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A new quantile regression forecasting model
Journal of Business Research 67 (2014) 779–784
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Journal of Business Research
A new quantile regression forecasting model☆ Kun-Huang Huarng ⁎, Tiffany Hui-Kuang Yu ⁎⁎ Feng Chia University, Taiwan
a r t i c l e
i n f o
Article history: Received 1 April 2013 Received in revised form 1 October 2013 Accepted 1 November 2013 Available online 13 December 2013 Keywords: Health care expenditure ICT New quantile information criterion Forecasting
a b s t r a c t Quantile regression is popular because it provides more information as well as comprehensive interpretations. To improve forecasting performance, this study proposes a new quantile information criterion (NQIC), on the basis of the coefficient of variation, and expects the NQIC to reflect whether a variable is predictable. The health care expenditure data determine the thresholds for the NQICs. The thresholds assist in forecasting the development of information and communication technology. From the empirical analyses, the NQICs and thresholds greatly improve the forecasting performance. © 2013 Elsevier Inc. All rights reserved.
1. Introduction The quantile regression model offers a more complete model than the conventional mean regression (Yu, Lu, & Stander, 2003). Studies apply the quantile regression model to interpret various problems, such as wages (Buchinsky, 1994; Machado & Mata, 2005; Martins & Pereira, 2004), survival analysis (Crowley & Hu, 1977; Koenker & Geling, 2001), financial analysis (Bassett & Chen, 2001), economic research (Hendricks & Koenker, 1992; Wang, Yu, & Liu, 2013), the study of the environment (Pandey & Nguyen, 1999), internet and communication technology (ICT) adoption (Yu, 2011), health care expenditure (Yu, Wang, & Chang, 2011), small business performance (Seo, Perry, Tomczyk, & Solomon, 2014), and so on. A number of studies advance the quantile regression model to forecasting. Granger, White, and Kamstra (1989) propose a method for combining the variety of possible interval forecasts on the basis of quantile regression techniques. Taylor (2007) forecasts the daily supermarket sales using exponentially weighted quantile regression (interval forecasts), which outperforms traditional methods. Banachewicz and Lucas (2008) use hidden Markov models to forecast the quantiles of ☆ The authors acknowledge and are grateful for the financial support provided by the National Science Council, Taiwan, ROC under grants NSC 101-2410-H-035-004-, NSC 100-2410-H-035-006-MY2, NSC 102-2410-H-035-038-MY2, and NSC 102-2410-H-035041-MY2. The authors would also like to thank several attendees at the 2013 Conference of the Global Innovation and Knowledge Academy for their valuable comments that have helped improve this article. ⁎ Correspondence to: K.-H. Huarng, Department of International Trade, Feng Chia University, 100 Wenhwa Road, Seatwen, Taichung 40724, Taiwan. ⁎⁎ Correspondence to: T. H.-K. Yu, Department of Public Finance, Feng Chia University, Taichung, Taiwan. E-mail addresses: [email protected] (K.-H. Huarng), [email protected] (T.H.-K. Yu). 0148-2963/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jbusres.2013.11.044
corporate default rates, which are important in financial risk management. Gerlacha, Chen, and Chan (2011) apply Markov chain Monte Carlo methods for the Bayesian time-varying quantile forecasting of Value-at-Risk in financial markets. Cai, Stander, and Davies (2012) propose a Bayesian approach to quantile autoregressive time series model estimation and forecasting and then apply the approach to currency exchange rate data. The empirical results show that an unequally weighted combining method outperforms other forecasting methodology. Yu (forthcoming) suggests using a quantile information criterion (QIC) to assist in forecasting. To improve forecasting performance, this study proposes a new QIC (NQIC) to identify if a variable is predictable. The health care expenditure data are in order to determine the thresholds for the NQICs. Then, the thresholds and the NQICs intend to forecast the ICT development. To that end, Section 2 reviews the concepts of the quantile regression model. Section 3 introduces the algorithms for the NQICs and determining their thresholds. Section 4 describes the variables of the two data sets, provides the empirical analyses for the two cases, and reveals the results of the estimation and forecasting performance. Section 5 concludes this paper. 2. Quantile regression model Koenker and Bassett (1978) propose quantile regression to infer the results of the conditional functions for different quantiles. Bao, Lee, and Saltoğlu (2006) consider that the main advantage of quantile regression is to provide better statistics by means of the empirical quantiles. Quantile regression can help “complete the picture” when we intend to understand the relationship between variables for which the effects may vary with outcome levels. In addition, quantile regression is more forgiving than ordinary least squares in that quantile regression is
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relatively insensitive to outliers and can avoid censoring problems (Conley & Galenson, 1998). As Bassett and Koenker (1982) extend the median to quantile regression so as to calculate various quantiles, the quantile regression does not require any distribution assumptions regarding the population and can estimate the parameters nonparametrically. Quantile regression models the conditional quantiles, which are quantiles of the conditional distribution of the response variable in the expression of functions of the covariates of observations. Quantile regression models use the least absolute deviations method to minimize the absolute values of the errors. The model for a median linear regression is: yi ¼ xi βθ þ εθ;i
ð1Þ
where the assumption is median (εθ,i|xi) = 0. This concept is extendable to any quantile, such as the 75th percentile, 95th percentile, etc. We can define the estimate by minimizing the sum of asymmetrically weighted residuals. 2 min 4 β
X
α jyt −xt βj þ
tjyt ≥ xt β
X
3 ð1−α Þjyt −xt βj5
ð2Þ
tjyt b xt β
