# An empirical evaluation of two methods for estimating economic damages.

An Empirical Evaluation of Two Methods For Estimating Economic
Damages

A variety of approaches have been proposed and are used by economists to estimate economic damages in personal injury/wrongful death tort cases. These approaches vary from the straightforward offset or Alaska method, which enables one to calculate economic damages by multiplying the current earnings level by the number of years of worklife expectancy, to more complex approaches involving the use of real wage growth and discount rates and an age-earning's profile adjustment.

Although each approach has some theoretical foundation, no one has attempted to empirically evaluate which method does the best job in forecasting earnings losses for a large sample of workers. This article will attempt to fill this void in the economic damages literature by examining a longitudinal sample of workers whose earnings are available from 1968 through 1983. A longitudinal sample is ideal for this type of study since it can be assumed that, as of a certain age, workers either die or become permanently disabled. From the earnings of the individual prior to the hypothesized year of death or diablement, future wages may be projected and the present value obtained. The predicted present values may be compared with the actual present values using the individual's wages after this arbitrary year.

Two methods will first be presented that will be used to estimate economic losses. Next the data set to be employed along with the empirical methodology will be discussed. The following section will present the empirical results followed by conclusions.

Estimating Economic Damages

The methods used by economists to estimate economic damages vary considerably. An approach which has received a great deal of support in recent years and is required in some states is the offset method. This method simplifies the analysis by assuming that the rate of wage growth equals the rate of discount. (1) Because these two rates are equal, the present value of the earnings losses is simply the product of the current earnings level and the number of years of worklife expectancy.

The theoretical justification for this approach is that wage growth rates and interest rates are both affected by inflation implying that these two variables are highly correlated. One major benefit of this approach is simplicity of presentation to a jury of non-economists. Disadvantages include the fact that these two rates are not exactly equal so this method represents an approximation at best. Furthermore, this method can over estimate earnings losses when one considers income and social security taxes that will be paid on these gross earnings amounts. (2)

The approach that will be compared to the offset uses an age-earnings profile to adjust earnings for each year. (3) It is well known that earnings for the typical person follow an inverted U pattern with respect to age. (4) Earnings increase rapidly earlier in the life cycle due to the continued investments in human capital, level off in the middle years, and then decline due to the depreciation of human capital as well as declines in hours of work.

The rationale for using this approach is that in addition to the increases in wages due to inflation and aggregate increases in productivity, individual's earnings may increase or decrease depending upon their age. For example, a typical 25-year old's earnings increase quite rapidly. Therefore, the offset approach alone may underestimate the loss for this individual. Conversely, an individual at the upper end of the age distribution may have decreasing earnings so that a slight overestimation may occur.

Data and Models

The data for this study come from the random portion of the Panel Study of Income Dynamics (PSID) for 1968 through 1983. The PSID is a longitudinal survey with detailed information on the financial and demographical characteristics of the sample families. (5) The sample for this study is restricted to male family heads who continuously participated in the labor force from 1968 through 1983, had not retired by 1983, and for whom information was available on age and education level. These restrictions yielded a sample of 812 usable observations.

The present values of actual earnings from 1971 through 1983 are compared to the present values estimated using two approaches to estimating earnings loses. (6) Earnings for 1968 through 1970 are inflated to 1970 dollars and are then averaged to produce the baseline earnings estimate, [Y.sub.0]. (7)

In the offset approach, it is assumed that the future rate of wage growth equals the rate of discount. Therefore, the present value in this case, [PV.sub.0], can be expressed as:

[PV.sub.0] = [TY.sub.0]

where [Y.sub.0] is the baseline earnings of the individual and T is the number of years of worklife expectancy; in this study, T = 13.

In the age-earnings adjustment approach, earnings are adjusted each year by a factor determined by the age-earnings profile. (8) Figure 1 shows the age-earnings profile for five educational levels. Five-year averages in earnings are calculated and are used to calculate the annual average rate of growth for each of the educational categories. These annual rates of growth are called age-earnings profile adjustment factors and are reported in Table 1. For example, column 3 shows that the earnings of a high school graduate who was 32 years old in 1971 would be increased by a factor of 1.023 for each of the next five years, which in turn in followed by a factor of 0.992 for the next five years. Algebraically, this may be represented as:

[Mathematical Expression Omitted]

where [A.sub.t] is the cumulative adjustment factor calculated from Table 1, and [PV.sub.A] is the present value obtained with the age-earnings profile method.

