Appendix
B

Analytical Approach and Econometric Results

For this analysis, the Congressional Budget Office (CBO) modeled preferred driving speeds at a given location as a function of the price of gasoline, time of day, month (time of year), average value of time, fixed physical characteristics of the freeway at that location (including grade, curvature, speed limit, and distance to nearest on- and off-ramps), and the overall demand for weekend travel in that month at that location. The time-of-day and time-of-year factors control for the effects of variation in the amount of daylight and in average weather conditions, as well as possible variations in the types of trips motorists make at different times of day or season. The average wage rate is a proxy for motorists’ value of time.¹

The travel demand term captures changes over time in average traffic density, or the median number of vehicles per day on weekends at each location, per month. That factor controls for the effect that overall traffic density, or proximity to other vehicles, might have on motorists’ preferred speeds even under relatively free-flowing weekend driving conditions. Thus, the model estimates the effect of gasoline prices on vehicle speeds independently of the effects of increased travel demand and other factors. The model also includes dummy variables for high and low outliers associated with imputed data and for speeds that are slow enough to indicate possible congestion. Finally, as a measure of data quality, the model includes the percentage of time the vehicle detection equipment was online that month, in case the measurements the equipment provides are correlated with the fraction of time that the equipment is functioning properly.

The data are organized as a panel, with each location–hour constituting a cross section. The main analysis examines the median (or 5th or 95th percentile) speeds for 11 one-hour periods of the day, observed over every Saturday and Sunday each month, at each of three locations. Thus the panel comprises 33 cross-sectional observations, with a time-step of 1 month and 48 months of observations. The percentile speed statistic for one cross section (summarizing observed speeds within a given hour of the day at a given location) might not be independent of that for another cross section (a different hour at the same location, or the same or a different hour at another location). CBO fit the data to an ordinary least-squares (OLS) model of the following form:

 

 

(1)

but computed panel-corrected standard errors (PCSE) û_it that allow for such a structure among the errors.² In Equation (1), the y_it term is the Qth percentile (for example, the median) vehicle speed on the weekend at location–hour i in month t. N is the number of location–hour cross sections (here, 33), T is the number of months observed (48), K is the number of exogenous regressors X in the model, and β is a vector of parameters to be estimated. The statistical significance of the fitted parameters depends on the PCSE term û_it, which contains the square roots of the diagonal terms in the following expression:

 

(2)

In Equation (2), is an NT × NT block-diagonal matrix formed from the panel structure of N cross sections and T time periods, with each block comprising an N × N matrix of terms of the form e_ite_jt, each term the product of the OLS residuals for cross sections i and j at time t:

 

(3)

The results indicate that the model fits the data reasonably well, with R2 values in excess of 0.5. Sample statistics are reported in Table B-1; results are reported in Table B-2.³

Table B-1. 

Sample Means and Vehicle Speeds, January 2003 to December 2006

		Mean		Standard Deviation		Minimum		Maximum
		Hour-Specific Monthly Percentiles (Analyzed Hours Only)
5th Percentile Speeda		62.8		8.3		8.7		78.1
Median Speeda		67.8		4.1		40.1		78.8
95th Percentile Speeda		70.8		3.3		53.9		79.1
Median Traffic Densityb		4.2		1.72		0.47		7.86
95th Percentile Traffic Densityb		5.06		1.75		0.74		8.99
		Other Continuous Variables
Real Retail Gasoline Price (Dollars per gallon)c		2.38		0.40		1.75		3.28
Real Wages (Dollars per hour)d		16.46		0.26		15.78		16.85
Daily Percent Uptime, Detector		86.1		18.6		12.5		100
		Indicator Variables
Month Effects		1/12		0.3		0		1
Early Morning (6–8 a.m., by route)		0.06		0.24		0		1
Prime I (9 a.m.–1 p.m., by route)		0.12		0.33		0		1
Prime II (2–6 p.m., by route)		0.06		0.24		0		1
Evening (7–9 p.m., by route)		0.06		0.24		0		1
Night (10 p.m.–midnight., by route)		0.03		0.17		0		1
		Congestion and Data Anomaly Indicators
5th Percentile Speed <55 mph		0.12		0.32		0		1
Median speed <60 mph		0.02		0.14		0		1
95th Percentile Speed <65 mph		0.02		0.14		0		1
Low-Speed Outlierse		0.07		0.27		0		1
High-Speed Outlierse		0.06		0.23		0		1

 

Source: Congressional Budget Office based on data from the Freeway Performance Measurement Project, https://pems.eecs.berkeley.edu.