where α is a parameter (0 b α b 1) that represents the size of the quantile, and is also the quantile α of the explanatory variable that we intend to examine in the quantile regressions. This problem does not have an explicit form; however, linear programming methods can solve the problem (Armstrong, Frome, & Kung, 1979). When α = 0.5, the quantile regression is the median regression. Since on this occasion the values of α and (1 − α) are both 0.5, the above equation changes to ∑ jyt −xt βj, indicating that the observations above and below the medi-
Table 1 Quantile forecasting of 1998 by using 1992–1997 health care expenditure data. 1992–1997
1998
0.05 LOG(GDP) OLD DOC IM LE
0.497261 0.013379 −0.012920 −0.008170 −0.019490
1.390759 0.027361 0.031001 −0.003110 0.026669
1.34671 0.01405 0.03511 −0.00234 −0.01954
0.25 LOG(GDP) OLD DOC IM LE
1.056123 0.005542 0.025187 −0.007550 −0.017050
1.370417 0.015938 0.054073 −0.003070 0.006608
1.12976 0.00192 0.04980 −0.00262 0.01077
0.50 LOG(GDP) OLD DOC IM LE
1.209432 0.001473 0.044355 −0.009780 −0.02126
1.577748 0.011047 0.088145 0.001800 0.000844
1.33037 0.01145 0.04925 −0.00216 −0.00336
0.75 LOG(GDP) OLD DOC IM LE
1.391142 −0.006450 0.031638 −0.003990 −0.030450
1.818118 0.013389 0.091582 0.007852 −0.006490
1.52785 0.00933 0.01192 0.00090 −0.00700
0.95 LOG(GDP) OLD DOC IM LE
1.579464 −0.041320 0.005280 −0.024830 −0.055570
1.983476 0.010996 0.128820 0.010607 −0.015770
1.92039 −0.00434 0.07328 −0.00237 −0.03289
y
an values are of the same weights. 3. New quantile information criterion 3.1. Rationale Not all the variables are predictable by using quantile regressions due to their data characteristics. To identify if they are predictable, Yu (forthcoming) proposes a quantile information criterion (QIC). This study proposes a New QIC (NQIC) to provide a systematic method to improve the forecasting results. To measure the dispersion of a distribution, this study applies the coefficient of variation as the NQIC, which is the ratio of the standard deviation and mean of a variable (Lind, Marchal, & Wathen, 2006). The coefficient of variation is unitless and is most useful in comparing the variability of different data sets (Rosner, 1995). For example, Bloch (2007) uses the coefficient of variation to test if the coefficients of two data sets are different. Yu (forthcoming) considers that the variables with large variations are difficult to forecast. Following these studies, this study first intends to identify the extreme values of the coefficients of variation of a data set to form the thresholds. Furthermore, the thresholds are in order to determine whether the variables in the other data set are unpredictable with the coefficients of variation lying outside the range of the thresholds. 3.2. Algorithms There are two data sets: the one for obtaining the thresholds is the sample data set and the other is the target data set. Following the above rationale, this section proposes the algorithm for calculating the NQICs and their corresponding thresholds, and the algorithm for forecasting on the basis of the thresholds. We list the algorithms in Appendices 1 and 2.
The algorithm for calculating the NQICs and thresholds serves to calculate the thresholds from the sample data set. Step 1 separates the data into in-sample and out-of-sample data. The calculation starts with the in-sample data ranging from t = 1 to d − 1 to test the out-of-sample data of t = d. The next run moves one step further; in other words, the calculation starts with the in-sample data from t = 1 to d to test the out-of-sample data of t = d + 1. We continue the process until we exhaust all the out-of-sample data. Step 2 calculates the NQICs and then checks if the corresponding quantile intervals can cover more than 50% of the corresponding variable in the out-of-sample data. Only those with more than 50% coverage (which we name hits) can advance to the calculation for the thresholds. Step 3 finds the maximum (max) from all the positive NQICs of all the hits and calculates their standard deviation (stdev_p). Similarly, it finds the minimum (min) from all the negative NQICs and their standard deviation (stdev_n). To be conservative, we narrow the range between the minimum and maximum by one standard deviation, respectively. In the algorithm for forecasting on the basis of the thresholds, we also separate the data into in-sample as well as out-of-sample data for the target data set in Step 1 of the previous algorithm. Step 2 calculates the NQICs for the out-of-sample data. During forecasting, if any NQICvt falls within the thresholds, the algorithm considers the variable v at time t predictable; otherwise Table 2 The hits for all the years. Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
LOG(GDP) OLD DOC IM LE
5 3 3 3 3
4 3 4 1 5
5 4 3 2 3
5 3 3 1 2
4 4 3 1 3
3 3 3 2 3
3 4 3 2 4
5 5 5 5 5
2 3 0 2 2
4 3 1 2 3
Note: The numbers in bold are the hits.
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Table 3 The NQICs for all the variables. LOG(GDP) 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
1.4926 1.5347 1.6004 1.4801 1.4949 1.5154 1.5466 1.6122 1.4956 1.5018 1.5475 1.5662 1.5765 1.5266 1.4599 1.4783
NQIC
OLD
0.0264 0.0256 0.0326 0.0293 0.0273 0.0256 0.0255 0.0265 0.0199 0.0260
0.0038 0.0031 0.0023 0.0014 0.0055 0.0050 0.0046 0.0059 0.0078 0.0085 0.0103 0.0088 0.0138 0.0155 0.0160 0.0133
NQIC
DOC
0.4071 0.4071 0.4019 0.3777 0.2315 0.2884 0.2477 0.2689 0.2684 0.2532
0.0325 0.0523 0.0643 0.0648 0.0510 0.0531 0.0485 0.0507 0.0356 0.0247 0.0233 0.0318 0.0149 0.0119 0.0003 0.0027
NQIC
IM
0.2027 0.1156 0.1191 0.1687 0.2353 0.3095 0.2972 0.3737 0.3550 0.5713
−0.0031 −0.0038 −0.0039 −0.0049 −0.0036 −0.0022 −0.0011 0.0046 0.0055 0.0078 0.0102 0.0088 0.0109 0.0092 0.0080 0.0074
NQIC
LE
NQIC
−0.2297 −0.3813 −1.6909 −13.7541 2.3556 1.0812 0.6183 0.2897 0.2017 0.1215
−0.0098 −0.0161 −0.0221 −0.0133 −0.0136 −0.0101 −0.0109 −0.0110 −0.0029 −0.0026 −0.0043 −0.0077 −0.0069 −0.0076 −0.0026 0.0007
−0.2932 −0.2766 −0.3000 −0.3443 −0.4950 −0.5389 −0.5362 −0.5030 −0.4026 −0.4194
Note: The NQICs for the hits are in bold.
unpredictable. Hence, Step 3 sums the number of predictable variables that we can forecast correctly, and the number of unpredictable variables that we cannot forecast correctly for performance evaluation purposes.