Evaluation Criteria and Empirical Results

Each of the approaches presented above will create an estimated present value, PV. The rate chosen for discounting the workers' actual earnings from 1971 through 1983 is the nontaxable municipal bond yield average from 1961 through 1970, 4.10 percent. (9) When applied to the actual earnings of these individuals, [Y.sub.t], an actual present value is created which may be expressed as:

[Mathematical Expression Omitted]

Several criteria will be used to evaluate the accuracy of these approaches. In addition to the commonly used root mean square error, the mean error and Theil's inequality coefficient will also be presented. (10) The mean error, ME, may be presented as:

ME = PV - PV

where PV is the mean actual present value and PV is the mean estimated present value.

The next measure to be presented is the root mean square error, RMSE, which equals:

[Mathematical Expression Omitted]

where [PV.sub.i] is the actual present value for individual i, [PV.sub.i] is the estimated present value for individual i, and n is the number of individuals.

The final criteria to be used for evaluating these approaches is Theil's inequality coeffiecient, U, which is expressed as:

[Mathematical Expression Omitted]

This expression has a range of 0 to 1. If U = 0, the approach is perfect since the actual present value will alway equal the estimated present value. If U = 1, there is no relationship between the actual and estimated present values. This expression may be decomposed into terms reflecting bias or systematic error in the model, [U.sup.M], and two other terms reflecting the variability in the data. A value for [U.sup.M] (the square of the ratio of ME to RMSE) above roughly 0.15 indicates a large bias in the model. [U.sup.M] may be written as:

[Mathematical Expression Omitted]

The results indicate support for the age-earnings profile adjustment. The offset method alone yielded: ME = $38,597; RMSE = $79,425; U = .214, and [U.sup.M] = .24. The offset method together with the age-earnings profile adjustment produced: ME = $10,080, RMSE = $70,562, U = .183, and [U.sup.M] = .02. These results support using the age-earnings profile adjustment when estimating earnings losses. It is noteworthy that the mean error for the offset approach is nearly four times as large as in the age-earnings adjustment approach. And the systematic bias of the offset approach is 12 times as large as in the age-earnings adjustment approach. The results also indicate that while both methods underestimate the loss, the offset method underestimates the loss by a considerably greater margin than does the age-earnings adjustment approach.

Conclusions

Although many methods have been proposed to deal with the estimation of economic damages, none have been rigorously tested empirically to determine their relative accuracy. This study is the first to use data on workers to compare estimation methods. A longitudinal data base such as the one found in the PSID should be used more often by economists to evaluate their proposals for the estimation of economic damages.

Using data from the PSID, this study finds that the offset approach, although easy to understand, is not the most accurate method available. Adjusting earnings for the impact of a worker's place in the life-earnings cycle will produce estimates of earnings losses that are more accurate.

The data for this study are 13 year individual earnings histories for the period 1971 through 1983. Because this period spans three full business cycles, the results are not likely to reflect the influence of a particular stage of the business cycle. Because a 13 year span is a relatively short earnings history, these data are potentially limited by the influence of a particular state of the earnings life cycle. However, this limitation is mitigated by the consideration that the age of the individuals in the sample are distributed over the interval 17 years to 52 years with a mean age of 42 years.

(1) See Franz (1978) for a discussion of the rationale for the offset approach.

(2) See Harmon and Lambrinos (1989) for an estimation of the bias introduced by taxes.

(3) Other approaches using various combinations of real and nominal wage growth rates, calculated from both industry and occupational data, along with real and nominal rates of discount, were also tried. The results do not differ appreciably from the offset results and are available from the authors upon request.

(4) See, for example, Miller (1965).

(5) The magnetic computer tape is available from the University of Michigan Survey Research Center, Ann Arbor, Michigan.

(6) Although 13 years may not be adequate to judge the true impact of the age-earnings profile adjustment, additional years of data are not available.

(7) Two other base periods were used with no appreciable change in the empirical results.

(8) The age-earnings profile is described in more detail in Lambrinos (1984b, 1985, 1989).

(9) The treasury bond rate, 4.35 percent, was also used without any change in the results.

(10) Pindyck and Rubinfeld (1981) discuss these criteria and others.

References

[1] Abraham, Fred J., 1988, Pitfalls to using the Real Rates or Age-Earnings Profile Models in Calculating Economic Loss, Journal of Forensic Economics, 1: 77-81.

