University of Iowa
I congratulate the authors for a very careful, statistically sophisticated, and impartial study on the impact of recent speed-limit changes on fatal interstate crashes. The findings have important policy implications as states face considerable pressure to increase maximum speed limits.
Any assessment of the impact of maximum speed-limit changes on traffic safety is difficult, and many reports have been written on this subject. The Balkin/Ord paper is an important contribution to this literature since it is comprehensive and current, covering all 50 states through 1998. Its findings are 1) the 1987 speed-limit change increased fatal accidents on rural interstates by about 200 crashes each year and the 1996 change added another 200 fatal crashes annually, for a combined total of 400 fatal crashes per year and 2) the impact of the 1996 change on the number of fatal accidents on urban interstates was not as strong, amounting to about 80 fatal crashes per year. The benefits of accelerated interstate travel come at the expense of safety though not all states are affected equally.
My comments on this paper have two purposes. First, the structural time series models which Balkin/Ord uses for characterizing the serial correlation among successive observations may not be familiar to readers of this journal. My comments address the relationship between these models and the more familiar Box-Jenkins ARIMA time series intervention models. Second, the Balkin/Ord study deals with a nationwide analysis of data from all 50 states. It is understandable that such an analysis can not be as detailed as studies that focus on specific states. My comments offer recommendations for model improvements, in particular suggestions for incorporating traffic volume and for using actual travel speeds instead of speed-change indicators.
Balkin/Ord uses structural time series intervention models for assessing the impact of maximum-speed-limit changes. This differs from other studies that use the ARIMA time series intervention models proposed by Box and Tiao (1975). The following discussion illustrates that these two model families (structural time series intervention models and ARIMA time series intervention models) are closely related. Let us ignore, without loss of generality, the seasonal component in the structural model in equation (4),
After taking successive differences, where B is the backshift operator we obtain
The noise follows a first order moving average process.1 The solution of the above difference equation is
This is like a regression of the number of fatalities on the indicator variable zt with one important difference. The errors are no longer independent and follow an integrated first order moving average [ARIMA(0,1,1)] model, a very common, nonstationary time series model. Box and Tiao (1975) studies the estimate of the intervention effect under this particular model and show that it is a contrast between two weighted averages, one of observations before the intervening event and the other of observations afterwards. The weights are symmetric and decay exponentially according to the time distance of the observations from the intervening event. The rate of decay depends on the moving average parameter or, in terms of the parameters in the structural model, on the ratio of the variances of and . Note that this estimate differs from the ordinary regression estimate in the model with independent errors, which is the difference of two (unweighted) averages; the observations before and after the intervening event are weighted equally. The equal weighting is inappropriate if observations are autocorrelated, and the analysis must be adjusted for the serial correlation in the observations. This adjustment can be achieved through ARIMA models as proposed by Box/Tiao or through the structural time series approach adopted by Balkin/Ord. Both approaches should lead to similar conclusions.
Balkin/Ord analyzes the number of traffic fatalities but fails to incorporate in its model a measure of risk exposure. Risk exposure can be measured through vehicle-miles traveled (VMT). Most states do have reliable estimates of traffic volume, typically obtained by sampling traffic flow at various continuous measurement stations spread throughout the state. An analysis of the ratio, the number of fatal accidents divided by VMT, is more meaningful since it adjusts accident numbers for traffic volume. The decreasing time trend in these ratios expresses the safety improvements of cars and roads.
The 1987 maximum-speed-limit increase was uniform across states. About one fifth of the states, mostly in the East, decided not to raise the maximum speed limit on rural interstates. The other states increased the maximum speed limit on rural interstates by 10 miles per hour (mph), from 55 mph to 65 mph. The response to the National Highway System Designation Act in 1995 was more varied. Appendix A of the National Highway Traffic Safety Administration Report to Congress (USDOT NHTSA 1998) gives a good overview of how states responded. About one fifth of the states did not increase the speed limit on rural interstates beyond the prior 65 mph limit. Most states raised the speed limits on rural interstates by 5 mph, to 70 mph. However, several other, mostly western, states raised it by 10 mph, to 75 mph. The fact that not all speed-limit changes were of the same magnitude was not incorporated in the analysis. The use of a single intervention dummy variable appears to be an oversimplification; it may have been better to incorporate the magnitude of the change.
