Question

The data in 401K are a subset of data analyzed by Papke (1995) to study the relationship between participation in a 401(k) pension plan and the generosity of the plan. The variable prate is the per- centage of eligible workers with an active account; this is the variable we would like to explain. The measure of generosity is the plan match rate, mrate. This variable gives the average amount the firm contributes to each worker's plan for each $1 contribution by the worker. For example, if mrate = 0.50, then a $1 contribution by the worker is matched by a 50¢ contribution by the firm. (1) Find the average participation rate and the average match rate in the sample of plans. (ii) Now, estimate the simple regression equation prate = Bo + B, mrate, and report the results along with the sample size and R-squared. (iii) Interpret the intercept in your equation. Interpret the coefficient on mrate. (iv) Find the predicted prate when mrate = 3.5. Is this a reasonable prediction? Explain what is happening here. (v) How much of the variation in prate is explained by mrate? Is this a lot in your opinion?

Answer 1

To solve this problem, you need to calculate the average participation rate and match rate, estimate the simple regression equation, interpret the intercept and coefficient, find the predicted participation rate, and determine the amount of variation explained by the match rate.

Step-by-step explanation:

The average participation rate and match rate in the sample of plans can be found by calculating the mean of the respective variables. The average participation rate is the mean of prate, and the average match rate is the mean of mrate.

To estimate the simple regression equation prate = Bo + B*mrate, you can use statistical software or a spreadsheet program that can perform linear regression analysis. The results of the regression analysis will provide you with the intercept coefficient (Bo), the slope coefficient (B), the sample size, and the R-squared value.

The intercept in the regression equation represents the expected value of prate when mrate is equal to zero. In this context, it would indicate the expected participation rate when there is no matching contribution from the firm.

The coefficient on mrate represents the change in the average participation rate for a 1-unit increase in the match rate. For example, if the coefficient is 0.5, it means that for every 1% increase in the match rate, the average participation rate increases by 0.5%.

When mrate = 3.5, you can substitute this value into the regression equation to find the predicted prate. This prediction is reasonable as long as it falls within the range of values for prate in the sample and the relationship between mrate and prate is linear.

R-squared represents the proportion of the variation in prate that is explained by the variation in mrate. A higher R-squared value indicates a stronger relationship between the variables. Whether a certain R-squared value is considered a lot depends on the context and the field of study.