National Center for Statistics and Analysis 400 Seventh St., S.W., Washington, D.C. 20590

March 2004 DOT HS 809 718

Time Series Analysis and Forecast of Crash 

Fatalities during Six Holiday Periods

Cejun Liu* and Chou-Lin Chen

Traffic Safety Facts

Research Note

Summary

This research note uses two Time Series techniques, HoltWinters (HW) Algorithm and Autoregressive Moving 

Average Model (ARMA), to predict motor vehicle crash 

fatalities during six holiday periods. Based on the data from 

1983 to 2001, the estimated fatalities in 2002 are: 564 and 564 

(New Year), 501 and 514 (Memorial), 748 and 755 (4th of July), 

486 and 498 (Labor), 572 and 565 (Thanksgiving), and 156 

and 145 (Christmas). Incorporating the actual 2002 fatality 

counts in the data series, the forecasts for the 2003 holiday 

periods are: 141 and 141 (New Year), 508 and 504 (Memorial), 

537 and 564 (4th of July), 526 and 545 (Labor), 547 and 564 

(Thanksgiving), and 548 and 576 (Christmas).

1. Introduction

Generally, there are higher fatality rates during holiday 

periods than during non-holiday periods. In 2001 and 2002, 

the overall average fatalities were 116 and 117 per day respectively. In comparison, the average fatalities during six 

major holiday periods were 153 and 156 per day, respectively 

[1, 2]). Analysis and forecasting of the fatality rates during 

holiday periods are useful for providing warnings that may 

reduce fatalities. In this research note, time series techniques 

are employed to analyze the fatality data during six holiday 

periods. The fatalities in 2002 are forecasted using the data 

from 1982 to 2001 and then compared with the actual observations in 2002. Fatalities in 2003 during six holiday 

periods are also forecasted when the observations in 2002 

are included in the analysis. Two forecasting techniques are 

used: Holt-Winters algorithm and ARMA (autoregressive 

moving average) models. Data from the Fatality Analysis 

Reporting System (FARS) were used. The FARS database is 

a national census of police-reported motor vehicle crashes 

resulting in fatal injuries. It is conducted by the National 

Center for Statistics and Analysis (NCSA) in the National 

Highway Traffic Safety Administration (NHTSA). 

2. Methodologies

In the current study, the extrapolation method will be used. 

That is, the forecast is based on an inferred study of past 

general data behavior over time (time series).

A time series model for the observed data {x(t)} is a specification of a sequence of random variables {X(t)} of which {x(t)} 

is postulated to be a realization. In this work, the stationary 

time series model is an appropriate model to be used to perform the analysis and forecast. Definitions and properties of 

stationary time series models can be found in Appendix 4.1. 

2.1. The Holt-Winters (HW) Algorithm 

The Holt-Winters algorithm is an effective forecasting technique that has less emphasis on the construction of a model 

for the time series data. In this process, two smoothing parameters, α and β with α, β ∈ [0, 1] are needed. They can be 

fixed or be chosen in a way to minimize the sum of squares 

of the one-step errors. See Appendix 4.2 for description of 

this technique.

2.2. ARMA Models

The family of ARMA processes plays a key role in the modeling of time series. Appendix 4.3 shows us the definition 

and some properties of ARMA (p, q) process. In this study, 
National Center for Statistics and Analysis 400 Seventh St., S.W., Washington, D.C. 20590 

Table 1 

Number of Killed Persons during New Year, Memorial, and Fourth of July Holiday Periods and Forecast 