As for the use of the data, this study recommends a data set with more years or more variables as the sample data set because the data set tends to provide more information for which the thresholds tend to be more comprehensive than the data set with fewer years or variables. Appendix 3 defines all the variables of these two data sets.
4. Empirical analyses 4.2. Thresholds from the health care expenditure data This section begins by introducing the two data sets. There are two empirical analyses. First, we take health care expenditure to determine the thresholds for the NQICs. Then, we use the ICT development data to validate these thresholds. 4.1. Data sets This study uses two data sets to demonstrate the forecasting process and to show how the new model can improve the forecasting results. The sample data set is the health care expenditure, whose data are from Taiwan and 24 OECD countries, ranging from the years 1992 to 2007 (OECD Health Data, 2009). Chang (2010) uses this data set to study the heterogeneous effects of the health care expenditure variables. The target data set is the ICT development, which is from the World Telecommunication/ICT Indicators 2008 database by the International Telecommunication Union (ITU, 2002). There are a total of 163 economies in the analysis over the period from 1999 to 2007. Yu (2011) uses this data set to analyze the heterogeneous effects of different variables on global ICT adoption.
Following the algorithm for calculating the NQICs and thresholds, we separate the data on health care expenditure into in-sample data from 1992 to 1997 and out-of-sample data from 1998 to 2007. From the in-sample data, we obtain the quantile intervals at different τs (0.05, 0.25, 0.50, 0.75, 0.95) for t = 1992 to 1997 together to forecast the quantile coefficients for 1998. In Table 1, the quantile coefficient of LOG(GDP) for 1998 falls between the quantile intervals of 1992–1997 for all different τs. Hence, the variable LOG(GDP) for 1998 is a hit. In addition, the quantile coefficient of OLD for 1998 falls between the quantile intervals for 1992–1997 at τ = 0.05, 0.75, and 0.95, correspondingly. In other words, we can forecast the variable OLD for 1998 correctly three out of five times. The variable OLD for 1998 is a hit. Similarly, we perform the forecasts of the quantile coefficients for 1999 using those for 1992 to 1998, those for 2000 using those for 1992 to 1999, and so on. Table 2 shows all the hits in bold. We calculate all the NQICs as in Table 3. Then we use the corresponding NQICs for all the hits in Table 2 to determine the thresholds.
Table 4 The NQICs for all the variables. LOG(GDP) 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
1.4926 1.5347 1.6004 1.4801 1.4949 1.5154 1.5466 1.6122 1.4956 1.5018 1.5475 1.5662 1.5765 1.5266 1.4599 1.4783
NQIC
OLD
0.0264 0.0256 0.0326 0.0293 0.0273 0.0256 0.0255 0.0265 0.0199 0.0260
0.0038 0.0031 0.0023 0.0014 0.0055 0.0050 0.0046 0.0059 0.0078 0.0085 0.0103 0.0088 0.0138 0.0155 0.0160 0.0133
NQIC
DOC
0.4071 0.4071 0.4019 0.3777 0.2315 0.2884 0.2477 0.2689 0.2684 0.2532
0.0325 0.0523 0.0643 0.0648 0.0510 0.0531 0.0485 0.0507 0.0356 0.0247 0.0233 0.0318 0.0149 0.0119 0.0003 0.0027
NQIC
IM
0.2027 0.1156 0.1191 0.1687 0.2353 0.3095 0.2972 0.3737 0.3550 0.5713
−0.0031 −0.0038 −0.0039 −0.0049 −0.0036 −0.0022 −0.0011 0.0046 0.0055 0.0078 0.0102 0.0088 0.0109 0.0092 0.0080 0.0074
NQIC
LE
NQIC
−0.2297 −0.3813 −1.6909 −13.7541 2.3556 1.0812 0.6183 0.2897 0.2017 0.1215
−0.0098 −0.0161 −0.0221 −0.0133 −0.0136 −0.0101 −0.0109 −0.0110 −0.0029 −0.0026 −0.0043 −0.0077 −0.0069 −0.0076 −0.0026 0.0007
−0.2932 −0.2766 −0.3000 −0.3443 −0.4950 −0.5389 −0.5362 −0.5030 −0.4026 −0.4194
Note: Different fonts represent various statuses of the NQICs: Predictable/unpredictable: bold/normal. Forecast correctly/cannot forecast correctly: italics/normal.
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Table 5 The estimation results for health care expenditure with NQIC thresholds. Actual
Estimate
Correct Incorrect
Predictable
Unpredictable
24 6
13 7
Following Eq. (A1-2), we determine the thresholds for the NQICs in Table 3 as follows: −0:421800339 ≤ NQIC ≤ 0:270079048: To demonstrate the appropriateness of the thresholds, we use the thresholds to estimate the data set of health care expenditure as in Table 4. The amount of predictable with correct estimation is 24 and the amount of unpredictable with incorrect estimation is 7. The amount of predictable with incorrect estimation is 6 and the amount of unpredictable with correct estimation is 13. Table 5 displays the estimation results. Following Eq. (A2-1), the correctness is: correct ¼
24 þ 7 ¼ 62%: 50
4.3. ICT development Next we can proceed to use the thresholds to forecast the ICT development. Following the Algorithm for Forecasting on the Basis of the Thresholds, we separate the data on ICT development into insample data from 1999 to 2004 and out-of-sample data from 2005 to 2007. We calculate all the NQICs for all the variables as in Table 6. The numbers in bold are within the thresholds. We plot the forecasts as in Figs. 1 to 3. The bars in each figure represent the quantile intervals from the insample data and the dots represent the quantile coefficients to forecast. When a bar covers a dot, it means that a quantile interval can correctly forecast a quantile coefficient. When we use the thresholds to check against the ICT development, Table 7 shows the analysis. Following Eq. (A2-1), the correctness is correct ¼
5þ3 ¼ 88:89%: 9
4.4. Discussion Yu (forthcoming) proposes a QIC with an ad-hoc ceiling to exclude those extreme values as unpredictable variables. The QIC requires only one data set. However, the determination of the ceiling values for different data sets can be a problem. On the other hand, the NQIC uses
Table 6 The NQIC for population density (x6), GDP per capita (x7), and telephone lines per 100 inhabitants (x8) in ICT development. t
cx6 t
1999 2000 2001 2002 2003 2004 2005 2006 2007
0.000463 0.000505 0.000808 0.000755 0.000879 0.000992 0.000955 0.000915 0.000706
NQICx6 t
cx7 t
0.2606 0.2525 0.2390
0.000165 0.000223 0.000257 0.000269 0.000240 0.000208 0.000200 0.000185 0.000165
NQICx7 t
cx8 t
0.1510 0.1484 0.1532
0.122257 0.198924 0.249607 0.281433 0.296592 0.324865 0.357558 0.398806 0.436500
NQICx8 t
Fig. 1. The forecasts of ICT development in 2005 by the quantile regressions of 1999–2004.
