Residual variance and asset pricing in the art market

In this paper, we compute residual variance of art prices to examine asset pricing in contemporary art market. Our empirical work shows a few interesting results. First, we discover that the residual variance is significantly and positively related to the average price level achieved by an artist. Second, the residual variance has additional explanatory power in terms of how often the artist’s works are cited and exhibited, even after we control for artist fixed (reputation) effects. Third, collectors tend to value more those artworks with higher residual variance. Artworks by those artists with high residual variance tend to outperform the market within and out of sample. One possible explanation of our results is that residual variance could be a proxy for creative risk taking by contemporary artists. The most creative artists dare to take more risks, which results in higher residual price volatility of their artworks.


Introduction
This paper computes residual variance of art prices to examine asset pricing in contemporary art market. Our work is motivated by two influential papers by Galenson and Weinberg ((2000) and (2001), GW afterwards). 1 These studies address the difference in the career peaks of 19th-century artists and 20th-century artists. They discover that 20th-century artists peaked significantly younger, in their late 30s and early 40s, while 19th-century artists peaked around their 50s. GW use the market price of art to measure creativity; they are most interested in at what age the artist produces the most expensive work. However, their measure of creativity uses only the information contained in the first moment of art prices.
It is natural to ask whether information based on the second moment of art prices would generate additional insights into the art creation process. There is a large body of the literature in finance that examines the volatility of asset price. Researchers such as Hirshleifer et al. (2012) have used the volatility of stock prices to examine risk taking by firms and the creativity of CEOs. As noted in Campbell et al. (2001), one may gain additional insight about individual stock risk by decomposing stock returns into market return and idiosyncratic return and by examining the time variation of the residual variance after adjusting for the market risk component. They note that this approach of examining the second moment of residuals brings us insights into the time-varying nature of firm volatility. Bartram et al. (2012) further find that the higher idiosyncratic volatility of U.S. firms is driven by higher firm innovation-higher R&D spending than comparable firms in foreign countries.
We build on a large body of the literature on measuring art market returns to compute idiosyncratic risk for artworks. Baumol (1986) analyzes 640 repeated sales records from 1652 to 1961 from Reitlinger's book and reaches the conclusion that the average annual rate of return for art is 0.55%. Goetzmann (1993) extends Baumol (1986)'s data, and he finds that the return of the art market depends on the time period; the return in the second half of the 20th century rivals the stock market. Mei and Moses (2002) search catalogs for all American, 19th-century and Old Master, Impressionist, and Modern paintings sold at Sotheby's and Christie's and collected close to 5,000 pieces of repeated sales data covering the period from 1875 to 2000. They find that the annualized art market return was about 10.1%, comparable to the U.S. equity return. Goetzmann et al. (2011) find that art-market returns are determined by economic growth and distribution of wealth. This paper will combine the hedonic models of GW and the repeated sales model of Goetzmann (1993) to compute the residuals of art prices. This approach has the advantage of being able to use the greatest amount of transaction data from the auction market. Our objective is to isolate those price innovations that reflect deviation from the regular art market and career-price path of an artist. And as we will argue in a later section of the paper, the average squared residuals (or residual variance) reflect price diversity as a result of risk taking by artists. We further employ the GARCH (1,1) model of Engle and Bollerslev (1986) to investigate the persistence of the residual variance.
Our empirical work has yielded a few interesting results. First, we discover that the residual variance is significantly and positively related to the average price level achieved by an artist. Second, residual variance has additional explanatory power in terms of how often the artist's works are cited and exhibited, even after we control for artist fixed (reputation) effects. Third, collectors tend to value more those artworks with higher residual variance. Moreover, artworks by those artists with high residual risk tend to outperform the market within and out of sample.
While our results may be subject to several interpretations, we will speculate in a later section of the paper that the average squared residuals (or residual variance/ residual variance) could reflect price diversity as a result of risk taking by artists, which is an important part of the creative process. It is consistent with the fact that art creation is a process full of uncertainties. Ex ante, the most creative artists need to take the considerable risks that their newly created art could be misunderstood or hated by the market. The most creative, relatively speaking, are also likely to have the most to lose. But they must continue to innovate despite this risk, even though are they are not always rewarded for their efforts.
