Model Specification And Data In Applied Economics Business Essay



In the study of Goodwin et al. (2004), long-run elasticities are proved to be 2 to 3 times greater than short-run elasticities. Keeping this interesting finding in mind, we are motivated to examine econometric relationship between car travel demand and its main determinants with a view towards determining short- and long-term elasticity of car travel demand in the UK for the period 1987-2009. Particularly fuel price is employed as a central explanatory variable. In this report, we would like to implement co-integration technique and error correction model in order to estimate the related elasticities. Additionally, a forecasting is performed to observe the corresponding change of car travel demand resulted from a rise of fuel prices for the years 2010-2020 on the basis of the sample data from 1987-2009.

Model specification and data

Some studies have been done on the topic in recent year. Odeck and Brathen (2007) utilises time-series data to estimate the short- and long-run elasticities of travel demand with respect to tolls in the case of Norway. Dargay (2007) applies pseudo-panel methodology to analyse the factors that have substantial impact on household car travel in the UK. Taking a cue from the previous literature, we formulate our econometric model as follow:

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Where dt is car travel demand; ft is fuel prices; and pt is car purchase costs. The elasticity coefficients β1 and β2 are long-run fuel price and purchase cost elasticity, respectively. They are obtained through co-integration analysis. The short-run elasticities are provided by the error correction model.

Having summarising those earlier reviews carried out by some researchers, Goodwin et al. (2002) presents that the elasticities of vehicle kilometres with respect to fuel price lie in the range -0.15 to -0.26 in short term while it is in the region of -0.3 to -0.58 in long run. More specifically, de Jong and Gunn (2001) offer evidence on fuel price elasticities of car travel across the EU. They find that the short-run fuel price elasticity is -0.16 for car-km and it is -0.26 in the long run, which is "broadly consistent with values quoted in previous surveys by Graham and Glaister (2002a) and Goodwin (1992)" (Graham and Glaister, 2004, p.262). These results seem plausible from the perspective of the conventional economic theory as a rise in fuel prices should generally deter people from using transports. The sign of β1 is thus expected to be negative and the price elasticity tends to be inelastic (less than one).

In the aspect of car-related expenses other than fuel prices, Dargay (2007) considers car purchase costs as an explanatory variable in his car travel model, and consequently finds that the cost elasticity of car travel is -0.35 in the short run and -0.46 in the long run. He also remarks that car travel responds more to car purchase prices than it is to fuel prices in a long run. In the light of the results, the sign of β2 is predicted to be negative and the value is assumed to be smaller than one.

However, in order to derive the constant ratios for variables, we take the logarithm on both sides of equation (1) and it gives:


Where Dt, Ft and Pt are Ln(dt), Ln(ft) and Ln(pt) respectively; and ut is the stochastic residual term. The data used for estimation are on an annually basis for the period 1987-2009, except when forecasting where the period is extended to 2020 from 2010. All data is culled from the Office for National Statistics (ONS) and the Department for Transport in the UK. Car travel demand is measured in billion vehicle-kilometres. The combined petrol and diesel price index is used as proxy for fuel prices, and we employ purchase of motor vehicles as proxy for car purchase costs. All prices are 100 in 1987.

Empirical results

Before estimating the short- and long-run elasticity of car travel demand, correlation matrix is utilised to understand the degree of association between bi-variables. Table 1 presents the matrix of correlation among the variables as follows:

Table 1: Correlation Matrix of Variables













(Refer to Excel Workbook - Correlation Matrix)

It can be observed that car travel distance and fuel prices had an extremely strong correlation (0.97) and that it was positive, which contradicted the results generated from the previous investigations as well as the conventional economic theory. Moreover, although correlation between P and D (-0.39) is not as strong as the case of F and D (0.97), the outcome appears that the car purchase costs are negatively associated with car travel distance, which is in line with our anticipation. However, these are not sufficient enough to make a solid conclusion without further investigation.

