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Hedonic price indexes of computers : an empirical comparison Duan, Hui 2006

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Hedonic Price Indexes of Computers: An Empirical Comparison by H u i Duan A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Science in Business Administration in The Faculty of Graduate Studies (Management Information Systems) The University of British Columbia April 2006 ©Hui Duan, 2006 Abstract This thesis compares the price changes of different computer platforms (desktop computers, notebooks, servers and workstations) in the 1998—2002 period using sev-eral hedonic methods to calculate price indexes on quarterly basis. It then discusses the differences among hedonic price indexes and their properties in terms of bias, vari-ance and resource. It further examines the effect of weight on certain price indexes. The findings should provide empirical insight into the hedonic quality-adjustment in price indexes, as well as the construction of C P I / P P I of ICT products for statistical agencies in O E C D countries. i i T a b l e o f C o n t e n t s Abstract ii Table of Contents iii List of Tables v List of Figures vi Acknowledgements vii 1 Introduction 1 2 Matched-Model Approach 2 3 Hedonic Approach 4 3.1 The Hedonic Theory 5 3.2 Hedonic Price Index Methods 7 4 Review of Relevant Literature 11 4.1 Mainframes 11 4.2 Desktops and Laptops 13 4.3 Workstations and Servers 16 5 Empirical Price Index Methods ' 18 5.1 The Pooled Regression Method 18 5.2 The Characteristics Price Method 19 5.3 The N O A Y R Method 20 5.4 The C C C Method 23 5.5 The Three-Period-Moving Method 25 6 The Database 26 7 The Methodology 29 8 Resulting Price Indexes 31 8.1 Desktop Price Indexes 31 8.2 Laptop Price Indexes 44 8.3 Workstation Price Indexes 56 8.4 Server Price Indexes 69 iii 9 Discussions 82 9.1 Computer Price Changes 82 9.2 Hedonic Price Index Methods 85 9.3 The Effect of Weight 91 10 Contributions 1 0 1 11 Future Research 1 ° 2 References 104 iv L i s t o f T a b l e s 1 Number of Observations (Desktop) 27 2 Number of Observations (Laptop) 27 3 Number of Observations (Workstation) 28 4 Number of Observations (Server) 28 5 Variables Description (Desktop) . 35 6 Desktop T P M Regression Results 35 7 Desktop Pooled Regression Results 36 8 Means of Selected Variables (Desktop) 37 9 Desktop Price Indexes 38 10 95% Confidence Intervals of Desktop Price Indexes 40 11 Variables Description (Laptop) 46 12 Laptop T P M Regression Results 46 13 Means of Selected Variables (Laptop) 47 14 Laptop Pooled Regression Results 48 15 Laptop C C C Regression Results 49 16 Laptop Price Indexes 50 17 95% Confidence Intervals of Laptop Price Indexes 52 18 Variables Description (Workstation) 59 19 Workstation T P M Regression Results 59 20 Workstation Pooled Regression Results 60 21 Workstation C C C Regression Results 61 22 Means of Selected Variables (Workstation) 62 23 Workstation Price Indexes 63 24 95% Confidence Intervals of Workstation Price Indexes 65 25 Variables Description (Server) 72 26 Server T P M Regression Results 72 27 Server Pooled Regression Results 73 28 Server C C C Regression Results 74 29 Means of Selected Variables (Server) 75 30 Server Price Indexes 76 31 95% Confidence Intervals of Server Price Indexes 78 32 A A G R s of Computer Price Indexes 84 33 Comparison of Hedonic Price Index Methods 90 34 Biases of Hedonic Price Indexes 90 35 Desktop Unweighted Price Indexes 93 36 95% Confidence Intervals of Desktop Unweighted Price Indexes . . . 94 37 A A G R Biases of Unweighted Desktop Price Indexes 100 v L i s t o f F i g u r e s 1 Desktop Price Indexes 39 2 95% Confidence Intervals of Desktop Pooled Index 41 3 95% Confidence Intervals of Desktop N O A Y R Index 41 4 95% Confidence Intervals of Desktop Divisia T P M Index 42 5 95% Confidence Intervals of Desktop Char T P M Index 42 6 95% Confidence Intervals of Desktop Divisia C C C Index 43 7 95% Confidence Intervals of Desktop Char C C C Index 43 8 Laptop Price Indexes 51 9 95% Confidence Intervals of Laptop Pooled Index 53 10 95% Confidence Intervals of Laptop N O A Y R Index ' 53 11 95% Confidence Intervals of Laptop Divisia T P M Index 54 12 95% Confidence Intervals of Laptop Char T P M Index 54 13 95% Confidence Intervals of Laptop Divisia C C C Index 55 14 95% Confidence Intervals of Laptop Char C C C Index 55 15 Workstation Price Indexes 64 16 95% Confidence Intervals of Workstation Pooled Index 66 17 95% Confidence Intervals of Workstation N O A Y R Index 66 18 95% Confidence Intervals of Workstation Divisia T P M Index . . . . 67 19 95% Confidence Intervals of Workstation Char T P M Index 67 20 95% Confidence Intervals of Workstation Divisia C C C Index 68 21 95% Confidence Intervals of Workstation Char C C C Index 68 22 Server Price Indexes 77 23 95% Confidence Intervals of Server Pooled Index 79 24 95% Confidence Intervals of Server N O A Y R Index 79 25 95% Confidence Intervals of Server Divisia T P M Index 80 26 95% Confidence Intervals of Server Char T P M Index 80 27 95% Confidence Intervals of Server Divisia C C C Index 81 28 95% Confidence Intervals of Server Char C C C Index 81 29 Desktop Unweighted Price Indexes 95 30 Desktop Weighted and Unweighted Divisia T P M Indexes 96 31 Desktop Weighted and Unweighted Char T P M Indexes 96 32 Desktop Weighted and Unweighted Divisia C C C Indexes 97 33 Desktop Weighted and Unweighted Char C C C Indexes 97 34 95% Confidence Intervals of Desktop Unweighted Divisia T P M Index 98 35 95% Confidence Intervals of Desktop Unweighted Char T P M Index . 98 36 95% Confidence Intervals of Desktop Unweighted Divisia C C C Index 99 37 95% Confidence Intervals of Desktop Unweighted Char C C C Index . 99 vi Acknowledgements I would like to thank my supervisor, Paul Chwelos, for providing guidance, pa-tience and financial assistance for the project. Your insights and suggestions con-tributed to the development and improvement of the thesis. I would cherish my enjoyable experience in this program. I would also appreciate Professors Ernst R. Berndt and Iain M . Cockburn for sharing the valuable data on computer markets. I thank Edison Chua, who made efforts to help organize the databases. Finally, I am indebted to my parents for their continuous support throughout my lifetime. vii 1 I n t r o d u c t i o n The value of Information Technology (IT) output and investment is of great impor-tance for companies that utilize IT equipment to obtain competitive advantage and achieve effective and efficient delivery of products and services. Measuring the return on IT investment and its productivity becomes a critical issue faced by companies while evaluating the performance of IT capital. Nevertheless, it usually takes a long period of time (from a half year to several years) for the IT investment to take effect on the entire organization. Such effect of IT investment requires the consideration of technological innovations to adjust the quality of IT products and yield a "more accurate" assessment. One way of making adjustments is to use price indexes which are "quality-adjusted" to deflate the prices in all periods and make the evaluation based on both the real inputs and real outputs. Because of the feature of rapidly changing technology inher-ited in information and communication (ICT) products, the accuracy of price indexes estimated by statistical agencies may have significant impact on the magnitude (or even the direction) of investment performance measurement, productivity measure-ment and the added value in national accounts. Especially in national accounts, the error in the deflator will lead to the equivalent error of opposite sign in real out-put and any other related real measures. In this sense, "discussing the problems of quality change in price indexes is the same as discussing the problems of quality change in quantity indexes and the productivity measures as well" (Triplett, 2004). However, quality change has been considered the most serious measurement problem in estimating price indexes. Different quality adjustment methodologies employed by O E C D 1 countries generate largely different trends of price movement of the ICT products. In Europe, the price declines, measured by national computer deflators, ranged from 10% to 47% for the early 1990s. France had the largest price declines, 1 Organization for Economic Co-operation and Development 1 and it was a hedonic price index. Consequently, such variation in price declines could have influence on G D P growth rates for about 0.2% to 0.3% per year, which could make the international price movements incomparable. The incomparability of ICT deflators could also create limitations on comparing the impacts of the ICT sector on economic growth across different O E C D countries (Moreau, 1996 and Konijn et al., 2003). Therefore, how to construct price indexes has been heavily investigated and researched in recent years. In the following sections, I shall discuss the conventional "matched-model" approach employed by statistical agencies in many O E C D coun-tries, and the hedonic approach that has been proposed by researchers for estimating price indexes. 2 M a t c h e d - M o d e l A p p r o a c h Most of the statistical agencies that estimate price indexes employ, a fundamental methodological principle to adjust quality changes. The agency chooses a sample of sellers2 and the corresponding products. A price in the initial period is collected for each selected products. It also collects the price for exactly the same product at some later period, from the same seller, which was selected in the initial period. The price index is then computed by matching the prices for both periods, observation by observation, or "model by model" (Triplett, 2004). These price indexes constructed by using this methodology are therefore called "matched-model indexes". Holding constant the quality of goods and services for comparison is one of the advantages of "matched-model" approach. However, Triplett (2004) suggested that problems occur when there is some undetected change in the product that makes the match not exactly the same, or the product observed in the initial period disappears, and it cannot be matched in the later period. Another disadvantageous aspect of 2They are retail outlets for constructing consumer price indexes ( C P I ) , or producers for producer price indexes ( P P I ) 2 this methodology is holding constant other factors which cannot be directly observ-able but may determine prices, such as customer service and location in the case of CPI, and reliability and manufacturers' reputation in the PPI case. Therefore, such methodology holds constant not only the items selected but other nonobservable aspects of the transactions, which might bias the measure of quality changes in the products and the estimated price indexes as well. As with almost all quality-adjustment price index methodologies, the "matched-model" approach faces two major problems: 1. "Inside-the-sample" problem: How to solve the consequence of situations where the old model and its replacement cannot be matched within the pricing spec-ification? 2. "Outside-the-sample" problem: How to adjust when quality change occurs on some products that have not been included in the sample for the index? For ICT equipment (especially computer), the "outside-the-sample" problem al-ways exists due to the rapid introduction of new models and improved characteris-tics/functions into the market. This problem cannot be satisfactorily solved by the "matched-model" approach. In his handbook on hedonic indexes, Triplett (2004) remarked that, statistical agencies generally employ the following four methods to deal with the "inside-the-sample" quality problem: 1. Overlapping link: The price index for computers available in periods t+ l / t is linked to the price index for computers available in periods t/t-1. All the available models and only the matched model prices are used. The quality ad-justment is equal to the price ratio between the initial item and the replacement item in the overlapping period t. 3 2. Direct comparison: This method assumes there is no quality difference between the initial computer and its replacement, and the difference is due to inflation. Such method biases the price indexes in case of any quality change, and it is often used when there is no significant quality change. 3. Link-to-show-no-price-change: Unlike the direct comparison method, this method has the opposite assumption, i.e., it attributes all the difference between prices of the initial item and its replacement to quality change rather than inflation. It has the bias in the opposite direction of the direct comparison method. 4. Imputed price change-implicit quality adjustment (IP-IQ): Both the "unmatched models" are dropped from the index for the period t. The price change for the replacement is imputed from price changes (IP) of other items whose quality did not change, which also implies an implicit quality adjustment (IQ). Each of the four methods have its advantages and limitations and they are used by statistical agencies in different rather than similar situations. 3 H e d o n i c A p p r o a c h A n alternative for estimating quality-adjusted price indexes is the hedonic approach, which is also "the most promising" approach to deal with quality adjustment ex-plicitly (Triplett, 1986). Although it was initially resisted that the use of hedonic approach in the construction of price indexes mainly for the reason of ungrounded theory, with more research and investigation being done in this area, a lot of the understanding has been obtained (Triplett, 1990 and Chwelos, 1999). So far the he-donic price indexes have already been implemented by many statistical agencies in O E C D countries for automobiles, computer goods and housing, especially the ICT components of CPI (e.g., Ohta and Griliches (1986), Berry et al. (1995) and Laferrere 4 (2003)). 3.1 The Hedonic Theory A hedonic price index is the price index that is estimated by using a hedonic function. The hedonic hypothesis, which is the theoretical basis for the hedonic function states: "a heterogeneous good can be treated as an aggregation of homogenous attributes". In mathematical terms, P = h(c), where P is an n-dimension vector of prices of heterogeneous goods (models), and c is a k x n matrix of the attributes, which are homogeneous by assumption. Waugh (1928) is known as the first one to use hedonic function in his study on vegetable quality and prices. In general, the form of the hedonic function is an empirical matter in any partic-ular case, since "it is an envelope function of either the users' value function or the producers' cost function", and "it is determined by the distribution of buyers and sellers across characteristics space" (Rosen, 1974). As a result, the observed hedonic function is the outcome of an equilibrium in a complex market, and it reflects neither the willingness-to-pay of consumers nor the marginal costs of producers. If we further assume the consumer's utility function to be U = U(c, M), where U is utility, c represents the characteristics which the heterogeneous good is composed of, and M is a vector of other homogeneous goods with PM as their prices. The minimum cost of attaining utility level U* can be expressed as: C* = C[PM, h(c), U*] = min [PMM + h(c) : U(c, M) = U*}. (1) c,M In the above formula, the cost of maintaining a specific utility level (or standard of living) is dependent on the prices of other homogeneous goods and the hedonic price as we assume there is only one heterogeneous good, for simplicity. Given the 5 information about prices, The consumer minimizes the cost of attaining that specific level of utility by choosing M and the characteristics c (Pollak, 1983). Therefore, the cost-of-living (COL) index between period t and period 0 with the inclusion of hedonic function becomes: C[PMt,h(c)t,U*} C O L ° ' ~ c{pM0,h(c)0,u*y ( 2 ) From economic perspective, the C O L index should be the ideal price index for CPI, since C O L indicates the minimum change in cost between two periods while keeping the utility level unchanged. Diewert (1976) suggests that the true cost-of-living index is bounded by Laspeyres3 and Paasche4 price indexes, where the Laspeyres price index is usually computed in CPI and serves as an approximation and upper bound to the COL. The full C O L index is too complicated to be tractable if there are many heteroge-neous goods. Generally we are not interested in the C O L index with full specification, we are rather more concerned with the price changes in computers, cars or houses for research reasons. However, the theory of C O L does provide underlying conceptual framework for constructing a practical consumer price index, as well as the measure of inflation (Triplett, 2001). The utility function defined above can be "separable", i.e., the consumer's mini-mizing decision on the choice of characteristics does not depend on other M goods. Then an "exact" subindex which involves only the goods that we are concerned with can be computed (Pollak, 1975). If the cost function d is defined as: d = d[h{c), q*] = min [h(c) : q(c) = q*], (3) 3The formula for Laspeyres price index is: (2~X=i pitQio) / (52i=i PioQio)-4The formula for Paasche price index is: (X)"=i pitQit)/(Y%=i pioQit)-6 accordingly, the exact characteristics price index is: j _ d[h(c)t, q*\ The above index is a theoretical index since it cannot be computed solely from the hedonic function; and it cannot be applied empirically because the knowledge of the utility function is also required. Feenstra (1995) showed how to construct bounds on the exact hedonic price indexes and the potential bias in the measure of the marginal value of characteristics given specific utility function. Diewert (1976) also showed that a superlative index5 can provide a close approx-imation to the C O L index in goods space, as the corresponding utility function is a second-order approximation to the true utility function. As a result, such superlative index does not require explicit demand estimation or utility function, but rather, it uses only the information about prices and quantities. The superlative index theory is of good value as it demonstrates the empirical implementation of theoretical objective with less strict assumptions. In fact, the usual price index such as the fixed-weight Laspeyres price index, does not either provide and estimate of the C O L index that is defined in characteristics space; instead, it is also an approximation to the C O L index (Triplett, 2004). 