Question

In Example 4.7 . we used data on nonunionized manufacturing firms to estimate the relationship between the scrap rate and other firm characteristics. We now look at this example more closely and use all available firms. i. The population model estimated in Example 4.7 can be written as $$\log (\operatorname{scrap})=\beta_{0}+\beta_{1} \text {hrsemp}+\beta_{2} \log (\text {sales})+\beta_{3} \log (\text {employ})+u$$ Using the 43 observations available for 1987 , the estimated equation is $$\begin{aligned} \widehat{\log (\text {scrap})}=& 11.74-.042 \text { hrsemp}-.951 \log (\text {sales})+.992 \log (\text {employ}) \\ &(4.57)(.019) \\ n=& 43, R^{2}=.310 \end{aligned}$$ Compare this equation to that estimated using only the 29 nonunionized firms in the sample. ii. Show that the population model can also be written as $$\log (s c r a p)=\beta_{0}+\beta_{1} h r s e m p+\beta_{2} \log (s a l e s / e m p l o y)+\theta_{3} \log (e m p l o y)+u$$ where \(\left.\theta_{3}=\beta_{2}+\beta_{3} . \text { [Hint: Recall that } \log \left(x_{2} / x_{3}\right)=\log \left(x_{2}\right)-\log \left(x_{3}\right) .\right]\) Interpret the hypothesis \(\mathrm{H}_{0}: \theta_{3}=0\) iii. When the equation from part (ii) is estimated, we obtain $$\begin{aligned} \widehat{\log (\text {scrap})}=& 11.74-.042 \text { hrsemp}-.951 \log (\text {sales/employ})+.041 \log (\text {employ}) \\ &(4.57)(.019) \\ n=& 43, R^{2}=.310 \end{aligned}$$ Controlling for worker training and for the sales-to-employee ratio, do bigger firms have larger statistically significant scrap rates? iv. Test the hypothesis that a \(1 \%\) increase in sales/employ is associated with a \(1 \%\) drop in the scrap rate.

Short answer

The equation shows that bigger firms don't have statistically larger scrap rates; the impact per 1% increase in the sales-to-employ ratio is slightly less than a 1% scrap rate reduction.

Step-by-step solution

Analyzing the Original Equation

The given estimated equation for the scrap rate using all available firms is: \( \widehat{\log (\text{scrap})} = 11.74 - 0.042 \, \text{hrsemp} - 0.951 \, \log (\text{sales}) + 0.992 \, \log(\text{employ}) \). This equation uses data from 43 firms and gives the relationship between the scrap rate and variables such as hours spent on employee training (hrsemp), and logs of sales and employment.

Comparing Sample Sizes

Compare this with the equation estimated using only 29 nonunionized firms. Examine the coefficient changes and statistical significance between these two samples (43 firms vs 29 nonunionized firms) to observe any changes. However, the coefficients from the 29 nonunionized firms are not provided directly but should be considered when comparing the two samples.

Rewriting the Population Model

To rewrite the population model, use the property \( \log(x/y) = \log(x) - \log(y) \) to express the model as: \( \log (scrap) = \beta_{0} + \beta_{1} \, \text{hrsemp} + \beta_{2} \, \log (sales/employ) + (\beta_{2} + \beta_{3}) \, \log(employ) + u \). This shows that \( \theta_{3} = \beta_{2} + \beta_{3} \).

Interpreting the Hypothesis \(\mathrm{H}_{0}: \theta_{3} = 0\)

The hypothesis \(\mathrm{H}_{0}: \theta_{3} = 0\) implies that the combined effect of \( \log (sales/employ)\) and \( \log(employ) \) on \( \log(scrap)\) is zero. If \( \theta_{3} = 0 \), it suggests \( \beta_{2} + \beta_{3} = 0\), meaning no combined impact from these two log variables.

Analyzing Part (iii) Equation

The equation from part (iii) is \( \widehat{\log (\text{scrap})} = 11.74 - 0.042 \, \text{hrsemp} - 0.951 \, \log (sales/employ) + 0.041 \, \log(employ) \). Notice that the coefficient of \( \log(employ)\) (0.041) is positive but small, and its significance will determine if bigger firms have larger scrap rates controlling other factors.

Testing the Hypothesis for Sales/Employ Impact

To test \( \theta_{3}=0 \), check the coefficient of \( \log(sales/employ) \), which is -0.951. If the hypothesis is that a 1% increase in sales/employ corresponds to a 1% reduction in scrap rate, we need to test if this coefficient is statistically different from -1, indicating a slightly lesser impact than hypothesized.

Key concepts

Regression Analysis

Regression analysis is a powerful statistical tool used to understand the relationship between one dependent variable and several independent variables.
In the context of manufacturing firms, we often use regression to predict certain outcomes, like scrap rate, based on measurable factors such as employee training hours and the logarithm of sales.
Through regression analysis, we can determine how changes in these factors influence the dependent variable, allowing firms to make informed decisions based on data. The accuracy of a regression model can be further enhanced by ensuring that all relevant variables are considered and that the sample size is adequate.
This method not only helps in prediction but also in finding the importance and significance of these variables through coefficients obtained in the analysis.

Logarithmic Models

Logarithmic models are often utilized in econometrics when studying relationships between variables that exhibit constant relative changes.
In our manufacturing firm analysis, we use the Natural Logarithm in the model to transform variables like sales and employment. This transformation stabilizes the variance, linearizes relations, and makes interpretation of coefficients more straightforward.
For example, a coefficient of \(0.041\) on \( ext{log(employ)}\) implies that a \(.041\)% increase in scrap is associated with a 1% increase in the number of employees.
This transformation enables us to capture elasticities where percentage changes in one variable lead to percentage changes in another, providing a clearer picture of the economic relationships.

Hypothesis Testing

Hypothesis testing in econometrics allows us to statistically infer whether certain effects observed in the sample hold true in the population.
In the problem of nonunionized manufacturing firms, we are interested in testing hypotheses like \(H_0: \theta_3 = 0\), which questions whether the sales-to-employee ratio and employee logging have a significant combined effect on scrap rates.
If \(H_0\) cannot be rejected, it suggests these factors do not impact the dependent variable, the scrap rate, as a whole.
The testing process involves calculating a test statistic and considering its p-value to decide if the hypothesis should be rejected or not.
This process is crucial for making informed business strategies based on statistically significant findings.

Manufacturing Firms

Manufacturing firms, especially in econometric studies, provide a rich ground for analyzing economic relationships between production factors and outputs.
In this exercise, we look closely at how characteristics within a firm, like training hours and operational size, influence the scrap rate—a key performance metric.
By using regression models, we can derive insights into how modifications in practices or scales impact production efficiency.
These analyses can help improve decision-making in manufacturing settings, leading to process optimizations and cost savings.

Nonunionized Firms

Nonunionized firms are those where the workforce is not represented by trade unions.
This can result in different dynamics in terms of labor relations, cost structures, and overall business efficiencies compared to unionized firms.
For economists, studying nonunionized firms often offers insights into these different dynamics and how they impact measurable outcomes like productivity and scrap rates.
Such analyses highlight the importance of labor arrangements in firms and inform strategies that may improve organizational practices and profitability.