one data set to generate the thresholds and uses the thresholds to determine the unpredictable variables in the other data set. The thresholds from the first data set serve to replace the ceiling values. To demonstrate the effectiveness of the NQIC, Table 8 lists the forecasting results of the QIC for comparison purposes. First, the overall performance increases from 7/9 = 78% to 8/9 = 89%. The number of unpredictable quantiles declines from 4 to 3 by using the NQIC. In addition, the forecasting performance of the predictable quantiles increases from 3 to 5 by using the NQIC. 5. Conclusion
0.276015 0.282813 0.296875
Note: Different fonts represent various statuses of the NQICs: Predictable/unpredictable: bold/normal. Forecast correctly/cannot forecast correctly: italics/normal.
This study proposes a new quantile information criterion (NQIC) and its thresholds to improve the quantile forecasting performance where the coefficient of variation serves as the NQIC. The coefficient of variation measures the dispersion of a variable and is unitless. Hence, we
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Fig. 2. The forecasts of ICT development in 2006 by the quantile regressions of 1999–2005. Fig. 3. The forecasts of ICT development in 2007 by the quantile regressions of 1999–2006.
can obtain the corresponding NQICs of different data sets. One data set (health care expenditure) can determine the thresholds for the NQICs; and the thresholds can assist in the forecasting of the other data set (ICT development). Two data sets are domain independent of each other. The thresholds of NQICs identify whether a variable is predictable and hence play an important role in improving the forecasting results. The application of quantile regression to estimation problems is very popular. Managers analyze more information from quantile regression to assist in decision making. Advancing quantile regression to forecasting can further assist managers to predict volumes or foresee problems in depth. As a result, the forecasting results from quantile models can enhance the quality of decisions. Future studies could apply the NQICs and their thresholds to other data sets to validate their feasibility. The rendering of other QICs may need to consider the balance between the performance and complexity of calculations. Regarding the unpredictable variables, we are eager to see if there is any approach to forecasting them.
Appendix 1. Algorithm for calculating the NQICs and thresholds Step 1 In the sample data set, separate the data into in-sample (t = 1 to d − 1) and out-of-sample data (t = d to d + x). Step 2 FOR idx = −1 to x. Step 2.1 Obtain the quantile intervals for all the variables at different τ using data for t = 1 to d + idx together.
Table 7 The forecasting results for ICT development with NQIC thresholds. Actual
Forecast
Correct Incorrect
Note: There are a total of 9 forecasts.
Predictable
Unpredictable
5 1
0 3
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Table 8 The forecasting results for ICT development with QIC thresholds (Yu, forthcoming).
The second data set is that for ICT development, and includes dependent (y2) and independent variables (x6 ~ x8) as follows:
Actual
Forecast
Correct Incorrect
Predictable
Unpredictable
3 0
2 4
(y2) Internet users per 100 inhabitants ¼
total ðfixedÞ Internet users ×100 population
population land area GDP (x7) GDP per capita ¼ population (x6) population density ¼
Note: There are a total of 9 forecasts.
(x8) telephone lines per 100 inhabitants ¼ Step 2.2 Calculate the NQIC as: v
NQIC t ¼
stdev cv1 ; …; cvt−1 v average c1 ; …; cvt−1
References ðA1 1Þ
where cvt − 1 represents the OLS coefficient for variable v at time t − 1. Step 2.3 Check if the quantile interval at different τs can cover more than half the amount of the corresponding quantile coefficients for t = d + idx + 1. If yes, we consider t a hit, and take NQICvt for further analysis. END_FOR Step 3 Find the maximum (max) from all the positive NQICs of all the hits and calculate their standard deviation (stdev_p). Find the minimum (min) from all the negative NQICs and their standard deviation (stdev_n). We define the thresholds as min þ stdev n ≤ NQIC ≤ max−stdev p:
ðA1 2Þ
Appendix 2. Algorithm for forecasting on the basis of the thresholds Step 1 Repeat step 1 in the algorithm for calculating the NQICs and thresholds for the target data set. Step 2 Calculate the NQICs for the out-of-sample data by using Eq. (A1-1). Step 3 The following equation reflects the correctness: correct ¼
total main lines ×100: population
#forecast the predictable þ #cannot forecast the unpredictable #overall
ðA2 1Þ where # forecast_the_predictable means the number of predictable variables that we can correctly forecast; # cannot_ forecast_the_unpredictable represents the number of unpredictable variables that we cannot forecast correctly; and #overall represents the total number of forecasts. Appendix 3. Definition of variables in two data sets This study uses two data sets. The first data set is that for health care expenditure, and includes dependent (y1) and independent variables (x1 ~ x5) as follows: (y1) growth of health care expenditures: LOG(HCE) (x1) income: LOG(GDP) (x2) the populations of elderly more than 65 years of age: OLD (x3) the number of physicians per 1000 population: DOC (x4) infant mortality: IM (x5) life expectancy: LE.