3.2 Hedonic Price Index Methods As defined earlier, a hedonic price index is the price index which involves a hedo-nic function. In the case of computer and ICT goods, Triplett (2004)'s handbook on hedonic indexes provides a comprehensive understanding. In this section, I shall sum-marize his discussion on the four major methods developed to calculate the hedonic price indexes by statistical agencies: 5 A n example is the Ideal Fisher index number formula. 7 1. Time dummy variable method: Most research hedonic price indexes use this method. As its name indicates, a time dummy variable is introduced into the hedonic regression to reflect the price difference between two periods. For example, if the regression equation is: lnPit = a0 + ailn(ci) + a2ln(c2) H h &i A + i + e*t, (5) the percentage change in computer prices between periods t and t + 1 can be computed as ebl, holding constant other computer characteristics6. It is the "pure price change" that is independent from the characteristics. Although conventionally statistical agencies use a fixed sample for all periods, the time dummy variable method can correct for some outside-the-sample biases by in-cluding a more comprehensive sample, especially for goods with rapidly chang-ing qualities. 2. Characteristics price index method: Instead of adding time dummy variables into the regression equation, this method treats the regression coefficients of the computer characteristics (i.e., c i , c 2, etc.) as their "implicit" prices. The price index for the whole computer good is then calculated based on the values of separate hedonic functions in the two periods. Conventional price indexes such as Laspeyres index or Fisher index can be applied, depending on the assump-tions about hedonic functions. This method provides an explicit way to link hedonic indexes to the conventional indexes, and it is also more consistent with the hedonic theory of characteristics. Nevertheless, the regression coefficients of the characteristics are very important and will have significant impacts on the subsequently constructed price indexes. Whereas there will not be a big concern when using the time dummy variable method if we are only interested in the price indexes. 6 I n this case, they are c\, c2, ... 8 3. Hedonic price imputation method: In this method, suppose we have the follow-ing regression equation: lnPit ~a0 + ailn(ci) + a2ln(c2) H h eit, (6) the imputed price for a computer in period t is calculated as: _ e{ao+ailn(ci)+a,2ln(c2)+~•} ^ The imputation happens only when a computer is introduced in period t + 1 and the price for period t becomes unavailable, since the actual price in period t + 1 can be observed. The price index for the model is then (Pt+i/Pt). Some researchers propose "double imputation", i.e., the index is calculated as (Pt+i/Pt). Although this "double imputation" has been criticized for the reason that the observed actual price is more accurate than the imputed (estimated) price, Pakes (2003) proposed double imputation for sample exits only in order to compensate the consumer for the consumer in the initial period. 7 In his opinion, such method "does not incur the additional estimation variance of the hedonic prices for the goods which are sold in the comparison period, and does not incur the selection bias of the matched model index for the g o o d s that d o not survive". 4. Hedonic quality adjustment method: This is an alternative imputation method as it estimates a hedonic quality adjustment rather than the unavailable price. Different from other hedonic methods, the hedonic quality adjustment method makes it possible to separate the estimation of hedonic function from the price index construction itself. That is, the databases used to carry out these two calculations are not necessarily required to be from the same sources (periods). 7For discussion of model entry and exit, refer to Stavins (1995). 9 In this method, the imputed price of the introduced computer (Pn,t) is calculated as the actual price of its replacement (Pm,t) adjusted by the estimated quality change (A(h)), i.e., Pntt = PmttA(h), and the price index is then calculated as Pn,t+\/Pn,t-8 Each of the following four methods differs in the use of price information from the hedonic function. The first two methods are "direct" methods in the sense that all their price information is drawn from the hedonic function without alternative source. Accordingly, the last two methods are referred as "indirect" methods because they are used to impute prices or adjust for quality changes only when the matched-model approach does not apply because of "unmatched comparisons", while the rest of calculation is based on conventional matched-model methods. While the above four methods can be adopted by statistical agencies to estimate the price indexes, each of them requires different databases as well as other resources. As mentioned earlier, the time dummy variable method is used in most research hedonic price indexes probably due to the simplicity and comprehensiveness inherited in the method. However, not so many statistical agencies use such method because of the requirement of producing indexes in a timely manner and the restriction of the available databases. Researchers, on the other hand, are generally not faced by these constraints, and most of their research purposes involve the comparison of different price indexes and constructing the "more comprehensive" indexes (Triplett, 2004). Among the four methods, the time dummy variable method is the simplest one, while the "indirect" methods (hedonic price imputation method and hedonic quality adjustment method) demand much more resources and databases than direct ones. However, one of the biggest criticism on the time dummy variable method is the holding of regression coefficients (characteristics prices) over the whole periods since it has been indicated by statistical tests that those coefficients do change over time. 8For more description of this method, please refer to the OECD handbook on hedonic indexes (Triplett, 2004, pp. 74-84). 10 This drawback could be overcome by estimating coefficients in separate regressions, which will be discussed later. 4 R e v i e w o f R e l e v a n t L i t e r a t u r e Many researchers investigated hedonic methods to measure general price changes and technological innovations of computer products. A thorough understanding of using hedonic, methodology to construct price indexes was obtained. In this section, relevant literature is reviewed to describe recent developments in this field. 4.1 Mainframes One of the classic studies is Chow (1967) 's research on mainframe computers. Chow (1967) used both performance measures and technical measures (such as access time and memory size) in his hedonic model and found that an average annual growth rate (AAGR) of -21% between 1960 and 1965 in quality-adjusted rental prices. While he also found that the computer stocks grew by an average of 78% per year between 1954 and 1965, which reflected "a moving equilibrium due to the comparative statics of price change". Cole et al. (1986) published reports on hedonic price indexes on individual com-puter components instead of the whole computer system between 1972 and 1984. Compared with an A A G R of -8.5% of the matched-model indexes of computer proces-sors, the hedonic price indexes had an A A G R of -19.2%. For disk drives, the matched model indexes declined at a rate of 6.9%, while the hedonic price indexes declined by 16.9%. Such discrepancies demonstrated the "inappropriateness of a matched-model index for computers" because "the matched-model index misses the replacement of old, higher priced models by new models, manufactured by improved methods and introduced at lower quality-adjusted prices" (Cole et al., 1986). By aggregating equa-11 tions of separate price indexes for computer components, Cartwright (1986) reported an A A G R of -13.8% in mainframe computer price indexes between 1972 and 1984, which became the basis of the B E A ' s 9 new official price indexes for mainframe com-puters. In 1986 the B E A began adjusting mainframe computer prices for quality-change, using hedonic pricing methods. In later 1990, the BLS 1 0 published its first experimental quality-adjusted PPI for the mainframes, which was also based on he-donic price methods (Sinclair, 1990). Similar work was applied to the 1983-1988 period by Cartwright and Smith (1988). However, the data set only includes IBM-compatible computers, which might create a bias as the data set could not represent the entire computer market. Gordon (1989) used four data sources to estimate hedonic price indexes for com-puters from 1951-1984. The data sets cover mainframes until the late 1970s and some "super-mini" computers and PCs in the 1980s. The results suggested an A A G R of -19.8% over the 33 years. Gordon (1989) also placed emphasis on problems of weight-ing price indexes and the issues with data selection and methodology. The purpose of price index is important for data selection: only new models are necessary to trace technological frontier, while comprehensive data are needed to deflate computer pur-chases. Caudill and Gropper (1997) reapplied the method used by Cole et al. (1986) to fill the gap of the mainframes price indexes between 1984 and 1994. The authors estimated price index based on a single "performance index" in the Computer Price Watch data set. The model with "performance index" as the regressor had very high R 2 and good fit, but the index itself was not explicitly described. The estimated price indexes based on four specifications indicated that the price of I B M and plug-compatible computers fell at an average annual rate of 18% and 19%, which is close to the rate (19.2%) estimated by Cole et al. (1986) in the 1972-1984 period. 9U.S. Bureau of Economic Analysis 1 0U.S. Bureau of Labor Statistics 12 4.2 Desktops and Laptops The hedonic price indexes of desktops and laptops were not constructed and inves-tigated until recently. In 1990, Berndt and Griliphes (1990) pulled data from four difference sources (Byte, P C Magazine, P C World and N Y Times) and estimated price indexes of microcomputers between 1982 and 1988. The database comprised an unbalanced panel for 1265 model observations. The authors also developed an empirical specification test for the selection of hedonic functions. In the study, time and vintage variables were used as regressors in the hedonic equations. Moreover, predicted prices for exiting and new models were used to calculate the Divisia price index. 1 1 Based on the constructed price indexes, the prices of microcomputers in the U.S. declined about 28% per year over the 1982-1988 period. Nelson et al. (1994) used three separate hedonic models to estimate price indexes of personal computers over the period 1984-1991 using 1841 observations in the data set. The authors developed and estimated a non-linear model that is parsimonious in parameters and allows time-varying prices of attributes, and it could be estimated using a pooled data. The data set in the study was based on I B M - P C compatible machines, which made the hedonic function specification cleaner. Considering the effect of CPI, the authors found that the nominal mail-order prices declined at an annual average rate of 24.62%, and the real prices declined at an annual average rate of 27.48%. The methodology employed in the paper of Berndt et al. (1995) was very similar to that in Berndt and Griliches (1990). The time and vintage variables were also used as regressors. Both papers suggested that the unmeasured price change should be uncorrelated to the vintage of the computer model. In the paper of Berndt et al. (1995), new and surviving models were included in the estimation of hedonic price equations. The authors also found the difference between parameters of hedonic price 11The Divisia index formula will be discussed in later sections. 