Values in 2002 and 2003

New Year Memorial Fourth of July

 Year Number (Days) One Day Number (Days) Number (Days) One Day

 1982 NA NA 498(3) 600(3) 200

 1983 375(3) 125 539(3) 620(3) 207 

 1984 346(3) 116 527(3) 223(1) 223 

 1985 496(4) 124 557(3) 689(4) 173 

 1986 223(1) 223 616(3) 611(3) 204 

 1987 535(4) 134 519(3) 556(3) 186 

 1988 407(3) 136 529(3) 631(3) 211 

 1989 443(3) 148 594(3) 748(4) 187 

 1990 421(3) 141 589(3) 268(1) 268 

 1991 441(4) 111 533(3) 718(4) 180 

 1992 164(1) 164 438(3) 535(3) 179 

 1993 370(3) 124 454(3) 525(3) 175 

 1994 372(3) 124 482(3) 519(3) 173 

 1995 392(3) 131 483(3) 661(4) 166 

 1996 420(3) 140 514(3) 627(4) 157 

 1997 190(1) 190 511(3) 508(3) 170 

 1998 545(4) 137 393(3) 479(3) 160 

 1999 354(3) 118 500(3) 509(3) 170 

 2000 469(3) 157 466(3) 717(4) 180 

 2001 357(3) 119 515(3) 206(1) 206 

 2002 570(4) 143 491(3) 683(4) 171

 HW 564(4) 141 501(3) 748(4) 188 

 2002 564(4) 141 514(3) 755(4) 189

 Forecast ARMA 95% C.L. 95%C.L. 95% C.L. 95% C.L. 95%C.L.

 (352,772) (88,193) (421,607) (556, 956) (139, 239)

 HW 141(1) 141 508(3) 537(3) 179

 2003 141(1) 141 504(3) 564(3) 188 

 Forecast ARMA 95%C.L. 95%C.L. 95% C.L. 95% C.L. 95%C.L.

 (89, 192) (89, 192) (413, 595) (417, 711) (139, 237)

Source: NCSA FARS 1982-2002

visualizations of six time series are shown to be stationary 

and confirmed by statistical criteria. The ARMA models are 

employed for the analysis. 

3. Results 

3.1. Data Manipulation

Fatalities during six holiday periods over 1982-2002 are listed in Tables 1 and 2. For the fatalities, i.e. New Year, Fourth 

of July and Christmas, which have different numbers of 

whole days during the holiday period, we first obtain the 

average number of fatalities for each day of the period, and 

used these numbers to perform a time series analysis and 

forecast, in the end, multiplied by whole days of the holiday period to get the total fatalities for each holiday period. 

Figure 1 shows the time series of the killed persons during 

six holiday periods over 1982-2001. From Figure 1, we can see 

there is a weak downtrend in Memorial Day and Labor Day 

series. Indeed, different ARMA models for those two data 

series from four other holidays’ series can be found.
National Center for Statistics and Analysis 400 Seventh St., S.W., Washington, D.C. 20590 

Table 2

Number of Killed Persons during Labor, Thanksgiving, and Christmas Holiday

Periods and Forecast Values in 2002 and 2003 

 Labor Thanksgiving Christmas

 Year Number (Days) Number (Days) Number (Days) One Day

 1982 628(3) 601(4) 458(3) 153 

 1983 636(3) 533(4) 352(3) 118 

 1984 609(3) 558(4) 643(4) 161 

 1985 605(3) 566(4) 152(1) 152 

 1986 663(3) 598(4) 508(4) 127 

 1987 630(3) 659(4) 409(3) 137 

 1988 592(3) 601(4) 511(3) 171 

 1989 588(3) 561(4) 553(3) 185 

 1990 599(3) 563(4) 567(4) 142 

 1991 577(3) 546(4) 135(1) 135 

 1992 460(3) 403(4) 410(3) 137 

 1993 522(3) 569(4) 402(3) 134 

 1994 494(3) 575(4) 455(3) 152 

 1995 511(3) 527(4) 358(3) 120 

 1996 525(3) 579(4) 166(1) 166 

 1997 507(3) 571(4) 480(4) 120 

 1998 464(3) 602(4) 364(3) 122 

 1999 485(3) 581(4) 485(3) 162 

 2000 529(3) 509(4) 442(3) 148 

 2001 482(3) 585(4) 601(4) 151 

 2002 541(3) 543(4) 130(1) 130

 HW 486(3) 572(4) 156(1) 156

 2002 498(3) 565(4) 145(1) 145

 Forecast ARMA 95% C.L. 95% C.L. 95% C.L. 95% C.L.

