### Introduction

The Vehicle Quota System (VQS) was implemented in May ‘90 by the Land Transport Authority of Singapore (LTA) to curb the rapidly increasing vehicle population growth. The VQS allocates a fixed number of Certificates of Entitlement (COE) available for competitive bidding during the bidding exercises conducted bi-monthly3. This report studies the factors affecting the monthly prices of COE in Category B and considers the best model to forecast future COE prices. (1-Feb-09, $689) ) (1-Feb-09, $689) ) A gentle decline in prices from 15 Mar ‘02 to 15 Jan ’09, and an increasing trend afterwards.

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Irregularities in the overall trend are due to external factors such as the consumer demand for cars and the overall state of the economy.

After a regression analysis of COE Prices against Time, the percentage error between actual and fitted values reveals 2 outliers in the dataset (as circled) that occur on 15 Jan and 1 Feb ‘09. Corresponding values falling on these dates were removed from the data set to ensure forecast accuracy.

This results in a discrepancy in Lag1 prices.

By definition, the Lag1 Price of a current period should be the actual Price of its immediate previous period. However, the original Lag1 Price on 15 Feb ‘09 is now $3089 instead of $689, due to the omission of the outliers. To avoid this, we may replace the 2 outlying prices using a centered moving average model, which is based on the existing prices before 15 January 2009 and after 1 February 2009. However, this method is biased in assuming that the prices at those points are going to follow the same trend. 2.

## Analysis and Forecasts of Explanatory

Variables Running a Multiple Regression (MR) of Price against Bids, Quota, Bid Ratio, and Lag1 (of Price), Quota and Bids were found to be insignificant by a t-test. We suspect that the insignificance of Quota and Bids reflects the Omitted Variable Bias phenomenon, which occurs when a regression analysis incorrectly omits one or more important independent variables2. This is because Quota and Bids are the root variables in Bid Ratio and thus determinants of COE price. Hence, we chose to exclude Bid Ratio as a variable and consider Quota and Bids instead.

The MR model obtained is Price=2203. 41+0. 97Lag1+9. 30Bids-13. 55Quota, and the adjusted R2 value indicated that the model accounted for 98. 35% of the variation in Price. Furthermore, the F-statistic for the MR model of 4955. 96 was large, suggesting that we could easily reject the null hypothesis. The 3 remaining independent variables were significant as their p-values of 6. 54E–156, 2. 05E–15, and 4. 80E–14 for Lag1, Quota, and Bids respectively were all less than 0. 05. The scatter plot of residuals against predicted values reveals moderately constant variance which hints that the residuals are random.

Furthermore, the Autocorrelation Function (ACF) output table of the residuals fails to reject 17 out of 20 values and the remaining 3 rejected values are not significantly far away from the Upper Bound (UB) value. This implies that a large proportion of the residuals are in fact random, and our MR model is robust enough to conduct long term forecasts. The variables Quota and Bids will first be forecasted separately and then input back into the MR model stated above to derive future COE prices. 3. 1. Quota 2 Gelpi (2002). Intermediate Statistical Methods: Omitted Variable Bias.

## Intermediate Statistical Methods

Omitted Variable Bias. Quota is an independent variable determined by the LTA. The amount of Quota made available during each bidding exercise is calculated based on 3 components:

- the provision for vehicle growth,
- replacement COEs for vehicles de-registered between the previous 6 months, and
- adjustments for over-projection of vehicle.

The quota for Aug ‘12 to Feb ‘13 was announced to be 701 per month. We will thus use the values in our estimation for Price. Although these values do not account for adjustments for error in the form of over-projection of vehicle de-registrations, they are close enough estimates for the actual values. If one desires to forecast COE prices beyond the 6-month period that announced Quota values provide for, a Random Walk model may be used instead. However, this model requires values to be stationary but an Augmented Dicky-Fuller’s unit root (ADF) test on Quota values shows that they are non-stationary (Pr < Rho values are greater than 0. 05).

Although this limits the accuracy of the naive forecasts provided by the Random Walk model, we will still be able to obtain a rough gauge of future Quota values. The graph of Quota against Time reveals an overall increase from 15 Mar ‘02 to 15 Apr ‘09, followed by a sharp decrease (indicated by the red arrow) from 1 May 2009 onwards. The decrease in Quota was due to the reduction in provision for growth reduced from 3% to 1. 5% per annum. A further reduction of the growth rate to 1% was implemented on 1 Aug ‘12, and a planned reduction of the growth rate to 0. % will take place in Feb ‘13. This seems to imply that Quota will continue its downward trend on the graph. However, the large number of COEs awarded between 2005 and 2006 (values circled in red) will expire after 10 years and since these expired COEs will be replaced in the Quota pool, Quota may temporarily increase between 2015 and 2016. The 10-year ‘lifespan’ of COE suggests that the values of Quota might follow a 10-year cycle and that they are dependent on historical values. The entire dataset of historical Quota values is thus vital in providing an accurate forecast of future Quota values. An ADF test on Bids showed that not all Pr < Rho values were less than 0. 05, indicating non-stationarity. Thus, a forecast model based on the original Bid values alone is not appropriate. Differencing Bid values once, an ADF test reflects identical Pr & Rho values of 0. 0001, indicating stationarity.