13 equations. Although little evidence supported parameter stability in desktops, the evidence did suggest stability of parameters in market of laptops. For laptops, the 1989-1992 A A G R was -23.2% based on results of adjacent year regressions, and -23.9% based on the Divisia index with annual parameters. For desktops, the A A G R was -31% over the same period. Finally, price indexes of desktops and laptops were then aggregated using Divisia weighting procedures, and the overall quality-adjusted price index of microcomputers declined at 30% per year, with a particularly large decrease in 1992. Baker (1997) investigated the hedonic price indexes of laptops from 1990 to 1995. In his study, the data set was from an annual review of laptops in PC Magazine. He used the common regressors such as weight, resolution and battery life in the hedonic functions to model performance of laptops. Three methods were used to construct price indexes: linear model with pooled data, linear model with adjacent years and non-linear model. The rate of price change across all methods was -29%, which was comparable with the estimated -24% in Berndt et al. (1995). In two of his papers, Chwelos (1999 and 2003) constructed a processor performance index based on published benchmark tests. The indexes had good explanatory power in the estimation model. Moreover, he constructed a novel set of technical prox-ies for performance. In the regression of laptop performance indexes on technical proxies, the R 2 was 0.9819 (Chwelos, 1999). Compared with an A A G R of -39.6% in constructed price indexes based on performance indexes and the A A G R of -40% in the indexes based on technical proxies, he found that these proxies not only "re-produce the performance index", but "demonstrate to be a nearly equivalent way of operationalizing performance in the hedonic function". The results suggested the feasibility of using technical proxies in larger and more general data sets where the performance measures were not available. Although there was a very strong correlation between the performance indexes and technical proxies in desktops (R2=0.9925), the estimated A A G R s were slightly 14 different: -32% based on performance indexes and -35% based on technical proxies (Chwelos, 1999). Berndt and Rappaport (2001) did a quarter-century review of technological progress of desktops and laptops in the U.S. market. Based on various indexes (pooled, ad-jacent years, Laspeyres and Paasche), they found that the prices of microcomputers declined at an average rate of 25% each year between 1976 and 1999. The price declines became larger over the period: 1990s had larger price declines than 1970s and 1980s, and the declines were larger in late 1990s than early part. As for para-meter stability, the coefficients of characteristics differ between desktops and laptops, beginning about 1987. Particularly for desktops, the characteristics coefficients had significant differences by year beginning in 1987, and for laptops, by 1993. However, there did not appear to be a monotonic trend in these coefficients over the period. Recently, some researchers compared the empirical similarities and differences between matched-model indexes and hedonic price indexes. Aizcorbe et al. (2003) constructed price indexes for desktops with Intel microprocessors from 1993-1999 using both methods. The authors found that "when the available data are a panel of high-frequency data on models whose characteristics are constant over time, matched-model price indexes can also be used to obtain constant quality price measures". The findings suggest that, certain matched-model indexes could yield price indexes that are numerically close to those obtained from hedonic techniques. The matched-model price indexes generally do not account for enough quality change in computers because of "unrepresentativeness" of the sample (or sample selection bias). It was thought that such bias could be reduced by including high-frequency data. However, when Deltas and Zacharias (2004) used the P C data from PC Magazine between 1993 and 1995, and compared the quantity-weighted matched-model indexes and hedonic price indexes with different frequencies, the results showed that the bias in matched-model price index increased with the sampling frequency, except for long-lived models. The rapidly deteriorating performance/price ratio of 15 short-lived models could be one of the probable reasons for the increased bias. Aizcorbe and Pho (2005) further examined the weights that could lead to the differences between matched-model indexes and hedonic price indexes. The authors used monthly scanner data for over 60 segments of ICT goods between 2001 and 2004. The unweighted geometric mean price index fell faster than the weighted Fisher and Divisia indexes. Part of the reason is: goods with low market shares declined faster than those with high market shares. Moreover, the dummy variable price index, which was unweighted, fell faster than other weighted indexes. Therefore, the authors concluded that the weights matter in the construction of price indexes. 4.3 Workstations and Servers There has not been much research on hedonic prices for workstations and servers in the literature. Rao and Lynch (1993) examined workstation prices and their hedonic attributes in 1989, without constructing a price index. The paper covers major at-tributes that affect price of a workstation: speed (MIPS), R A M (memory), hard disc space, colour or mono display, etc. One of the surprising results is the preference of linear model over double-log specification, as suggested by a Box-Cox test. Interest-ingly, the workstation speed (MIPS), which was considered as an extremely important determinant of price, had small effect. Moreover, the prices of major vendors were comparable to the market prices. Aizcorbe et al. (2000) used quarterly data from multiple sources (Dataquest, M i -croDesign Resources, etc.) to construct price indexes for high technology goods (PCs, notebooks, workstations and servers) and their processors between 1993 and 1998. The authors constructed geomean and Fisher matched-model indexes, as well as time dummy hedonic price indexes. The A A G R s of Fisher matched-model price indexes for servers and workstations lie between -16.4% and -26.2%, and the estimated time dummy hedonic price indexes for workstation and server (Xeon) CPUs had an A A G R 16 of -20.3%. However, "a comparison of the matched-model indexes compiled using a superlative index number formula with those generated using a hedonic regression technique suggests that the hedonic approach yields noisy and imprecise period-by-period measures of price change" (Aizcorbe et al., 2000). In 2004, Reenen (2004) conducted an empirical analysis of the hedonic prices and demand for work group (low-end) servers using I D C 1 2 data from the first quarter of 1996 to the first quarter of 2001 in three regions. The hedonic price indexes were based on adjacent quarter regressions with attributes including speed, memory, symmetric multiprocessing, operating system, etc. 'In an example of regression results, the coefficient of time dummy variable suggested the price declined by 7% between the first and second quarter of 2000 in Western Europe. 'International Data Corporation. 17 5 E m p i r i c a l P r i c e I n d e x M e t h o d s In practice, there are several hedonic methods that could be applied to constructing price indexes across a certain period of time. Some of them are "direct" hedonic price indexes, while others are regarded as the "imputation" or "composite" indexes. I shall discuss their principles and applications in this section. 5.1 The Pooled Regression Method Pooled regression, by its name, is based on all the observations in whole database, and it is the simplest way of using hedonic functions to construct price indexes. It uses time dummy variables for the N — 1 periods in the total N periods (generally no dummy variable is used for the base period), i.e., P = h(c;Dn), (8) where n — 2, • • • , N. If the specification of the hedonic function is double-log, the equation would be: lnPit = a0 + ailn(ci) + a2ln(c2) H h b2D2 H h bNDN + eit (9) In equation 9, the coefficients of characteristics (c) will capture the change of price due to quality improvement, while other uncontrolled price change will be measured by the set of time dummy variables (D). Taking the anti-logs of the coefficients b2, • • • , 6/v will generate the price indexes from period 2 to period N. However, since it is well known that the anti-log of the coefficient b2 is a biased estimate of the anti-log of b2, a correction factor is needed to calculate the accurate price index for period 2. 1 3 Thus, the price index for period 13Similiar adjustments apply to other time dummy variable coefficients. 18 t will be calculated by the following formula: I i t = ehH(Se(bt))^ (t = 2,---,N), (10) where \(Se(bt))2 is the correction factor for coefficient bt. As mentioned earlier, the pooled regression method is the simplest one among hedonic regression methods. The multicollinearity between independent variables (in this case, the set of hedonic characteristics c) is not of great importance in the sense of calculating price indexes, since the regression coefficients of time dummy variables show the "pure" price changes holding constant the characteristics. As a matter of fact, it is the explanatory ability of the set of independent variables (usually K2 or adjusted-i?2) that will influence the stability of the indexes. (Triplett, 2004) However, the pooled regression method also has some serious limitations: first, the coefficients of characteristics are constrained to be equal across all the periods; second, it is sensitive to sample selection as this method does not weight price data by quantity (sales); last, it does not have solid background in index number theory (Griliches, 1971a and Triplett, 1989). Therefore, the pooled regression method becomes the "least preferred way to make use of a hedonic function in the construction of a price index" (Chwelos, 1999). 5.2 The Characteristics Price Method Another empirical method to construct price indexes is the characteristics price method. As mentioned before, it treats the coefficients of characteristics (c) as the "implicit prices" and applies conventional index number formulas on these "implicit prices" and quantities to calculate price indexes. If the hedonic function is assumed to be linear, the corresponding Fisher price index of period t for characteristics can 19 be calculated as: /:; 7 > : , . ^ E ^ V ( t = 2,... i A 0 > ( n ) where j stands for the jth characteristics in the hedonic function and q is the quantity of the characteristics. Generally, in the case that the hedonic function is not linear, the Fisher price index of period t can be written as: where h(q) is a hedonic function with the characteristics quantity vector q (Triplett, 1989 and Dulberger, 1989). Because of the fact that the characteristics price index corresponds to conventional index number theory, it was the first government-adopted hedonic method, and one of the preferred methods for constructing price indexes (Triplett, 1989). 5.3 The NOAYR Method One way to solve the problem of constrained characteristics coefficients in pooled regression method is to estimate the coefficients in each period separately. Suppose the hedonic function is specified similar to equation 6, that is, lnPitt = a0,t + aittln(cltt) + a2itln(c2tt) H h akttln(ck,t) + %t- (13) The unbiased estimates of the intercept ao,t and the K slope coefficients aitt, • • • , ak,t for each period t can be obtained by performing N regressions over the periods. As an alternative, if the time dummy variables are used, these unbiased estimates can 20 also be obtained by pooling the data from all iV periods: K N K N InPij = a0 + ^ 2 akln(ck) + ^ btDt + ^ $Z $k,tDtln(ck) + e i i t, (14) k=l t=2 k=l t=2 where bt captures the differences between the value of intercept in period t and the value in the base period, and 6k,t measures the differences between the values of slope coefficients. Both equation 13 and equation 14 produce the same unbiased estimates of the coefficients. If we assume the slope coefficients do not change (remain constant) over the N periods, the regression function could be reduced to: K N lnPitt = a0 + ^ 2 akln(ck) + ^ btDt + ei>t, (15) k=l t=2 which becomes the same regression used in the pooled regression method (Griliches, 1971b). Although the slope coefficients do not change in every period, it is not reasonable to assume that they remain constant over the all N periods. Therefore, in order to obtain unbiased estimates of bt and 8k,t for all the periods, the hedonic function must be specified as equation 14. But unfortunately, in reality there is not enough amount of data in each period (cross-sectional data) to perform such regression. Some researchers have proposed using scanner data as a solution to this problem. However, scanner data is not only "prohibitively expensive", but unavailable when estimating the long-term price trends (APE 2005). To solve the data shortage problem, the Adjacent Year Regression (AYR) method was suggested for estimation. This method pools the observed data in adjacent periods by assuming that the intercept coefficient; varies, while the slope coefficient is assumed to remain constant for the two adjacent periods. A n example of hedonic function of the A Y R method is equation 5, where b\ is the difference between the 21 intercept coefficients in period 1 and period 0. To make it general, suppose the hedonic function is: The slope coefficients ak are assumed to be equal "in both period t — 1 and t. Under this assumption, these coefficients are unbiased slope estimates for both periods; ao is the unbiased intercept estimate for period t—1, and (a0+bt) is the unbiased intercept estimate for period t. In the case that the slope coefficients vary in the two periods, unbiased estimates could only be obtained by performing the regression function in equation 14 by in-cluding the time dummy variables for the slope coefficients. As a result, the estimates (a 0, ak and bt) obtained from the hedonic function 16 could be biased due to omitted variables. It could also happen that the slope coefficients may remain constant for more than two periods or even for all the N periods. Under this circumstance, unbiased estimates could be obtained by performing the pooled regression for all the N periods (such as equation 9). Although the estimates obtained from the A Y R method are still unbiased, they could be inefficient in this.case (APE 2005). The biased estimates problem could be solved by conducting F-tests that the coefficients 6ktt are zeros. If the null hypothesis cannot be rejected, use equation 16 to obtain unbiased estimates for each adjacent periods. If they are not close to zeros, use equation 14 instead. This method is referred to as Flexible A Y R method. The Flexible A Y R method, however, does not solve the inefficiency problem. To obtain both unbiased and efficient estimates for all the N periods, one should first test whether the slope coefficients vary over the two or more periods. If the slope coefficients are tested to remain constant, a single regression should be performed by pooling the data in all of the periods and including time dummy variables for K (16) fe=i 22 intercepts. If the slope coefficients are tested to vary in two periods, separate regres-sions should be performed for each period (like equation 13). Such method is also called "non-overlapping" A Y R (NOAYR) method. Estimates of intercept and slope coefficients obtained from this method are both unbiased and efficient ( A P E 2005). 5.4 The C C C Method Apart from the biased and inefficient estimates problems, other theoretical shortcom-ings of the A Y R method were pointed out by Auer (2004). Although the A Y R method reduces the number of coefficients to be estimated and solves the data shortage problem to some extent, it is not sufficient because there are still many coefficients needed to be estimated from two adjacent periods. As a result, some of the estimated coefficients are not efficient, and they could be significantly affected by the addition and deletion of a single observation (Auer, 2004). Another significant shortcoming of the A Y R method is the negligence of some important information contained in the entire database. Some of the models may appear more than two periods. Instead, their prices could be recorded more than twice or three times, which generates more observations in the database. Auer (2004) claimed that the price changes of a single "matched" model provide quite valuable information for estimating long-term price trends, as these changes can be attributed purely to the time passage and this is exactly the "quality-adjusted" price index. However, the A Y R method treats such difference as a disturbance which affects the error term, rather than additional information on price changes over time. Furthermore, Auer (2004) suggested that "the slope coefficients are not likely to change largely either upward or downward". In fact, evidence shows that these changes happen gradually and systematically rather than fluctuate randomly. To avoid the theoretical shortcomings inherited in the A Y R method, Auer (2004) proposed a "superior hedonic regression technique" which could use all the avail-23 able information in the database. Such technique is called "Continuously Changing Coefficients" (CCC) method. To illustrate the C C C method, Equation 13 could be rewritten as: K lnPitt = a0>t + ^  akttln(ck) + e i ) t, t=l,---,N, (17) fc=i where the at represent the intercept and slope coefficients of the hedonic regression. There is assumed to be a gradual change of these coefficients (akti,ctk£, • • • ,ak,N) over time. Auer (2004) used a polynomial of degree Zk to model the gradual change: ctk,t = Oik + X>,^ , k = 0,---,K, (18) z=l ) where 9kjZ are the coefficients of the polynomial and t is the trend variable from 1 to N. The time trend can be approximated very well by the polynomials of degree Zk = 5. Using the relationships of time series intercept and slope coefficients, the hedonic regression function can be pooled into a single C C C equation: lnPit = / 7 2 [ a k + /\2 Qk,ztz J ln(ck) + €ift, k=0 \ 2=1 / or, K K Zk lnPit = akln{ck) + ^  £ ^, 2[t z/n(c f c)] + c M , (19) fe=0 fe=0 z=l where ln(co) = 1 (the constant term). By applying appropriate index number for-mulas, the estimates of ak, Q\ (z = 1, • • • , Zk) could be used to calculate the general price trends in the N periods. In his study, Auer (2004) estimated the monthly price indexes for laser printers from 1992 to 2003. The results indicate that polynomials of degree 3 is adequate 24 for all the coefficients, with most of the coefficients having a polynomial of degree 1. Comparing the official laser printer matched-model price indexes published by the Statistisches Bundesamt and the price indexes constructed by the C C C method, Auer (2004) found that the C C C generated indexes declined by more than 90%, while the official indexes declined by only 50%. Therefore, Auer (2004) suggested that the official indexes did not accurately account for the changes in quality and performance characteristics of laster printers in over the periods. 