A variable P value rolling Grey forecasting model for Taiwan semiconductor industry production

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A variable P value rolling Grey forecasting model for Taiwan

semiconductor industry production

Shih-Chi Chang

a,b,

*, Hsien-Che Lai

a

, Hsiao-Cheng Yu

a

a_{Institute of Management of Technology, College of Management, National Chiao-Tung University,} Hsinchu 300, Taiwan

b_{Department of International Trade, Ta Tung Institute of Technology, Chiayi 600, Taiwan} Received 24 July 2003; received in revised form 1 September 2003; accepted 5 September2003

Abstract

The semiconductor industry plays an important role in Taiwan’s economy. In this paper, we constructed a rolling Grey forecasting model (RGM) to predict Taiwan’s annual semiconductor production. The univariate Grey forecasting model (GM) makes forecast of a time series of data without considering possible correlation with any leading indicators. Interestingly, within the RGM there is a constant, P value, which was customarily set to 0.5. We hypothesized that making the P value a variable of time could generate more accurate forecasts. It was expected that the annual semiconductor production in Taiwan should be closely tied with U.S. demand. Hence, we let the P value be determined by the yearly percent change in real gross domestic product (GDP) by U.S. manufacturing industry. This variable P value RGM generated better forecasts than the fixed P value RGM. Nevertheless, the yearly percent change in real GDP by U.S. manufacturing industry is reported after a year ends. It cannot serve as a leading indicator for the same year’s U.S. demand. We found out that the correlation between the yearly survey of anticipated industrial production growth rates in Taiwan and the yearly percent changes in real GDP by U.S. manufacturing industry has a correlation coefficient of 0.96. Therefore, we used the former to determine the P value in the RGM, which generated very accurate forecasts.

Keywords: Grey forecasting; RGM; Semiconductor industry

* Corresponding author. Tel.: +886-3-571-2121x57520; fax: +886-3-572-6749. E-mail address: chi.mt90g@nctu.edu.tw (S.-C. Chang).

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1. Introduction

Forecasting of total production revenue in high-tech industries is useful for companies to prepare marketing strategies/production capacity planning and for financial institutions to make investment decisions. However, volatility, cyclic, and vulnerability to technology enhancements and demand shifts are common characteristics in high-tech industries. Therefore, production forecasting in high-technology industries, like the semiconductor industry, is much more difficult than in traditional industries, such as food manufacturing[1].

Industry practitioners use both quantitative and qualitative methods[2]to do forecasting. Qualitative forecasting methods include the expert system, the Delphi method, etc. Quantitative forecasting methods include regression analysis, time series analysis, exponential smoothing, neural networks, and Grey forecasting model (GM).

Several studies have proposed time series models for industrial production and revealed the applicability of time series models to industrial production forecasting [3–6]. These methods typically require large amounts of data to construct the forecasting model. Hsu [7] showed that the GM requires minimal data and is the best among all existing models for short-term predictions. Xu and Wen [8] applied the GM to accurately forecast the number of passengers on international air transportation. Lin and Wang [9] used the model to predict the number of engineering officers for a ship in Taiwan and showed that the GM has higher prediction validity than the Markov chain model. Lin and Yang [10] also applied the GM to forecast Taiwan optical-electronics industry production. Because GM can work with as few as four observations, it is an ideal model to forecast Taiwan semiconductor industry production based on data only in the past decade.

A rolling Grey forecasting model (RGM) reconstructs itself whenever a new observation rolls in. RGM takes advantage of the latest information in updating a forecasting model. The original univariate RGM has the limitation of making forecast of a time series of data without considering possible correlation with any leading indicators. Within the RGM, there is a constant, P value, which was customarily set to 0.5. It is a hypothesis of this paper to make this P value a variable of time and shall be determined by a leading indicator. Because the annual semiconductor industry production in Taiwan is closely tied with U.S. demand, we hypothesized that the yearly percent change in real gross domestic product (GDP) by U.S. manufacturing industry could be a good representation of the U.S. demand change. We let the P value be affected by the yearly percent change in real GDP by U.S. manufacturing industry. This variable P value RGM did generate better forecast than RGM model with P value fixed at 0.5. Nevertheless, the yearly percent change in real GDP by U.S. manufacturing industry is not a leading indicator for the same year’s U.S. demand.