However, the IACF graph reveals a decaying trend, implying over-differencing. Hence, we deduce that any forecast model based on the original Bids alone, or the Bids (differenced once) alone, is inappropriate and there is thus a need to introduce other predictor variables for Bids. It is observed that Bids and Quota are directly proportional. When the supply of quota is low, potential bidders will expect prices to be high and this leads to fewer people bidding.

With this supply and demand relationship in mind, we deduce that the number of bids is dependent on Quota. Hence, Quota is used as one of the independent variable in the forecast of Bids. Past bid values reflect the past demand for COEs.

Given that consumer expectations is a determinant of demand1, consumers will first examine the Bids of the previous round before placing their bid in the current round. The number of bidders in the current round also includes unsuccessful bidders of the previous round. Thus, past bid values affect the behavior of future bids and hence, Lag1 of Bids is also a variable in the prediction of Bids. Using the 6 values of Quota obtained in Section 2. 1, an MR of Bids against Quota and Lag1 of Bids is run, where Bids=92. 14+0. 66Quota+0. 42lag1. The adjusted R2 value indicates that the model accounts for 87. % of the variation in Bids. The scatter-plot of the residuals reveals a relatively constant variance. Furthermore, the ACF test of the residuals failed to reject 18 out of 20 values and the remaining 2 rejected values were not significantly far away from the UB value. This implies very low autocorrelation of the residuals and hence the forecasted values based on the MR model are accurate, with a low MAPE of 8. 18%.

Based on the analysis of randomness of residuals in the beginning of Section 2, our model is not only effective in forecasting in the short term, but also robust enough to conduct longer-term forecasts. However, it is important to note that the forecasted prices of the current period are used as the Lag1 price values in the forecast of the following period.

An accumulation of the forecast error is inevitable as time proceeds. Furthermore, the values predicted by the MR model converge to the mean since the forecasted value of the previous period is assumed to be the actual value of the current period, which might compromise the accuracy of the forecasts.

## Integrated Process Of Order

PRice Forecasts Due to the inherent limitations of the MR model, we chose to run a suitable time series model to forecast Price and to cross-reference our previous MR results. The Box-Jenkins methodology was used to determine the best model to be used.

First, a visual analysis of the graph of Price against Time revealed an absence of seasonality and hence, deseasonalising the data was not required. Second, stationarity of the data was tested by running an AR(1) model on Price. A coefficient of 1 showed that the values violate the stationarity condition. An ADF test on Price further confirmed that the values were non-stationary since Pr< Rho values were greater than 0. 05. Price was thus differenced with many integrated processes of different orders to make the data stationary. The integrated process of order 1, I(1), was found to be the most suitable. An ADF test showed that I(1) values are stationary, with identical Pr > Rho values of 0. 0001. In addition, the IACF graph of I(1) shows no decays which implies that there was no under or over differentiation. The stationarity of the I(1) model ensures that the forecasts will not converge.

Third, optimum p and q values (order of the autoregressive and moving average terms respectively) were identified by choosing the model which gives us the least forecast errors, with low AIC and MAPE. The ARIMA(0,1,0) model with p = 0 and q = 0 yielded values with the least forecast errors and the lowest MAPE of 9. 78% compared to other models including ARIMA (1,1,0)-10. 09% and ARIMA (1,1,1)-9. 99%. Furthermore, the PACF graph indicates that there is no correlation between the data set and any possible lags that could have gone unexplained.

Finally, the ACF test on I(1) shows that there is little correlation of the residuals. Thus, we conclude that the residuals are random which proves the robustness of the model. Therefore we concluded that I(1) was the most suitable model with a constant estimate of 184. 7 and a variance estimate of 10070477. The advantages of I(1) is that it stabilizes the original times series and makes the forecasts possibly more accurate. Furthermore, such methods are highly common in business time series. However, a limitation of the I(1) model is that it gives a naive forecast. 4.

FINAL FORECAST and Evaluation Although the MR model in Section 2. 3 well-accounts for the important explanatory variables (Bid, Quota, Lag1 Price), the prices it forecasts converge to a mean. On the other hand, the I(1) model in Section 3 for Price alone is a stationary model and is more robust in forecasting long-term. Also, its forecasted prices do not converge to a mean like how the MR’s forecasted prices do. Our final forecasting model, ‘MRI(1)’ is thus a weightage of these 2 models, where the weights placed are inversely proportionate to the MAPEs of the respective models. I(1), with a higher MAPE (9. 8%), receives a lower weightage of 0. 45, while the MR model of lower MAPE (7. 86%), receives a weightage of 0. 54312.