5.5 The Three-Period-Moving Method The data shortage problem in estimating hedonic price indexes could not be solved when we need the construct the indexes more frequently. For instance, when we need to construct quarterly or monthly price indexes, obtaining sufficient number of observations in each quarter/month would be one of the critical factors that may affect the hedonic regression and the price indexes. Even though the A Y R method reduces the number of coefficients to be estimated, the price indexes are sensitive to the change of small number of observations in the database due to fewer degrees of freedom. Neither does the A Y R method take into account of the quantity (sales) data in the database. As we know, the quantity of a certain computer model sold in the period could provide relevant and valuable information on the computer market composition and the share of that model in the market, which in turn, would affect accuracy of the constructed price indexes. The A Y R method, however uses time dummy variables to measure the price trends and ignores the both the quantity information and the models that are "matched" between periods. Therefore, another method is proposed in this thesis to construct hedonic price indexes. It is called the "Three-Period-Moving" (TPM) Method. Instead of pooling the data in two adjacent periods, the T P M method pools data in three (3) adjacent periods in one regression and moves one period forward for the next regression. By 25 doing so, we increase the number of observations in each regression and also increase the degrees of freedom to obtain more robust estimated coefficients. The hedonic regression of this method can be expressed as the following equation: K lnPijt = a0,t + ^ 2 ak,tln(ck,t) + u,u t=(n- 1), n, (n + 1), (20) fc=i where n represents a single period in the database. The prices in period n — 1 and period n +1 (i.e., Pi>n-i and Pi,n+i) can be imputed from the above equation. Both of the periods are included in the regression, which could reduce the prediction errors while imputing the prices. Furthermore, if Auer (2004) 's assumption on the system-atical and gradual change in regression coefficient's is correct, the bias of coefficients in period n could be offset by including period n — 1 and period n + 1. When the regression moves one period forward, the regression becomes: Similarly, prices in period n and period n + 2 (p^n and pi,n+2) can be imputed from the regression "centered on" period n+1. If certain index number formula is applied, price indexes could be constructed based on the imputed prices and quantity (sales) data. The data in this study were from the Gartner web site. 1 4 Gartner is the world's lead-ing provider of research and analysis about the global information technology indus-1 4 T h e data was provided by Professors Ernst R. Berndt and Iain M . Cockburn through the financial support of the National Science Foundation to the National Bureau of Economic Research (SES-0219235). K (21) 6 T h e D a t a b a s e 26 try. Gartner Dataquest program offers statistics, forecasts and analysis of more than 35 major IT and telecom products in the global market. It also provides "compari-son columns" to compare different IT products. This study focuses on four computer platforms including desktops (PCs), laptops (notebooks), servers and workstations, as classified by Gartner. Table 1 describes the number of observations in desktops. There are a total of 5188 observations in the desktop category with study period from the first quarter of 1998 to the first quarter of 2002 (17 quarters). Table 1: Number of Observations (Desktop) Quarter Year 1 2 3 4 Total 1998 187 525 259 44 1015 1999 496 303 402 338 1539 2000 285 123 430 329 1167 2001 210 637 83 207 1137 2002 330 0 0 0 330 The number of laptop (notebook) observations in each quarter is listed in Table 2. The total number is 1912 with the time spans from the second quarter of 1998 to the second quarter of 2002 (14 quarters). Table 2: Number of Observations (Laptop) Quarter Year 1 2 3 4 Total 1998 0 58 173 0 231 1999 248 125 109 0 482 2000 50 65 85 276 476 2001 83 239 164 153 639 2002 0 84 0 0 84 As for workstations, there are a total of 1620 observations from the fourth quarter of 1998 to the fourth quarter of 2001 (10 quarters, as shown in Table 3). 27 Table 3: Number of Observations (Workstation) Quarter Year 1 2 3 4 Total 1998 0 0 0 131 131 1999 182 0 0 231 413 2000 274 221 0 191 686 2001 200 138 26 26 390 For desktops, laptops and workstations, the mimber of observations in each quar-ter are sufficient to perform hedonic regression analysis and hence, calculate quarterly price indexes. Table 4: Number of Observations (Server) Quarter Year 1 2 3 4 Total 1995 30 32 24 21 107 1996 60 33 10 3 106 1998 0 0 27 1 28 1999 20 29 31 0 80 2000 17 0 56 10 83 2001 0 46 42 16 104 2002 6 56 28 12 102 2003 77 4 0 21 102 Table 4 shows that the server database has records from the first quarter of 1995 to the fourth quarter of 2003 (712 records in total). However, due to the data shortage problem in this category, it is impossible to perform hedonic regression for each quarter. As a result, the quarterly data were pooled to be annual data from 1995 to 2003. But unfortunately, the data in 1997 are still unavailable for this study. Quantity (sales) data are available in desktop and laptop databases. Although such data are absent in servers and workstations, one (1) unit of quantity of was assigned to each observation in order to apply the methods which demand quantity data for constructing price indexes. The adjustment is thought to be reasonable given the absence of data and the requirements of these methods. 28 7 T h e M e t h o d o l o g y Little work has been done to investigate the performance and properties of various he-donic price indexes for different computer platforms. Although Aizcorbe et al. (2000) did similar research for PCs, notebooks, workstations and servers, the constructed indexes were only matched-model indexes and time dummy variable hedonic price indexes. More importantly, the statistical properties of price indexes constructed by different methods could possibly affect the selection of hedonic methods for statistical agencies. Therefore, the purpose of the thesis is to compare different hedonic price indexes resulted from the above-mentioned methods and their biases and variances. The pooled regression method is a "direct" method, hence the index is calculated by taking the anti-log of the coefficient of the time dummy variable plus the correction factor, which is the easiest to implement. To construct the characteristics price indexes, the implicit prices of the charac-teristics were obtained from the C C C method and the T P M method. The means of the characteristics were weighted by their quantities to represent the database more accurately. The Fisher index formula was applied to each index between adjacent periods, and the "chain rule" was used to transform the indexes of adjacent periods to indexes across all the periods. Since both the T P M method and the C C C method are "indirect methods", the "matched-model" method is applied when calculating the indexes. Price of a model is imputed only if its "matched model" does not exist. Therefore, the indexes were calculated by the following steps: 1. Perform hedonic regressions and obtain predicted price for each model. 2. Use a unique "identifier" to identify the "matched model" in the comparison period. 3. Replace the predicted price and quantity by price of the matched model and 29 its quantity. 4. Apply the Tornquist-Divisia index number formula to calculate price index of the adjacent periods. 5. Use the "chain rule" to generate price indexes for all the periods. Specifically, the Tornquist-Divisia index between period t and period 0 is given by: where sln is the model's market share in period t, and s° denotes the model's market share in period 0 (Berndt and Griliches, 1990).15 The reason why applying the Tornquist-Divisia index number formula is because it "preserves all of the desirable qualities of the composite index while avoiding the Paasche/Laspeyres problem" (Chwelos, 1999). In the Paasche/Laspeyres formula, if a model exits or a new model enters the market, no quantity could be associated with its price. Thus the calculated Paasche/Laspeyres index becomes meaningless. While in the Divisia formula, for new and exiting models which do not have "matched models" in the study period, their market shares are set to zeros for the missing period. By averaging their market shares across two periods, the "zero quantity" problem, which exists in the Paasche/Laspeyres index formula, is avoided. A n N O A Y R price index was constructed for each platform in this study, as the regression coefficients given by this method are considered unbiased and efficient. As a result, the N O A Y R index is also a "composite" index. For the consecutive periods where the slope coefficients of the hedonic regression remain unchanged, the pooled regression method was applied. For the two adjacent periods where the slope coefficients are different, prices were imputed for each period by performing separate 30 hedonic regressions. The price index between the two adjacent periods was then calculated by following steps used in the C C C and T P M methods. In order to assess the bias and variance of each price index, bootstrap technique was applied to calculate 95% bias-corrected confidence intervals based on 1000 repli-cations. Bootstrap is a useful nonparametric resampling method to let us infer the distribution of each estimated variable. Generally, 1000 replications is enough to cal-culate bias-corrected confidence intervals, i.e., the process of calculating price indexes was replicated 1000 times. 8 R e s u l t i n g P r i c e I n d e x e s In this section, the price indexes of the four computer platforms are presented for comparison of the differences among the various hedonic methods in each of the plat-form. The differences among the price indexes generated by these hedonic methods are significant over the periods. To investigate statistical properties of these indexes, bootstrap statistics are presented for better understanding of their biases and distri-butions. 8.1 Desktop Price Indexes The study period of desktops is from the first quarter of 1998 to the first quarter of 2002. The quantity-weighted means of selected variables are presented in Table 8. Among the means of variables, the processor clock speed increases by more than 5 times during the 17 quarters; the standard memory increases by more than 3 times; the size of the hard disk increases by about 8 times. Beginning from the second quarter of 2001, the percentage of desktops with Pentium IV CPUs increased in each quarter, and by the first quarter of 2002, 2/3 of the desktop computers had Pentium IV CPUs. Pentium IV processor architecture is Intel's first all-new C P U design called 31 the Netburst architecture, which runs faster and perform much less work per C P U cycle (Wikipedia, 2006a).16 A l l these changes reflect significant quality improvements in desktop systems. Although the percentage of desktops with Celeron CPUs did not change much over the periods, it did reflect the quality change as the Celeron processor usually acts as the "low-end" of each generation of Pentium processor. A summary of variables in the database and their expected effects is presented in Table 5. Some of the attributes are typically used in prior empirical work: C P U clock speed, C P U level 2 cache memory, standard memory, hard disk size, dummy variables for C P U architectures. In this study, most of the hedonic regressions include the above variables for all the methods. As for the form of hedonic function, although previous research shows that the functional form has little effect on the price indexes (Hoffmann, 1998), a box-cox test suggests that the double-log specification is not strongly preferred over the semi-log and linear specifications (A = 0.12). The pooled regression results are presented in Table 7. There are 40 records which do not have either "lhd" or "lsl2" values, and they were automatically dropped from the regression sample by the software. The results suggest that the model fit is good (R 2 = 0.625) with all the estimated coefficients are significant at the 1% level. The quarter dummy variables have expected signs and magnitudes across the periods. Although the Pentium IV C P U dummy variable is not significant in the pooled regression, it is significant at the 1% level in some T P M regressions (for example, Table 6 is a T P M regression with the fourth quarter of 2000 being the center period). It implies the larger percentage of Pentium IV CPUs at later stage and the limitation of constrained coefficients in the pooled regression. The average R 2 is 0.60 in the T P M regressions over the periods, and the variables in the regressions differ from quarter to quarter as the quantities of attributes change at different rates. 1 6 T h e design objective is to sacrifice instructions per clock cycle in order to achieve a greater number of cycles per second (i.e., greater frequency or clock speed) 32 In the C C C regression, R 2 is 0.646 and the following variables are significant at the 1% level with polynomials of degree 5: intercept, lprospd, lsl2, lsmem, lhd, dcdrom, dPent3. The variables lwt and dPent2 are significant at the 1% level with polynomials of degree 3. While performing the A Y R regressions, two quarters (the first quarter of 2000 and the first quarter of 2001) were tested to be distinct. Hence the sample time frame was divided into three periods. Accordingly, hedonic regressions were performed separately based on the structural changes. The desktop price indexes are shown in Table 9 and Figure 1. According to the results, the characteristics price indexes based on C C C method (Char CCC) have the lowest average annual growth rates (AAGR) of -38.6%, while the Divisia price indexes based on T P M method (Divisia T P M ) have the highest A A G R of -24.7%. Interestingly, both Divisia T P M indexes and Divisia C C C indexes increased in four quarters (i.e., the second quarter of 1998 and the first three quarters of 2000). Such increases could result from the calculation of price indexes in these two methods. In the four quarters, there exists a portion of "matched-models" whose prices increased over the period, which influenced the estimated price indexes. Since observed prices were used in the index formula, the imputed prices from hedonic regressions, however, could have no effect on price indexes in this scenario. On the contrary, two of the characteristics price indexes (Char T P M and Char CCC) reflect smooth price changes and did not increase in the four quarters. Although quantity-weighted characteristics units were used in the two methods, implicit char-acteristics prices were estimated for calculating indexes, rather than observed prices. The Fisher index formula could be an explanation for smooth price indexes. Moreover, the two distinct quarters in the N O A Y R regression made the N O A Y R price indexes diverge from the Pooled price indexes, simply because the price indexes of the distinct quarters were calculated by employing the Divisia index formula, as well as the hedonic functional specifications changed after these quarters. 