Stamp returns and economic factors.

I. Introduction

In the last two decades, a number of prominent financial economists have considered the extent to which assets provide hedges against inflation and/or other sources of systematic risk. Motivated by the high inflation rates of the 1970s, Fama and Schwert [5] conducted the seminal analysis on asset returns and inflation, investigating the extent to which the returns to various assets, including common stock, government bonds, treasury bills, real estate, and human capital, were hedges against anticipated and unanticipated inflation. Fama and Schwert found that bonds, bills, and real estate provided effective hedges against anticipated inflation for the period 1953-1971, but only real estate provided a hedge against unanticipated inflation. In fact, most of the assets had a negative sensitivity to unanticipated inflation, with common stock showing a negative relationship to both sources of inflation.

Modern theories of asset pricing claim that asset prices should depend on their exposures to the state variables that describe the economy [10; 14; 2]. Chen, Roll, and Ross [1], in an attempt to identify the relevant state variables, conducted an exploratory investigation into the systematic factors priced by the stock market. They found that prices in the stock market during the 1958-1984 period reflected the levels of unanticipated industrial production, default and term premiums, and, to some degree, unanticipated inflation. In a subsequent paper, Roll and Ross [12] reported the sensitivities of stock returns for forty-four industries to the above-mentioned factors. Their results were mixed for unanticipated production, with positive sensitivities for twenty-four of the industries and negative sensitivities for the remaining twenty. However, their results for the other factors were strikingly uniform.

In particular, Roll and Ross [12] found that sensitivities to the default premium (as measured by the difference between low-grade and high-grade bond returns) were positive for all industries, while sensitivities to the term premium (as measured by the difference between long-term government bond and treasury bill returns) were positive for all but one industry. Since Roll and Ross [12] measured both default and term premiums ex post (that is, using actual returns), the positive sensitivities indicate that stock returns decline when increases occur in both the aggregate default risk and in the slope of the term structure of interest rates. On the other hand, similar to the findings of Fama and Schwert [5], Roll and Ross [12] found that stock return sensitivities to unanticipated inflation, as well as to the change in anticipated inflation, were negative for all forty-four industries studied. Long positions in common stock, then, did not provide hedges against the inflation, default, and term structure factors of systematic risk.

Sophisticated investors may still be able to hedge against the various sources of risk by taking well-calculated short positions in certain securities. However, alternative investments such as antique furniture, art, coins, firearms, jewelry, stamps, wine, and other tangibles [3; 6; 9; 11; 13; 16] may exist that give investors the desired exposure to systematic risk sources without the expense of short selling. Although investigation of the risk characteristics of tangible investments is normally seriously limited by the scarcity of reliable data, we have access to an extensive database of U.S. stamp prices covering the period 1947 through 1988. In this study we use these data to estimate and report the sensitivities of U.S. stamp prices to the systematic economic risk factors investigated by Fama and Schwert [5], Chen, Roll, and Ross [1], and Roll and Ross [12], as well as to stock and bond returns. We find that stamps have positive sensitivities to the inflation factors and negative sensitivities to the default and term structure factors. With the exception of production, then, stamp sensitivities are of opposite sign to the stock sensitivities reported by Roll and Ross [12]. Though not perfect hedges in and of themselves, we have evidence that this type of tangible asset may be useful in conjunction with securities for structuring portfolios in such a way as to give alternative exposures to systematic economic risks. In section II we discuss our data and empirical tests. Our results are presented in sections III, IV, and V, with conclusions in section VI.

II. Data and Empirical Methodology

The stamp data for this study come from a large sample of auction prices covering the years 1947 through 1988. The beginning year (1947) was chosen because in that year the Scott Standard Postage Stamp Catalogue, the standard reference numbering system for the industry, made a major change in its numbers. The prices are actual transactions taken from auction catalogs representing 87 U.S. auction firms in 1,404 auctions. The entire data set consists of 224,696 auction prices for 2002 different stamps (which includes varieties) issued between 1847 and 1930. Most libraries do not have this information. Even the oldest auction houses do not have price information going back very far. Where the information does exist, it may be owned by private individuals or by organizations for which membership is required for access. Although this study does not cover the 1929-46 period, stamp auction prices are rare during that period, being more available from 1900 to 1920.