The paper proceeds as follows. We describe the models in Sect. 2 and our data in Sect. 3. In Sect. 4, we first compute the residual variance of art prices and then examine the relationship between the residual variance and the importance of artists as perceived by art critics and historians. In Sect. 5, we use an asset pricing approach to see if the art market values those artists whose artworks have high residual price risks. In Sect. 6, we provide an analysis to address a sample selection bias. We provide some possible explanations for our findings Sect. 7. In Sect. 8, we then conclude the paper.

Methodology
Our first model is a hedonic regression of the logarithm of price: In this model, i denotes the artist i, j denotes the jth painting, and t is the time of the transaction. s ij denotes the age of the artist when the jth painting is created by the ith artist. This model says that the logarithm of the price is a polynomial function of the age of the artist when the painting is created. The time dummy (t) stands for the return of the art market index, and the artist dummy (i) denotes the fixed effect of the artist. X represents the vector of the hedonic variables, such as the height, width, medium (oil or watercolor, etc.), shape (rectangular, oval or other shapes), (1) whether or not the painting is signed, or whether the transaction took place in one of the three major auction houses-Christie's, Sotheby's, or Phillips. We use nominal prices, as the time dummies would automatically pick up inflation. We use log prices to mitigate the artificial mean effects since high prices also tend to have a high variance. Eq. 1 is quite similar to that used in Galenson and Weinberg ((2000) and (2001)). It says that the log price of an artwork is a function of the age of the artist, the art market movement, the artist fixed effect, and some characteristics related to the artwork. GW discover that the value of an artwork is related to the age of the artists when it was produced. They find the quality of the artwork declines precipitously for successful modern American artists as their age increases. Goetzmann et al. (2011) find that the prices of individual artworks are largely influenced by art market returns. Thus, ijt measures the residual value of the artwork that is essentially the relative (or percentage) deviation from his regular market and age determined price path. Normally, ijt could reflect innovations in artist creativity or random demand forces in the art market. Since we assume in this paper that the random demand forces in the market are the same across all artists, ijt chiefly reflects innovations in artist creativity. Moreover, the variance of ijt , 2 i , measures the price diversity produced by the artist. In general, the more diverse the price, the more diverse the artist's creative production. It could reflect the artist's willingness to try many different things, thus taking more risks in the creative process.
In previous studies, 2 i is assumed to be constant across different artists; in our paper, 2 could vary across artists. We will compute the unconditional 2 i as well as the conditional 2 is , where s stands for the age of the artist. Our objective is to measure the price diversity over the lifetime of the artist. This allows us to numerically compare which artist has more price diversity as well as when the artist has the most price diversity over his lifetime. It further allows us to examine whether these residual variances are related to success in the art market as well as in art history. Since we have heteroscedasticity in Eq. 1, we will estimate the model using the generalized least square (GLS) method.
Our second model is the repeated sale model: It is easy to see from Eq. 1 that the second model can be derived easily from a simple difference between prices in two subsequent sales and that it is simply made of the market time dummies and the difference between two residuals. Thus, estimating residual variance in Eq. 2 is equivalent to estimating Eq. 1 multiplied by a factor of 2. Thus, just like the residual variance of returns to large stocks, a priori, 2 i could be higher (or lower) for high-priced artists than for low-priced artists. If there is no sample selection bias, 2 i could also have no relationship to the fixed effects in equation (1). We will address possible sample selection bias in Sect. 6. Equation 2 can be estimated using the repeated sales approach of Goetzmann (1993) using GLS. One can easily use it to do mark-to-market in order to estimate Ln(P ijT )(T = 2012) -which is the market value of the painting estimated at the end of 2012. (2) One of the caveats for the above two models is that the sample may be subject to the Heckman selection bias problem as the observed samples are the more marketable ones. A recent paper by Korteweg and Sorensen (2012) extends Heckman's model by explicitly examining the selection bias. Using transactions of residential properties in Alameda, California, they have estimated the Eq. (2) using an MCMC Bayesian approach. KS discover that if the holding period is relatively long, the bias problem is quite small. Since the average holding period of our sample is 10.6 years, the time far exceeds the 5.1 years in Korteweg and Sorensen (2012). Thus, we think the selection bias should be modest in our case.
