#### 3.1. Results of Unit Roots and Co-Integration Tests

When testing for unit roots and co-integration, we have used a 0.05 threshold in this study. As discussed above, we employ PP test for the unit root test for the series of variables. The results are shown in

Table 2. The

p-values of PP values are smaller than 0.05. Thus, the null hypothesis of non-stationarity can be rejected for first differences of these series. All time-series are I (1).

The co-integration is to test whether a linear combination of each individually non-stationary time series is itself stationary. The results of the Johansen co-integration test for the series (

D,

Y,

P) are reported in

Table 3 (The numbers inside the brackets are the number of lags. The values in parentheses are

p-values calculated under the null hypothesis of non-stationarity). The likelihood ratio tests show that the null hypothesis of absence of co-integrating relation (

R = 0) can be rejected at 5% level of significance, and that the null hypothesis of existence of at most one co-integrating relation (

R ≤ 1) and two co-integration relation (

R ≤ 2) also cannot be rejected in

Table 4 (The optimal lag length is chosen as four by using Akaike’s information criterion described in Pantula

et al. [

25]. The

p-values are calculated under the corresponding null hypothesis). This implies that there is one or more co-integrating equation at 5% level of significance. The optimal lag length is chosen as four by using Akaike’s information criterion described in Pantula

et al. [

25].

**Table 3.**
Results of Phillips-Perron unit root tests.

**Table 3.**
Results of Phillips-Perron unit root tests.
| Levels | First-differences |
---|

D | −20.954 [6] (0.058) | −108.355 [5] * (0.000) |

Y | −10.658 [4] (0.393) | −80.052 [3] * (0.000) |

P | −7.814 [3] (0.599) | −84.742 [2] * (0.000) |

The co-integration vector equation is estimated to

Z =

D – 1.4784

Y + 0.5469

P by using canonical co-integrating regression (CCR) suggested in Park [

27]. Thus, the long-run income elasticity of diesel demand is 1.4784 and the long-run price elasticity is −0.5469. Both the two are statistically significant at the 1% level. The signs for the elasticity are alignment with economic theory, and diesel demand is elastic with respect to income, and is inelastic with respect to price. Furthermore, we can identify ECT,

${\epsilon}_{t-1}={D}_{t-1}-1.4784{Y}_{t-1}+0.5469{P}_{t-1}$ from estimated co-integration equation. In addition, we test Durbin-Watson statistic to detect the presence of autocorrelation in the residuals from the regression analysis. The estimated Durbin-Watson statistic give a value of 1.603 close to 2. Therefore, the CCR has no autocorrelation.

**Table 4.**
Results of Johansen co-integration tests.

**Table 4.**
Results of Johansen co-integration tests.
Null hypotheses | Likelihood ratio test statistic | p-values |
---|

H_{0} (R = 0) | 35.372 * | 0.039 |

H_{0} (R ≤ 1) | 17.609 | 0.060 |

H_{0} (R ≤ 2) | 2.816 | 0.086 |

#### 3.2. Results of Error-Correction Model

In the ECM, the optimal lag lengths in Equation (2) are chosen by using AIC. The lag lengths of L_{11}, L_{12} and L_{13} in Equations (2) are chosen as 4, 3, and 4, respectively. The short-run price elasticity of diesel demand is estimated to be −0.3566 and the short-run income elasticity 1.5887. Both the two are statistically significant at the 1% level. The signs for elasticities are consistent with economic theory, and diesel demand is elastic with respect to income and in-elastic with respect to price.

In order to determine the most appropriate model, we conduct specification error tests. First, one important point to consider in estimating the ECM is that there might be a structural break. If there are any structural breaks, necessary adjustment to reflect the structural break should be made. We checked the model stability using CUSUM and CUSUMSQ tests suggested by Brown

et al. [

28]. The tests are appropriate for time series data and might be used if one is uncertain about when a structural break might have taken place. Moreover, the tests are quite general in that they do not require a prior specification of when the structural break takes place [

29]. The null hypothesis is that the coefficients are the same in every period,

i.e., there is no structural break.