A potential problem in analyzing these data (as for most tangibles) is that each auction price is for an asset of potentially different quality than stamps with the same Scott number sold at this or other auctions. Grading standards differ over time and between auction houses. Most auction houses have multiple grades for centering, with stamps that are centered and have large margins receiving the highest grade. Stamp prices vary directly with overall conditions, including centering, paper quality, type and clarity of cancel (for used only), size of margins, faults, freshness, etc. The variability in condition is most evident between nineteenth and twentieth century stamps and between mint versus used stamps. Nineteenth century stamps have been handled more frequently and thus damaged; the paper, ink and glue (for most stamps) is more aged and differs between stamps; and the printing uniformity and centering are, on average, more erractic than for twentieth century stamps. The mint stamps have been subjected to less abuse than used stamps. The latter have passed through the mails, have frequently been canceled with heavy hand-made cork cancels, and have then been soaked off the envelope in a variety of ways.

Table I. List of Auction Firms and Their Locations

Auction Firm                Location

Matthew Bennett             Baltimore, Maryland
John A. Fox                 Floral Park, New York
William Fox                 Springfield, New Jersey
H.R. Harmer                 New York, New York
Steve Ivy                   Dallas, Texas
John W. Kaufman             Washington, D.C.
Daniel F. Kelleher          Boston, Massachusetts
Greg Manning                Montville, New Jersey
Metro                       Middle Village, New York
Metro-Simmy                 Cranbury, New Jersey
Robert Siegel               New York, New York
Simmy's                     Boston, Massachusetts
Suburban                    Springfield, Massachusetts
Richard Wolffers            San Francisco, California

This problem of "quality variation" can cause the mean and variance of stamp prices to change over time and across stamps for reasons other than economic. To minimize the quality variation problem, we restricted our sample by using the following criteria:

A. Only stamps issued by the U.S. Postal Service are included.

B. Only stamps sold off cover (i.e., not on the envelope) as a single or set are included.

C. Used stamps with unusual cancels that change the character and value of the stamps are excluded.

D. Only the price for the first lot of each Scott-numbered stamp is used. The highest quality stamp is usually offered first.

E. Only the most prestigious auction firms, i.e., those that consistently attract only the highest quality stamps, are represented. In all, fourteen firms are represented (see Table I). This procedure was chosen primarily to guarantee that the average quality of the stamps in our data set remains constant over time.

F. Only stamps traded in at least 50% of all auctions conducted by the fourteen auction firms are included.

Criteria B and C were chosen to guarantee that a particular Scott-numbered stamp would be similar to others with the same number. Criteria D and E were chosen to guarantee that most stamps in our data would be of high quality. Use of these criteria substantially reduces the amount of quality variation. Criterion F results in only frequently-traded stamps being included, thus excluding very rare stamps that might be of more variable quality or have less well-established value. Criterion F also results in the mix of stamps in our data remaining largely unchanged over time.

Another problem is that some auction houses will report a stamp as sold when in fact it did not meet a reserve price, a minimum price that must be met for the stamp to be sold. Reserve prices are more prevalent for expensive and infrequently-traded stamps and also at smaller less reputable auction houses, especially those selling their own stamps. By restricting our analysis to the more prestigious auction houses and the more frequently traded stamps, the problem is substantially reduced. Beyond that, we assume the problem is constant over time.

In our regression models we include a dummy variable for each Scott-numbered stamp. Combined with the above list of criteria, this should reduce the variance in quality and eliminate any problem of the mean quality changing across time or stamps. We further reduce the effect of quality variation across stamps with different Scott numbers and across time by estimating models using weighted least squares, where the weights come from a variance function that includes dummy variables representing each Scott-numbered stamp as well as trend variables.

The selection procedure resulted in a sample of 20,693 prices representing 43 different individual stamps or sets for the years 1947 through 1988, as shown in Table II. Counting the stamps in each set individually, there are 50 distinct Scott-numbered stamps. In all, 777 auctions are represented in this sample, for an average of 19 auctions per year. The actual number of auctions per year fluctuates, with a low of 7 in 1947 and a high of 33 in 1983. Usually, multiple auctions do not take place on the same day and there is usually at least one auction each month. The annual means, medians, and standard deviations of prices for this sample are plotted in Figures 1-4. (The standard deviations are for the natural logs of prices.) Notice that the level of prices increased through 1980 and then declined sharply. The standard deviation for the log of prices also increased through 1980, but leveled off for the remainder of the sample.