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Contents lists available at ScienceDirect
Journal of Business Research
A new quantile regression forecasting model☆ Kun-Huang Huarng ⁎, Tiffany Hui-Kuang Yu ⁎⁎ Feng Chia University, Taiwan
a r t i c l e
i n f o
Article history: Received 1 April 2013 Received in revised form 1 October 2013 Accepted 1 November 2013 Available online 13 December 2013 Keywords: Health care expenditure ICT New quantile information criterion Forecasting
a b s t r a c t Quantile regression is popular because it provides more information as well as comprehensive interpretations. To improve forecasting performance, this study proposes a new quantile information criterion (NQIC), on the basis of the coefficient of variation, and expects the NQIC to reflect whether a variable is predictable. The health care expenditure data determine the thresholds for the NQICs. The thresholds assist in forecasting the development of information and communication technology. From the empirical analyses, the NQICs and thresholds greatly improve the forecasting performance. © 2013 Elsevier Inc. All rights reserved.
1. Introduction The quantile regression model offers a more complete model than the conventional mean regression (Yu, Lu, & Stander, 2003). Studies apply the quantile regression model to interpret various problems, such as wages (Buchinsky, 1994; Machado & Mata, 2005; Martins & Pereira, 2004), survival analysis (Crowley & Hu, 1977; Koenker & Geling, 2001), financial analysis (Bassett & Chen, 2001), economic research (Hendricks & Koenker, 1992; Wang, Yu, & Liu, 2013), the study of the environment (Pandey & Nguyen, 1999), internet and communication technology (ICT) adoption (Yu, 2011), health care expenditure (Yu, Wang, & Chang, 2011), small business performance (Seo, Perry, Tomczyk, & Solomon, 2014), and so on. A number of studies advance the quantile regression model to forecasting. Granger, White, and Kamstra (1989) propose a method for combining the variety of possible interval forecasts on the basis of quantile regression techniques. Taylor (2007) forecasts the daily supermarket sales using exponentially weighted quantile regression (interval forecasts), which outperforms traditional methods. Banachewicz and Lucas (2008) use hidden Markov models to forecast the quantiles of ☆ The authors acknowledge and are grateful for the financial support provided by the National Science Council, Taiwan, ROC under grants NSC 101-2410-H-035-004-, NSC 100-2410-H-035-006-MY2, NSC 102-2410-H-035-038-MY2, and NSC 102-2410-H-035041-MY2. The authors would also like to thank several attendees at the 2013 Conference of the Global Innovation and Knowledge Academy for their valuable comments that have helped improve this article. ⁎ Correspondence to: K.-H. Huarng, Department of International Trade, Feng Chia University, 100 Wenhwa Road, Seatwen, Taichung 40724, Taiwan. ⁎⁎ Correspondence to: T. H.-K. Yu, Department of Public Finance, Feng Chia University, Taichung, Taiwan. E-mail addresses: [email protected] (K.-H. Huarng), [email protected] (T.H.-K. Yu). 0148-2963/$ – see front matter © 2013 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jbusres.2013.11.044
corporate default rates, which are important in financial risk management. Gerlacha, Chen, and Chan (2011) apply Markov chain Monte Carlo methods for the Bayesian time-varying quantile forecasting of Value-at-Risk in financial markets. Cai, Stander, and Davies (2012) propose a Bayesian approach to quantile autoregressive time series model estimation and forecasting and then apply the approach to currency exchange rate data. The empirical results show that an unequally weighted combining method outperforms other forecasting methodology. Yu (forthcoming) suggests using a quantile information criterion (QIC) to assist in forecasting. To improve forecasting performance, this study proposes a new QIC (NQIC) to identify if a variable is predictable. The health care expenditure data are in order to determine the thresholds for the NQICs. Then, the thresholds and the NQICs intend to forecast the ICT development. To that end, Section 2 reviews the concepts of the quantile regression model. Section 3 introduces the algorithms for the NQICs and determining their thresholds. Section 4 describes the variables of the two data sets, provides the empirical analyses for the two cases, and reveals the results of the estimation and forecasting performance. Section 5 concludes this paper. 2. Quantile regression model Koenker and Bassett (1978) propose quantile regression to infer the results of the conditional functions for different quantiles. Bao, Lee, and Saltoğlu (2006) consider that the main advantage of quantile regression is to provide better statistics by means of the empirical quantiles. Quantile regression can help “complete the picture” when we intend to understand the relationship between variables for which the effects may vary with outcome levels. In addition, quantile regression is more forgiving than ordinary least squares in that quantile regression is
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relatively insensitive to outliers and can avoid censoring problems (Conley & Galenson, 1998). As Bassett and Koenker (1982) extend the median to quantile regression so as to calculate various quantiles, the quantile regression does not require any distribution assumptions regarding the population and can estimate the parameters nonparametrically. Quantile regression models the conditional quantiles, which are quantiles of the conditional distribution of the response variable in the expression of functions of the covariates of observations. Quantile regression models use the least absolute deviations method to minimize the absolute values of the errors. The model for a median linear regression is: yi ¼ xi βθ þ εθ;i
ð1Þ
where the assumption is median (εθ,i|xi) = 0. This concept is extendable to any quantile, such as the 75th percentile, 95th percentile, etc. We can define the estimate by minimizing the sum of asymmetrically weighted residuals. 2 min 4 β
X
α jyt −xt βj þ
tjyt ≥ xt β
X
3 ð1−α Þjyt −xt βj5
ð2Þ
tjyt b xt β