We found out that the correlation between the yearly survey of anticipated industrial production growth rates in Taiwan and the yearly percent change in real GDP by U.S. manufacturing industry has a high correlation coefficient of 0.96. Fortunately, the former is a leading indicator because it is done at the end of each year and publicized at the beginning of the next year. Therefore, we used the yearly survey of anticipated industrial production growth rates in Taiwan to determine the P value in the RGM. The forecasts of the variable P value RGM (1,1) were superior to those of a fixed P value RGM (1,1) as well as to the forecasts made by the industry watchers in Industrial Technology Research Institute, Taiwan.

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Fig. 1. The production of T aiwan’ s semiconductor industry .

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This paper is organized as follows. Section 2 gives a brief introduction to the semiconductor industry in Taiwan. Section 3 reviews of RGM (1,1). Section 4 describes production forecast by RGM (1,1). Section 5 presents the variable P value RGM (1,1), and Section 6 gives the conclusions.

2. The semiconductor industry in Taiwan

Taiwan’s semiconductor industry has grown and evolved mainly over the past 20 years. The overall semiconductor production revenue reached NTD 652.9 billion in 2002, equivalent to US$18.65 billion calculated at the exchange rate of 35 NTD to 1 dollar. In the rest of this paper, forecasts are all based on the currency of NTD. In 2001, Taiwan semiconductor industry consists of more than 100 design companies, 20 firms producing wafers, over 40 packaging firms, and 30 testing firms. Taiwan Semiconductor Manufacturing (TSMC) and United Microelectronics (UMC) have become the number 1 and 2 IC foundry operators in the world. Fig. 1 shows Taiwan’s semiconductor industry production from 1994 to 2002 [11].

3. Rolling GM (1,1)

Grey forecasting requires only few observations for model construction [7,12–15]. This section briefly reviews the RGM to illustrate the method used to construct the model we adopted by creating a sequence of one-order linear moving[10,14,16].

The data sequence is given as {x(0)(k), 1 V k V n}. Assume that x(0)(k) is positive for all k. Let r denote the length of the rolling interval. Suppose we wish to forecast x(0) at time t0,

r + 1 V t0Vn. The one-step-ahead RGM (1,1) means that we forecast the outcome at time t0

based on r observations occurring before t0. The first-order differential equation for the model

is:

dXð1Þ dt þ aX

ð1Þ_{¼ b} _ð1Þ

where t denotes the independent variables in the system, a represents the developed coefficient, and b is the Grey controlled variable. The parameters to be determined in the model are a and b. The primitive sequence used to build the model to forecast the output at time t0 is:

xð0Þði; kÞ ¼ ½xð0ÞðiÞ; xð0Þði þ 1Þ; xð0Þði þ 2Þ; . . . ; xð0ÞðkÞ; where r ¼ k i þ 1 ð2Þ

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In constructing the model, the Grey system must apply a one-order accumulated generating operation (AGO) to the primitive sequence to provide the middle message for building a model and to weaken the tendency toward variation. The AGO of x(0)(i;k) is defined as x(1)(i;k). That is,

xð1Þði; kÞ ¼ ½xð1ÞðiÞ; xð1Þði þ 1Þ; xð1Þði þ 2Þ; . . . ; xð1ÞðkÞ

¼ X i j¼i xð0ÞðjÞ;X iþ1 j¼i xð0ÞðjÞ;X iþ2 j¼i xð0ÞðjÞ; . . . ;X k j¼i xð0ÞðjÞ ! ð3Þ

From Eqs. (1) and (3) and the ordinary least-square method, coefficient aˆ becomes:

ˆa¼ a b 2 4 3 5 ¼ ðBT_BÞ1 BTYN ð4Þ

Furthermore, the accumulated matrix B is:

B¼ ½Pxð1ÞðiÞ þ ð1 PÞxð1Þði þ 1Þ 1 ½Pxð1Þ_{ði þ 1Þ þ ð1 PÞx}ð1Þ_{ði þ 2Þ} ₁ ] ] ½Pxð1Þ_{ðk 1Þ þ ð1 PÞx}ð1Þ_ðkÞ ₁ 2 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 5

where P is equal to 0.5 in the original model. The constant vector YN is:

YN ¼ ½xð0Þði þ 1Þ; xð0Þði þ 2Þ; . . . ; xð0ÞðkÞT

The approximate relationship can be obtained by substituting aˆ obtained in the differential equation and solving Eq. (1) as follows:

ˆxð1Þðt þ 1Þ ¼ xð0Þð1Þ b a

eatþb

a ð5Þ

When ˆx(1)(1) = ˆx(0)(1), the sequence one-order inverse-accumulated generating operation (IAGO) is acquired. The sequence that must be reduced as Eq. (6) can be obtained.

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Given t= 1, 2,. . ., k, obtain the sequence of reduction as follows: ˆ

Xð0Þði; kÞ ¼ ðˆxð0Þð1Þ; ˆxð0Þð2Þ; . . . ; ˆxð0Þðk þ 1ÞÞ ð7Þ

where ˆx(0)(k + 1) is the Grey elementary forecasting value for ˆx(0)(k + 1).

To obtain the forecast of xˆ(0)(t) for each r + 1 V t V n, repeat the above procedure systematically. After generating and developing the model, further tests are necessary to understand the error between the forecasted value and the actual value. To demonstrate the efficiency of the proposed forecasting model, we adopted the residual error test to compare the actual value and forecasted value. e(k + 1) is the residual percentage between the forecasted value and actual value when the model at tk + 1. To calculate

the residual percentage, we used:

eðk þ 1Þ ¼ x ð0Þ_{ðk þ 1Þ ˆx}ð0Þ_{ðk þ 1Þ} xð0Þ_{ðk þ 1Þ} 100%; k þ 1Vn ð8Þ

As the forecasting model adopted in this paper is a rolling model, the model will be reconstructed when a new observation rolls in. The RGM is then a reasonable method for gaining the time varying tendency because the RGM parameters are updated continuously with time. The average residual error, e, is the RGM accuracy assuming that the data adopted in the RGM have four points. We used Eq. (9) to compute the model accuracy.

e¼ 1 n 4

Xn1 k¼4

eðk þ 1Þ% ð9Þ

4. Forecast Taiwan semiconductor production with RGM (1,1)

The data adopted in this paper were from ITRI, the most preeminent research institute in the Taiwan semiconductor industry.

According to RGM (1,1) and the data adopted (in million NTD), this study forecasted Taiwan semiconductor industry production from 1998 to 2002 in the following procedures:

Obtain the data sequence from year 1994 to 2002 as:

Xð0Þ¼ ð1019; 1720; 1882; 2479; 2834; 4235; 7144; 5269; 6529Þ

Set the length of the rolling interval as 4. From Eq. (2), we obtain the primitive sequence x(0)(1;4) as

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From Eq. (3), we obtain the one-order AGO sequence x(1)(1;4) as xð1Þð1; 4Þ ¼ ð1019; 2739; 4621; 7100Þ

Accumulate matrix B and constant vector YNas follows when P is equal to 0.5:

B¼ 1879 1 3680 1 5860:5 1 2 6 6 6 6 4 3 7 7 7 7 5 YN ¼ 1720 1882 2479 2 6 6 6 6 4 3 7 7 7 7 5

From Eq. (4) then solve aˆ to get

ˆa¼ a b 2 4 3 5 ¼ 0:19352 1290:4 2 4 3 5

From Eq. (5), acquire the forecasting model as

ˆxð1Þðt þ 1Þ ¼ xð0Þð1Þ 1290:4 ð0:19352Þ

eð0:19352Þtþ 1290:4 ð0:19352Þ

From Eq. (6), obtain the forecasting value for the year 1998 (t = 4) as

ˆxð0Þð4 þ 1Þ ¼ ˆxð1Þð4 þ 1Þ ˆxð1Þð4Þ ¼ 10 002 7069 ¼ 2933

According to the data sequence, we acquire five primitive sequences when the length of the rolling interval is set at four. By repeating the above procedures, we obtained the forecasted semiconductor industry production from 1998 to 2002. From Eqs. (8) and (9), we calculated and compared the absolute value of the residual errors and the average residual error of RGM (1,1), the results were shown in Table 1.

