A Quantitative Approach to Detect Structural Breaks in the Trend of Bid Prices

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1 A Quantitative Approach to Detect Structural Breaks in the Trend of Bid Prices Mohammad Ilbeigi, Baabak Ashuri, Ph.D., and Soheil Shayegh Georgia Institute of Technology Atlanta, Georgia Construction of transportation projects are suffering from uncommon increasing prices and costs of projects during last decade. Although the prices show an upward trend, they are subject to considerable fluctuation and sometimes significant sudden changes or structural breaks. These structural breaks can lead to considerable forecasting errors. Furthermore, detecting the structural breaks and estimating their dates can help transportation agencies and contractors to identify effective factors that have led to those sudden changes in the prices. In this paper, by using a quantitative approach based on linear regression models, presence of the structural breaks and estimation of the break dates in the submitted bid price of one of the most important and common line items in transportation projects are investigated. It is expected that the results can help transportation agencies and contractors to improve forecasting models, identify the effective factors and events which may lead to the structural breaks and consequently develop appropriate strategies during the unexpected increase or decrease of the prices. Key Words: Bid Price, Structural Break, Linear Regression Models, Chow Test Introduction Transportation agencies across the nation are facing great challenges of rising costs for construction of new highway projects, as well as maintenance and modernization of existing infrastructure systems. The National Highway Construction Cost Index, which is recorded by the Federal Highway Administration (FHWA), shows significant increase particularly in States, such as Georgia, Florida and California (FHWA 2008). The FHWA National Bid- Price Index (BPI) increased 47.7% from , 3 times greater than its largest growth in any 3-year period between 1990 and A similar pattern was observed at the State level across the country. For instance, the Highway Cost Index (HCI) of Texas Department of Transportation (TxDOT) was doubled between 1997 and 2006, an increase much higher than the growth in Consumer Price Index (CPI) within the same period (Bohuslav 2007). Oregon Department of Transportation experienced 163% increase in construction costs for structural systems between 1987 and 2007 (Holmgren et al. 2010). Although cost and price escalation is not a new event, price escalation in 2000s was dramatic and uncommon and most highway contractors and State DOTs did not expect it. For instance, in highway projects, the eleven-year average growth rate of the Construction Cost Index (CCI) from 1990 through 2001 was 1.5% per year while since 2001, the average growth rate has been 8% per year (Dayton 2006). While a rate of 2.5 to 3.5% was typical for material price escalation in 2000, the recent Engineering News-Records (ENR) construction material cost index shows an increase at the rate of 10.3% in November 2008 compared to a year before. Construction costs show an upward trend during the last years, however, it is subject to considerable fluctuations and sometimes sudden sharp changes. Volatility in the prices increases the need for accurate cost forecasting models. However, significant sudden changes may decrease the accuracy of the forecasting models considerably. For example, time series forecasting models may not perform well when a considerable sudden change occurs (Ashuri and Lu 2010). Also, presence of a sudden change might affect the accuracy of regression models. A sudden sharp change can be considered as a structural break. A structural break is defined as an unexpected shift in a sequence of observations which can lead to considerable forecasting errors and unreliability of the models (Gujarati 2007). Thus, the forecasted results might be misleading and unreliable if the presence and time of the possible structural breaks are not investigated. Furthermore, accurately detecting and analyzing the structural breaks in the trends of prices can be helpful to identify the effective factors which may change the prices suddenly and create the