Our third model is a simple GARCH extension of the hedonic model (1): where we assume a GARCH(1,1) model for the conditional variance h is : Here h is denotes the conditional variance h is for artist i where s is the age of the artist at the time of creating the art (the date signed). This model specifies that the conditional variance of artistic innovation follows a GARCH(1,1) process, where measures the persistence of artistic innovation, while measures the impact of a single innovation on future variance. It is easy to see that the model (4) is an extension of the GARCH(p,q) model by Engle and Bollerslev (1986), which can be estimated using maximum likelihood estimation.

Data and descriptive analysis
We obtain the list of contemporary artists from the contemporary art sales catalogs of Christie's and Sotheby's auction houses. We then collect auction data for these contemporary artists from the online database of artinfo.com, artprice.com and websites of various auction houses. Our data contain transactions of major auction houses around the world from 1980 to 2012. Our data for each artwork contain the following information: the name of the artist, the title of the artwork, the year the painting was created, the year the painting was auctioned, the sale price at auction, the lowest and highest estimates of the painting by the auction house, the height, width, medium and shape of the painting, whether or not the painting is signed, and the name of the auction house where the painting was auctioned. To be included in the sample, we require an artist to have at least ten artworks. In the end, we are left with 81,567 observations from 275 artists. Table 1 in the Appendix reports the frequency distribution of the sample by the alphabetic order of the first names of the artists. The distribution across artists is relatively even. Most of the artists' works comprise less than 0.5% of the data. There are a few exceptions. Andy Warhol's works account for 3% of the total observations, and Jean Dubuffet's works make up 2.43% of the sample. We have also constructed repeated-sales data based on auctions from Sotheby's and Christie's as their sales The table reports the frequency distribution by artists. We obtain the list of the contemporary artists from Christie's and Sotheby's auction houses. We then collect auction data information for these contemporary artists from the database of artinfo.com. The database contains the transaction information of all major auction houses from 1980 to 2012. We require an artist to have at least 10 observations. The sample has 81,567 observations from 275 artists. These observations are the ones which were auctioned only once in history or the last occurrence of a repeated sale. The table reports two columns for the frequency distribution, the first is the number of observations, and the second is the probability distribution catalog makes it easier to track repeated sales and information on exhibition and literature citation.
In order to measure the importance of artworks as well as artists, we collect data on artwork citations in major art history books as well as major exhibitions. We then compile the total citation and exhibition counts by artists based on our sample of repeated sales data. Sotheby's and Christie's auction catalogs list separately literature citations and the exhibitions they could find for each piece they put up for sale in their catalogs. We count each citation and each exhibition listed for the most recent auction to develop the values available in our database. This information is only available from Sotheby's and Christie's online catalogs. Their history goes back only as far as 1998. Earlier sales information on these two variables has to be found and hand collected in a library with a rich collection of catalogs, such as the one at the Metropolitan Museum in New York. In addition, we also measure the importance of artists directly by compiling artist rankings made by several art history Ph.D. students from Florida State University. Table 2 reports the frequency distributions of the sample by the characteristics of the artworks. All panels include two columns for the frequency distribution. The first column is the number of observations. The second column is the percentage frequency distribution. Panel A is for the medium of the paintings, and oil paintings make up about 40% of the sample. Panel B is for the shape of the paintings. More than 95% of the paintings are rectangular. Panel C shows the signature status of the paintings, and more than 65% of the paintings are signed by the artists. Panel D displays the auction house status of the paintings: the top three auction houses are Christie's, Sotheby's and Phillips. About 40% of the paintings were auctioned in the top three auction houses. Panel E is the frequency distribution by calendar years. The more recent years, the more observations. There are because more artworks by contemporary artists are available for sale. Table 3 reports the estimation results for Eq. 1. Similar to GW, we show that the price of an artwork is related to the age of the artist when the work is created and that this is a nonlinear relationship. Because of the nonlinearity, a one-year increase in age has different impacts on the price of an art work, depending on the age of an artist. For example, holding other variables constant, when the artist is 