 (421, 575) (470, 660) (109, 181) (109, 181)

 HW 526(3) 547(4) 548(4) 137

 2003 545(3) 564(4) 576(4) 144 

 Forecast ARMA 95% C.L. 95% C.L. 95% C.L. 95% C.L.

 (467, 623) (470, 657) (436, 720) (109, 180)

Source: NCSA FARS 1982-2002
National Center for Statistics and Analysis 400 Seventh St., S.W., Washington, D.C. 20590

Figure 1: 

Fatalities during Six Holiday Periods over 1982-2001

 

Source: NCSA FARS 1982-2001

3.2. Forecasts by HW Algorithm

Using 1982-2001 data, the Holt-Winters algorithm is implemented to predict 2002 fatalities. Forecast values in 2003 are 

also obtained when actual observations in 2002 are included 

in the time series. Table 3 shows detailed information for 

the two optimized smoothing parameters α and β in the 

exponential smoothing recursive process and the forecast 

values by the Holt-Winters algorithm. For 2002 fatalities, the 

Holt-Winters algorithm performs forecasts very well from 

the comparisons of the actual observations with the forecast 

values listed in Tables 1 and 2.

3.3. Forecasts by ARMA Model

Sample autocorrelation function (ACF) and sample partial 

auto-correlation function (PACF), preliminary and maximum likelihood estimation procedures with minimum 

AICC criteria are used to find the best model for each data 

series. The ARMA models and the forecast values in 2002 

and 2003 are shown as follows (mean-corrected). For the

forecast value, the 95% confidence limit (C.L.) can be constructed assuming approximate normality in terms of meansquared error (MSE) (i.e. C.L = measure ± 1.96 √ MSE). For 

350

300

250

200

150

100

50

1980 1985 1990 1995 2000 2005

Fatal Number

750

600

450

300

1980 1985 1990 1995 2000 2005

Fatal Number

400

300

200

100

1980 1985 1990 1995 2000 2005

Fatal Number

Year Year Year

800

700

600

500

400

300

1980 1985 1990 1995 2000 2005

Fatal Number

Year

800

700

600

500

400

300

1980 1985 1990 1995 2000 2005

Fatal Number

Year

300

250

200

150

100

50

1980 1985 1990 1995 2000 2005

Fatal Number

Year

New Year

Labor

Memorial

Thanksgiving

Fouth of July

Christmas

Table 3: Forecasted Fatalities of Six Holidays in 2002 

and 2003 by Holt-Winters Algorithm. 

 Holiday α β Forecast

New Year 2002 0.24 0.27 140.15

 2003 0.24 0.27 140.50

Memorial 2002 0.30 0.78 500.30

 2003 0.30 0.78 507.08

Fourth 2002 0.46 0.14 187.08

of July 2003 0.42 0.16 178.97

Labor 2002 0.53 0.20 485.85

 2003 0.39 0.50 525.34

Thanks- 2002 0.76 0.24 571.08

Giving 2003 0.75 0.25 546.66

Christmas 2002 0.64 0.40 155.70

 2003 0.65 0.38 136.14

most cases, the value forecasted by ARMA model is a little 

bit higher than the one obtained by Holt-Winters algorithm. 
National Center for Statistics and Analysis 400 Seventh St., S.W., Washington, D.C. 20590 

New Year

2002: 

MA (0): X (t) = Z(t), Z(t) ~ WN(0, 726.094). 

Forecast: 140.106, √ MSE = 26.946.

2003:

MA (0): X(t) = Z(t), Z(t) ~ WN(0, 690.188). 

Forecast: 140.25, √ MSE = 26.271.

Memorial

2002:

AR (1): X(t) = 0.4005 X(t-1) + Z(t), 

 Z(t) ~ WN(0, 2243.63).