Ideally, we would like to have return data with which to estimate the sensitivities to the various economic factors. We could then simply run regressions of stamp returns on the factors, with the regression coefficients representing the sensitivities. However, since the stamp auctions are not equally spaced in time, and since each stamp is not traded each month, returns would have to be calculated over varying intervals. Furthermore, return data would require first taking differences for prices of the same Scott-numbered stamp. Thus, the effect of quality differences among stamps with the same Scott number would be aggravated. Therefore, instead of using return data, we estimate regressions in the natural logarithm of stamp prices. As discussed below, the log of the stamp price represents the sum of the trend in the stamp price and the unexpected return. We treat each price as if it were an end-of-month transaction price regardless of the day of the month on which the transaction actually took place. We assume the difference between the observed log stamp price and the hypothetical end-of-month log price to be random measurement error in the log price. It is well known that such random measurement errors in the dependent variable do not bias the coefficients, but, on average, reduce the statistical significance of the estimates. Thus, our results can be viewed as conservative. All the economic variables we use are available on a monthly basis, but none are available on a daily basis. In the following discussion, we denote the price of stamp i at time t as [STP.sub.it].

The economic factors we related to stamp prices were discussed in the introduction and are those used in the studies by Fama and Schwert [5], Chen, Roll, and Ross [1], and Roll and Ross [12]. We also relate stamp prices to indices of stock and bond returns. Table III lists the variables and their symbols, definitions, means, and standard deviations. All variables in Table III are monthly, not seasonally adjusted, and cover the period 1947 through 1988. The expected inflation (EI) variable was constructed using a procedure outlined in Fama and Gibbons [4], and the unanticipated production (UP), unanticipated default premium (UDP), and unanticipated term [TABULAR DATA FOR TABLE II OMITTED] [TABULAR DATA FOR TABLE III OMITTED] premium (UTP) variables were constructed as in Chen, Roll, and Ross [1]. The production data were supplied by the Federal Reserve Board and the inflation data by the Department of Labor, and all financial data came from CRSP and Ibbotson Associates.

The regressions studied in Fama and Schwert [5] regressed returns on anticipated and unanticipated inflation (EI and UI), whereas the regressions studied in Roll and Ross [12] regressed returns on UI and change in anticipated inflation (DEI), as well as on unanticipated production (UP) and unanticipated default and term premia (UDP and UTP). Notice that, by construction, the explanatory variables in these studies are generally non-trending. This poses no problem in the Fama-Schwert and Roll-Ross regressions because the dependent variables (asset returns) are also non-trending. In our regressions, however, the dependent variable (log stamp price) has a pronounced trend and possibly a cycle that must be accounted for in the regressions; otherwise, they will show up in the error term. We model the trend-cycle term as a function of deterministic variables. In this way, the expected log stamp price E([LSTP.sub.it+1]) for time t + 1 is conditional on information known at time t, with deviations around this expectation, i.e., [LSTP.sub.it+1] - E([LSTP.sub.it+1]), accounted for by the unanticipated systematic economic factors and an idiosyncratic error term. Notice that since stamp prices are converted to natural logs, the above deviation is a measure of unexpected stamp returns: [[LSTP.sub.it+1] - E([LSTP.sub.it+1])] = [([LSTP.sub.it+1] - [LSTP.sub.it]) - (e([LSTP.sub.it+1]) - [LSTP.sub.it])] = unexpected return. Thus the regression coefficients on the unanticipated factor values can be thought of as factor sensitivities for stamps.