33 . The 95% bias-corrected confidence intervals for each index are presented in Ta-ble 10 and Figure 2 to Figure 7. From the results, we may notice that the large standard errors of a few quarter dummy variable coefficients in pooled regression led to wider confidence intervals for the price index in that quarter. For example, al-though the calculated index for the fourth quarter of 1998 declines, the upper bound of its confidence interval is even higher compared with the previous quarter, which implies the instability of price index in that quarter. It is also worth notice that the indexes constructed by applying Divisia formula (Divisia T P M and Divisia CCC) demonstrate relatively large variances and even biases in the quarters where some matched-models have price increases. 34 Table 5: Variables Description (Desktop) Variable Description Effect lprice Log of price n/a lprospd Log of processor clock speed (MHz) + lsl2 Log of standard level 2 cache memory (KB) + lml2 Log of maximum level 2 cache memory (KB) + lsmem Log of standard memory (MB) + lmmem Log of maximum memory (MB) + lhd Log of hard disk size (MB) + lwt Log of warranty (month) + dcdrom Dummy variable for CD-ROM + dCele Dummy variable for Celeron CPU + dPent Dummy variable for Pentium CPU + dPent2 Dummy variable for Pentium II CPU + dPent3 Dummy variable for Pentium III CPU + dPent4 Dummy variable for Pentium IV CPU + dPentM Dummy variable for Pentium,M CPU + dPentP Dummy variable for Pentium Pro CPU + Table 6: Desktop T P M Regression Results Variable Coefficient (Std. Err.) lprospd 0.552** (0.048) lsl2 0.009 (0.054) lsmem 0.307** (0.020) lhd 0.132** (0.016) lwt 0.123** (0.019) dcdrom 0.065** (0.020) dPent3 0.094* (0.038) dPent4 0.144** (0.056) Intercept 0.097 (0.344) N 1176 R 2 0.679 F (8,1167) 308.85 Significance levels : f 10% * : 5% ** : 1% 35 Table 7: Desktop Pooled Regression Results Var iable Coefficient (Std. E r r . ) lprospd 0.710** (0.022) lsl2 0.054** (0.013) lsmem 0.217** (0.010) lhd 0.068** (0.009) dcdrom 0.060** (0.009) dPentP 0.939** (0.153) dPent2 0.176** (0.024) dPent3 0.112** (0.013) d982 -0.303** (0.019) d983 -0.520** (0.022) d984 -0.529** (0.037) d991 -0.630** (0.025) d992 -0.702** (0.028) d993 -0.862** (0.029) d994 -0.863** (0.031) dOOl -0.920** (0.032) d002 -0.959** (0.037)' d003 -1.002** ' (0.033) d004 -1.222** (0.035) dOl l -1.261** (0.035) d012 -1.322** (0.034) d013 -1.308** (0.041) d014 -1.782** (0.041) d021 -1.893** (0.043) Intercept 1.545** (0.115) N 5148 R 2 0.625 F (24,5123) 355.193 Significance levels : f : 10% * : 5% ** : 1% 36 Table 8: Means of Selected Variables (Desktop) Year Qtr price prospd 12std memstd hdsize wt dcdrom dCele dPent2 dPent3 dPent4 1998 1 2391.79 297.70 460.98 52.12 4132.63 33.31 0.75 0 0.75 0 0 1998 2 3261.03 367.45 512.00 77.34 6107.61 33.54 0.89 0.03 0.97 0 0 1998 3 1589.33 414.45 512.00 78.20 7166.66 33.66 0.82 0 1 0 0 1998 4 1570.80 396.54 440.15 88.00 11088.97 28.27 0.81 0.19 0.81 0 0 1999 1 1278.46 411.58 404.01 66.80 8850.49 32.52 0.60 0.28 0.61 0.11 0 1999 2 1297.99 463.16 420.97 75.63 7735.40 32.66 0.67 0.24 0.34 0.42 0 1999 3 1376.84 541.96 405.46 92.29 9053.41 32.59 0.81 0.27 0.02 0.72 0 1999 4 1363.65 597.32 253.95 90.90 10806.49 34.49 0.81 0.30 0 0.70 0 2000 1 1601.20 624.56 263.05 111.20 11646.67 33.28 0.89. 0.24 0 0.76 0 2000 2 2061.57 761.88 235.85 109.17 11207.17 35.37 0.96 0.17 0 0.83 0 2000 3 1837.67 714".22 242.63 112.91 11467.36 33.94 0.90 "0.19 0 0.81 0 2000 4 1424.06 795.02 227.92 113.07 12346.73 35.03 0.92 0.19 0 0.81 0 2001 1 1088.32 742.95 213.00 106.37 14919.97 32.88 0.94 0.32 0 0.68 0 2001 2 1336.91 926.81 227.49 117.98 16998.53 35.16 0.96 0.22 0 0.58 0.20 2001 3 1363.37 1001.66 223.11 113.42 16893.49 36.00 1.00 0.26 0 0.39 0.35 2001 4 1158.15 1371.42 220.19 155.03 30099.87 35.48 0.97 0.27 0 0.18 0.55 2002 1 1165.36 1590.95 242.33 165.72 30102.15 35.90 0.99 0.29 0 0.04 0.67 Table 9: Desktop Price Indexes Year Qtr Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1998 1 1 1 1 1 1 1 1998 2 0.739 1.252 1.034 0.821 0.942 0.785 1998 3 0.595 0.828 0.570 0.658 0.926 0.619 1998 4 0.590 0.813 0.510 0.649 0.808 0.541 1999 1 0.533 0.765 0.461 0.580 0.747 0.505 1999 2 0.496 0.732 0.421 0.518 0.718 0.457 1999 3 0.422 0.693 0.365 0.448 0.684 0.401 1999 4 0.422 0.670 0.342 0.447 0.658 0.379 2000 1 0.399 0.698 0.350 0.457 0.648 0.365 2000 2 0.384 0.740 0.357 0.437 0.622 0.344 2000 3 0.367 0.752 0.364 • 0.420 0.556 0.312 2000 4 0.295 0.532 0.254 0.437 0.531 0.279 2001 1 0.284 0.434 0.214 0.388 0.442 0.250 2001 2 0.267 0.415 0.196 0.359 0.429 0.223 2001 3 0.271 0.410 0.193 0.362 0.377 0.189 2001 4 0.168 0.329 0.143 0.359 0.311 0.151 2002 1 0.151 0.309 0.130 0.209 0.299 0.136 CO -I X •73 co CO CN -I 1 1 1 2 .3 1 1 4 5 —I 1 1 1 1 1 -6 7 8 9 10 11 Time —1 1 r-12 13 14 — — - Pooled — Divisia TPM Divisia CCC - NOAYR "-.—- -H Char TPM Char CCC Figure 1: Desktop Price Indexes Table 10: 95% Confidence Intervals of Desktop Price Indexes Year Qtr Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1998 1 1 1 1 1 1 1 1 1 1 1 1 1 1998 2 0.698 0.779 0.957 1.460 0.766 1.297 0.768 0.873 0.922 0.955 0.698 0.881 1998 3 0.561 0.631 0.754 0.899 0.484 0.665 0.612 0.706 0.896 0.949 0.537 0.703 1998 4 0.541 0.642 0.736 0.891 0.433 0.602 0.592 0.703 0.760 0.848 0.471 0.618 1999 1 0.503 0.561 0.690 0.844 0.388 0.543 0.541 0.620 0.702 0.784 0.438 0.586 1999 2 0.466 0.526 0.657 0.805 0.355 0.496 0.478 0.555 0.675 0.756 0.393 0.532 1999 3 0.397 0.449 0.625 0.764 0.304 0.427 0.412 0.483 0.642 0.721 0.346 0.466 1999 4 0.394 0.450 0.602 0.737 0.287 0.405 0.407 0.484 0.616 0.696 0.324 0.440 2000 1 0.368 0.430 0.625 0.778 0.290 0.423 0.414 0.510 0.607 0.689 0.311 0.428 2000 2 0.354 0.413 0.622 0.872 0.290 0.452 0.392 0.486 0.578 0.664 0.291 0.402 2000 3 0.342" 0.392 0.653 0.942 0.301 0.481 0.379 0.466" 0.517 0.592 0.263 0.368 2000 4 0.274 0.318 0.455 0.608 0.203 0.308 0.392 0.486 0.492 0.566 0.235 0.330 2001 1 0.260 0.305 0.368 0.500 0.171 0.258 0.342 0.440 0.411 0.472 0.211 0.296 2001 2 0.246 0.286 0.344 0.477 0.156 0.237 0.319 0.404 0.399 0.459 0.188 0.264 2001 3 0.249 0.293 0.347 0.476 0.157 0.239 0.320 0.406 0.347 0.407 0.157 0.225 2001 4 0.154 0.183 0.273 0.382 0.115 0.177 0.319 0.404 0.284 0.340 0.126 0.181 2002 1 0.138 0.165 0.256 0.358 0.104 0.160 0.185 0.236 0.273 0.326 0.113 0.163 1 — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — [ — 1 2 3 4 G G 7 8 9 10 11 12 '13 14 15 16 17 Figure 2: 95% Confidence Intervals of Desktop Pooled Index •I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Figure 3: 95% Confidence Intervals of Desktop N O A Y R Index 41 I - i n 1 r- i 1 - i 1 1 1 1 1 -1 2 3 4- 5 6 7. 8 .9 10 11 12 13- 14 15~ 16 17-Figure 4: 95% Confidence Intervals of Desktop Divisia T P M Index If* $1, 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r -1 2 3 - 4 5 6 7 8 9 10 11 12 13' 14 15. 16 17 Time Figure 5: 95% Confidence Intervals of Desktop Char T P M Index 42 C CD 1 2 3 4 5 6 7 8 '9 10 "11 12 13 14 15 16 17-Figure 6: 95% Confidence Intervals of Desktop Divisia C C C Index is T 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 - : 1 2 3' 4 5- 6 7 8 :9 10 '11 12 131 14 15 16 17 .Time Figure 7: 95% Confidence Intervals of Desktop Char C C C Index 43 8.2 Laptop Price Indexes The study period of laptops is from the second quarter of 1998 to the second quarter of 2002. The quantity-weighted means of selected variables are presented in Table 13. Significant quality improvements are reflected in the table. For instance, hard disk size increased by almost 5 times over the periods; the mean standard memory in the second quarter of 2002 is 3 times more than what it was 4 years ago. Table 11 describes the variables used in hedonic regressions and their expected effects on prices. These variables are widely used in prior empirical research. Different from desktops, laptops have some unique features which are valuable for users: the life of battery and the weight. Since laptops are designed for mobile computing, how long the battery can last and its weight largely affect the quality of a laptop. Moreover, a dummy variable for Lithium-ion battery was used because such type of battery can provide more power and recharge more quickly than the Nickel-cadmium battery. Based on Box-Cox test results, the double-log specification was strongly preferred over linear and semi-log functional forms (A = —0.038 with 95% confidence interval ranging from -0.100 to 0.025). The pooled regression results are presented in Table 14. Overall, the model ex-plains almost 80% of the variance and most of independent variables are significant at 1% level. About 300 records do not have values of battery life and were dropped from the regression sample. A l l the quarter dummy variables had expected signs and magnitude. The variables "lsl2" and "lrescolors" were not significant in the pooled regression. The R 2 of all T P M regressions ranged from 0.51 to 0.85. Table 12 shows an example of T P M regression results with the first quarter of 2001 being the center quarter in the sample. In the regression, the R 2 was 0.679 and was considered to have sufficient prediction ability. 44 The results of laptop C C C regression are presented in Table 15, where "tr" rep-resents the polynomial degree. The model fit was good with R2=0.799 and a few variables were significant at polynomial of degree 1. Particularly, the Pentium III dummy variable was significant at polynomial of degree 3. Similar to desktops, two quarters were tested to be distinct from the previous quarter in the A Y R regressions (the first quarter of 2000 and the first quarter of 2001). Therefore, N O A Y R regressions were performed on three different periods. Separate regressions were also performed on the two distinct quarters and their ad-jacent quarters to obtain predicted prices. The resulting laptop price indexes are presented in Table 16 and Figure 8. Two of the six indexes have lowest A A G R s : Pooled (-33.5%) and N O A Y R (-33.3%) over the periods. Two indexes based on T P M regressions (Divisia T P M and Char T P M ) not only have close A A G R s (-15.4% and -14.7%), but overlap each other for several quarters. The Divisia C C C index and Char C C C index also show price changes that are not quite different and similar A A G R s (-26.4% and -28% respectively). Due to structural changes of hedonic specifications in two distinct quarters, the N O A Y R indexes deviated from the Pooled indexes starting from the first quarter of 2000, but the differences between the two indexes narrowed as the Pooled indexes declined faster in later quarters. Compared with desktop price indexes, the laptop price indexes have smaller vari-ances and demonstrate good stability over the periods, as suggested by Table 17 and Figure 9 to Figure 14. However, similar to their counterparts of desktops, the price indexes constructed using Divisia formula have skewed distributions and the calcu-lated indexes are biased in some quarters, although their variances are close to those of characteristics price indexes. 45 Table 11: Variables Description (Laptop) Variable Description Effect lprice Log of price n/a lprospd Log of processor clock speed (MHz) + lsl2 Log of standard level 2 cache memory (KB) + lsmem Log of standard memory (MB) + lmmem Log of maximum memory (MB) + lhd Log of hard disk size (MB) + lrescolors Log of number of pixels in maximum resolution + ldispsize Log of size of display (inches) + lweight Log of weight (pounds) -lbatlife Log of battery life (minutes) + batlion Dummy variable for Lithium-ion battery + dCele Dummy variable for Celeron CPU + dPent2 Dummy variable for Pentium II CPU + dPent3 Dummy variable for Pentium III CPU + dPent4 Dummy variable for Pentium IV CPU + Table 12: Laptop TPM Regression Results Variable Coefficient (Std. Err.) lprospd 0.016 (0.091) lsl2 0.070 (0.108) lsmem 0.109** (0.034) lhd 0.185** (0.027) lrescolors 0.028* (0.013) ldispsize 0.907** (0.200) lweight -0.332** (0.060) lbatlife 0.088* (0.043) batlion 0.142** (0.053) dPent3 0.168* • (0.082) Intercept 1.900** (0.645) N 340 R 2 0.659 F (10,329) 63.618 Significance levels : f : 10% * : 5% ** : 1% 46 Table 13: Means of Selected Variables (Laptop) Year Qtr price prospd 12std memstd hd colour batlife weight dCele dPent2 dPent3 1998 2 3079.42 255.87 512.00 55.28 5035.74 7.35 3.12 7.01 0 1 0 1998 3 3103.51 283.66 505.51 59.17 5296.03 7.40 2.92 6.77 0 1 0 1999 1 2974.38 350.54 302.58 64.32 6608.39 9.52 2.80 6.52 0.05 0.89 0 1999 2 2702.41 371.70 317.88 63.18 6613.75 12.39 2.74 6.02 0.06 0.87 0 1999 3 2466.70 396.60 264.50 59.61 5971.46 11.70 3.03 6.62 0.22 0.68 0 2000 1 2380.65 554.64 273.83 60.87 7671.33 15.70 2.98 5.98 0.10 0 0.78 2000 2 2647.94 630.58 238.78 68.49 8454.19 14.00 3.37 5.78 0.13 0 0.87 2000 3 2206.54 565.30 237.92 54.47 6142.43 13.29 2.82 5.86 0.14 0.01 0.85 . 2000 4 2383.26 587.95 237.54 71.83 .8303.07 15.07 3.01 5.28 0.14 0 0.85 2001 1 2361.02 661.59 239.02 85.80 13815.46 16.00 3.65 5.95 0.12 0 0.87 2001 2 1810.68 691.12 228.59 81.49 10256.63 15.65 2.42 5.53 0.21 0 0.79 2001 3 2032.96 884.69 259.95 111.73 15768.12 15.15 2.73 5.98 0.18 0 0.82 2001 4 1948.13 922.21 326.51 154.19 17451.38 14.86 2.68 5.70 0.16 0 0.84 2002 2 1936.08 1148.43 456.51 180.66 25355.16 15.71 2.67 5.19 0.16 0 0.61 Table 14: Laptop Pooled Regression Results Variable Coefficient (Std. Err.) lprospd 0.261** (0.039) lsmem 0.142** (0.013) lmmem 0.071** . (0.014) lhd 0.198** (0.012) dPent2 0.170** (0.021) dPent3 0.161** (0.015) dPent4 0.069 (0.045) ldispsize 1.149** (0.081) lweight -0.420** (0.023) lbatlife 0.062** (0.015) batlion 0.070** (0.020) d983 -0.173** (0.024) d991 -0.331** (0.025) d992 -0.504** (0.028) d993 -0.542** (0.031) dOOl -0.785** (0.045) d002 -0.785** (0.046) d003 -0.815** (0.044) d004 -0.899** (0.043) dOl l -1.085** (0.049) d012 -1.202** (0.048) d013 -1.355** • (0.051) d014 -1.471** (0.054) d022 -1.647** (0.064) Intercept 1.364** (0.206) N 1592 R 2 0.794 F (24,1567) 251.775 Significance levels : f : 10% * : 5% ** : 1% 48 Table 15: Laptop CCC Regression Results Variable Coefficient (Std. Err.) lprospd 0.481** (0.096) lsmem 0.045 (0.036) lhd 0.177** (0.012) dPent2 0.170** (0.021) dPent3 -5.968** (0.701) dPent4 0.186** (0.069) ldispsize 2.075** (0.177) lweight -0.361** (0.046) lbatlife 0.159** (0.040) batlion -0.161** (0.047) tr 0.200** (0.050) trspd -0.013 (0.009) trsl2 -0.017** (0.006) tr2sl2 0.001** (0.000) trsmem 0.043** (0.009) tr2smem -0.003** (0.001) trdisp -0.082** (0.017) trP3 1.575** (0.177) tr2P3 -0.130** (0.015) tr3P3 0.003** (0.000) trweight -0.008* (0.005) trbatlife -0.027** (0.009) tr2batlife 0.002** (0.001) trbatlion 0.023** (0.005) Intercept -1.590** (0.546) N 1592 R 2 0.799 F (24,1567) 259.526 Significance levels : ] : 10% * : 5% ** : 1 % 49 Table 16: Laptop Price Indexes Year Qtr Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1998 2 1 1 1 1 1 1 1998 3 0.841 0.970 0.890 0.836 0.999 0.874 1999 1 0.718 0.943 0.707 0.692 0.953 0.686 1999 2 0.604 0.870 0.626 0.585 0.911 0.615 1999 3 0.582 0.817 0.572 0.545 0.852 0.552 2000 1 0.457 0.806 0.587 0.405 0.832 0.554 2000 2 0.457 0.797 0.590 0.399 0.801 0.557 2000 3 0.443 0.764 0.563 0.385 0.769 0.526 2000 4 0.407 0.748 . 0.515 0.362 0.757 0.478 2001 1 0.338 0.686 0.454 0.318 0.681 0.423 2001 2 0.301 0.599 0.385 0.285 0.602 0.370 2001 3 0.258 0.566 0.352 0.246 0.552 0.324 2001 4 0.230 0.529 0.316 0.224 0.540 0.289 2002 2 0.193 0.507 0.279 0.196 0.524 0.257 Time • • - • Pooled — — Divisia TPM B — Divisia CCC — A — - NOAYR — - * — Char TPM Char CCC Figure 8: Laptop Price Indexes Table 17: 95% Confidence Intervals of Laptop Price Indexes Year Qtr Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1998 2 1 1 1 1 1 1 1 1 1 1 1 1 1998 3 0.803 0.890 0.941 0.996 0.862 0.924 0.794 0.880 0.984 1.015 0.857 0.891 1999 1 0.686 0.759 0.912 0.970 0.673 0.751 0.653 0.733 0.936 0.973 0.654 0.718 1999 2 0.577 0.641 0.840 0.898 0.592 0.669 0.549 0.625 0.890 0.933 0.580 0.651 1999 3 0.552 0.624 0.782 0.848 0.538 0.618 0.509 0.592 0.824 0.877 0.517 0.590 2000 1 0.424 0.496 0.767 0.840 0.534 0.659 0.361 0.453 0.797 0.861 0.502 0.617 2000 2 0.423 0.502 0.757 0.836 0.530 0.668 0.357 0.450 0.767 0.830 0.495 0.628 2000 3 0.413 0.481 0.726 0.801 0.505 0.647 0.347 0.433 0.731 0.800 0.464 0.596 2000 . 