ITRI releases yearly production forecasts for the semiconductor industry at the beginning of each year [17–21]. Based on the forecasts, RGM (1,1) is better than ITRI’s forecast for 1998 and 2002. Although ITRI’s forecast outperformed RGM, RGM is a much simpler and more economical

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compared to ITRI’s forecast based on ITRI experts’ industry experience and domain knowledge.

Table 2 shows the forecasts of ITRI. The average residual error of ITRI’s forecast is 21.27%, which was smaller than 30.24% of RGM.

5. Forecast Taiwan semiconductor production with variable P value RGM (1,1)

From Table 1, we found that the RGM (1,1) forecast for 2001 deviated obviously from the actual value. The September 11 attacks decreased the market demand resulting in an unexpected downturn in the semiconductor industry. What is more important is that the original RGM (1,1) was based only on past data without considering the economic factors in the forecasted year. In other words, if we take the leading factor into RGM consideration, the accuracy of the RGM forecast could be significantly improved. We therefore make the P value a variable of time, a parameter that influences matrix B in Eq. (4). To construct the leading factor and P value relationship, the first step is to determine the best P value. We experimented a series of P values from 0.1 to 0.9 to determine and then put these values into the forecasting model to obtain the forecasts. We selected the best P value that generated the smallest residual errors to be the one for our variable P value RGM (1,1). Appendix A shows the results of such

Table 2

Forecasts of ITRI

ITRI

Forecast value Residual percentage

1998 3348 18.14

1999 3523 16.81

2000 6753 5.47

2001 8545 62.17

2002 6283 3.77

Average residual error 21.27

Table 1

Forecasts of RGM (1,1)

Year aˆ Actual

value Forecast value Residual percentage 1998 a = 0.19352, b = 1290.4 2834 2933 3.49 1999 a = 0.19455, b = 1425.6 4235 3483 17.75 2000 a = 0.28968, b = 1424.1 7144 5447 23.76 2001 a = 0.47141, b = 896.2 5269 10 851 105.93 2002 a = 0.08104, b = 4673.0 6529 6512 0.25

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experiments. The optimal P values from 1998 to 2002 are 0.4, 0.9, 0.9, 0.1, and 0.5, respectively.

The critical point for improving the RGM (1,1) accuracy is how to determine the appropriate P value. Because the annual semiconductor industry production in Taiwan is closely tied with U.S. demand, we hypothesized that the yearly percent change in real GDP by U.S. manufacturing industry [22] could be a good representation of the U.S. demand change. The yearly percent changes in real GDP by U.S. manufacturing industry from 1998 to 2002 are 4.12, 4.82, 4.72, 6.00, and 1.80, respectively. According to the data, we constructed the relationship between P value ( P) and the yearly percent change in real GDP by U.S. manufacturing industry (m) in Eq. (10).

P¼ 0:441813 þ 0:062467m ð10Þ

We also used the P values figured out from Eq. (10) to make forecasts. The average residual error was 12.55%, better than 21.27% of ITRI’s forecast and 30.24% from the fixed P value RGM (1,1).Table 3

shows the forecasts. Table 3

Forecasts of variable P value RGM (1,1)

Variable P value RGM (1,1)a Variable P value RGM (1,1)b P value Forecasted value Residual error (%) P value Forecasted value Residual error (%) 1998 0.70 3132 10.52 0.59 3020 6.56 1999 0.74 3775 10.86 0.72 3749 11.48 2000 0.74 6298 11.84 0.80 6546 8.37 2001 0.07 6811 29.27 0.04 6624 25.72 2002 0.55 6511 0.28 0.66 6498 0.48