Stationarity and Co-integrating analysis

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In the present report, the long-run elasticity of car travel demand is determined through co-integration analysis. Nonetheless, it is known that there is a risk of suffering bias by regressing non-stationary time-series data. The regression is not spurious unless the related variables have the same order of integration. This is also an essential feature to have prior to attempt undertaking the co-integrated regression. (Greene, 2003)

Both of the Augmented Dickey-Fuller (ADF) and the Dickey-Fuller GLS test are used. The EViews 6 is employed to perform these tests and all other estimations in the remainder of this report. Table 2 demonstrates the test for stationarity of the selected series.

Table 2: Unit Root Test (Refer to EViews Output File - Appendix A)







1st Differences


1st Differences



























Notes: Number of lags in parentheses. The critical values are according to MacKinnon's statistics. *and** denotes significance at the 1, 5 percent level respectively.

Having compared the reported t-statistics with the critical values, we cannot reject the null hypothesis that the variables in the level form are non-stationary, regardless of which test is performed. On the other hand, the resulting t-statistic of each considered series with first difference is statistically significant enough to reject the null hypothesis of non-stationarity, implying that they are all integrated of order one. Once the pre-condition is satisfied, we proceed with examining the long-run equilibrium relationship. Based on E-G procedure, equation (2) is estimated by OLS regression. Table 3 summarises the results of co-integrated regression below:

Table 3: Co-integration Analysis (Refer to EViews Output File - Appendix B)



Dependent variable: D



Explanatory variables


















D.W. statistic


DF-GLS test for residual (ut)


ADF test for residual (ut)



Notes: t-values in brackets. *and** indicates significance at 1 and 5 percent level respectively.

D, F, P and LD, LF, LP is interchangeable.

The high value of R2 0.926 indicates that about 92.6% of the variation in car travel demand can be explained by fuel prices and car purchase costs. Notwithstanding, the Durbin-Watson statistic has not been successful to support the rejection of the null hypothesis of no co-integration. However, this could be compensated to some extent by the t-values generated from DF-GLS and ADF test on the residual ut, which are statistically significant at 5% and 1% level respectively. The residual ut is thus proved to be stationary in level. This suggests that there exists the long-term relationship among the variables. Therefore, the long-run fuel price elasticity is 0.27 with the strong t-value (15.555) at 1% significance level, and the long-run cost elasticity is 0.117 with t-value of 2.19 (at 5% level).

The Error Correction Model (ECM)

According to Asteriou and Hall (2007), the short-run elasticities can be obtained via the ECM when the variables are found to be co-integrated. Thus, we perform OLS regression based on the modified version of equation (2):


Where Δ represents that the corresponding variables are first differenced; the value of ecmt-1 refers to the residual ut derived from the co-integrating regression and then lags one period; and εt is a white noise with mean zero. λ1, λ2 and γ are to be estimated, and the result is reported in table 4.

Table 4: The Error Correction Model (Refer to EViews Output File - Appendix C)

Dependent variables: ΔD



Independent variables





















D.W. statistic



Note: t-values in parentheses.**denotes significance at 5 percent level.

The diagnostics presented above are not satisfactory. The value of R2adj 0.376 is rather weak, and the low value of Durbin-Watson statistic indicates the emergence of residual autocorrelation. Fortunately, the parameter estimate of purchase cost (ΔP) is statistically significant at the 5% level. Yet, the coefficient of fuel price (ΔF) has not been found to be significantly different from zero. However, the computing t-value -2.467 of the parameter of ecm(t-1) exceeds the critical t-value at the 5% level, implying that the adjustment coefficient γ is statistically significant enough to allow the co-integrating variables restoring to their long-term stable path at a moderate rate of 0.414 even though there are drifts in a short-run.


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A comparison of short-term and long-term elasticities of demand for car travel between the present report and those earlier studies is tabulated in table 5 below:

Table 5: Comparison of Short-term and Long-term Elasticities






Fuel price


Car Purchase Cost

Present Report





Dargay (2007)





de Jong and Gunn (2001)







1Short-term elasticities; 2Long-term elasticities; Values for present report are extracted from table 3 and table 4. Other values are obtained from the corresponding studies.