4 0.380 0.445 0.706 0.785 0.459 0.583 0.325 0.409 0.719 0.789 0.423 0.540 2001 1 0.309 0.377 0.647 0.721 0.406 0.521 0.282 0.361 0.645 0.712 0.376 0.479 2001 2 0.278 0.331 0.558 0.629 0.343 0.434 0.253 0.323 0.563 0.633 0.328 0.414 2001 3 0.235 0.287 0.525 0.598 0.318 0.401 0.219 0.281 0.518 0.581 0.291 0.364 2001 4 0.207 0.258 0.481 0.558 0.286 0.361 0.197 0.255 0.503 0.570 0.258 0.325 2002 2 0.169 0.218 0.462 0.536 0.250 0.325 0.170 0.227 0.488 0.552 0.224 0.294 ... 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Time Figure 9: 95% Confidence Intervals of Laptop Pooled Index \ \ V -1 ^ a i . ^ „ 'l 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Time Figure 10: 95% Confidence Intervals of Laptop N O A Y R Index 53 T 1 1- ~1 1 1 1 1 1 1 1 1 -1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17: Time Figure 11: 95% Confidence Intervals of Laptop Divisia T P M Index 1: 8 9 10 11 12 13 14 15 16 17 Time Figure 12: 95% Confidence Intervals of Laptop Char T P M Index 54 • < » ^ ^ . . . . . T 1 r~ - 1 1 1 1 1 1 1 1 1 r-1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Figure 13: 95% Confidence Intervals of Laptop Divisia C C C Index :3r. i i i 1 r~ —\ n 1 1 1 1 1 1 1 r-1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Time Figure 14: 95% Confidence Intervals of Laptop Char C C C Index 55 8.3 Workstation Price Indexes Workstation is a general-purpose computer designed to be used by one person at a time and which offers higher performance than normally found in a personal com-puter, especially with respect to graphics, processing power and the ability to carry out several tasks at the same time. Workstations lie between personal computers (PCs) and minicomputers in terms of computing power. There is no clear line that distinguishes desktops from workstations. However, usually high-end desktop com-puters are equivalent to low-end workstations (Wikipedia, 2006b). Table 22 lists selected means of some variables in the workstation database. The processor clock speed and the hard disk size increased by 3 times over the three-year period. The increasing percentages of Pentium III and Pentium IV CPUs indi-cates technological innovations. Different from normal desktops, fast workstaions are equipped with Intel Xeon CPUs. Xeons are high-end Intel C P U chips that include error checking memory, system management features and fast system buses and level 2 caches, which can improve system performance in carrying out tasks. What also makes workstations different from desktops is the support of SMP (Symmetric Multiprocessing) feature. SMP is a computer architecture that provides fast performance by making multiple CPUs available to complete individual processes simultaneously (multiprocessing). Unlike asymmetrical processing, any idle processor can be assigned any task, and additional CPUs can be added to improve performance and handle increased loads. SMP uses a single operating system and shares common memory and disk resources. Table 18 describes the variables used in hedonic regressions. Since workstations usually have higher performance in graphics than normal desktops, it is reasonable to include the variables for video memory in the analysis. By conducting a Box-Cox test, the double-log specification was preferred over the other two functional forms (A=0.054 with 95% confidence interval ranging from 56 -0.034 to 0.142). Table 20 shows the workstation pooled regression results. Overall, the model explained more than 60% of total variance, and all the coefficients of quarter dummy variables had expected signs. However, probably because of constrained coefficients of the characteristics, the magnitudes of some quarter dummy variable coefficients fluctuated rather than followed a general downward trend. The average R 2 of T P M regressions was around 0.5, and they ranged from 0.4 to 0.9 depending on each sample period. Table 19 is an example of T P M regression with the fourth quarter of 1994 being the center quarter. The R 2 is 0.5 and the dummy variable for Pentium III C P U is negative, indicating its relatively low performance compared with Xeon C P U . The C C C regressions results are presented in Table 21. The overall R 2 is 0.585. The constant term and "lprospd" had very significant coefficients in polynomials of degree 3. The coefficients of some other variables were not significant after the trend variables were included. Based on F-tests of A Y R regressions on workstations, two quarters were distinct from their previous quarters: the first quarter of 2000 and the second quarter of 2001. Therefore, the sample time frame was divided into three different periods and an N O A Y R regression was performed accordingly. The resulting workstation price indexes are presented in Table 23 and Figure 15. Divisia T P M indexes and Char T P M indexes have nearly equal A A G R s (-17% and -16.5% respectively), but the Divisia T P M indexes fluctuated in a wider range. The other four indexes declined faster (around -27%) over the periods. These indexes diverged in the fourth quarter of 1999, probably due to two missing previous quarters. The methods which used Divisia index formula (Divisia T P M and Divisia CCC) and time dummy variables (Pooled and NOAYR) generated indexes of 1.0 between the third quarter and fourth quarter of 2001. The unchanged indexes accurately reflect no price movements in that period, as all the 52 records are "matched-models" with 57 equal prices. The 95% confidence intervals are presented in Table 24 and Figure 16 to Figure 21. Figure 18 and Figure 19 reflect the fact that, the calculated Divisia T P M price indexes are seriously biased in certain quarters, while the characteristics indexes using same T P M regressions show smooth price changes and unbiased estimates. Such discrepancy could result from the differences between underlying employed index formulas (matched-model Divisia v.s. Fisher). 58 Table 18: Variables Description (Workstation) Variable Description Effect lprice Log of price n/a lprospd Log of processor clock speed (MHz) + lsl2 Log of standard level 2 cache .memory (KB) + . lml2 Log of maximum level 2 cache memory (KB) + lsmem Log of standard memory (MB) + lmmem Log of maximum memory (MB) + lhd Log of hard disk size (MB) + smpsupt Number of Symmetric Multiprocessing CPUs + lstdvm Log of standard video memory (MB) + lmaxvm Log of maximum video memory (MB) + dCele Dummy variable for Celeron C P U + dPent2 Dummy variable for Pentium II C P U + dPent3 Dummy variable for Pentium III C P U + dPent4 Dummy variable for Pentium IV C P U + dXeon Dummy variable for Intel Xeon C P U + Table 19: Workstation T P M Regression Results Var iable Coefficient (Std. E r r . ) lprospd 1.212** (0.131) lsl2 0.421** (0.046) lsmem 0.350** (0.045) lmmem 0.272** (0.038) lhd 0.119* (0.046) dPent3 -0.304** (0.039) Intercept -6.564** (0.845) N 687 R 2 0.505 F (6,680) 115.714 Significance levels : f : 10% * : 5% ** : 1% 59 Table 20: Workstation Pooled Regression Results Variable Coefficient (Std. E r r . ) lprospd 0.678** (0.079) lsl2 0.482** (0.036) lsmem 0.410** (0.022) lmmem 0.213** . (0.018) lhd 0.053* (0.024) dPent2 0.373** (0.079) dPent3 0.330** (0.066) dPentP 1.571** (0.252) dXeon 0.560** (0.067) d991 -0.316** (0.039) d994 -0.550** (0.049) dOOl -0.445** (0.050) d002 -0.494** (0.057) d004 -0.868** (0.066) dOl l -0.828** (0.066) d012 -1.006** (0.071) d013 -1.058** (0.091) d014 -1.058** (0.091) Intercept -2.995** (0.661) N 1614 R 2 0.585 F (18,1595) 124.692 Significance levels : f : 10% * : 5% ** : 1% 60 Table 21: Workstation C C C Regression Results Var iable Coefficient (Std. E r r . ) lprospd -1.312** (0.208) lsl2 0.295** (0.086) lsmem 0.423** (0.022) lmmem 0.225** (0.018) lhd 0.063** (0.023) dPent3 -0.083* (0.038) dXeon 0.280** (0.081) tr -8.612** (0.687) tr2 1.273** (0.114) tr3 -0.052** (0.006) trspd 1.311** (0.109) tr2spd -0.200** (0.018) tr3spd 0.008** (0.001) trsl2 0.040* (0.016) trXeon -0.020* (0.010) Intercept 10.661** (1.248) N 1614 R 2 0.585 F (15,1598) 150.267 Significance levels : f:10% * : 5% ** : !% 61 Table 22: Means of Selected Variables (Workstation) Year Qtr price prospd 12std memstd hdsize stdvm smpsupt dPent2 dPent3 dPent4 dXeon 1998 4 3823.01 378.69 512 100.43 6431.45 9.12 2.04 0.98 0 0 0 1999 1 3942.73 447.53 635.78 133.98 8072 9.28 2 0.65 0.26 0 0.05 1999 4 3850.96 521.65 547.46 149.61 9984.44 19.56 2.10 0.09 0.72 0 0.19 2000 1 4014.97 561.49 475.56 148.09 10266.47 22.68 2.09 0.08 0.77 0 0.15 2000 2 3936.24 673.89 348.67 167.38 13690.01 31.02 2.12 0 0.84 0 0.16 2000 4 3239.18 842.82 256 228.86 15181.76 39.15 2 0 0.82 0 0.18 2001 1 3481.54 842.47 256 . 230.72 15528.22 39.78 2 0 .0.84 0 0.16 2001 2 2864.64 987.68 256 234.67 16933.97 27.65 2 0 0.72 0.17 0.10 2001 3 2376.77 1049.81 256 192 17734.92 20.31 2 0 0.62 0.31 0.08 2001 4 2376.77 1049.81 256 192 17734.92 20.31 2 0 0.62 0.31 0.08 Table 23: Workstation Price Indexes Year Qtr Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1998 4 1 1 1 1 1 1 1999 1 0.729 0.873 0.761 0.750 0.988 0.760 1999 4 0.578 0.835 0.657 0.638 0.947 0.643 2000 1 0.641 0.859 0.679 0.688 0.896 0.635 2000 2 0.611 0.725 0.566 0.635 0.835 0.608 2000 4 0.421 0.592 0.423 0.431 0.774 0.482 2001 . 1 0.438 0.605 0.432 0.448. 0.695 0.421 2001 2 0.367 0.561 0.375 0.371 0.689 0.378 2001 3 0.349 0.561 0.378 0.341 0.643 0.363 2001 4 0.349 0.561 0.378 0.341 0.572 0.394 Figure 15: Workstation Price Indexes Table 24: 95% Confidence Intervals of Workstation Price Indexes Year Qtr Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1998 4 1 1 1 1 1 1 1 1 1 1 1 1 1999 1 0.677 0.783 0.843 0.888 0.718 0.811 0.696 0.804 0.963 1.009 0.714 0.799 1999 4 0.524 0.644 0.788 0.853 0.601 0.742 0.582 0.710 0.910 0.987 0.585 0.710 2000 1 0.580 0.726 0.793 0.901 0.610 0.777 0.601 0.788 0.847 0.943 0.577 0.708 2000 2 0.540 0.697 0.642 0.763 0.478 0.625 0.545 0.713 0.782 0.890 0.551 0.689 2000 4 0.366 0.485 0.531 0.621 0.357 0.477 0.366 0.485 0.718 0.837 0.426 0.556 2001 1 0.379 0.502 0,524 0.649 0.372 0.499 0.382 0.507 0.635 0,755 0.369 0.488 2001 2 0.314 0.432 0.475 0.605 0.321 0.440 0.309 0.432 0.629 0.751 0.328 0.440 2001 3 0.295 0.414 0.469 0.614 0.325 0.445 0.283 0.396 0.575 0.702 0.314 0.427 2001 4 0.294 0.408 0.485 0.624 0.323 0.448 0.286 0.400 0.510 0.629 0.337 0.476 3 7 8 Time 10 11 12 13 Figure 16: 95% Confidence Intervals of Workstation Pooled Index Figure 17: 95% Confidence Intervals of Workstation N O A Y R Index 66 -.•i.*.-,^,lfs-. -1 1 1 r- - | 1 -1 2 3 4 5 6 7 8 9 10 11 12 13 Figure 18: 95% Confidence Intervals of Workstation Divisia T P M Index -IN 1 1 1 r- - | 1 1 1 1 1 1 1 -1 2 3 •4-"- 5 6 r-7> 8 9 ;10- 11 -12 13 Figure 19: 95% Confidence Intervals of Workstation Char T P M Index 67 -<?> 9 -1 1 1 r- ~i 1 1 1 1 1 ! 1 2 3 4 5 6 7 8 9 10 11 12 13 Time Figure 20: 95% Confidence Intervals of Workstation Divisia C C C Index 1 2 3 4 5 B 7. 8 9 10 11 12 13 Time Figure 21: 95% Confidence Intervals of Workstation Char C C C Index 68 8.4 Server Price Indexes Server is a type of computer which provides some service for other computers (clients) connected to it via a network (Wikipedia, 2006c). The most common example is a file server which has a local disk and services requests from remote clients to read and write files on that disk. Servers can be classified in different ways. In terms of the type of provided services, it could be proxy server, F T P server, mail server, etc. In terms of the level of performance and capacity, it could be classified as an entry-level server, a mid-range server, an enterprise server or a datacenter server. Since the purpose of a server is to provide service and manage data and resources on the network, how fast the data could be transmitted becomes one of the important factors that influence its performance. Usually this is measured as the "Data Transfer Rate" in Mbps (million bits per second). Unlike ordinary computers, servers require not only high performance in transmit-ting data, but stable and reliable performance that can consistently provide services and storage. In order to function as a system that satisfies strict requirements, a server is usually equipped with more processors, and they have different features from those used in a desktop or laptop. As a result, more types of CPUs are manu-factured by different vendors to meet such demands (for example, Intel Itanium, I B M Powerpc series and Sun SPARC series). Table 29 lists means of selected variables in the server database. Technological innovations over the 9-year period are reflected in the increases of processor clock speed, standard memory, hard disk size. The variables used in hedonic regressions and their expected effects are described in Table 25. Dummy variables were created for different types of processors and server categories. To choose the proper functional form, a Box-Cox test was conducted and the 69 double-log specification was preferred over linear and semi-log specifications (A=0.09 and 95% confidence interval ranged from 0.06 to 0.12). Table 27 shows the server pooled regression results. The R 2 is over 0.8, which suggests the model fits the data quite well. However, due to missing values of "ldtr" and some server type dummy variables (dSAIA, dWeb) in certain periods, they could not be included in the regression. The variables ended with "spd" denote interaction terms of processor clock speed and C P U dummy variables, since the architectures of processors manufactured by various venders are deemed quite different. The coeffi-cients of year dummy variables had expected signs and followed a downward trend over the period. A l l the T P M regressions had R 2 over 0.6, and the highest R 2 was over 0.9. Table 26 shows the T P M regression results with year 1998 as the center year. The R 2 is 0.663, which is sufficient for predicting prices. The C C C regression results are presented in Table 28, and "tr" represents the trend variable. Particularly, "lprospd" and "lsmem" had significant coefficients at polynomials of degree 3. With R 2 over 0.8, the model could accurately predict prices for each year and provide a good basis for constructing price indexes subsequently. In A Y R regressions, 1999 is the only year that was tested to be distinct from its previous year (1998). Therefore, a N O A Y R regression was performed on the two separate periods (1995-1998 and 1999-2003). The resulting price indexes are presented in Table 30 and Figure 22. The Char C C C price indexes have the lowest average annual growth rate (AAGR) of -32%, and the highest A A G R is -12.8% (Divisia T P M ) . It is interesting that the Pooled index and N O A Y R index did not change much between 1996 and 1998, while other indexes declined by at least 38% (Divisia T P M ) . The relatively small number of records in 1998, and the resulting large standard error of the regression coefficient could be the factors which lead to such difference, as the price index was calculated by the anti-log of coefficient plus the correction factor. 70 Such large standard errors could also lead to the wider confidence intervals of the two indexes as shown in Table 31, while the other four indexes behave stably over the periods. The Divisia T P M index and Char T P M index also diverged in 1998, which resulted from the differences between the imputed prices and observed prices in small sample size. Figure 23 and Figure 28 show the distributions of calculated price indexes in each year. The results suggest that, compared with pooled price index, although the N O A Y R price index has similar trends in terms of price changes, it has much larger variances particularly in later periods. What is interesting is the almost identical trends and variances in two pairs of indexes (Divisia T P M index, Char T P M index and Divisia C C C index, Char C C C index). The reason for the similarity could be the assigned unit (1) quantity to each model and the relatively small number of matched models in adjacent years. The biased indexes in later periods might imply that the C C C regression could not accurately model the structural changes in the coefficients of characteristics. 