Average residual error 12.55 10.52

a

P value figured out from Eq. (10). b _{P value figured out from Eq. (11).}

Table 4

All forecasts compared with actual values

Actual ITRI RGM (1,1) RGM (1,1)a RGM (1,1)b 1998 2834 3348 2933 3132 3020 1999 4235 3523 3483 3775 3749 2000 7144 6753 5447 6298 6546 2001 5269 8545 10 851 6811 6624 2002 6529 6283 6512 6511 6498

Average residual error (%) 21.27 30.24 12.55 10.52

a _{Variable P value RGM (1,1) using the P value figured out from Eq. (10).} b

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Fig. 2. The curves of the actual and forecast data.

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Nevertheless, the yearly percent change in real GDP by U.S. manufacturing industry is not a leading indicator for the same year’s U.S. demand. The yearly survey of anticipated industrial production growth rates in Taiwan is a leading indicator because it is done at the end of each year and publicized at the beginning of next year. We found out that its correlation with the yearly percent changes in real GDP by U.S. manufacturing industry has a correlation coefficient of 0.96. Therefore, we used the yearly survey of anticipated industrial production growth rates [23–27] in Taiwan to determine the P value in the GM and the data from 1998 to 2002 are 4.13, 7.00, 8.60, 7.40, and 5.60, respectively.

According to the data, we constructed the following equation to show the relationship between the P value ( P) and the anticipated industrial production growth rate ( g).

P¼ 0:389138 þ 0:047647g ð11Þ

Eq. (11) helps determine the appropriate P value when the coming year is forecasted. To test the accuracy of the regression, use Eq. (11) to determine the P values. The average residual error of the variable P value RGM (1,1) using Eq. (11) was 10.52%, which was still better than the fixed P value RGM (1,1) and the ITRI’s forecast. Table 3 shows the forecasts.

We also compared all forecasts with the actual values in Table 4. From Table 4, we found the forecasts of the variable P value RGM (1,1) adopting Eqs. (10) or (11) are superior to the fixed P value RGM (1,1) or the ITRI’s forecast, especially in 2001.Fig. 2shows the curves of the actual and forecast data.

6. Conclusions

Original univariate GM has the limitation of making forecast of a time series of data without considering possible correlation with any leading indicators. We believe that the annual semiconductor industry production in Taiwan should be closely tied with the U.S. demand. This study made an enhancement to the traditional GM, which customarily fix the P value at 0.5, by changing the P value in accordance to the yearly percent change in real GDP by U.S. manufacturing industry. This variable P value RGM (1,1) did generate better forecast than the fixed P value RGM (1,1). Nevertheless, the yearly percent change in real GDP by U.S. manufacturing industry is not a leading indicator for the same year’s U.S. demand.

We found out that the yearly survey of anticipated industrial production growth rates in Taiwan and the yearly percent change in real GDP by U.S. manufacturing industry are highly correlated. Fortunately, the former is a leading indicator because it is done at the end of each year and publicized at the beginning of the next year. Therefore, we used the yearly survey of anticipated industrial production growth rates in Taiwan to determine a new P value for next year in the variable P value RGM. The forecasts of the variable P value RGM (1,1) were superior to those of a fixed P value RGM (1,1) as well as to the forecasts made by the industry watchers in Industrial Technology Research Institute, Taiwan.

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P 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a 0.18049 0.18359 0.1868 0.1901 0.19352 0.19704 0.20069 0.20446 0.20835 b 1193.6 1216.5 1240.2 1264.8 1290.4 1316.9 1344.4 1373.1 1402.8 Forecasted value 2594 2672 2755 2841 2933 3030 3132 3240 3355 Actual value 2834 2834 2834 2834 2834 2834 2834 2834 2834 Error rate (%) 8.45 5.70 2.80 0.26 3.49 6.90 10.51 14.34 18.38 Appendix A

A.1. 1998 Production forecast for the semiconductor industry under different P values