where α is a parameter (0 b α b 1) that represents the size of the quantile, and is also the quantile α of the explanatory variable that we intend to examine in the quantile regressions. This problem does not have an explicit form; however, linear programming methods can solve the problem (Armstrong, Frome, & Kung, 1979). When α = 0.5, the quantile regression is the median regression. Since on this occasion the values of α and (1 − α) are both 0.5, the above equation changes to ∑ jyt −xt βj, indicating that the observations above and below the medi-
Table 1 Quantile forecasting of 1998 by using 1992–1997 health care expenditure data. 1992–1997
1998
0.05 LOG(GDP) OLD DOC IM LE
0.497261 0.013379 −0.012920 −0.008170 −0.019490
1.390759 0.027361 0.031001 −0.003110 0.026669
1.34671 0.01405 0.03511 −0.00234 −0.01954
0.25 LOG(GDP) OLD DOC IM LE
1.056123 0.005542 0.025187 −0.007550 −0.017050
1.370417 0.015938 0.054073 −0.003070 0.006608
1.12976 0.00192 0.04980 −0.00262 0.01077
0.50 LOG(GDP) OLD DOC IM LE
1.209432 0.001473 0.044355 −0.009780 −0.02126
1.577748 0.011047 0.088145 0.001800 0.000844
1.33037 0.01145 0.04925 −0.00216 −0.00336
0.75 LOG(GDP) OLD DOC IM LE
1.391142 −0.006450 0.031638 −0.003990 −0.030450
1.818118 0.013389 0.091582 0.007852 −0.006490
1.52785 0.00933 0.01192 0.00090 −0.00700
0.95 LOG(GDP) OLD DOC IM LE
1.579464 −0.041320 0.005280 −0.024830 −0.055570
1.983476 0.010996 0.128820 0.010607 −0.015770
1.92039 −0.00434 0.07328 −0.00237 −0.03289
y
an values are of the same weights. 3. New quantile information criterion 3.1. Rationale Not all the variables are predictable by using quantile regressions due to their data characteristics. To identify if they are predictable, Yu (forthcoming) proposes a quantile information criterion (QIC). This study proposes a New QIC (NQIC) to provide a systematic method to improve the forecasting results. To measure the dispersion of a distribution, this study applies the coefficient of variation as the NQIC, which is the ratio of the standard deviation and mean of a variable (Lind, Marchal, & Wathen, 2006). The coefficient of variation is unitless and is most useful in comparing the variability of different data sets (Rosner, 1995). For example, Bloch (2007) uses the coefficient of variation to test if the coefficients of two data sets are different. Yu (forthcoming) considers that the variables with large variations are difficult to forecast. Following these studies, this study first intends to identify the extreme values of the coefficients of variation of a data set to form the thresholds. Furthermore, the thresholds are in order to determine whether the variables in the other data set are unpredictable with the coefficients of variation lying outside the range of the thresholds. 3.2. Algorithms There are two data sets: the one for obtaining the thresholds is the sample data set and the other is the target data set. Following the above rationale, this section proposes the algorithm for calculating the NQICs and their corresponding thresholds, and the algorithm for forecasting on the basis of the thresholds. We list the algorithms in Appendices 1 and 2.
The algorithm for calculating the NQICs and thresholds serves to calculate the thresholds from the sample data set. Step 1 separates the data into in-sample and out-of-sample data. The calculation starts with the in-sample data ranging from t = 1 to d − 1 to test the out-of-sample data of t = d. The next run moves one step further; in other words, the calculation starts with the in-sample data from t = 1 to d to test the out-of-sample data of t = d + 1. We continue the process until we exhaust all the out-of-sample data. Step 2 calculates the NQICs and then checks if the corresponding quantile intervals can cover more than 50% of the corresponding variable in the out-of-sample data. Only those with more than 50% coverage (which we name hits) can advance to the calculation for the thresholds. Step 3 finds the maximum (max) from all the positive NQICs of all the hits and calculates their standard deviation (stdev_p). Similarly, it finds the minimum (min) from all the negative NQICs and their standard deviation (stdev_n). To be conservative, we narrow the range between the minimum and maximum by one standard deviation, respectively. In the algorithm for forecasting on the basis of the thresholds, we also separate the data into in-sample as well as out-of-sample data for the target data set in Step 1 of the previous algorithm. Step 2 calculates the NQICs for the out-of-sample data. During forecasting, if any NQICvt falls within the thresholds, the algorithm considers the variable v at time t predictable; otherwise Table 2 The hits for all the years. Year
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
LOG(GDP) OLD DOC IM LE
5 3 3 3 3
4 3 4 1 5
5 4 3 2 3
5 3 3 1 2
4 4 3 1 3
3 3 3 2 3
3 4 3 2 4
5 5 5 5 5
2 3 0 2 2
4 3 1 2 3
Note: The numbers in bold are the hits.
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Table 3 The NQICs for all the variables. LOG(GDP) 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
1.4926 1.5347 1.6004 1.4801 1.4949 1.5154 1.5466 1.6122 1.4956 1.5018 1.5475 1.5662 1.5765 1.5266 1.4599 1.4783
NQIC
OLD
0.0264 0.0256 0.0326 0.0293 0.0273 0.0256 0.0255 0.0265 0.0199 0.0260
0.0038 0.0031 0.0023 0.0014 0.0055 0.0050 0.0046 0.0059 0.0078 0.0085 0.0103 0.0088 0.0138 0.0155 0.0160 0.0133
NQIC
DOC
0.4071 0.4071 0.4019 0.3777 0.2315 0.2884 0.2477 0.2689 0.2684 0.2532
0.0325 0.0523 0.0643 0.0648 0.0510 0.0531 0.0485 0.0507 0.0356 0.0247 0.0233 0.0318 0.0149 0.0119 0.0003 0.0027
NQIC
IM
0.2027 0.1156 0.1191 0.1687 0.2353 0.3095 0.2972 0.3737 0.3550 0.5713
−0.0031 −0.0038 −0.0039 −0.0049 −0.0036 −0.0022 −0.0011 0.0046 0.0055 0.0078 0.0102 0.0088 0.0109 0.0092 0.0080 0.0074
NQIC
LE
NQIC
−0.2297 −0.3813 −1.6909 −13.7541 2.3556 1.0812 0.6183 0.2897 0.2017 0.1215
−0.0098 −0.0161 −0.0221 −0.0133 −0.0136 −0.0101 −0.0109 −0.0110 −0.0029 −0.0026 −0.0043 −0.0077 −0.0069 −0.0076 −0.0026 0.0007
−0.2932 −0.2766 −0.3000 −0.3443 −0.4950 −0.5389 −0.5362 −0.5030 −0.4026 −0.4194
Note: The NQICs for the hits are in bold.
unpredictable. Hence, Step 3 sums the number of predictable variables that we can forecast correctly, and the number of unpredictable variables that we cannot forecast correctly for performance evaluation purposes.