71 Table 25: Variables Description (Server) Variable Description Effect lprice Log of price n/a lprospd Log of processor clock speed (MHz) + lsmem Log of standard memory (MB) + lmmem Log of maximum memory (MB) + lhd Log of hard disk size (MB) + ldtr Log of data transfer rate (Mbps) + maxprocs Number of supported maximum processors + dPent Dummy variable for Pentium C P U + dPent3 Dummy variable for Pentium III C P U + dPentP Dummy variable for Pentium Pro C P U + dAlpha21264 Dummy variable for D E C Alpha 21264 C P U + dPowerpc Dummy variable for IBM Powerpc C P U + dUSPARC2 Dummy variable for Sun UltraSPARC II C P U + dSAIA Dummy variable for SAIA server -dUnixEp Dummy variable for Unix Enterprise server + dUnixEn Dummy variable for Unix Entry-level server + dUnixMd Dummy variable for Unix Mid-range server + dUnixS Dummy variable for general Unix systems server + dWeb Dummy variable for Web server -dDataEp Dummy variable for Datacenter Enterprise server + Table 26: Server T P M Regression Results Var iable Coefficient (Std. Er r . ) lprospd 0.426* (0.177) lsmem 0.101 (0.089) lmmem 0.487** (0.070) maxprocs 0.001* (0.000) dUnixEp 1.176** (0.265) dUnixMd 0.606* (0.257) dUnixS 1.060** (0.203) dWeb 1.323** (0.364) dPentPspd 0.102 (0.077) dPentspd -0.079 (0.059) Intercept 2.896** (0.967) N 200 R 2 0.663 F (10,189) 37.17 Significance levels : f : 10% * : 5% ** : 1% 72 Table 27: Server Pooled Regression Results Var iable Coefficient (Std. E r r . ) lprospd 0.027 (0.102) lsmem 0.237** (0.044) lmmem 0.301** (0.030) lhd 0.044 (0.031) dUnixEp 1.486** (0.149) dUnixMd 0.804** (0.096) dUnixS 0.713** (0.176) dDataEp 2.078** (0.173) dPent3spd -0.149* (0.062) dAlphaspd 0.059** (0.018) dPowerspd -0.029* (0.015) dUSPARC2spd -0.069** (0.018) d96 -0.703** (0.115) d98 -0.769** (0.261) d99 -1.223** (0.207) dOO -1.478** (0.218) dOl -2.147** ' (0.231) d02 -2.353** (0.237) d03 -2.768** (0.251) Intercept 4.902** (0.569) N 448 R 2 0.816 F (19,428) 94.452 Significance levels : t : 10% * : 5% ** : 1% 73 Table 28: Server C C C Regression Results Var iable CoefRcient (Std. E r r . ) lprospd -0.451* (0.213) lsmem 0.617** (0.222) lmmem 0.967** (0.096) dSAIA -0.422** . . (0.145) dUnixEp 1.537** (0.145) dUnixMd 0.869** (0.100) dUnixS -1.005** (0.287) dWeb -2.024** (0.547) dDataEp 1.950** (0.169) dAlphaspd -0.346** (0.103) dPowerspd -0.022 (0.014) trspd 0.607** (0.167) tr2spd -0.134** (0.038) tr3spd 0.008** (0.002) trsmem -0.450* (0.186) tr2smem 0.125** (0.041) tr3smem -0.009** (0.003) trmmem -0.288** (0.042) tr2mmem 0.025** (0.004) trAlpha 0.054** (0.014) trWeb 0.659* (0.303) Intercept 5.441** (0.446) N 693 R 2 0.814 F (21,671) 140.265 Significance levels : t : 10% * : 5% ** : 1% 74 Table 29: Means of Selected Variables (Server) Year price prospd memstd hdsize dtr dUnixEn dUnixMd dDataEp dPowerpc dUSPARC2 1995 71086.67 131.60 67.39 27575.73 151.25 0 0 0 0.11 0 1996 46177.59 160.87 81.19 29590.12 371.51 0 0 0 0.09 0 1998 216883.8 288.04 662.86 279561.4 393.11 0 0.04 0 0.18 0.25 1999 94647.89 283.94 456.8 233998.6 364 0 0.29 0 0.36 0.19 2000 205202.5 420.94 1219.08 557944.5 363 0.29 0.19 0 0.23 0.11 2001 79753.55 651.14 1167.08 499280 439.81 0.14 0.31 0.15 0.12 0.11 . 2002 87044 730.25 1789.49 500421.4 978.89 0.27 0.35 0.10 0.16 0.07 2003 93693.66 801.22 1769.96 1537846 1024.10 0.47 0.37 0.16 0.13 0.09 Table 30: Server Price Indexes Year Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1995 1 1 1 1 1 1 1996 0.498 0.706 0.521 0.560 0.687 0.416 1998 0.480 0.435 0.056 0.557 0.334 0.056 1999 0.301 0.348 0.042 0.447 0.264 0.034 2000 0.234 0.336 0.037 0.388 0.231 0.027 2001 0.12Q 0.310 0.027 0.196 0.220. 0.025 2002 0.098 0.295 0.024 0.191 0.192 0.022 2003 0.065 0.300 0.021 0.164 0.193 0.015 1 3 1 1 1 4 5 * 6 i IBBHIS Time Pooled Divisia TPM Divisia CCC — NOAYR X Char TPM Char CCC Figure 22: Server Price Indexes Table 31: 95% Confidence Intervals of Server Price Indexes Year Pooled Divisia T P M Divisia C C C N O A Y R Char T P M Char C C C 1995 1 1 1 1 1 1 1 1 1 1 1 1 1996 0.394 0.637 0.620 0.832 0.407 0.684 0.397 0.716 0.583 0.747 0.286 0.554 1998 0.313 0.734 0.328 0.562 0.023 0.135 0.292 0.913 0.215 0.482 0.022 0.121 1999 0.216 0.410 0.258 0.435 0.018 0.099 0.241 0.823 0.147 0.401 0.013 0.073 2000 0.159 0.332 0.249 0.449 0.016 0.085 0.194 0.727 0.120 0.363 0.010 0.055 2001 0.080 0.169 0.217 0.423 .0.013 0.058 0.098 0.355 0.116 0.354 0.009 .0.048 2002 0.061 0.141 0.187 0.408 0.011 0.056 0.098 0.354 0.094 0.310 0.009 0.045 2003 0.040 0.100 0.195 0.415 0.009 0.049 0.085 0.314 0.090 0.314 0.005 0.035 4-- 1—J ~1 r~ •-5 *Tjme Figure 23: 95% Confidence Intervals of Server Pooled Index Figure 24: 95% Confidence Intervals of Server N O A Y R Index 79 Figure 25: 95% Confidence Intervals of Server Divisia T P M Index Figure 26: 95% Confidence Intervals of Server Char T P M Index 80 1:- 4—4-- — — -J J j Time Figure 27: 95% Confidence Intervals of Server Divisia C C C Index J 1 j. 5 - I 1 1 1 ! 5 Time Figure 28: 95% Confidence Intervals of Server Char C C C Index 81 9 Discussions As described earlier, the purpose of this study is to apply six different hedonic price index methods on four major computer platforms (desktops, laptops, servers and workstations), and use the resulting price indexes as empirical evidence for measuring the price movements of ICT products, especially computers. In this section I shall compare the trends of technological innovations in the computer industry, as well as the properties and features of each hedonic price index method. 9.1 Computer Price Changes Over the period of 1998-2002, continual technological progress in computer manufac-turing industry led to steady and fast declines of price indexes measured by hedonic methods. Table 32 shows the average annual growth rate(AAGR) of each price in-dex for each computer platform. Based on these price indexes, the average A A G R s of indexes for the four platforms lie between -20% and -30%, which suggests that consumers benefited from technological innovations, and enjoyed faster and more powerful computers with less spending. The estimated average A A G R s lies within the range of A A G R s estimated by most researchers. Among the four platforms, the prices of desktops dropped fastest with an average A A G R of -32.7%. The highest desktop A A G R is -24.7%, which is estimated using Divisia T P M method. However, it is still the lowest compared with the same price indexes of other three platforms. Such differences cbuld be attributed to the consistent and stable technological improvements in manufacturing desktop computers. As the variables means in Table 8 indicate, the quantities of most characteristics either remain constant or change in a gradual manner over time. Therefore, the quality-adjusted price changes of desktops have a stable and gradual declining trend in the long-term, as suggested by the smooth Char T P M index and Char C C C index. 82 On the contrary, there is no monotonic and stable movement of technological advancement in laptop computers. A l l the price indexes indicate that the quality of laptops improved at a lower rate in year 2000, which could also be inferred from Table 13, where most of the variables remained nearly unchanged during the period. However, they also illustrate that the technological innovations in laptops followed a similar trend to that of desktops after year 2000. The A A G R s of Pooled index and N O A Y R index are lower than those of other indexes for laptops mainly because of the significant structural changes in characteristics between the third quarter of 1999 and the first quarter of 2000. The figures in Table 13 suggest that in the first quarter of 2000, while other characteristics changed gradually, almost 80% of laptops were equipped with Pentium III CPUs instead of Pentium II CPUs, and the proportion of Celeron processors dropped by more than half. Such jumps in technological inno-vations were accurately modeled by the N O A Y R method. Although the coefficients were held constant in Pooled regression (which could bias the coefficients), the unde-tected price changes were attributed to the time dummy variable for the first quarter of 2000 with a high R 2 . As a contrast, the methods that assume gradual and smooth changes of characteristics (the C C C method and T P M method) generated indexes which showed constant or increasing price changes in that quarter. Such difference in price indexes could be considered as empirical evidence for Melser (2004) 's comments on the C C C method. 1 7 From the resulting price indexes and the changes of characteristics in Table 22, we may infer that, the movement of price changes in workstations was largely influ-enced by the changes in the proportion of computer processors. Starting from the fourth quarter of 1998 t i l l the fourth quarter of 1999, the Pentium II CPUs used by workstations were largely replaced by Pentium III and Xeon processors, which could result from the increasing demand of processing power for large amount of work. As 1 7 Melser (2004) commented that, "If the technological innovations do not happen smoothly, but rather occur in jumps, the C C C price index could not accurately reflect the price changes and wil l lead to 'artificially' smooth changes". 83 a result, the quality-adjusted price indexes demonstrated a declining trend. However, from the first quarter of 2000 to the first quarter of 2001, not only were the market shares of different workstation CPUs quite stable, .but other characteristics remained almost unchanged. It could explain the flat and increasing price movements in some indexes over the period. In the server market, the prices followed a downward trend. The largest decline of prices happened from 1996 to 1998, as suggested by four price indexes (Divisia T P M , Char T P M , Divisia C C C and Char CCC) . Such obvious decline of prices reflect significant advancement of technological innovations in server industry over that period. In Table 29, the figures show that the processor speed nearly doubled, and the standard memory increased by more than 8 times during the period. Judging from the resulting price indexes, the pace of technological innovations appeared to be steady and slower starting from 1999. Table 32: A A G R s of Computer Price Indexes Index Desktops Laptops Workstations Servers Pooled -36.9% -33.5% -28.8% -26.8% N O A Y R -31.1% -33.3% -29.9% -18.0% Divisia T P M -24.7% -15.4% -17.0% -12.8% Divisia C C C -39.1% -26.4% -26.9% -28.0% Char T P M -25.7% -14.7% -16.5% -16.7% Char C C C -38.6% -28.0% -26.3% -32.0% Average -32.7% -25.2% -24.2% -22.4% 84 9.2 Hedonic Price Index Methods In previous sections, six different hedonic methods were used to construct price in-dexes of four computer platforms. The resulting price indexes over the periods imply the underlying differences in these hedonic methods. For better understanding of their performances, I shall discuss and compare these methods in terms of bias, variance and resource: To calculate accurate price indexes, statistical agencies are usually concerned with the size of bias in the indexes estimated by a particular hedonic method. In general, I classify the bias in an estimated hedonic price index into three categories: • Estimation Bias: Bias could occur in the process of estimating the hedonic function. If the method holds constant the coefficients of existing characteristics over time, the estimated coefficients could be biased because of the "constant" constraint. Similarly, as new technology emerges, if the method does not allow new characteristics to be included in the hedonic function, the coefficients could also be biased due to missing important variables in estimation. • Market Bias: Because of rapid technological innovations in the computer mar-ket, old models were replaced by new ones in a short period of time. Moreover, models manufactured by certain vendors had much larger market shares than other vendors. For the purpose of calculating the price index as a deflator, accounting for the market share of each model becomes important to reflect general price changes. Thus, if the hedonic method does not make such adjust-ment, it could lead to bias in the resulting price index. • Bootstrap Bias: Bootstrap bias of an estimate is defined as the difference be-tween the expectation of the estimator and the value of the estimate. In this case, the bootstrap bias measures the deviation of observed price index from the mean after 1000 replications. The bootstrap bias is usually undesirable as 85 we do not want the variation of price index to be overwhelmingly on either the low side or the high side (Efron and Tibshirani, 1993). Table 33 shows the sizes of potential biases in the six hedonic price index meth-ods. Among them, the N O A Y R method has small estimation bias as it allows the coefficients of characteristics to change over time, and new characteristics could be included in the hedonic function as well. The Divisia T P M and Char T P M methods have medium estimation biases as they allow new characteristics could be included in the hedonic function. However, the three-period pooled regressions could possi-bly cause biased coefficients. The Divisia C C C and Char C C C methods explicitly model smooth changes in the coefficients of characteristics. However, they do not allow the inclusion of new characteristics as they are based on the estimation of a single hedonic function. Therefore, these two methods also have medium estimation biases. On the other hand, the Pooled regression method has large estimation bias because it neither allows the characteristics coefficients to change nor includes new characteristics in later period. Compared with other methods, the Pooled regression method has large market bias as it makes no adjustment on the market share of a computer model. The N O A Y R method weights the price ratios by market shares (Divisia index formula) when there is significant difference between hedonic functions in two adjacent periods. It does not make adjustment in periods where the hedonic function specification remains unchanged. Therefore, the N O A Y R method has medium market bias. On the contrary, the other four hedonic methods have small biases as they weight the price ratios by either market shares, or quantities of characteristics (Fisher index formula). As the figures in section 8 indicate, the price indexes estimated by Pooled re-gression method, Char T P M method and Char C C C method demonstrated small bootstrap biases. On the other hand, the price indexes estimated by Divisia T P M and Divisia C C C methods had relatively large bootstrap biases, especially in periods 86 where the price indexes increased. As discussed earlier, the N O A Y R method has medium bootstrap bias because it employs two different ways to calculate indexes depending on the significance of change in hedonic function. Among all the biases, the estimation bias has the biggest effect on price indexes as the imputed prices (or regression coefficients) are determinant factors in index calcula-tions. However, due to limited sample sizes and lack of quantity data on workstations and servers, it becomes difficult to assess the performance of each hedonic method for these two platforms. In most cases, technological innovations happen in a smooth and systematic man-ner, and a new computer technology becomes gradually prevalent in the market. Therefore, taking all types of biases into account, the Char C C C price index is least biased and could be used as reference for other indexes. For comparison purpose, Table 34 shows the "relative" bias of each price index per year for desktops and laptops based on the A A G R s in Table 32. In Table 34, the plus sign ("+") indicates that the index overestimates quality improvement, while the minus sign ("-") means the index understates quality improvement. Besides the size of potential bias in a hedonic method, variance and resource are the factors that need attention for statistical agencies: • Variance: Variance is another undesirable aspect of the price index and large variance usually indicates inefficiency of an estimator. In this case, 95% confi-dence intervals of price indexes reported by bootstrap results reflect the size of variance. • Resource: In order to calculate price indexes, a hedonic method requires a com-prehensive database on computer market, and certain number of calculations and operations are needed as well. They could be considered as the resource or cost of implementing the hedonic method. Table 33 also lists the size and variance and the