[2] Bryan, William R. and Charles M. Linke, 1988, Estimating Present Values of Future Earnings: Experience with Dedicated Portfolios, Journal of Risks and Insurance, 55: 271-86.

[3] Bureau of the Census, 1983, Lifetime Earnings Estimates for Men and Women in the United States: 1979, Washington, D.C.: U.S. Government Printing Office.

[4] Economic Report of the President, 1988, Washington, D.C.: U.S. Government Printing Office.

[5] Franz, Wolfgang W., 1978, A Solution to Problems Arising from Inflation When Determining Damages, Journal of Risk and Insurance, 45: 323-33.

[6] Harmon, Oskar and James Lambrinos, 1989, Taxes and the Bias of the Offset Methods of Estimating Economic Damages, Unpublished Manuscript.

[7] Hosek, William R., 1982, Problems in the Use of Historical Data in Estimating Economic Loss in Wrongful Death and Injury Cases, Journal of Risk and Insurance, 49: 300-08.

[8] Lambrinos, James, 1984a, On the Use of Nominal v. Real Economic Variables in Forecasting Future Earnings Losses, Journal of the Missouri Bar, 40: 389-393.

[9] Lambrinos, James, 1984b, The Importance of Age in the Estimation of Economic Losses, Trial Lawyers Quarterly, 16: 48-52.

[10] Lambrinos, James, 1985, On the Use of Historical Data in the Estimation of Economic Losses, Journal of Risk and Insurance, 52: 464-76.

[11] Lambrinos, James, 1989, Maximizing Economic Loss Damages, New York: John Wiley and Sons, Inc.

[12] Miller, Herman P., 1965, "Lifetime Income and Economic Growth" American Economic Review, 55: 834-44.

[13] Pindyck, Robert S. and Daniel L. Rubinfeld, 1981, Econometric Methods and Economic Forecasts New York: McGraw-Hill Book Company.

James Lambrinos is Associate Professor and Director of the Graduate Management Institute at Union College in Schenectady, New York. Oskar R. Harmon is Assistant Professor of Economics at the University of Connecticut.

A variety of approaches have been proposed and are used by economists to estimate economic damages in personal injury/wrongful death tort cases. These approaches vary from the straightforward offset or Alaska method, which enables one to calculate economic damages by multiplying the current earnings level by the number of years of worklife expectancy, to more complex approaches involving the use of real wage growth and discount rates and an age-earning's profile adjustment.

Although each approach has some theoretical foundation, no one has attempted to empirically evaluate which method does the best job in forecasting earnings losses for a large sample of workers. This article will attempt to fill this void in the economic damages literature by examining a longitudinal sample of workers whose earnings are available from 1968 through 1983. A longitudinal sample is ideal for this type of study since it can be assumed that, as of a certain age, workers either die or become permanently disabled. From the earnings of the individual prior to the hypothesized year of death or diablement, future wages may be projected and the present value obtained. The predicted present values may be compared with the actual present values using the individual's wages after this arbitrary year.

Two methods will first be presented that will be used to estimate economic losses. Next the data set to be employed along with the empirical methodology will be discussed. The following section will present the empirical results followed by conclusions.

Estimating Economic Damages

The methods used by economists to estimate economic damages vary considerably. An approach which has received a great deal of support in recent years and is required in some states is the offset method. This method simplifies the analysis by assuming that the rate of wage growth equals the rate of discount. (1) Because these two rates are equal, the present value of the earnings losses is simply the product of the current earnings level and the number of years of worklife expectancy.

The theoretical justification for this approach is that wage growth rates and interest rates are both affected by inflation implying that these two variables are highly correlated. One major benefit of this approach is simplicity of presentation to a jury of non-economists. Disadvantages include the fact that these two rates are not exactly equal so this method represents an approximation at best. Furthermore, this method can over estimate earnings losses when one considers income and social security taxes that will be paid on these gross earnings amounts. (2)

The approach that will be compared to the offset uses an age-earnings profile to adjust earnings for each year. (3) It is well known that earnings for the typical person follow an inverted U pattern with respect to age. (4) Earnings increase rapidly earlier in the life cycle due to the continued investments in human capital, level off in the middle years, and then decline due to the depreciation of human capital as well as declines in hours of work.