As for the use of the data, this study recommends a data set with more years or more variables as the sample data set because the data set tends to provide more information for which the thresholds tend to be more comprehensive than the data set with fewer years or variables. Appendix 3 defines all the variables of these two data sets.
4. Empirical analyses 4.2. Thresholds from the health care expenditure data This section begins by introducing the two data sets. There are two empirical analyses. First, we take health care expenditure to determine the thresholds for the NQICs. Then, we use the ICT development data to validate these thresholds. 4.1. Data sets This study uses two data sets to demonstrate the forecasting process and to show how the new model can improve the forecasting results. The sample data set is the health care expenditure, whose data are from Taiwan and 24 OECD countries, ranging from the years 1992 to 2007 (OECD Health Data, 2009). Chang (2010) uses this data set to study the heterogeneous effects of the health care expenditure variables. The target data set is the ICT development, which is from the World Telecommunication/ICT Indicators 2008 database by the International Telecommunication Union (ITU, 2002). There are a total of 163 economies in the analysis over the period from 1999 to 2007. Yu (2011) uses this data set to analyze the heterogeneous effects of different variables on global ICT adoption.
Following the algorithm for calculating the NQICs and thresholds, we separate the data on health care expenditure into in-sample data from 1992 to 1997 and out-of-sample data from 1998 to 2007. From the in-sample data, we obtain the quantile intervals at different τs (0.05, 0.25, 0.50, 0.75, 0.95) for t = 1992 to 1997 together to forecast the quantile coefficients for 1998. In Table 1, the quantile coefficient of LOG(GDP) for 1998 falls between the quantile intervals of 1992–1997 for all different τs. Hence, the variable LOG(GDP) for 1998 is a hit. In addition, the quantile coefficient of OLD for 1998 falls between the quantile intervals for 1992–1997 at τ = 0.05, 0.75, and 0.95, correspondingly. In other words, we can forecast the variable OLD for 1998 correctly three out of five times. The variable OLD for 1998 is a hit. Similarly, we perform the forecasts of the quantile coefficients for 1999 using those for 1992 to 1998, those for 2000 using those for 1992 to 1999, and so on. Table 2 shows all the hits in bold. We calculate all the NQICs as in Table 3. Then we use the corresponding NQICs for all the hits in Table 2 to determine the thresholds.
Table 4 The NQICs for all the variables. LOG(GDP) 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007
1.4926 1.5347 1.6004 1.4801 1.4949 1.5154 1.5466 1.6122 1.4956 1.5018 1.5475 1.5662 1.5765 1.5266 1.4599 1.4783
NQIC
OLD
0.0264 0.0256 0.0326 0.0293 0.0273 0.0256 0.0255 0.0265 0.0199 0.0260
0.0038 0.0031 0.0023 0.0014 0.0055 0.0050 0.0046 0.0059 0.0078 0.0085 0.0103 0.0088 0.0138 0.0155 0.0160 0.0133
NQIC
DOC
0.4071 0.4071 0.4019 0.3777 0.2315 0.2884 0.2477 0.2689 0.2684 0.2532
0.0325 0.0523 0.0643 0.0648 0.0510 0.0531 0.0485 0.0507 0.0356 0.0247 0.0233 0.0318 0.0149 0.0119 0.0003 0.0027
NQIC
IM
0.2027 0.1156 0.1191 0.1687 0.2353 0.3095 0.2972 0.3737 0.3550 0.5713
−0.0031 −0.0038 −0.0039 −0.0049 −0.0036 −0.0022 −0.0011 0.0046 0.0055 0.0078 0.0102 0.0088 0.0109 0.0092 0.0080 0.0074
NQIC
LE
NQIC
−0.2297 −0.3813 −1.6909 −13.7541 2.3556 1.0812 0.6183 0.2897 0.2017 0.1215
−0.0098 −0.0161 −0.0221 −0.0133 −0.0136 −0.0101 −0.0109 −0.0110 −0.0029 −0.0026 −0.0043 −0.0077 −0.0069 −0.0076 −0.0026 0.0007
−0.2932 −0.2766 −0.3000 −0.3443 −0.4950 −0.5389 −0.5362 −0.5030 −0.4026 −0.4194
Note: Different fonts represent various statuses of the NQICs: Predictable/unpredictable: bold/normal. Forecast correctly/cannot forecast correctly: italics/normal.
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Table 5 The estimation results for health care expenditure with NQIC thresholds. Actual
Estimate
Correct Incorrect
Predictable
Unpredictable
24 6
13 7
Following Eq. (A1-2), we determine the thresholds for the NQICs in Table 3 as follows: −0:421800339 ≤ NQIC ≤ 0:270079048: To demonstrate the appropriateness of the thresholds, we use the thresholds to estimate the data set of health care expenditure as in Table 4. The amount of predictable with correct estimation is 24 and the amount of unpredictable with incorrect estimation is 7. The amount of predictable with incorrect estimation is 6 and the amount of unpredictable with correct estimation is 13. Table 5 displays the estimation results. Following Eq. (A2-1), the correctness is: correct ¼
24 þ 7 ¼ 62%: 50
4.3. ICT development Next we can proceed to use the thresholds to forecast the ICT development. Following the Algorithm for Forecasting on the Basis of the Thresholds, we separate the data on ICT development into insample data from 1999 to 2004 and out-of-sample data from 2005 to 2007. We calculate all the NQICs for all the variables as in Table 6. The numbers in bold are within the thresholds. We plot the forecasts as in Figs. 1 to 3. The bars in each figure represent the quantile intervals from the insample data and the dots represent the quantile coefficients to forecast. When a bar covers a dot, it means that a quantile interval can correctly forecast a quantile coefficient. When we use the thresholds to check against the ICT development, Table 7 shows the analysis. Following Eq. (A2-1), the correctness is correct ¼
5þ3 ¼ 88:89%: 9
4.4. Discussion Yu (forthcoming) proposes a QIC with an ad-hoc ceiling to exclude those extreme values as unpredictable variables. The QIC requires only one data set. However, the determination of the ceiling values for different data sets can be a problem. On the other hand, the NQIC uses
Table 6 The NQIC for population density (x6), GDP per capita (x7), and telephone lines per 100 inhabitants (x8) in ICT development. t
cx6 t
1999 2000 2001 2002 2003 2004 2005 2006 2007
0.000463 0.000505 0.000808 0.000755 0.000879 0.000992 0.000955 0.000915 0.000706
NQICx6 t
cx7 t
0.2606 0.2525 0.2390
0.000165 0.000223 0.000257 0.000269 0.000240 0.000208 0.000200 0.000185 0.000165
NQICx7 t
cx8 t
0.1510 0.1484 0.1532
0.122257 0.198924 0.249607 0.281433 0.296592 0.324865 0.357558 0.398806 0.436500
NQICx8 t
Fig. 1. The forecasts of ICT development in 2005 by the quantile regressions of 1999–2004.