amount of required resource for 87 • each hedonic method. The tables and figures in section 8 demonstrated that the price indexes constructed by Pooled regression and Char T P M methods have relatively small variances. Since the data in all periods were pooled in a single regression, the Pooled regression method had very efficient estimates. As for Char T P M indexes, data in three periods were pooled by T P M regressions, and the employed Fisher index formula reduced the fluctuations of price indexes. On the other hand, variances of the two Divisia price indexes (Divisia T P M and Divisia CCC) are large., particularly in periods where the indexes increased. Although the Char C C C price indexes have small bootstrap bias, its variances are medium due to the C C C hedonic regression. Compared with the Pooled regression, the C C C regression usually have more coefficients to be estimated, and the coefficients of polynomials could significantly affect the imputed prices and the variation of price indexes. As for required resources, among all the methods, the Pooled regression method is the simplest one to implement and needs small amount of resource, since all the indexes are calculated directly from the coefficients of time dummy variables in one regression. Furthermore, it does not need the quantity (sales) data to weight price ratios. On the other hand, the Divisia T P M and Divisia C C C methods require large amount of resource to implement as they not only weight price ratios by quantity, but match the sample model by model. Differently, the Char T P M and Char C C C methods do not match the sample model by model, but rather calculate the quantity-weighted mean prices based on regression coefficients, and applies the Fisher index formula. In order to calculate the N O A Y R indexes, hedonic function need to be estimated for each period and be compared against the hedonic function in previous period. In the periods where hedonic functions are different, quantity data are needed to apply the Divisia index formula. Therefore, the three hedonic methods require medium amount of resource to implement. For statistical agencies, the properties listed in Table 33 are worth consideration 88 while choosing an appropriate hedonic method to measure quality-adjusted price movements. For instance, although it rarely happens, if all the characteristics change at the same rate over time (i.e., the hedonic functions have the same coefficients), the Pooled regression method is appropriate because of its unbiased and efficient estimates, as well as the small amount of resource. If there are "jumps" in technological innovations rather than smooth changes, the N O A Y R method could be appropriate to construct price indexes. In this scenario, the coefficients of characteristics in hedonic regressions are unbiased and efficient, and the price indexes have medium bootstrap bias and variance. On the other hand, if many characteristics change gradually or if the sample contains high-frequency data so that the N O A Y R method cannot detect significant changes, the Char C C C method and the Char T P M method could be appropriate as smooth changes are explicitly modeled in these methods. These two methods also need medium amount of resource and have small market biases and bootstrap biases. If the statistical agencies attach importance to market shares in the computer industry, and assume observed prices to be better than imputed prices, the Divisia T P M method and Divisia C C C method could be used to construct price indexes. However, statistical agencies should also consider much more required resources for these methods and pay attention to the potential large bootstrap biases and variances of the price indexes. 89 Index Table 33: Comparison of Hedonic Price Index Methods Estimation Bias Market Bias Bootstrap Bias Variance Resource Pooled NOAYR Divisia T P M Divisia CCC Char T P M Char CCC Large Small Medium Medium Medium Medium Large Medium Small Small Small Small Small Medium Large ' Large Small Small Small Medium Large Large Small Medium Small Medium Large Large Medium Medium Table 34: Biases of Hedonic Price Indexes Index Desktops Laptops Pooled -1.7% +5.5% N O A Y R -7.5% +5.3% Divisia T P M -13.9% -12.6% Divisia C C C +0.5% -1.6% Char T P M -12.9% -13.3% Char C C C — — 90 9.3 The Effect of Weight Aizcorbe et al. (2005) studied hedonic and matched-model price indexes and found that the weight could lead to differences between them. To further investigate the effect of weight on price indexes, I calculated four unweighted price indexes (Divisia T P M , Divisia C C C , Char T P M and Char CCC) for desktop computers. Different from previous weighted price indexes, the unweighted indexes were constructed us-ing unit (1) quantity assigned to each model, while the hedonic regressions remained unchanged in order to examine only the effect of weight. Table 35 and Figure 29 illus-trate the resulting unweighted desktop price indexes. The comparisons of weighted and unweighted price indexes are presented from Figure 30 to Figure 33. To as-sess their biases and variances, the 95% confidence intervals of the unweighted price indexes are listed in Table 36 and from Figure 34 to Figure 37. From Table 35 we may notice the higher A A G R s of unweighted indexes than weighted ones. The differences could be completely attributed to the effect of weights on price indexes. Table 37 lists the bias of each unweighted price index compared to the weighted one. It shows that the unweighted price indexes underestimate computer price changes by average 2.7%-4.6% per year. Such market biases reflect the fact that computer models with faster quality improvement have generally larger market shares than those models with slower improvement. By treating all the models equally, the unweighted price indexes were not be able to capture the higher rate of technological innovation in the desktop computer industry. As the comparison figures suggest, the unweighted Divisia T P M and Divisia C C C indexes differ largely from weighted indexes, especially in periods where the prices of some models increased. The weighted price indexes showed upward price movements in those periods, whereas the unweighted indexes showed a downward trend. More-over, as we could notice from their 95% confidence intervals, the unweighted price indexes have smaller bootstrap biases and variances. 91 On the other hand, the unweighted Char TPM'and Char C C C indexes are similar to weighted indexes not only in price movements but in the bootstrap biases and variances. As the figures suggest, the unweighted Char T P M and Char C C C indexes also reflect smooth price changes and have small bootstrap biases and variances. Based on the comparisons of weighted and unweighted price indexes, we may infer that using quantity as weight could have large influence on the Divisia price indexes and relatively small effect on the characteristics price indexes. One possible explanation is that, the Divisia index formula attaches more importance to weight, as any change in weight could lead to exponential effect on the resulting indexes. Thus the weighted Divisia T P M and Divisia C C C price indexes are more sensitive to price increases in some models. On the contrary, the Char T P M and Char C C C price indexes employ Fisher index formula, which weights the price ratios by quantity and takes the geometric mean of Laspeyres and Paasche price indexes. Therefore, the Char T P M and Char C C C indexes explicitly reduce the effect of weight and the fluctuations of price movements. Due to the difficulty of obtaining quantity data for computer models, it is not possible for every statistical agency to construct weighted hedonic price indexes. The results show that, it becomes feasible to construct unweighted price indexes to approximate weighted price indexes which employ Fisher index formula, as quantity (weight) does not have large effect on the resulting indexes. However, if the statistical agency needs to calculate price indexes that are based on Divisia index formula, quantity (weight) plays an important role in determining the magnitude or even the direction of price movements. 92 Table 35: Desktop Unweighted Price Indexes Year Qtr Divisia T P M Divisia C C C Char T P M Char C C C 1998 1 1 1 1 1 1998 2 0.976 0.865 0.957 0.854 1998 3 0.873 0.661 0.947 0.692 1998 4 0.838 0.601 0.824 0.604 1999 1 0.789 0.531 0.770 0.549 1999 2 0.751 0.482 0.740 0.496 1999 3 0.695 0.424 0.700 0.445 1999 4 0.687 0.407 0.677 0.420 2000 1 0.681 0.395 0.672 0.405 2000 2 0.658 0.369 0.647 0.382 2000 3 0.633 0.358 0.578 0.348 2000 4 0.528 0.293 0.548 0.312 2001 1 0.479 0.274 0.458 0.280 2001 2 0.462 0.258 0.445 0.252 2001 3 0.462 0.254 0.422 0.226 2001 4 0.383 0.195 0.362 0.190 2002 1 0.366 0.178 0.345 0.169 A A G R -22.0% -34.5% -23.0% -35.4% Table 36: 95% Confidence Intervals of Desktop Unweighted Price Indexes Year Qtr Divisia T P M Divisia C C C Char T P M Char C C C 1998 1 1 1 1 1 1 1 1 1 1998 2 0.919 1.024 0.777 0.962 0.945 0.966 0.783 0.925 1998 3 0.825 0.916 0.588 0.740 0.929 0.963 0.628 0.769 1998 4 0.781 0.887 0.530 0.677 0.791 0.857 0.547 0.671 1999 1 0.734 0.828 0.468 0.606 0.739 0.805 0.490 0.619 1999 2 0.699 0.792 0.419 0.553 0.710 0.772 0.438 0.564 1999 3 0.645 0.734 0.371 0.484 0.669 0.732 0.392 0.506 1999 4 0.641 0.727 0.352 0.465 0.646 0.712 0.369 0.484 2000 1 0.635 0.728 0.338 0.461 0.638 0.705 0.355 0.468 2000 2 0.610 0.702 0.318 0.427 0.614 0.684 0.335 0.442 2000 3 0.593 0.686 0.311 0.417 0.548 0.611' 0.304 0.402 2000 4 0.488 0.561 0.253 0.339 0.519 0.581 0.274 0.362 2001 1 0.440 0.513 0.235 0.320 0.431 0.482 0.245 0.326 2001 2 0.430 0.497 0.225 0.306 0.419 0.469 0.221 0.293 2001 3 0.431 0.505 0.223 0.304 0.399 0.445 0.199 0.262 2001 4 0.356 0.416 0.170 0.230 0.338 0.382 0.167 0.222 2002 1 0.338 0.396 0.156 0.211 0.323 0.364 0.149 0.197 co Ox CN ~i 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r 1 2 3 4 5 6 7 8 . 9 10 11 12 13. 14 15 16 17 : -—-• - D i v i s i a T P M Divisia C C C • - Char T P M — Char C C C Figure 29: Desktop Unweighted Price Indexes 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 -1 JZ 3' 4 5 6 '7/ 8 9 10 11 >12 '13 14 15 16 17 i l i l l l l l l l ^ ^ Weighted Unweighted Figure 30: Desktop Weighted and Unweighted Divisia T P M Indexes T 1 1 1 r~ Tl I I I I I I 1~ 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Weighted Unweighted Figure 31: Desktop Weighted and Unweighted Char T P M Indexes 96 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Time •*-—-Weighted - - 4 - - - Unweighted Figure 32: Desktop Weighted and Unweighted Divisia C C C Indexes Figure 33: Desktop Weighted and Unweighted Char C C C Indexes 97 fell 1 .2' 3' 4 5 6 -7; 8 \9 10 -11 12 13 14 15 16 17 Figure 34: 95% Confidence Intervals of Desktop Unweighted Divisia T P M Index t v 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 r-1 2 3' 4 .5 6 7 8 9 10 11 12 13 14 15 16 17 Time Figure 35: 95% Confidence Intervals of Desktop Unweighted Char T P M Index 98 -1 T 1 1 1 ~l 1 1 1-1 .2 3," 4- 5 6 7 8 ".9- 10 1  12 13 14 15 16 7 Figure 36: 95% Confidence Intervals of Desktop Unweighted Divisia C C C Index 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 Figure 37: 95% Confidence Intervals of Desktop Unweighted Char C C C Index 99 Table 37: A A G R Biases of Unweighted Desktop Price Indexes Index Divisia T P M Divisia C C C Char T P M Char C C C Weighted -24.7% -39.1% -25.7% -38.6% Unweighted -22.0% -34.5% -23.0% -35.4% Bias -2.7% -4.6% -2.7% -3.2% 100 10 Contributions This thesis makes contributions to the research literature of hedonic price indexes in the following aspects: First, the C C C hedonic regression method and the N O A Y R method were applied to four categories of computer platforms other than laser printers. The resulting C C C regressions indicate that slope coefficients changed gradually and systematically. Moreover, the C C C price indexes accounted for significant "quality changes" in the characteristics of computers. Nevertheless, the N O A Y R index does not differ much from the Pooled regression index. One of the reasons could be the high-frequency sample in the data set that make the changes in the hedonic function not significant enough to be tested different from its previous period. The numerically close price indexes reinforce Auer (2004)'s assumption on the gradual changes of characteristics prices. On the other hand, it gave us more understanding of the sampling frequency that matters in previous research work by Aizcorbe et al. (2003) and Deltas and Zacharias (2004). Second, the hedonic price indexes of workstations and servers, which did not involve enough research, were estimated by employing several hedonic methods. The resulting price indexes could provide empirical reference for future research on the two computer platforms. What is more important, this thesis presents an overview of biases and variances of different empirical hedonic price indexes. The results indicate that statistical agencies should pay attention to the statistical properties while selecting a hedonic price index for measurement of price changes. Particularly, although the Divisia index formula has good and desirable properties (avoid the Paasche/Laspeyres problem and put more weights on observed prices), two of the Divisia indexes (Divisia T P M and Divisia CCC) showed generally larger biases and variances than other superlative indexes, and they could be very sensitive if the prices of some matched models increase in the 101 sample, which could be described as "noisy and imprecise" in Aizcorbe et al. (2000). For comparison purpose, this thesis also evaluates the six hedonic methods in terms of resource based on required database and time, as well as the potential biases and variance of the index measurement. Finally, the thesis examines the effect of quantity (weight) on price indexes and shows that it is feasible to calculated unweighted price indexes to approximate weighted indexes which employ Fisher index formula. Therefore, it provides deeper understand-ing of hedonic methods, as well as the insight on the selection of a particular method for construction of price indexes by statistical agencies. 11 Future Research The lack of quantity data in workstation and server databases increases the potential biases in estimated price indexes, especially for indexes that utilize market shares for calculation. Future studies could examine the price changes of the two types of computers based on complete quantity information. It is also an interesting finding that, although the Pooled regression index and the Divisia C C C index have similar long-term A A G R s , the underlying methods and index formulas are quite different from each other. The Pooled regression method is the simplest hedonic method: it does not utilize quantity data and has loose connection to conventional index number formulas. On the other hand, the Divisia C C C method is a composite method: it utilizes data on "matched models" and applies the Divisia formula. Hence, it requires more time to implement, and is much more resource-intensive. In this sense, the resulting similar A A G R s of the two indexes necessitate further investigation. In fact, these two indexes deviated in the movement of price changes of each quarter/year. The N O A Y R index is numerically close to the Pooled index, but it has larger 102 variances probably because of the method employed to calculate price indexes for the periods where the hedonic function showed structural changes. Therefore, the appli-cation of N O A Y R method could be further explored based on infrequent sampling in future research work. This thesis also proposed a new T P M method which could solve problems in the A Y R method, use data on quantities and have connection to conventional index num-ber formulas. However, compared to other indexes, the price indexes constructed by T P M regressions declined at a lower rate, which indicates that the T P M regressions did not "account for enough quality improvements" over the periods. In some quar-ters of the desktop database where a portion of records are "matched" models with increasing prices, the Divisia T P M indexes increased more than the Divisia C C C in-dexes. 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