The rationale for using this approach is that in addition to the increases in wages due to inflation and aggregate increases in productivity, individual's earnings may increase or decrease depending upon their age. For example, a typical 25-year old's earnings increase quite rapidly. Therefore, the offset approach alone may underestimate the loss for this individual. Conversely, an individual at the upper end of the age distribution may have decreasing earnings so that a slight overestimation may occur.

Data and Models

The data for this study come from the random portion of the Panel Study of Income Dynamics (PSID) for 1968 through 1983. The PSID is a longitudinal survey with detailed information on the financial and demographical characteristics of the sample families. (5) The sample for this study is restricted to male family heads who continuously participated in the labor force from 1968 through 1983, had not retired by 1983, and for whom information was available on age and education level. These restrictions yielded a sample of 812 usable observations.

The present values of actual earnings from 1971 through 1983 are compared to the present values estimated using two approaches to estimating earnings loses. (6) Earnings for 1968 through 1970 are inflated to 1970 dollars and are then averaged to produce the baseline earnings estimate, [Y.sub.0]. (7)

In the offset approach, it is assumed that the future rate of wage growth equals the rate of discount. Therefore, the present value in this case, [PV.sub.0], can be expressed as:

[PV.sub.0] = [TY.sub.0]

where [Y.sub.0] is the baseline earnings of the individual and T is the number of years of worklife expectancy; in this study, T = 13.

In the age-earnings adjustment approach, earnings are adjusted each year by a factor determined by the age-earnings profile. (8) Figure 1 shows the age-earnings profile for five educational levels. Five-year averages in earnings are calculated and are used to calculate the annual average rate of growth for each of the educational categories. These annual rates of growth are called age-earnings profile adjustment factors and are reported in Table 1. For example, column 3 shows that the earnings of a high school graduate who was 32 years old in 1971 would be increased by a factor of 1.023 for each of the next five years, which in turn in followed by a factor of 0.992 for the next five years. Algebraically, this may be represented as:

[Mathematical Expression Omitted]

where [A.sub.t] is the cumulative adjustment factor calculated from Table 1, and [PV.sub.A] is the present value obtained with the age-earnings profile method.

Evaluation Criteria and Empirical Results

Each of the approaches presented above will create an estimated present value, PV. The rate chosen for discounting the workers' actual earnings from 1971 through 1983 is the nontaxable municipal bond yield average from 1961 through 1970, 4.10 percent. (9) When applied to the actual earnings of these individuals, [Y.sub.t], an actual present value is created which may be expressed as:

[Mathematical Expression Omitted]

Several criteria will be used to evaluate the accuracy of these approaches. In addition to the commonly used root mean square error, the mean error and Theil's inequality coefficient will also be presented. (10) The mean error, ME, may be presented as:

ME = PV - PV

where PV is the mean actual present value and PV is the mean estimated present value.

The next measure to be presented is the root mean square error, RMSE, which equals:

[Mathematical Expression Omitted]

where [PV.sub.i] is the actual present value for individual i, [PV.sub.i] is the estimated present value for individual i, and n is the number of individuals.

The final criteria to be used for evaluating these approaches is Theil's inequality coeffiecient, U, which is expressed as:

[Mathematical Expression Omitted]

This expression has a range of 0 to 1. If U = 0, the approach is perfect since the actual present value will alway equal the estimated present value. If U = 1, there is no relationship between the actual and estimated present values. This expression may be decomposed into terms reflecting bias or systematic error in the model, [U.sup.M], and two other terms reflecting the variability in the data. A value for [U.sup.M] (the square of the ratio of ME to RMSE) above roughly 0.15 indicates a large bias in the model. [U.sup.M] may be written as:

[Mathematical Expression Omitted]

The results indicate support for the age-earnings profile adjustment. The offset method alone yielded: ME = $38,597; RMSE = $79,425; U = .214, and [U.sup.M] = .24. The offset method together with the age-earnings profile adjustment produced: ME = $10,080, RMSE = $70,562, U = .183, and [U.sup.M] = .02. These results support using the age-earnings profile adjustment when estimating earnings losses. It is noteworthy that the mean error for the offset approach is nearly four times as large as in the age-earnings adjustment approach. And the systematic bias of the offset approach is 12 times as large as in the age-earnings adjustment approach. The results also indicate that while both methods underestimate the loss, the offset method underestimates the loss by a considerably greater margin than does the age-earnings adjustment approach.

Conclusions

Although many methods have been proposed to deal with the estimation of economic damages, none have been rigorously tested empirically to determine their relative accuracy. This study is the first to use data on workers to compare estimation methods. A longitudinal data base such as the one found in the PSID should be used more often by economists to evaluate their proposals for the estimation of economic damages.