Note: Hours analyzed are 6, 8, 9, and 11 a.m.; noon; and 1, 4, 5, 7, 8, and 10 p.m. Analysis results are not dependent on that specific set of hours.

a. Miles per hour.

b. Thousands of vehicles per hour.

c. Average monthly retail price for all grades and formulations, adjusted for inflation (base period January 2006). Data from the Department of Energy, Energy Information Administration.

d. Adjusted for inflation (base period January 2006). Data from the Department of Commerce, Bureau of Labor Statistics.

e. Denotes sustained periods of low- or high-speed anomalies in the data (congestion indicators capture brief, temporary slowdowns only).

Table B-2. 

Vehicle Speeds and Gasoline Prices, Primary Econometric Results

(Miles per hour)

		5th Percentile					Median					95th Percentile
		Speed		Std. Error			Speed		Std. Error			Speed		Std. Error
Intercept		85.2	**	17.4			55.3	**	16.3			61.6	**	14.0
Real Retail Gasoline Pricea		-0.024	**	0.005			-0.015	**	0.004			-0.001		0.004
Traffic Densityb		-0.33	*	0.15			-0.49	**	0.10			-0.14		0.10
Real Wages		-1.04		1.06			0.91		1.00			0.54		0.86
		(Time-of-Day × Route) Effects
		Early Morning (6–8 a.m.)
I-680, San Ramon		1.20		0.92			0.94		0.60			2.44	**	0.46
I-405, Westminster		3.61	**	0.66			2.39	**	0.56			3.77	**	0.60
I-8, San Diego		3.56	**	0.81			1.21	*	0.61			1.64	**	0.64
		Prime I (9 a.m.–1 p.m.)
I-680, San Ramon		-0.14		0.63			-0.46		0.33			1.08	*	0.45
I-405, Westminster		0.23		0.83			0.90		0.65			1.54	*	0.64
I-8, San Diego		2.79	**	0.82			1.65	**	0.59			1.27	*	0.59
		Prime II (2–6 p.m.)
I-680, San Ramon		(Omitted Factor)
I-405, Westminster		-0.85		1.14			0.13		0.71			0.95		0.64
I-8, San Diego		3.34	**	0.82			1.85	**	0.57			1.21	*	0.59
		Evening (7–9 p.m.)
I-680, San Ramon		0.35		0.59			0.45		0.24			0.02		0.30
I-405, Westminster		2.32	**	0.81			1.30	*	0.65			1.45	*	0.62
I-8, San Diego		2.94	**	0.81			0.99		0.57			0.48		0.64
		Night (10 p.m.–Midnight)
I-680, San Ramon		0.75		0.77			-0.15		0.47			0.19		0.43
I-405, Westminster		1.81	*	0.86			1.47	*	0.66			2.48	**	0.63
I-8, San Diego		1.78		0.94			-0.58		0.59			-0.03		0.66
		Significance
Month Effects		Jointly significant					Jointly significant					Not significant
Congestion, Outlier Flags			**					**					**
Detector Percent Online		0.02	*	0.01			0.02	*	0.01			0.003		0.009
R-squared		0.718					0.613					0.523

 

Source: Congressional Budget Office based on data from the Freeway Performance Measurement Project, https://pems.eecs.berkeley.edu.

Notes: Std. = standard; ** = significant at 1 percent; * = significant at 5 percent; I = Interstate.

