Question

Demand estimation

Early in 1993, the Southeastern Transportation Authority (STA), a public agency responsible for serving the commuter rail transportation needs of a large Eastern city, was faced with rising operating deficits on its system. Also, because of a fiscal austerity program at both the federal and state levels, the hope of receiving additional subsidy support was slim. The board of directors of STA asked the system manager to explore alternatives to alleviate the financial plight of the system. The first suggestion made by the manager was to institute a major cutback in service. This cutback would result in no service after 7:00 pm, no service on weekends, and a reduced schedule of service during the midday period Monday through Friday. The board of STA indicated that this alternative was not likely to be politically acceptable and could only be considered as a last resort. The board suggested that because it had been over five years since the last basic fare increase, a fare increase from the current level of $1 to a new level of $1.50 should be considered. Accordingly, the board ordered the manager to conduct a study of the likely impact of this proposed fare hike. The system manager has collected data on important variables thought to have a significant impact on the demand for rides on STA. These data have been collected over the past 24 years and include the following variables. Price per ride (in cents) - This variable is designated P in Table 1. Price is expected to have a negative impact on the demand for rides on the system. Population in the metropolitan area serviced by STA - It is expected that this variable has a positive impact on the demand for rides on the System. This variable is designated T in Table 1. Disposable per capita income - This variable was initially thought to have a positive impact on the demand for rides on STA This variable is designated I in Table 1. Parking rate per hour in the downtown area (in cents) this variable is expected to have a positive impact on demand for rides on the STA. It is designated H in Table Below

Year	Weekly Riders (Y)	Price (P) Per	Population	Income (I)	Parking Rate
	(X1,000)	Ride (Cents)	(T)(x1,000)		(H) (Cents)
1966	1200	15	1800	2900	50
1967	1190	15	1790	3100	50
1968	1195	15	1780	3200	60
1969	1110	25	1778	3250	60
1970	1105	25	1750	3275	60
1971	1115	25	1740	3290	70
1972	1130	25	1725	4100	75
1973	1095	30	1725	4300	75
1974	1090	30	1720	4400	75
1975	1087	30	1705	4600	80
1976	1080	30	1710	4815	80
1977	1020	40	1700	5285	80
1978	1010	40	1695	5665	85
1979	1010	40	1695	5800	100
1980	1005	40	1690	5900	105
1981	995	40	1630	5915	105
1982	930	75	1640	6325	105
1983	915	75	1635	6500	110
1984	920	75	1630	6612	125
1985	940	75	1620	5883	130
1986	950	75	1615	7005	150
1987	910	100	1605	7234	155
1988	930	100	1590	7500	165
1989	933	100	1595	7600	175
1990	940	100	1590	7800	175
1991	948	100	1600	8000	190
1991	955	100	1610	8100	200

1. What is the Durbin-Watson statistic for this regression? What does this indicate about the presence of autocorrelation in the data?

2. Based on an analysis of the correlation matrix of the independent variables, what can you say about the presence of multicollinearity in the model?

Answer 1

1. To calculate d statistics, we need to regress the independent variables - price, parking rate, income and population on weekly rides (dependent variable).

Year	t	Y	Price	Population	Income	Parking Rate	et	et2	[e-e(t-1)]	[et-e(t-1)]2
1966	1	1200	15	1800	2900	50	18.12	328.51
1967	2	1190	15	1790	3100	50	24.22	586.63	6.10	37.16
1968	3	1195	15	1780	3200	60	22.51	506.51	-1.71	2.94
1969	4	1110	25	1778	3250	60	-42.33	1792.20	-64.84	4204.26
1970	5	1105	25	1750	3275	60	-22.71	515.67	19.63	385.18
1971	6	1115	25	1740	3290	70	-22.67	513.96	0.04	0.00
1972	7	1130	25	1725	4100	75	26.46	700.35	49.13	2414.24
1973	8	1095	30	1725	4300	75	7.39	54.54	-19.08	364.01
1974	9	1090	30	1720	4400	75	10.43	108.85	3.05	9.29
1975	10	1087	30	1705	4600	80	18.26	333.45	7.83	61.27
1976	11	1080	30	1710	4815	80	15.25	232.52	-3.01	9.07
1977	12	1020	40	1700	5285	80	-1.78	3.17	-17.03	290.01
1978	13	1010	40	1695	5665	85	-2.53	6.40	-0.75	0.56
1979	14	1010	40	1695	5800	100	-25.86	668.49	-23.33	544.09
1980	15	1005	40	1690	5900	105	-32.30	1043.42	-6.45	41.56
1981	16	995	40	1630	5915	105	8.99	80.89	41.30	1705.36
1982	17	930	75	1640	6325	105	9.16	83.92	0.17	0.03
1983	18	915	75	1635	6500	110	-4.42	19.54	-13.58	184.45
1984	19	920	75	1630	6612	125	-19.40	376.29	-14.98	224.33
1985	20	940	75	1620	5883	130	-28.29	800.49	-8.89	79.12
1986	21	950	75	1615	7005	150	-9.17	84.17	19.12	365.51
1987	22	910	100	1605	7234	155	-0.07	0.00	9.11	82.91
1988	23	930	100	1590	7500	165	23.79	565.76	23.85	569.06
1989	24	933	100	1595	7600	175	7.39	54.62	-16.40	268.81
1990	25	940	100	1590	7800	175	26.26	689.54	18.87	356.02
1991	26	948	100	1600	8000	190	4.96	24.63	-21.30	453.52
1991	27	955	100	1610	8100	200	-11.66	135.93	-16.62	276.30
Total								10310.48		12929.04

d = 12929.04 / 10310.48

d = 1.25

the durbin watson hypothesis is as follows-

H₀ : ρ = 0 i.e. no auto-correlation

H₁ : ρ > 0 i.e. positive auto-correlation

d = 1.25 < 2 implying positive auto-correlation means that the error terms are positively correlated.

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2. Correlation Matrix of Independent variable

	Price	Population	Income	Parking Rate
Price	1
Population	-0.93605	1
Income	0.934054	-0.96556	1
Parking Rate	0.957645	-0.93482	0.946939	1

All the independent variables are highly correlated.

the results of regression analysis shows the following-

	Coefficients	Standard Error	t Stat	P-value
Intercept	-299.12	464.47	-0.64	0.53
Price	-1.66	0.52	-3.16	0.00
Population	0.85	0.25	3.40	0.00
Income	-0.04	0.01	-3.32	0.00
Parking Rates	1.90	0.37	5.12	0.00

As can be seen from the table above, income has a negative impact on weekly rides. this is as against of what was expected. The results, thus indicate the presence of multi-collinearity.

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