Using data from the PSID, this study finds that the offset approach, although easy to understand, is not the most accurate method available. Adjusting earnings for the impact of a worker's place in the life-earnings cycle will produce estimates of earnings losses that are more accurate.

The data for this study are 13 year individual earnings histories for the period 1971 through 1983. Because this period spans three full business cycles, the results are not likely to reflect the influence of a particular stage of the business cycle. Because a 13 year span is a relatively short earnings history, these data are potentially limited by the influence of a particular state of the earnings life cycle. However, this limitation is mitigated by the consideration that the age of the individuals in the sample are distributed over the interval 17 years to 52 years with a mean age of 42 years.

(1) See Franz (1978) for a discussion of the rationale for the offset approach.

(2) See Harmon and Lambrinos (1989) for an estimation of the bias introduced by taxes.

(3) Other approaches using various combinations of real and nominal wage growth rates, calculated from both industry and occupational data, along with real and nominal rates of discount, were also tried. The results do not differ appreciably from the offset results and are available from the authors upon request.

(4) See, for example, Miller (1965).

(5) The magnetic computer tape is available from the University of Michigan Survey Research Center, Ann Arbor, Michigan.

(6) Although 13 years may not be adequate to judge the true impact of the age-earnings profile adjustment, additional years of data are not available.

(7) Two other base periods were used with no appreciable change in the empirical results.

(8) The age-earnings profile is described in more detail in Lambrinos (1984b, 1985, 1989).

(9) The treasury bond rate, 4.35 percent, was also used without any change in the results.

(10) Pindyck and Rubinfeld (1981) discuss these criteria and others.

References

[1] Abraham, Fred J., 1988, Pitfalls to using the Real Rates or Age-Earnings Profile Models in Calculating Economic Loss, Journal of Forensic Economics, 1: 77-81.

[2] Bryan, William R. and Charles M. Linke, 1988, Estimating Present Values of Future Earnings: Experience with Dedicated Portfolios, Journal of Risks and Insurance, 55: 271-86.

[3] Bureau of the Census, 1983, Lifetime Earnings Estimates for Men and Women in the United States: 1979, Washington, D.C.: U.S. Government Printing Office.

[4] Economic Report of the President, 1988, Washington, D.C.: U.S. Government Printing Office.

[5] Franz, Wolfgang W., 1978, A Solution to Problems Arising from Inflation When Determining Damages, Journal of Risk and Insurance, 45: 323-33.

[6] Harmon, Oskar and James Lambrinos, 1989, Taxes and the Bias of the Offset Methods of Estimating Economic Damages, Unpublished Manuscript.

[7] Hosek, William R., 1982, Problems in the Use of Historical Data in Estimating Economic Loss in Wrongful Death and Injury Cases, Journal of Risk and Insurance, 49: 300-08.

[8] Lambrinos, James, 1984a, On the Use of Nominal v. Real Economic Variables in Forecasting Future Earnings Losses, Journal of the Missouri Bar, 40: 389-393.

[9] Lambrinos, James, 1984b, The Importance of Age in the Estimation of Economic Losses, Trial Lawyers Quarterly, 16: 48-52.

[10] Lambrinos, James, 1985, On the Use of Historical Data in the Estimation of Economic Losses, Journal of Risk and Insurance, 52: 464-76.

[11] Lambrinos, James, 1989, Maximizing Economic Loss Damages, New York: John Wiley and Sons, Inc.

[12] Miller, Herman P., 1965, "Lifetime Income and Economic Growth" American Economic Review, 55: 834-44.

[13] Pindyck, Robert S. and Daniel L. Rubinfeld, 1981, Econometric Methods and Economic Forecasts New York: McGraw-Hill Book Company.

James Lambrinos is Associate Professor and Director of the Graduate Management Institute at Union College in Schenectady, New York. Oskar R. Harmon is Assistant Professor of Economics at the University of Connecticut.

Printer friendly Cite/link Email Feedback | |

Author: | Lambrinos, James; Harmon, Oskar R. |
---|---|

Publication: | Journal of Risk and Insurance |

Date: | Dec 1, 1989 |

Words: | 2104 |

Previous Article: | The transactions cost theory of insurance: contracting impediments and costs. |

Next Article: | Use of financial futures by life insurers. |

Topics: |