Balkin/Ord models the speed-limit changes through 0/1 indicator variables. The model could be improved by including the actually traveled speeds. Admittedly, it would be difficult to obtain reliable traffic-speed data, certainly more difficult than obtaining reliable VMT data. However, many states do collect data on traffic speed. Iowa, for example, takes 24-hour measurements on 1 day each quarter at 4 rural interstate and 2 urban interstate stations. In earlier papers, Ledolter and Chan (1994; 1996) analyzed average traffic speeds, as well as the proportion of cars exceeding 55, 60, and 65 mph. Iowa had increased the maximum speed limit on its rural interstates from 55 mph to 65 mph in May of 1987. The average traffic speed on rural interstates did not change abruptly but increased gradually from about 59 mph in 1985/1986 to about 66 mph in 1990/1991. Our study shows that traffic speed does not change abruptly with the passing of a new rule but adapts gradually over a period of several quarters. The average actual travel speed (or a certain percentile of the distribution) is more indicative of driver behavior than the posted change in the maximum speed limit, making models that incorporate the actually traveled speeds preferable.
Several states also raised the maximum speed limit on rural primary roads. The safety impact of this speed-limit change is not reflected in the Balkin/Ord study since their analysis focuses solely on interstates. Its impact is found among the numbers of fatal accidents on the rural primary system. The impact may be substantial since most fatal accidents occur on rural noninterstates.2 An investigation of the number of fatal accidents on rural primary roads, especially for those states that raised the speed limits on these road systems, is needed. Furthermore, speed increases may carry over to road systems not subject to the increased speed-limits. In our earlier analyses of the 1987 Iowa speed limit change on rural interstates, Ledolter and Chan (1994) and Ledolter and Chan (1996), we found small increases in the average actual travel speed on road systems that remained subject to the 55 mph limit. Hence, studies on the number of noninterstate fatal crashes for states that raised maximum speed limits on rural interstates but not on rural primary roads are also needed.
A brief comment on seasonality: for purposes of impact assessment, seasonality is a nuisance variable that needs to be excluded; seasonality by itself is of little interest. The number of fatal accidents is seasonal because traffic volume is seasonal. In addition, the number of fatal accidents per (million) VMT is seasonal because the risk of getting into an accident depends on seasonal weather and road conditions. Many studies model seasonality by including seasonal indicator variables or trigonometric functions (harmonics) of the seasonal frequency. The structural noise model in the Balkin/Ord paper goes one step further and allows for slowly changing seasonal components. This could have also been achieved by incorporating seasonal ARIMA components into a time series intervention model. Balkin/Ord's figure 1 shows that the seasonal fluctuations for Arizona are reasonably stable over time, indicating that a model with constant seasonal indicators would have been equally appropriate.
The Balkin/Ord paper analyzes aggregate data on the number of fatal interstate accidents. It does not use information to address if speed-related circumstances contributed to the accident, nor does it use such factors such as gender and age of the driver, road and weather conditions at the time of the accident, type of vehicle involved, and evidence of alcohol involvement. Aggregate analyses are always subject to the criticism that important factors have been overlooked. A follow-up analysis of individual fatal accidents would be worthwhile. Microdata on each fatal accident, with detailed information on various contributing factors, including whether speed was a contributing factor, is available for each accident. Admittedly, such analysis for 50 states would be an almost impossible task, and this comment is not a criticism of the Balkin/Ord study. Instead, it is offered as a recommendation that additional studies confirm the findings at the aggregate level with detailed analyses at the micro-level, at least for a few selected states.
Abraham, B. and J. Ledolter. 1983. Statistical Methods for Forecasting. New York: Wiley.
Box, G.E.P., G.M. Jenkins, and G.C. Reinsel. 1994. Time Series Analysis Forecasting and Control. Upper Saddle River, NJ: Prentice-Hall.
Box, G.E.P. and G.C. Tiao. 1975. Intervention Analysis with Applications to Economic and Environmental Problems. Journal of the American Statistical Association 70:70-9.
Ledolter, J. and K. Chan. 1994. Safety Impact of the Increased 65 mph Speed Limit on Iowa Rural Interstates. Final Report. Midwest Transportation Center, University of Iowa.
_____. 1996. Evaluating the Impact of the 65 mph Maximum Speed Limit on Iowa Rural Interstates. The American Statistician 50:79-95.
U.S. Department of Transportation (USDOT), National Highway Traffic Safety Administration (NHTSA). 1998. Report to Congress: The Effect of Increased Speed Limits in the Post-NMSL Era. Washington, DC.
Johannes Ledolter is the John F. Murray Research Professor of Management Sciences at the University of Iowa and Professor of Statistics at the Vienna University of Economics and Business Administration in Austria. He is a Fellow of the American Statistical Association and an elected member of the International Statistical Institute. Address: Department of Management Sciences, Tippie College of Business, University of Iowa, Iowa City, IA 52242. Email: email@example.com.