one data set to generate the thresholds and uses the thresholds to determine the unpredictable variables in the other data set. The thresholds from the first data set serve to replace the ceiling values. To demonstrate the effectiveness of the NQIC, Table 8 lists the forecasting results of the QIC for comparison purposes. First, the overall performance increases from 7/9 = 78% to 8/9 = 89%. The number of unpredictable quantiles declines from 4 to 3 by using the NQIC. In addition, the forecasting performance of the predictable quantiles increases from 3 to 5 by using the NQIC. 5. Conclusion
0.276015 0.282813 0.296875
Note: Different fonts represent various statuses of the NQICs: Predictable/unpredictable: bold/normal. Forecast correctly/cannot forecast correctly: italics/normal.
This study proposes a new quantile information criterion (NQIC) and its thresholds to improve the quantile forecasting performance where the coefficient of variation serves as the NQIC. The coefficient of variation measures the dispersion of a variable and is unitless. Hence, we
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Fig. 2. The forecasts of ICT development in 2006 by the quantile regressions of 1999–2005. Fig. 3. The forecasts of ICT development in 2007 by the quantile regressions of 1999–2006.
can obtain the corresponding NQICs of different data sets. One data set (health care expenditure) can determine the thresholds for the NQICs; and the thresholds can assist in the forecasting of the other data set (ICT development). Two data sets are domain independent of each other. The thresholds of NQICs identify whether a variable is predictable and hence play an important role in improving the forecasting results. The application of quantile regression to estimation problems is very popular. Managers analyze more information from quantile regression to assist in decision making. Advancing quantile regression to forecasting can further assist managers to predict volumes or foresee problems in depth. As a result, the forecasting results from quantile models can enhance the quality of decisions. Future studies could apply the NQICs and their thresholds to other data sets to validate their feasibility. The rendering of other QICs may need to consider the balance between the performance and complexity of calculations. Regarding the unpredictable variables, we are eager to see if there is any approach to forecasting them.
Appendix 1. Algorithm for calculating the NQICs and thresholds Step 1 In the sample data set, separate the data into in-sample (t = 1 to d − 1) and out-of-sample data (t = d to d + x). Step 2 FOR idx = −1 to x. Step 2.1 Obtain the quantile intervals for all the variables at different τ using data for t = 1 to d + idx together.
Table 7 The forecasting results for ICT development with NQIC thresholds. Actual
Forecast
Correct Incorrect
Note: There are a total of 9 forecasts.
Predictable
Unpredictable
5 1
0 3
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Table 8 The forecasting results for ICT development with QIC thresholds (Yu, forthcoming).
The second data set is that for ICT development, and includes dependent (y2) and independent variables (x6 ~ x8) as follows:
Actual
Forecast
Correct Incorrect
Predictable
Unpredictable
3 0
2 4
(y2) Internet users per 100 inhabitants ¼
total ðfixedÞ Internet users ×100 population
population land area GDP (x7) GDP per capita ¼ population (x6) population density ¼
Note: There are a total of 9 forecasts.
(x8) telephone lines per 100 inhabitants ¼ Step 2.2 Calculate the NQIC as: v
NQIC t ¼
stdev cv1 ; …; cvt−1 v average c1 ; …; cvt−1
References ðA1 1Þ
where cvt − 1 represents the OLS coefficient for variable v at time t − 1. Step 2.3 Check if the quantile interval at different τs can cover more than half the amount of the corresponding quantile coefficients for t = d + idx + 1. If yes, we consider t a hit, and take NQICvt for further analysis. END_FOR Step 3 Find the maximum (max) from all the positive NQICs of all the hits and calculate their standard deviation (stdev_p). Find the minimum (min) from all the negative NQICs and their standard deviation (stdev_n). We define the thresholds as min þ stdev n ≤ NQIC ≤ max−stdev p:
ðA1 2Þ
Appendix 2. Algorithm for forecasting on the basis of the thresholds Step 1 Repeat step 1 in the algorithm for calculating the NQICs and thresholds for the target data set. Step 2 Calculate the NQICs for the out-of-sample data by using Eq. (A1-1). Step 3 The following equation reflects the correctness: correct ¼
total main lines ×100: population
#forecast the predictable þ #cannot forecast the unpredictable #overall
ðA2 1Þ where # forecast_the_predictable means the number of predictable variables that we can correctly forecast; # cannot_ forecast_the_unpredictable represents the number of unpredictable variables that we cannot forecast correctly; and #overall represents the total number of forecasts. Appendix 3. Definition of variables in two data sets This study uses two data sets. The first data set is that for health care expenditure, and includes dependent (y1) and independent variables (x1 ~ x5) as follows: (y1) growth of health care expenditures: LOG(HCE) (x1) income: LOG(GDP) (x2) the populations of elderly more than 65 years of age: OLD (x3) the number of physicians per 1000 population: DOC (x4) infant mortality: IM (x5) life expectancy: LE.
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