Table 5 contains the results of the tests. Both tests suggest that the null hypothesis of absence of structural break cannot be rejected. Thus, the models are stable over time.

Second, it is important to consider whether there are any factors that are not included in this model but may affect the diesel demand. This aspect is a general specification error issue. Therefore, a fairly reasonable test to establish the specification error was conducted. It is interesting to determine whether there might be a significant bias in the estimates due to the omitted variables or non-exogeneity of the regressors. This is particularly important in this case since many scholars have tried to assess the effect on demand of a number of variables that are not included here because of the explicit focus on price and income.

Regression specification error test (called RESET) suggested by Ramey [

30] was used in this study. In its simplest form, the procedure involves running a test regression of ∆

D_{t} on the regressors included in Equation (2) along with the square of the predicted value of ∆

D_{t} from the original regression. If the coefficient of the squared predicted-∆

D_{t} term in the test regression is not significant, the null hypothesis of no specification error can be maintained. The

t-value associated with the squared predicted-∆

D_{t} term is 0.832, which is not statistically significant at any meaningful level. Thus, despite the simplicity of this model and the estimation procedure, there is no indication of any major specification problem.

Finally, this paper also used Durbin-Watson test statistic to detect the presence of autocorrelation in the residuals from the regression analysis. Furthermore, ECM include the lagged dependent variables, and therefore we employ the Durbin h- test. The estimated Durbin h-statistic give a value of −1.21 and p-value is 0.23. Therefore, the ECM model passes the test of autocorrelation.

Table 5 (The

p-values are calculated under the null hypothesis of absence of structural break) gives the computed values of short-run and long-run elasticities for income and price. This implies that diesel demand is elastic with respect to income and the long-run elasticity is smaller than the short-run. This may be explained by rapid adjustment of consumer’s automobiles stocks in the short run. The fact the long-run income elasticity was less than the short-run elasticity may suggest a problem with the model . This can be driven by omitted variable bias. For example, Liddle [

6,

7] considered vehicle stock in addition to income and fuel price. However, we cannot reject the null hypothesis of no specification error, as discussed above. Moreover, Rao and Rao [

14] also found that the long-run elasticity is less than the short-run elasticity in Fiji. Alves and Bueno [

10] detected that income elasticity does not differ from short-run and long-run in Brazil. This issue merits further investigation in the future. Furthermore, diesel demand is inelastic with respect to price and the long-run elasticity is larger than the short-run values in its absolute value. The rising prices will bring about a gradually decrease of diesel demand in both the short and long run. Finally, the coefficient of ECT,

β_{14}, is significant (at the 1% level) and has a absolute value of 0.238 suggesting the diesel consumption adjusts toward is long-run level with almost 23.8% of the total adjustment occurring within the first quarter.

In conclusion, the estimated results of the elasticities of diesel demand are presented at

Table 6. Comparing our results with the findings of Lim and Yoo’s [

17] study that deals with the gasoline demand function in Korea reveals that the diesel demand is less elastic to both price and income change than the gasoline demand.

**Table 5.**
Results of CUSUM and CUSUMSQ tests.

**Table 5.**
Results of CUSUM and CUSUMSQ tests.
| CUSUM test | CUSUMSQ test |
---|

Test statistic | p-values | Test statistic | p-values |
---|

Equation (2) | 0.796 | 0.142 | 0.117 | 0.451 |

**Table 6.**
Estimated results of the elasticities of diesel demand.

**Table 6.**
Estimated results of the elasticities of diesel demand.
| Short-run | Long-run |
---|

Income elasticity | 1.589 ** (5.12) | 1.478 ** (17.39) |

Price elasticity | −0.357 ** (−5.49) | −0.547 ** (−13.99) |