Hours analyzed are 6, 8, 9, and 11 a.m.; noon; and 1, 4, 5, 7, 8, and 10 p.m. Results are very similar for an analysis of entirely different hours, for the exclusion of prime hours, or both. Panel structure is 33 cross-sections (3 locations × 11 hours) for 48 months.

a. Average monthly price for all grades and formulations.

b. Thousands of vehicles per hour, either median (middle column) or 95th percentile densities (outside columns) for a given time of day on the weekend, in the same month, at a given location. Panel structure is 33 cross-sections (3 locations × 11 hours) for 48 months.

Total Trips

The vehicle count data have a substantially different structure from the data on vehicle speeds, which are expressed as a distribution summarizing the speeds that were observed each month and the frequency of occurrence for each speed. In contrast, vehicle counts are observed directly and pertain to travel on a single day. The data consist of total daily vehicle counts at each of a dozen locations around California on every Wednesday, Saturday, and Sunday from April 2003 through December 2006.⁴ Because of differences in capacity, local population density, and patterns of travel demand, vehicle counts vary substantially more from one location to the next than do vehicle speeds. To pool the data into a panel, CBO expressed the vehicle counts as a fraction of the baseline mean vehicle count at each location, using nonholiday Wednesdays in January 2003 (several months before the period covered in the analysis) as the baseline.

With vehicle counts, there is less likelihood of contemporaneous correlation between cross sections than there is with the speed data, because no two cross-sectional (location–day) counts are collected on the same day at the same location. Also, as the analysis shows, counts at different locations respond slightly differently to changes in gasoline prices, depending partly on the ease of access to public transportation at each location.

Thus, CBO fit a one-way, fixed-effects model to the data, with a separate fixed effect for each location–day. The fixed effects control not only for day-of-week differences in volume of traffic at each location relative to the baseline but also for differences among locations in population density, proximity to residential or employment locations, and existence of alternative routes or modes of travel. As an alternative specification, a two-way, fixed-effects model would fit separate effects also for each week (183 weeks total). However, CBO takes the more parsimonious approach described below.

The analysis models the daily relative vehicle count at each location as a function of the price of gasoline, with separate price effects estimated for weekdays and weekends and for locations where rail transit either is or is not accessible. The model also allows for quadratic location-specific trends in vehicle totals, and it includes fixed effects for the month and indicators for holiday travel (estimated separately as effects for the first, middle, or final day of multiple-day holiday travel periods, as appropriate); summer weekends (which can feature above-average travel); nonholiday outliers; and, to control for differences in data quality, a continuous measure of the percentage of time each vehicle detector station was not operating and an indicator for zero percent observed (meaning that the observation in question was imputed entirely).

Up to the error term, the model for daily total vehicles has same the form as that used for Equation (1). Here, the error term has the following form:

 

(4)

 

where, as before, the ε_it are mean-zero, independent, identically distributed errors (although in this model, the cross-term covariances are assumed to be zero), and is the fixed-effect term for each location–day cross section i.

The model fits the data well, with R2 terms of nearly 0.9. A standard statistical F test strongly rejects the hypothesis of no route–day fixed effects, and a Hausman test strongly rejects an alternative specification with random effects as opposed to fixed effects.⁵ Sample statistics are reported in Table B-3; results are reported in Table B-4.

Table B-3. 

Sample Means for Total Daily Vehicles, April 2003 to December 2006

		Mean		Standard Deviation		Minimum		Maximum
Total Vehicles
	(Relative to baseline average, 1:1 = 100)	88.6		13.2		28.2		184.2
Real Average Weekly Retail Gasoline Price
	(Relative to baseline period; all grades, all formulations; 1:1 = 100)	111.2		18.6		79.4		152.5
Real Average Weekly Retail Gasoline Price
	(All grades, all formulations, dollars per gallon)	2.42		0.40		1.73		3.32
Daily Percent Uptime, Detector Station		81.5		34.8		0		100
		Indicator Variables
Detector Offline All Day		0.13		0.33		0		1
Day 1 of 3-Day Holidaya		0.05		0.22		0		1
Day 2 of 3-Day Holidayb		0.05		0.22		0		1
Day 3 of 3-Day Holidayc		0.02		0.12		0		1
Veterans Day Weekendd		0.02		0.15		0		1
Summer Weekende		0.22		0.41		0		1

 

Source: Congressional Budget Office based on data from the Freeway Performance Measurement Project, https://pems.eecs.berkeley.edu.

