplicity and explanatory power. A well-specified model accurately captures the relationships among variables without including unnecessary elements or omitting important variables. In the exercise, extending the model by adding variables like `faminc`, `hhsize`, `educ`, and `age` refines the analysis by including potentially relevant factors. However, multicollinearity—a condition where independent variables are highly correlated—can distort the model, affecting the reliability of the coefficients. By dropping variables one by one and reassessing the model, you ensure clarity and accuracy in explanation.