Using the variables defined above, we study estimates of the following regressions and variations thereof:

(Models 1-3)

[LSTP.sub.it] = [a.sub.o] + [summation of [d.sub.j][D.sub.ij] + [a.sub.1]t + [a.sub.2][(t - 255).sub.2] + [a.sub.3](t [greater than] 80 : 12) + [a.sub.4] sin([Omega]t + [Phi]) + [b.sub.1]X[R.sub.t] + [[Epsilon].sub.it]; where j=2 to 43] (1)

(Model 5)

[LSTP.sub.it] = [a.sub.o] + [summation of [d.sub.j][D.sub.ij] + [a.sub.1]t + [a.sub.2][(t - 255).sub.2] + [a.sub.3](t [greater than] 80 : 12) + [a.sub.4] sin([Omega]t + [Phi]) + [c.sub.1]E[I.sub.t] + [c.sub.2]U[I.sub.t] [[Epsilon].sub.it]; where j=2 to 43] (2)

(Model 6)

[LSTP.sub.it] = [a.sub.o] + [summation of [d.sub.j][D.sub.ij] + [a.sub.1]t + [a.sub.2][(t - 255).sub.2] + [a.sub.3](t [greater than] 80 : 12) + [a.sub.4] sin([Omega]t + [Phi]) + [c.sub.1]U[I.sub.t] + [c.sub.2]DE[I.sub.t] + [c.sub.3]U[P.sub.t] + [c.sub.4]UD[P.sub.t] + [c.sub.5]UT[P.sub.t] [[Epsilon].sub.it]; where j=2 to 43] (3)

where:

i = stamp i, i = 1, . . ., 43;

t = time period t, t = 1 for 46 : 1, . . ., 509 for 88 : 5;

[D.sub.ij] = dummy variables that take the value of 1 for i = j and 0 for j [not equal to] i;

(t [greater than] 80 : 12) = time after December, 1980;

sin(x) = the sin function, with frequency [Omega] and phase [Phi];

XR = VWR, SCR, or BR; (as defined in Table III); and

[e.sub.it] = residual error term for stamp i at time t.

Model 4, not written, includes all variables in Models 1 through 3, and Model 7, also not written, includes all variables in Models 1 through 6. Model 5 is our counterpart to the Fama-Schwert regressions, and Model 6 is our counterpart to the Roll-Ross regressions.

We chose to estimate the systematic part (i.e., [E.sub.t-1] ([LSTP.sub.it])) based solely on dummy variables for distinct Scott numbers and deterministic functions of time. Preliminary exploration showed very similar fits with functions of time alone and trending variables such as log CPI, log Stock Wealth and log Bond Wealth (where the wealth variables represent the value of a one-dollar investment in stock and/or bonds in December, 1945). However, the coefficients of trending variables were unstable with respect to minor variations in specification. On the other hand, the coefficients of the return variables were not sensitive to changes in the specification of the trend variables. Examination of Figures 1 and 2 reveals the unique pattern of stamp prices followed over the period of our study. After the fact, we are not surprised that economic variables cannot predict this pattern. Certainly no other economic variable has a similar pattern. We view the significance of the variable TA80 as a test of the hypothesis that the stamp market underwent a structural change on or around 1980. TA80 was highly significant in all the regressions we tried.

The constant term, [a.sub.0], reflects the identifying normalization [d.sub.i] = 0. Thus [d.sub.i] for i = 2, . . ., 43 reflect the mean difference, other things equal, between the log price of the ith Scott number and the log price of the first Scott number. The adjusted variable [STP.sub.it][e.sup.-[d.sub.i]] is price corrected for Scott number. We use these corrected prices to form an annual price index for stamps [ILLUSTRATION FOR FIGURES 1 AND 2 OMITTED]. The deterministic functions of time allow for a pure trend, a quadratic deviation from a pure trend, a structural change beginning in 1981 (as suggested by the raw figures), and a cyclical effect. The variable [(t - 255).sup.2] is used for the quadratic term because t = 255 is the middle of the series. Thus, the coefficient of t (that is, [a.sub.1]) represents the mean trend over the period t = 1, . . ., 509, while [a.sub.2] measures the quadratic deviation from the mean trend. We used nonlinear least squares to estimate [Omega] and [Phi] in the cyclical variable.