AICC = 215.955. 

Forecast: 513.711, √ MSE = 47.367.

2003:

AR (1): X(t) = 0.4003 X(t-1) + Z(t), 

 Z(t) ~ WN(0, 2160.137).

AICC = 225.673. 

Forecast: 503.48, √ MSE = 46.677.

Fourth of July

2002:

MA (0): X(t) = Z(t), Z(t) ~ WN(0, 646.887).

Forecast: 188.75, √ MSE = 25.434.

2003:

MA (0): X(t) = Z(t), Z(t) ~ WN(0, 630.37).

Forecast: 187.905, √ MSE = 25.11.

Labor

2002:

AR (1): X(t) = 0.7861 X(t-1) + Z(t), 

 Z(t) ~ WN(0, 1559.86).

AICC = 209.473. 

Forecast: 497.6784, √ MSE = 39.495.

2003:

AR (1): X(t) = 0.7493 X(t-1) + Z(t), 

 Z(t) ~ WN(0, 1580.429).

AICC = 219.761. 

Forecast: 544.414, √ MSE = 39.755.

Thanksgiving

2002: 

MA (0): X(t) = Z(t), Z(t) ~ WN(0, 2358.03).

Forecast: 564.35, √ MSE = 48.56.

2003:

MA (0): X(t) = Z(t), Z(t) ~ WN(0, 2266.41).

Forecast: 563.33, √ MSE = 47.61.

Christmas

2002: 

MA (0): X(t) = Z(t), Z(t) ~ WN(0, 326.64).

Forecast: 144.65, √ MSE = 18.24. 

2003:

MA (0): X(t) = Z(t), Z(t) ~ WN(0, 326.71).

Forecast: 143.95, √ MSE = 18.08.

We can also see that those models perform forecasts pretty 

well in 2002 as compared to the actual observations.

4. Appendix

4.1. Stationary Time Series

Loosely speaking, a time series model for the observed data 

{x(t)} is a specification of a sequence of random variables 

{X(t)} of which {x(t)} is postulated to be a realization. A time 

series {X(t), t =0, ±1, ...} is said to be stationary if it has statistical properties similar to those of the “time-shifted” series 

{X(t+h), t=0, ±1, ...} for each integer h.

Two simple but very useful stationary models are IID (independently and identically distributed) noise and White 

noise. For IID noise, random variables X(t) are mean 0 and 

variance σ2 ( = E[X(t)2] ), specified as {X(t)}∼IID (0, σ2). If {X(t)} 

is a sequence of uncorrelated random variables, each with 

mean 0 and variance σ2, then it is referred to as white noise, 

specified as {X(t)}∼WN (0, σ2). Every IID (0, σ2) sequence is 

WN (0, σ2) but not conversely. 

For a time series {X(t)}, sample autocorrelation function 

(ACF) and sample partial auto-correlation function (PACF) 

are used in choosing an appropriate model to the observed 

time series.

4.2. Holt-Winters (HW) Algorithm

The Holt-Winters algorithm is an effective forecast technique that has less emphasis on the construction of a model 

for the time series. Giving time series {X(t)}, t=1,...,n from the 

following classical decomposition model 

 X(t) = m(t) + s(t) + Y(t), t=1,...n, (1)

where m(t) is a trend component, s(t) is a seasonal component with known period d (i.e. s(t+d)=s(t) and ∑j=1 s(j)=0)

and Y(t) is a random noise component which is stationary 

with E(Y(t))=0. The estimated component m(t) and s(t) at 

times t=1, 2,....., n can be computed in terms of exponential 

smoothing recursions schemes. In this process, two smoothing parameters, α and β with α, β ∈ [0, 1] are needed (no 

seasonal component ). Here, they are chosen in a way to 

minimize the sum of squares of the one-step errors ∑j=3

d

n
National Center for Statistics and Analysis 400 Seventh St., S.W., Washington, D.C. 20590 

(X(j)-P j-1 X(j))2, Pj is predictor operator. Details of the HW algorithm can be found in references [3-6]. The HW algorithm 

is one of the best-known forecasting techniques in time series theories [7].