2 structural breaks. However, there is little knowledge about detecting structural breaks in the price of important line items in transportation projects. The research objective of this paper is to check the presence of the structural breaks in the submitted bid prices for one of the most common line items in transportation projects in the state of Georgia and estimate the break dates accurately. It is expected that the results of this study help transportation agencies and contractors to identify effective factors for sudden changes in the prices, consider the detected structural breaks in their forecasting models and develop appropriate strategies for sudden changes in the prices. Methodology of Detecting Structural Break In this paper, the process of detecting and analyzing a structural break has two steps: 1) Testing whether a structural break has been occurred or not, 2) Detecting and estimating the break date. Different methods for investigation on the two steps have been developed. In this research, several methods and approaches based on the linear regression modeling implemented. Testing for the Presence of a Structural Break Chow test (1960) is one of the most famous methods to detect a structural break. Many literatures consider it as the classical test for the presence of a structural break. Chow test is based on the regression models. The procedure of this test is to split the sample into two sub-periods over the candidate break date, and then check whether the parameters of the regression models in the two sub-periods are statistically different or not. The null hypothesis of the test asserts that the parameters of the two models are same. Gregory Chow (1960) showed that his test statistic which is shown below follows the F distribution with k and N 1 +N 2-2k degree of freedom if there is no structural break. ( ) ( ) ( ) ( ) Where: S c : the sum of squared residuals from the combined data. S 1 : the sum of squared residuals from the first sub-period S 2 : the sum of squared residuals from the second sub-period N 1 : number of observations in the first sub-period N 2 : number of observations in the second sub-period K: total number of the parameter in the regression models (explanatory variables plus the intercept) Thus, the null hypothesis of no structural break is rejected with a predetermined significance level, if the Chow statistic is tested by the F distribution criterion and rejected. Estimating the Structural Break Date The most important limitation of the Chow test is the fact that the break date must be determined in advance. Even if a break date is considered based on some known feature of the data, the Chow test can be misleading, as the candidate break date is correlated with the data and the test may indicate a break falsely (Hansen 2001). Quandt (1960) proposed to calculate the Chow statistic for all possible break dates and choose the break date candidate with the largest statistics as the real break date. However, this approach is only valid in one especial case when the Chow test is constructed with the homoscedastic form of the covariance matrix (Hansen 2001). Later, researchers indicated that using the least squares can be an appropriate approach to detect the break dates. Chong (1995) and Bai (1997) showed that the sum of squared errors as a function of time can have a local minimum near each break date. Thus, the global minimum can be considered as the break date estimator and the other local minima can be contemplated as the other candidate break date estimators. Hansen (2001) noted that a strong and V shape minimum point for the sum of squared errors indicates a significant and strong structural break. It is not surprising if a data set consists of more than one structural break. A sequential method to detect multiple structural breaks was developed by Bai and Perron (1998). The method starts with checking for the presence of a single structural break. If the null hypothesis of no structural break is rejected, the sample is split in two subsamples at the date with the global minimum value of sum of squared errors. Then, the test is reapplied to each subsample.

3 This procedure continues until each subsample test fails to find evidence indicating a structural break. Furthermore, Bai (1997) showed that by iterative reestimation of break dates based on refined samples more accurate results can be obtained. Detecting Structural Breaks in the Bid Prices In this section, the aforementioned methodology for detecting the presence of a structural break and estimating the break dates is applied to the submitted bid prices for one of the most important line items in transportation projects. The line item is to construct a recycled superpave asphalt concrete with the thickness of 12.5 mm. The data set consists of 1177 observations of all submitted bid prices for this line item by winner bidders in transportation projects in the state of Georgia from January 1998 to July Figure 1 shows the scatter plot of the winning submitted bid prices for this line item. Figure 1: Scatter Plot of the Submitted Bid Price Since all the aforementioned procedure for detecting a structural break is based on the linear regression models, the first step is to model the variations of the submitted bid prices using regression analysis. An extensive literature review was conducted to identify possibly effective factors on submitted bid prices for this asphaltic line item. Six explanatory variables for the initial regression models for this line item were determined as follows: 1- Duration of the project: Duration of a project might be an important effective factor to determine the bid price. Considering the volatility of the material price, cost uncertainty and consequently risks may be higher for longer projects. The unit of the duration is days. 2- Quantity: Quantity of the line item can be an important factor to attract different size of contractors. Larger projects can be more attractive for big contractors and they may submit lower price for them. At the same time, large quantities may lead to higher risks. 3- Total Price of the contract: Total proposal bid price or contract amount may show the project size. Contractors may have various strategies to determine price for different line items based on the project size. 4- Relative value of the line item (Item%): This variable shows the relative dollar value of the line item compared to the total bid price of the project by dividing the total price of each item on the total bid price.

4 5- Number of the bidders: Number of bidders can be an indicator of competition. Level of competition may have significant effect on the submitted bid prices. Usually higher competition leads to lower prices. 6- Asphalt price index: Since liquid asphalt is the most expensive and volatile material in an asphalt mixture, its price should be consider in the model. Thus, a linear model as follow is developed: The second step is to split the data set at each month, consider two linear regression models before and after that month and then calculate the Chow statistic for each month. Figure 2 shows the results of this step. It should be noted that since there are not enough observations to determine the regression parameters for the months too close to the beginning or the end of the sample, break dates cannot be checked for those months. The conventional solution is to consider all break dates typically in the interior t percent to 1-t percent of the sample. The trimming parameter t is typically between five to fifteen percent (Hansen 2001). In this research five percent trimming point is used. Figure 2: Graph of Chow Statistic for Each Month Figure 2 indicates that since the graph of the Chow statistic is higher than the F distribution critical value for some months, the submitted bid prices have experienced a statistically significant structural break at least in one month. The largest value of the Chow statistic occurs in August Thus, if the assumption of the homoscedastic form of the covariance matrix in the linear regression model is valid, August 2005 is the first candidate of break date. Since the validity of this assumption is under question, the method based on the minimum sum of squared errors is conducted. Figure 3 shows the graph of the sum of squared errors for linear regression models at each month. Interestingly, the global minimum occurs in August Furthermore, there is a local minimum in May 2008.