As mentioned earlier, the conditional variance of stamp prices may change across time and across stamps for a variety of reasons. While we have tried to control for quality variation in our sample selection procedure, variation across stamps may still occur due to differences in quality and because some stamps may be viewed as more speculative than others. Any remaining quality variation is incorporated in the error term. Further, it is well known in the finance literature [15] that the condition variance of investment returns varies over time. To account for the potential change in stamp variance, we estimate Models 1 through 7 using weighted least squares. Our procedure is as follows:

First, Model 7 is estimated by ordinary least squares, and the residuals are saved and transformed to [Mathematical Expression Omitted], where [e.sub.it] is the empirical residual and [s.sup.2] = d[prime]d/(N - K). The log transformation is a natural way to analyze variances. We assume the obvious model of heteroskedasticity [[Epsilon].sub.it] [similar to] N(0, [e.sup.[Alpha]+[X.sub.it][Beta]]) for [X.sub.it], a suitable vector of independent variables. Under this model,

[Mathematical Expression Omitted]

where [Gamma] is Euler's constant and [[Eta].sub.it] is a mean zero iid random variable for which all moments exist. If we write our structural model as:

Y = X[Gamma] + [Epsilon], (5)

then

[e.sup.-X[Beta]/2]Y = ([e.sup.-X[Beta]/2]X)[Gamma] + ([e.sup.X[Beta]/2][Epsilon]) (6)

is homoskedastic. If we had a random sample of observed [[Epsilon].sub.it]'s, it would be simple enough to estimate [Beta] and use [Mathematical Expression Omitted] in an estimated generalized least squares (EGLS) procedure to obtain consistent and asymptotic efficient estimates of [Gamma]. However, here we cannot observe [[Epsilon].sub.it]. The simplest procedure would be to substitute [Mathematical Expression Omitted] for [Mathematical Expression Omitted]. However, the moments of [Mathematical Expression Omitted] may not exist in finite samples (for instance, it is well known that for some X matrices, [e.sub.it] = 0 for one or more i, t pairs) and a sufficiently small value for [e.sub.it] could yield a very influential outlier that would unduly perturb the value of [Mathematical Expression Omitted]. Motivated by the facts that

[Mathematical Expression Omitted]

under the null hypothesis of homoskedasticity (i.e., [H.sub.o]: [Alpha] = 2 log [Sigma], [Beta] = 0), and that K[s.sup.2]/N is asymptotically irrelevant, that is, for fixed i, t

[Mathematical Expression Omitted]

we substituted [Mathematical Expression Omitted] for [Mathematical Expression Omitted]. All moments of [Mathematical Expression Omitted] exist, and there are no extreme outliers for [Z.sub.it] in our data. (Under our assumptions, outliers in [Z.sub.it] are highly improbable unless there were outliers in [X.sub.it][Beta], which is improbable for our X variables and does not occur for [Mathematical Expression Omitted].) The transformed residuals are then regressed against the explanatory variables in Model 7, with the fitted values from this second stage regression used to form estimates of the conditional variance. Finally, Models 1 through 7 are estimated with EGLS, where the variance estimates from the second stage regression are used to form the variance-covariance matrix necessary for generalized least squares.

III. Estimates of Variance Function

As discussed above, stamp prices vary directly with overall conditions. Effects of differences in variability are shown in the parameter estimates for the coefficients of the dummy variables in Table IV. Mint stamps generally have negative estimated coefficients, indicating smaller variances. In contrast, used stamps generally have positive or slightly negative coefficients, indicating smaller variances. Then as individual groups, the average parameter estimates for the nineteenth century used, nineteenth century mint, and twentieth century mint are .09 (range: -.33 to +.38), -.41 (range: -.90 to -.10), and -.72 (range: -1.27 to -.32), respectively. This indicates that, as expected, the greatest variability is in nineteenth century used and the least variability is in twentieth century mint. Further, as can be seen from Table IV, anticipated inflation is a very important explanatory variable in the variance function. High anticipated inflation explains some of the increase in the variance that occurred between 1980 and 1985 [ILLUSTRATION FOR FIGURES 1 AND 2 OMITTED].

