Question

The variable rdintens is expenditures on research and development (R\&D) as a percentage of sales. Sales are measured in millions of dollars. The variable profmarg is profits as a percentage of sales. Using the data in RDCHEM for 32 firms in the chemical industry, the following equation is estimated: $$\begin{aligned} \widehat{\text {rdintens}} &=.472+.321 \log (\text {sales})+.050 \text { profmarg} \\ &(1.369)(.216) \\ n &=32, R^{2}=.099 \end{aligned}$$ i. Interpret the coefficient on \(\log (\text {sales} \text { ). In particular, if sales increases by } 10 \%,\) what is the estimated percentage point change in \(r\)dintens? Is this an economically large effect? ii. Test the hypothesis that R\&D intensity does not change with sales against the alternative that it does increase with sales. Do the test at the \(5 \%\) and \(10 \%\) levels. iii. Interpret the coefficient on profmarg. Is it economically large? iv. Does profinarg have a statistically significant effect on rdintens?

Short answer

The log of sales has a small, positive, but not statistically significant effect on R&D intensity. Profit margin does not have a statistically significant or economically large effect either.

Step-by-step solution

Interpretation of Coefficient on Log(Sales)

The equation provided is for estimating the variable `rdintens`. The coefficient for \( \log(\text{sales}) \) is 0.321, which indicates that for a 1% increase in sales, \( \text{rdintens} \) is expected to increase by approximately 0.321 percentage points. If sales increase by 10%, then \( 10 \times 0.321 = 3.21 \) percentage points is the expected increase in \( \text{rdintens} \). Given this percentage change, one would assess the economic significance based on industry standards and the firm's specific circumstances.

Hypothesis Test for Log(Sales) Coefficient

We test whether \( \text{rdintens} \) changes with sales. Null hypothesis \( H_0: \beta_1 = 0 \) (no change) against the alternative \( H_1: \beta_1 ot= 0 \). Using the coefficient of 0.321 and standard error 0.216, the test statistic \( t = \frac{0.321}{0.216} \approx 1.486 \). With 30 degrees of freedom \((n - k = 32 - 2)\), we compare to critical values for 5% and 10% levels: \( t_{0.05} \approx 2.042, t_{0.10} \approx 1.697 \). Since 1.486 < 1.697, we fail to reject \( H_0 \) and conclude \( \text{rdintens} \) is not significantly related to sales at both levels.

Interpretation of Profmarg Coefficient

The coefficient for `profmarg` is 0.050, suggesting a 1 percentage point increase in profit margin will increase \( \text{rdintens} \) by 0.050 percentage points. To evaluate economic significance, consider industry context. A small coefficient suggests it's not economically large, implying minor change in R&D intensity as profit margins change.

Statistical Significance of Profmarg Coefficient

Assess \( \text{profmarg} \)'s coefficient significance using its standard error. Assuming it's similar to \( \log(\text{sales}) \) errors, use the t-test \( t = \frac{0.050}{0.216} \approx 0.231 \). Hypothesis \( H_0: \beta_2 = 0 \) fails to be rejected since \( 0.231 < 1.697 \) across both levels of significance. Hence, `profmarg`'s contribution to \( \text{rdintens} \) isn't statistically significant.

Key concepts

Research and Development

Research and Development (R&D) is a crucial component for businesses, especially in industries like chemicals where innovation drives competitiveness and growth. R&D spending is often expressed as a percentage of sales, offering insights into how much a firm invests into developing new products or processes. This percentage, in turn, can influence a firm's long-term success and market standing.

In the provided exercise, the variable \( \text{rdintens} \) represents R&D expenditures relative to sales. Understanding how factors like sales and profit margins impact this percentage helps companies strategize their spending better. When exploring R&D intensity, it is important to consider:

• The industry's typical R&D spending norm, which helps to determine whether an investment is high or low.
• The company's objectives, as higher R&D spending may indicate a focus on innovation over immediate profit.
• Changes in market demand or regulations, which may necessitate increased R&D spending despite financial pressures.

Assessing R&D intensity also involves analyzing how much such investments contribute to creating value through innovations.

Regression Analysis

Regression Analysis is a statistical method used to explore the relationship between independent variables and a dependent variable. In our exercise, regression is used to understand how sales and profit margins impact R&D intensity.

Key concepts in regression analysis include:

• **Coefficients**: Each independent variable has a coefficient representing its estimated effect on the dependent variable. Here, the coefficient for \( \log(\text{sales}) \) is 0.321, meaning a 1% increase in sales likely results in a 0.321 percentage point increase in R&D intensity.
• **Standard Errors**: These measure the variability of the coefficient estimates. A larger standard error may suggest less confidence in the estimate's precision.
• **R-squared (\( R^2 \))**: This value tells us how much of the dependent variable’s variation is explained by the independent variables. An \( R^2 \) of 0.099 indicates that around 9.9% of R&D intensity changes can be explained by sales and profits—which is relatively low. This implies other factors might be at play.

Analyzing regression results helps in making informed decisions about resource allocation and strategic planning by quantifying likely outcomes of sales and profit changes.

Statistical Significance

Statistical Significance is used to determine if the results of an analysis are likely due to an actual underlying effect rather than random chance. Assessing statistical significance is crucial for interpreting regression results in our exercise.

Two main elements in determining statistical significance are:

• **Hypothesis Testing**: This involves setting a null hypothesis (no effect) and an alternative hypothesis (presence of effect). For instance, testing if \( \text{rdintens} \) changes with sales involves checking if the sales coefficient is indeed different from zero.
• **p-values and t-tests**: The calculation of a t-statistic helps determine if results are statistically significant. In the exercise, the t-value calculated for \( \log(\text{sales}) \) was approximately 1.486, which compared against the critical value, suggested a lack of significance at standard levels (5% and 10%).

The lack of statistical significance in the coefficient for both sales and profit margin doesn't necessarily mean they are unimportant, but rather, within the given sample size and variability, we couldn't establish a strong enough association. Understanding these concepts helps frame the limitations and confidence in the findings, guiding further research or data collection if needed.