S.-C. Chang et al. / T echnological For ecasting & Social Change 72 (2005) 623–640

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P 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a 0.18101 0.18422 0.18755 0.19098 0.19455 0.19824 0.20206 0.20603 0.21015 b 1319.6 1344.6 1370.6 1397.6 1425.6 1454.7 1484.9 1516.4 1549.1 Forecasted value 3077 3170 3269 3373 3483 3600 3723 3854 3994 Actual value 4235 4235 4235 4235 4235 4235 4235 4235 4235 Error rate (%) 27.34 25.14 22.81 20.35 17.75 15.00 12.09 8.99 5.69

A.2. 1999 Production forecast for the semiconductor industry under different P values

Chang et al. / T echnological For ecasting & Social Change 72 (2005) 623–640 635

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A.3. 2000 Production forecast for the semiconductor industry under different P values P 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a 0.26117 0.26779 0.27474 0.28203 0.28968 0.29773 0.30619 0.31511 0.3245 b 1264.7 1301.3 1339.9 1380.8 1424.1 1470 1518.8 1570.6 1625.9 Forecasted value 4393 4623 4873 5147 5447 5777 6142 6546 6994 Actual value 7144 7144 7144 7144 7144 7144 7144 7144 7144 Error rate (%) 38.50 35.29 31.79 27.96 23.76 19.13 14.03 8.37 2.10 S.-C. Chang et al. / T echnological For ecasting & Social Change 72 (2005) 623–640

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A.4. 2001 Production forecast for the semiconductor industry under different P values P 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a 0.39685 0.4132 0.43094 0.45027 0.47141 0.49463 0.52023 0.54862 0.58025 b 751.74 783.32 817.66 855.13 896.19 941.37 991.32 1046.8 1108.9 Forecasted value 7007 7732 8589 9613 10 851 12 366 14 249 16 626 19 686 Actual value 5269 5269 5269 5269 5269 5269 5269 5269 5269 Error rate (%) 32.98 46.74 63.01 82.44 105.93 134.70 170.42 215.55 273.62 Chang et al. / T echnological For ecasting & Social Change 72 (2005) 623–640 637

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A.5. 2002 Production forecast for the semiconductor industry under different P values P 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 a 0.09828 0.09436 0.09018 0.08574 0.08104 0.07609 0.07088 0.06544 0.05976 b 4268.5 4371.9 4474.1 4574.6 4673 4768.8 4861.5 4950.7 5035.8 Forecasted value 6416 6457 6487 6506 6512 6507 6489 6459 6417 Actual value 6529 6529 6529 6529 6529 6529 6529 6529 6529 Error rate (%) 1.73 1.10 0.64 0.36 0.25 0.34 0.61 1.07 1.71 S.-C. Chang et al. / T echnological For ecasting & Social Change 72 (2005) 623–640

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Shih-Chi Chang received a BS in International Trade from Cheng-Chi University, Taiwan, in 1995 and an MBA degree from Chung-Cheng University, Taiwan, in 1997. He is currently a PhD candidate at the Institute of Technology Management in

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Chiao-Tung University, Taiwan, and a lecturer at Ta-Tung Junior Technological College of Commerce, Chia-Yi, Taiwan. His research interests include production forecasting, industry analysis, and technology management.

Hsien-Che Lai received a BS in Business Administration from Cheng-Kung University, Taiwan, in 2000. He is now a PhD candidate at the Institute of Technology Management in Chiao-Tung University, Taiwan. His research interests include science and technology policy and national innovation systems.

Hsiao-Cheng Yu received a BS in Electronic Engineering from Chung-Yuan University, Taiwan, in 1972 and a PhD in Industrial and System Engineering from the Georgia Institute of Technology, Atlanta, GA. A telecommunications consultant with Contel Information Systems in Great Neck, NY, from 1981 to 1985, he then joined AT&T Bell Labs as a system engineer and architecture planner from 1985 to 1992. He is a professor at the Institute of Technology Management, Chiao-Tung University, Taiwan. His current research interests include telecommunication policy and production forecasting.