The time trend variables show that, when other factors are equal, stamp prices are becoming more variable over time. This pattern is verified by the aggregate curve shown in Figures 3 and 4. Over the period studied here, the total number of stamp auctions has increased considerably [TABULAR DATA FOR TABLE IV OMITTED] because many additional firms not meeting our Criterion E entered the auction business. In addition, discussion of stamps as an investment has greatly increased in the popular literature. It seems clear that, as compared to earlier years, more stamp purchases in recent years have been made by investors relative to hobbyists. Thus our results suggest that market prices have become more variable as stamps have become increasingly an investment. Initially we had worried that in spite of our precautions quality variation might have been larger in the earlier years. To the contrary, however, our results show a low stable variability in price from 1947 to 1964. We note that our EGLS procedure, by using the heteroskedasticity results, greatly reduces the weight on the stamps with the most variable prices, thus limiting the possible impact of quality variation. Before estimating the EGLS model we estimated unweighted regressions. These gave substantially similar results to our weighted regressions. If uncorrected quality variation were responsible for some of our results, then the weighted and unweighted regressions would be expected to give quite different results.

IV. Estimates of Models 1-7

Estimates of Models 1 through 7 are reported in Table V. Notice from Models 1 through 3 that the log stamp price (LSTP) is negatively related to stock and bond returns when considered individually. However, when considered jointly (Model 4), the coefficients for broad-based stock returns (VWR) and bond returns (BR) are still negative, while the coefficient for small capitalization stock returns (SCR) is positive. Since VWR is a value-weighted index, it gives more weight to large capitalization companies than to small capitalization companies. Thus, estimates of Model 4 imply that LSTP is negatively related to BR and large capitalization stocks, but positively related to SCR. Since stamps and small capitalization stocks are primarily owned and traded by individuals instead of institutions, it may come as no surprise that the two are positively related. Stamps and small stocks, then, may be viewed as substitute investments by some investors.

As noted in section II, Models 5 and 6 are our counterparts to the Fama-Schwert and Roll-Ross regressions, respectively. The estimates for Model 5 are positive for both anticipated (EI) and unanticipated (UI) inflation, though significant only for EI. Since a coefficient of one indicates a perfect hedge against anticipated inflation, the estimated EI coefficient value of 13.5 in Model 5 reveals that the deviation of LSTP from its long-term trend cycle is highly sensitive to expected inflation.

Estimates of Model 6 show significant sensitivities to all factors except unanticipated production (UP). Notice that UI and DEI (the change in anticipated inflation) are of opposite sign, [TABULAR DATA FOR TABLE V OMITTED] but the magnitude of DEI is about six times that of UI. According to Chen, Roll, and Ross [1], UI and DEI may both be considered "innovations," but DEI may contain information not present in UI "if inflation forecasts are influenced by economic factors other than past forecasting errors [1, 388]." The sensitivities to both unanticipated term premia (UTP) and unanticipated default premia (UDP) are negative and close to one in absolute value, which is opposite in sign to the sensitivities reported by Roll and Ross. As noted in the introduction, these negative signs imply that stamps provide some protection to unanticipated changes in term structure and default risk.

Taken together, Models 1 through 6 imply that stamps are related in an opposite way to the factors that negatively impact stock returns and, with respect to UDP and UTP, bond returns. As further investigation, we present in Table II the estimates of Model 7, which includes all variables used in Models I through 6. Consistent with Models 4 through 6, LSTPs are positively and significantly related to SCR, EI and DEI, and negatively and significantly related to VWR and BR. The coefficients of the variables, UI, UP, UTP, and UDP, are not significantly different from zero. While UP was insignificant in Model 6, UI, UTP, and UDP were all significant. The lack of significance in these variables may be the result of multicollinearity with VWR and BR, or may be due to the presence of EI helping to explain the long-term cycle in stamp prices.

V. Indices of Stamp Prices

Figures 3 and 4 graph four indices. These include the raw indices as well as indices based on the estimated model. Notice that all four tell basically the same story. The large runup in prices that occurred between 1971 and 1980 is particularly interesting. In 1980 through 1987, the mean adjusted price index exceeds the mean price index. For all indices, 1980/1981 is the peak period. It is interesting to note that the average raw price index and the average adjusted price index moved up in almost perfect proportion from 1971 to 1979, with the raw price index being higher. However, the adjusted price increased much faster than the raw price between 1979 and 1980, and slightly increased between 1980 and 1981, while the raw price index declined. After 1981 the price indices moved down in parallel with the adjusted price index being higher. The raw price index places greater weight on the highest value Scott-numbered stamps. Thus, the modest differences in the behavior of the indices primarily reflects the highest value stamps leading in both the price rise of 1971-1980 and the price decline of 1981-1986.