We notice that the long-term fuel price elasticity of car travel demand is approximately 3 times greater than its short-term elasticity, which confirms the finding of Goodwin et al. (2004) as stated at the outset of the report. The short-run fuel price elasticity 0.08 is relatively inelastic compared with others' results (-0.1 and -0.16). This is not unusual because the fuel price may not a unique determinant judged by travellers, at least in the short-term. It could be witnessed by R2adj obtained from ECM that there remains about 63% of the variability in car travel demand cannot be explained by fuel prices. Besides, the inelasticity of demand WRT fuel price (0.08) may also be attributed to the slow rate of adjustment towards the long-run level. On the contrary to the finding above, the car travel is found to be more responsive to the change of purchase cost in the short-run than long-run.

Further, when comparing the long-run cost and fuel price elasticities (0.117 and 0.27, respectively), the outcome is not in agreement with the conclusion made by Dargay (2007) that the sensitivity of car travel to purchase cost is higher than that to fuel price. Also, of important note is that although the magnitudes of elasticity are all less than one as anticipated, all the signs of the related elasticities are not in alignment with our expectations. For instance, it is hard to believe that 1% increase of car purchase cost will raise car travel demand by 0.283% whereas they would normally be expected to be negatively correlated.

Model amendments

Based on the results presented in the current report, there is a clear indication that variables, other than fuel price and purchase cost, are required to be included in the model so that to redirect the results in terms of the signs and magnitudes of the coefficients toward accuracy. Among many explanatory variables, household income level is often referred to on the topic. Generally speaking, a rise in income may lead to an increase in car travel demand. This is evident by Johansson and Schipper (1997), who found that the income elasticity of car travel demand lie in the range 0.65-1.25, suggesting income has a positive influence on the demand for car travel. In addition to income levels, car travel time is also of particular interest. If the level of traffic congestion is low, the total travel time will be reduced and thus the demand for car travel tends to be higher. Frank et al. (2008) also find that the mode choice is significantly affected by the travel time. Furthermore, some studies have been conducted on cross-elasticities of demand for travel (Accut and Dodgson (1996); Warden et al. (1994)). They observe that the demand for different modes of travel is related to the prices of other modes of transport. Moreover, there seems to appear a problem of autocorrelation. To deal with this, a lagged dependent variable should be included in the regression model.

Based on the foregoing, we extend the equation (2) as follows:


Where: Y t = household disposable income;

T t = car travel time;

X t = bus/rail fares.

Dt-1 = lagged car travel demand


The car travel demand is forecasted for the years 2010-2020 on the basis of the sample data from 1987-2009. The following equation is employed for the projection:


Thereafter, we apply the sample data 1987-2009 to perform the single OLS regression, and it gives:

(6)(Refer to Eviews Output Files - Appendix D, 4.1)

Assuming an increase of 2 percent in fuel prices per annum, we use the Excel to compute the value of fuel prices for the years 2010-2020(refer to Excel workbook). These values are then applied to equation (6) in order to get the predicted values of the car travel distance.

Figure 1: Forecasting of Car Travel Demand 2010-2020 using data 1987-2009 (2%)

(Refer to Eviews Output Files - Appendix D, 4.2)

The statistical results indicate that the forecasting seems perform well, with the small error and the Theil inequality coefficient (zero means a perfect fit). As can be seen in Figure 1, the car travel distance is likely to successively go up to 2020. It implies that fuel price is positively correlated with the car travel demand, which confirms the result from the correlation matrix as well as our empirical results.

Figure 2: Forecasting of Car Travel Demand 2010-2020 using data 1987-2009 (5%)

(Refer to Eviews Output Files - Appendix D, 4.4)

Having looked at the forecasting figures for the 5% increase in fuel price, the situation is roughly the same for the 2% rise in fuel price. That is, fuel price tends to have positive relationship with the car travel demand, at least in 10 years.