Note: n = 6,558

a. Includes Wednesdays on the eve of Thanksgiving, Christmas, and New Year’s Day. (There were no Wednesday holidays from April 2003 to December 2006.)

b. Includes Saturday of Thanksgiving week.

c. Includes Sunday of Thanksgiving week.

d. Veterans Day is a fixed-date holiday that was observed on a Friday in 2005 and 2006.

e. May 23 to September 6, the dates of the earliest Memorial Day Saturday and latest Labor Day Sunday from 2003 to 2006. Weekends constitute two-thirds of the data; one-third of the data fall between May 23 and September 6 because the data that CBO analyzed date from April 2003.

Table B-4. 

Total Trips and Gasoline Prices, Primary Econometric Results

		Dependent Variable
		Total Daily		Standard
		Vehiclesa		Error		Net Effect		F Test		(p >F)
Intercept		94.9	**	1.08
Real Relative Price of Gasolineb		-0.0002		0.011				0.00		0.99
Price × Weekend		0.0003		0.009		0.0001		0.00		0.99
Price × Rail		-0.034	**	0.013		-0.034		12.42	**	0.004
Price × Weekend × Rail		0.044	**	0.012		0.01		1.33		0.25
Summer Weekends		0.475		0.22
Day 1, Holiday Period		-2.65	**	0.26
Day 2, Holiday Period		-3.16	**	0.27
Day 3, holiday period		-3.02	**	0.48
Veterans Day Weekend		3.19	**	0.44		0.03 (Sat), 0.17 (Sun)
Percent Uptime		0.024	**	0.004
		Significance
Location Trends		Jointly significant						5.95	**	0.0001
Location Squared Trends		Jointly significant						14.78	**	0.0001
Month Effects		Jointly significant						27.49	**	0.0001
Cross-Section Effects		All significant at 1 percent

 

Source: Congressional Budget Office based on data from the Freeway Performance Measurement Project, https://pems.eecs.berkeley.edu.

Notes: p > F = statistical significance; **= significant at 1 percent; Sat = Saturday; Sun = Sunday.

Exclusion of squared-trend terms yields similar results. Panel structure is 36 cross sections (12 locations × 3 days) for 183 weeks.

a. Relative to baseline (1:1 = 100).

b. Average weekly retail price, all grades and formulations (relative to 2003 baseline price).

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1

Using the average gross hourly wage in the analysis yields the same outcome as would using the hourly wage, net of taxes (because the analysis is not done at the level of the individual motorist). Both are statewide values that vary only over time, at approximately the same rate: The effect of any changes in marginal tax rates on the average net hourly wage would be very small.

2

See Nathaniel Beck and Jonathan N. Katz, "What to Do (and Not to Do) with Time-Series Cross-Section Data," American Political Science Review, vol. 89 (September 1995), pp. 634–647.

3

CBO also analyzed the data using the Parks method, a related but older and now less common technique that produces unbiased parameter estimates but that can underestimate the standard errors (thus overestimating the precision of the estimates). See Richard Parks, "Efficient Estimation of a System of Regression Equations When Disturbances Are Both Serially and Contemporaneously Correlated," Journal of the American Statistical Association, vol.62 (1967), pp. 500–509. In CBO’s analysis, the Parks method yielded qualitatively similar results.

4

Earlier data can be analyzed for most routes, but at the cost of excluding, for computational reasons, routes lacking any earlier data. However, results are generally unchanged if such routes are excluded from the analysis.

5

See J.A. Hausman, "Specification Tests in Econometrics,"Econometrica, vol. 46, no. 6 (1978), pp. 1251–1271.

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