The adjusted indices depend on the regression results only via the estimated coefficients of the dummy variables for the Scott numbers. These coefficients are extremely robust to alternative specifications. All stamps in our data set were traded in at least one half the auctions. In all cases the trading was reasonably uniform over the period. Therefore, the stamp dummy variables were approximately orthogonal to the time-trending variables. Of course, the row indices involve no parametric assumptions. Therefore, our stamp price indices portray the pattern of the stamp market largely free of parametric assumptions. Similar comments apply to our indices of log-price standard deviations.

Figures 1-4 together provide an interesting summary of the development of the stamp market. Three very different periods are apparent: 1947-1975, 1976-1980, and 1981-1988. It is important to note that stamp prices increased fivefold between 1947 and 1968 and doubled between 1968 and 1975, all before the visible price explosion. From 1976 to 1980, prices again increased fivefold. Furthermore, between 1947 and 1982, no price retreat lasted more than two years. Considering inflation and the costs of storage, protection and insurance (or self-insurance) for stamps, a tenfold price increase over the 28-year period 1947 to 1975 is modest. By 1979, stamp prices had reached 90 percent of the 1980 peak. At the same time, as shown in Figures 3 and 4, the variability in those prices jumped to an unprecedented level. In 1980, variability peaked along with price levels, and, since 1980, the unexplained variability has remained well above the pre-1979 levels. Thus, the price peak of 1980/81 seems to have led to a permanent increase in the variability of stamp prices. This result supports the hypothesis that the stamp market has undergone a permanent structural change. Of course, the stamp market may some day return to a pattern of low price variability and slow but steady price increases. Currently we see no reason to expect such a development.

We find that stamp prices have a highly significant trend over time. The trend through 1980 is upward and accelerating. After 1980, the trend is downward and levels off. No combination of trending economic variables was able to explain this pattern without including deterministic functions of time. The magnitude of the 1979-1982 "bubble" is clearly visible in the graphs of our price indices. Throughout the period, our corrected and uncorrected indices show the same basic pattern, although we did uncover a slight tendency for the raw price index to lead the corrected price index.

VI. Conclusions

In this paper we have investigated the price and return behavior of an important market for tangible assets, United States stamps, implementing a methodology that can be applied to other markets with similar problems. The short term return on stamp investments is estimated to be a function of economic variables. Because tangibles generally are stores of value, our results indicate plausible behavior of the market for stamps considered as a financial asset. We find that stamps are related in an opposite way to many of the important systematic factors that influence stock and bond returns. For example, stamp returns are highly positively related to expected inflation. Stamps are also negatively related to large capitalization stocks and to bonds, but positively related to small capitalization stocks. This suggests that stamps may be a substitute for small capitalization stocks, while acting as a hedge against movements in the large stock and bond markets.

Our estimation technique includes a model for the conditional variance of unanticipated stamp returns. We find some interesting systematic patterns in the time series and cross-sectional behavior of the conditional variance. The variance has a pronounced trend and is highly related to anticipated inflation. The variance for mint stamps is lower than for used stamps, and the variance for older stamps is higher than for more recently issued stamps. These characteristics of the estimated variance function are consistent with our prior beliefs of stamp return variation.

We would like to thank the following individuals or organizations for their assistance on this project: Dr. Stanley Bierman (Bierman Philatelic Library), Bruce Rutherford (Collectors' Club of New York), Ginnie Horn (American Philatelic Research Library), Doug Kelsey (Postal History Museum), Bill Kitchner (former editor of Stamp Auction News), John J. MicGlire (General Manager of Stamp Auction News, later called Stamps Auction News), Nancy Pope (National Museum of American History, Smithsonian), Stanley Richmond (Daniel Kelleher Stamp Auctions), Robert A. Siegel (Robert A. Siegel Auction Galleries, Inc.), Jo Ann Vincent (former editor of Stamp Auction News), Les Winick (Collectors' Club of Chicago), and scores of auction agents, auction owners/employees, dealers, and collectors. We are also grateful to Shoamin Huang and Audrey Cardell for able research and editorial assistance, respectively.

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