## Info

Averages

Data source: Compustat PC+, September 2001.

A. A simple regression model with sales revenue as the dependent Y variable and R&D expenditures as the independent X variable yields the following results (t statistics in parentheses):

Sales;. = \$20.065 + \$6.062 R&D, R2 = 99.8%, SEE = 233.75, F = 8460.40 (0.31) (91.98)

How would you interpret these findings?

B. A simple regression model with net income (profits) as the dependent Y variable and R&D expenditures as the independent X variable yields the following results (t statistics in parentheses):

Profits; = -\$210.31 + \$2.538 R&Df, R2 = 99.3%, SEE = 201.30, F = 1999.90 (0.75) (7.03)

How would you interpret these findings?

C. Discuss any differences between your answers to parts A and B.

### ST3.2 Solution

A. First of all, the constant in such a regression typically has no meaning. Clearly, the intercept should not be used to suggest the value of sales revenue that might occur for a firm that had zero R&D expenditures. As discussed in the problem, this sample of firms is restricted to large companies with significant R&D spending. The R&D coefficient is statistically significant at the a = 0.01 level with a calculated t statistic value of 91.98, meaning that it is possible to be more than 99% confident that R&D expenditures affect firm sales. The probability of observing such a large t statistic when there is in fact no relation between sales revenue and R&D expenditures is less than 1%. The R&D coefficient estimate of \$6.062 implies that a \$1 rise in R&D expenditures leads to an average \$6.062 increase in sales revenue.

The R2 = 99.8% indicates the share of sales variation that can be explained by the variation in R&D expenditures. Note that F = 8460.40 > F* 13 a=0 01 = 9.07, implying that variation in R&D spending explains a significant share of the total variation in firm sales. This suggests that R&D expenditures are a key determinant of sales in the computer software industry, as one might expect.

The standard error of the Y estimate, or SEE = \$233.75 (million), is the average amount of error encountered in estimating the level of sales for any given level of R&D spending. If the u error terms are normally distributed about the regression equation, as would be true when large samples of more than 30 or so observations are analyzed, there is a 95% probability that observations of the dependent variable will lie within the range Y ± (1.96 X SEE), or within roughly two standard errors of the estimate. The probability is 99% that any given

Y will lie within the range Y ± (2.576 X SEE), or within roughly three standard errors of its predicted value. When very small samples of data are analyzed, as is the case here, "critical" values slightly larger than two or three are multiplied by the SEE to obtain the 95% and 99% confidence intervals.

Precise critical t values obtained from a t table, such as that found in Appendix C, are 113, a=o o5 = 2.160 (at the 95% confidence level) and t*13, a=0 01 = 3.012 (at the 99% confidence level) for df = 15 - 2 = 13. This means that actual sales revenue Y ; can be expected to fall in the range Y ± (2.160 X \$233.75), or Y ± \$504.90, with 95% confidence; and within the range

Y ± (3.012 X \$233.75), or Y ± \$704.055, with 99% confidence.

B. As in part A, the constant in such a regression typically has no meaning. Clearly, the intercept should not be used to suggest the level of profits that might occur for a firm that had zero R&D expenditures. Again, the R&D coefficient is statistically significant at the a = 0.01 level with a calculated t statistic value of 44.72, meaning that it is possible to be more than 99% confident that R&D expenditures affect firm profits. The probability of observing such a large t statistic when there is in fact no relation between profits and R&D expenditures is less than

1%. The R&D coefficient estimate of \$2.538 suggests that a \$1 rise in R&D expenditures leads to an average \$2.538 increase in current-year profits.

The R2 = 99.3% indicates the share of profit variation that can be explained by the variation in R&D expenditures. This suggests that R&D expenditures are a key determinant of profits in the aerospace industry. Again, notice that F = 1999.90 > F* 13 a=0 01 = 9.07, meaning that variation in R&D spending can explain a significant share of profit variation.

The standard error of the Y estimate of SEE = \$201.30 (million) is the average amount of error encountered in estimating the level of profit for any given level of R&D spending. Actual profits Y{ can be expected to fall in the range Y ± (2.160 X \$201.30), or Y ± \$434.808, with 95% confidence; and within the range Y ± (3.012 X \$201.30), or Y ± \$606.3156, with 99% confidence.

C. Clearly, a strong link between both sales revenue and profits and R&D expenditures is suggested by a regression analysis of the computer software industry. There appears to be slightly less variation in the sales-R&D relation than in the profits-R&D relation. As indicated byR2 the linkage between sales and R&D is a bit stronger than the relation between profits and R&D. At least in part, this may stem from the fact that the sample was limited to large R&D intensive firms, whereas no such screen for profitability was included.