4.3. ARMA Models

A stationary time series {X(t)} is called an ARMA(p, q) process if for every t 

X(t) = φ1X(t-1) +...+φp X(t-p) + 

 + Z(t) + θ1Z(t-1) + ...+ θ

q

 Z(t-q), (2)

where {Z(t)} ~ WN(0, σ2). {X(t)} is said to be an ARMA (p, q) 

model with mean μ if {X(t)-μ} is an ARMA (p, q) process defined by Eq.(2). A stationary solution {X(t)} of the Eq.(2) exists 

if and only if φ(z)=1- φ1z - ... - φp zp ≠ 0 , ∀ |z| =1. 

For pure autoregressive (AR) models, the Yule-Walker algorithm is used to implement the preliminary estimation of 

the models. For pure moving average (MA) models or mixed 

ARMA models, the Innovations algorithm is used to implement the preliminary estimation of the models. Final decisions with respect to order selection of the models are made 

on the basis of the Maximum Likelihood Estimator and the 

minimum AICC (bias-Corrected Information Criterion of 

Akaike) criterion. 

Once a well-fitted model for a time series is obtained, it can 

then be employed to predict X (n+h) (h>0) with known mean 

and auto-covariance function in terms of the values {X(t)}, 

t=1,...,n. Refer to [3-6] for details. ITSM and SAS are used in 

the calculations.

In this work, the stationarity issue of the data series can be 

confirmed by sample ACF and sample PACF of each data series and hence the ARMA models are used for the analysis. 

For non-stationary process, ARIMA (auto-regressive integrat

ed moving average) models should be used. For the data series 

in this study, there is no evidence to use ARIMA process.

5. References

[1] National Highway Traffic Safety Administration. Traffic 

Safety Facts 2001. Washington, DC: US Department 

of Transportation.

[2] 2002 Annual Assessment. National Center for Statistics 

and Analysis, NHTSA, http://www-nrd.nhtsa.dot.gov/

pdf/nrd-30/NCSA/Rpts/2003/Assess02.pdf

[3] D.C. Montgomery, L.A. Johnson and J.S. Gardiner, 

Forecasting and Time Series Analysis, 2nd, McGraw-Hill, 

Inc., New York (1990).

[4] W.S. Wei, Time Series Analysis, Addison-Wesley Publishing 

Company, Inc. New York (1990).

[5] P.J. Brockwell and R.A. Davis, Introduction to Time Series 

and Forecasting, New York: Springer (1996).

[6] H.Arsham, Time Series Analysis and Forecasting Techniques.

http://ubmail.ubalt.edu/~harsham.

[7] L. Richard, How Should Additive Holt-Winters Estimates be 

Corrected? International Journal of Forecasting, 14: 393 

(1998). 

*Cejun Liu is a Program Analyst employed by Rainbow Technology Inc., a contractor working for the Mathematical Analysis Division, National Center 

for Statistics and Analysis, NHTSA.

Chou-Lin Chen is a Mathematical Statistician and team leader in the Mathematical Analysis Division, National Center for Statistics and Analysis, 

NHTSA. 

Very useful suggestions and comments from Santokh Singh and Dennis Utter at NCSA, other reviewers at NHTSA and helpful communication with 

Professor, Dr. Peter J. Brockwell are greatly appreciated.

For additional copies of this research note, please 

call 1-800-934-8517 or fax your request to (202) 366-

3189. For questions regarding the data reported 

in this research, contact Cejun Liu [202-366-5354] 

or Chou-Lin Chen [202-366-1048]. Internet users 

may access this research note and other general 

information on highway traffic safety at: http:/

/www-nrd.nhtsa.dot.gov/departments/nrd-30/

ncsa/AvalInf.html