5 Figure 3: Least Squares Break Date Estimation: Sum of Squared Errors for Each Month Then, according to the proposed procedure, the data set is split at the global minimum (August 2005) and the procedure is reapplied to both sub-periods. The results for the sub-period of January 1998 to August 2005 indicate that a global minimum for this sub-period occurs in March However, this may not be a considerable structural break because the minimum value is not strong and V shape. Also, this point is near to the first break date candidate (August 2005) and might be affected by that point. For the sub-period of August 2005 to July 2013, the results indicate that there is a structural break in September 2008 (Figure 4 and 5). Although the minimum at this point is not very strong and V shape as August 2005, it can be a significant structural break since it is near to the local minimum of the whole data set (May 2008). Figure 4: Graph of Chow Statistic Sub-period of Aug 2005 to July 2013

6 Figure 5: Sum of Squared Errors for Sub-period of Aug 2005 to July 2013 Finally, the whole data set is split in September 2008 and the procedure reapplied for sub-periods of before and after that month. The results indicate that there is no evidence for presence of a structural break between September 2008 and July However, the Chow statistic is higher than the critical value and the sum of squared errors has a global minimum in August 2005 in the sub-period of January 1998 to September 2008 (Figure 6 and 7). Therefore, the presence of a significant structural break in August 2005 can be confirmed. Figure 6: Graph of Chow Statistic Sub-period of Jan 1998 to Sept 2008

7 Figure 7: Sum of Squared Errors for Sub-period of Jan 1998 to Sept 2008 Conclusion Transportation agencies and contractors are facing serious problems to accurately forecast the future construction costs because of sudden changes in the price of critical line items in transportation projects. Information about the presence of the structural breaks in the bid prices and accurate estimation of their dates can improve the results of forecasting models. For example, if the forecasting model is based on the linear regression analysis, separate models for observations before and after the structural breaks might be more helpful than a single regression model using the entire data set. Furthermore, some types of time series forecasting models are not accurate when a structural break occurs. Therefore, the validity of the forecasting models can be investigated by detecting structural breaks too. Moreover, detecting the structural breaks and estimating their dates can help transportation agencies to identify effective factors such as considerable shock in the price of critical input commodities or new conditions in the market that may lead to the sudden changes in the bid prices. In this paper the presence of the structural breaks in the submitted bid prices for one of the most important line items in transportation projects in the state of Georgia was investigated. The results indicate that in August 2005, the bid prices experienced a statistically significant structural break. Furthermore, some evidences of a structural break around September 2008 were detected. It is expected that the results of this study can help transportation agencies and contractors to improve the accuracy of their forecasting models and identify the effective factors and events which may lead to the sudden changes in the bid prices. References Ashuri, B., Lu, J. (2010). Time series analysis of ENR construction cost index. Journal of Construction Engineering and Management, 136(11), Bai, J. (1997). Estimating Multiple Breaks One at a Time. Econometric Theory. 13(3), Bai, J. Perron, P. (1998). Estimating and Testing Linear Models with Multiple Structural Changes. Econometrica. 66(1), Bohuslav, T. (2007). Cost Control Texas. Presented at Joint Conference on Cost Savings Methods, St. Louis, Missouri. Texas Department of Transportation, Austin. Chong, T. (1995). Partial Parameter Consistency in a Misspectified Structural Change Model. Economics Letter. 49(4),

8 Chow, G. C. (1960). Tests of equality between sets of coefficients in two linear regressions. Econometrica: Journal of the Econometric Society, Dayton, K. J., (2006). Recent Trends in Highway Construction Costs. in WASHTO Subcommittee on Construction Boise, Idaho. Federal Highway Administration (FHWA) (2007). Growth In Highway Construction And Maintenance Costs. Report Number: CR , U.S. Department of transportation. Federal Highway Administration (FHWA) (2008). Price Trends for Federal-Aid Highway Construction. URL: accessed on March 23, Gujarati, D. (2007). Basic Econometrics. New Delhi: Tata McGraw-Hill. pp Hansen, B. E. (2001). The new econometrics of structural change: Dating breaks in US labor productivity. The Journal of Economic Perspectives, 15(4), Holmgren, M., K. L. Casavant, and E. Jessup. (2010). Evaluation of Fuel Usage Factors in Highway Construction in Oregon. Final Report SPR668. School of Economic Sciences, Washington State University, Pullman. Quandt, R. (1960). Tests of the Hypothesis that a Linear Regression Obeys Two Separate Regimes. Journal of the American Statistical Association. 55, Wong, J. M., Chan, A. P., Chiang, Y. H. (2005). Time series forecasts of the construction labour market in Hong Kong: the Box Jenkins approach. Construction Management and Economics, 23(9),