25 years old, a one-year increase in age will increase the expected price of his work by about 5%. At age 34, a one-year increase in age will increase the price by about 0.05%, and at age 65, a one-year increase in age will decrease the price by about 0.36%. Fig. 1 plots the age-price profile for all artists in the sample, and the average peaking age for contemporary artists is around 35, which is somewhat older than what Galenson and Weinberg (2000) found for American artists born after 1920. In our paper, contemporary artists appear to peak about five years later than the modern American artists studied in their paper. Table 3 also reports other determinants of art prices. The brand name of the auction houses has a significant impact on the prices. Being auctioned at Journal of Cultural Economics (2023) 47:513-545 Christie's, Sotheby's or Phillips, compared to being auctioned at other auction houses, increases the price of an artwork by close to 30%. A one-inch increase in height increases the price by 2%; a one-inch increase in width increases the price by 1%. Oil paintings are generally more expensive, no matter what medium. The next most expensive is sculpture and water color paintings. Whether or not the artwork is signed by the artist is also important, which means a close to 10%

Residual variance and artist ranking
The table reports the frequency distribution of the sample by the characteristics of the paintings. We obtain the list of the contemporary artists from Christie's and Sotheby's auction houses. We then collect auction data information for these contemporary artists from the database of artinfo.com. The database contains the transaction information of all major auction houses from 1980 to 2012. We require an artist to have at least 10 observations. The sample has 81,567 observations from 275 artists. These observations are the ones which were auctioned only once in history or the last occurrence of a repeated sale. All panels report two columns for the frequency distribution. The first column is the number of observations. The second column is the probability distribution. Panel A is for the medium of the paintings. Panel B is for the shape of the paintings. Panel C is for the signature status of the paintings. Panel D is for the auction house status of the paintings. The top three auction houses are Christie's, Sotheby's and Phillips. Panel E is the frequency distribution by calendar years difference in price for an artwork with a signature versus one without. And we observe significant time fixed effects and the artist fixed effects which we will elaborate in Fig. 2 and Table 6. The artist fixed effect essentially captures the distinct effect on market prices by each artist, and thus, it reflects the lifetime achievement or reputation of each artist.
where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. This model assumes that the logarithm of price is a polynomial function of the age of the artist when the painting is created, hence the model captures the life cycle of residual variance. The time dummy (t) controls the market return of the art index, and the artist dummy (i) denotes the fixed effect of the artist. X represents the vector of the hedonic variables, such as the height, width, medium (oil or watercolor, etc.), shape (rectangular, or other shapes), whether or not the painting is signed, or whether the transaction was held in one of the three major auctions houses-Christie's, Sotheby's and Phillips (CSP). We use nominal prices, as the time dummies would automatically pick up inflation. We assume heteroscedasticity for the variance of the residual across artists and we estimate the model using the Generalized Least Square (GLS) method; then, we compute residual variance, based on age. We use log prices to mitigate the artificial mean effects since the variance of the residual is the linear function of price levels, which makes our results prone to the artificial inflation of variance if we use the price level itself. When we use the logarithm of the price, however, the artificial mean effect disappears. * denotes significant at 10%; * * denotes significant at 5%; * * * denotes significant at 1%  Fig. 2 plots both the contemporary art index based on repeated sales data and the year fixed effects estimated from Eq. 1. This graph shows that the time series of the year fixed effects tracks the repeated-sales index closely. Both indices show that the contemporary art market rose rapidly from 1980 to the early 1990s and then had a crash from the early 1990s to 1995, because the overall art market was heavily influenced by the exit of Japanese collectors at the time. The contemporary art market then rose steadily except for a short period of decline in 2008 and 2009. The focus of this paper is the residuals from Eq. 1 and their second moments. As an example, Fig. 3 plots the sorted residuals of the model by size and by five-year age intervals for Andy Warhol. As the residual ijt measures the relative (or percentage) deviation from his regular market and age-determined price path, our plots essentially show how his individual works at different times were valued by the market. We can see that Warhol had many below average works between 20 and 30 years old, while he produced some of his most expensive works when he was in his 30s. This is consistent with art history. Warhol was born in 1928. His first exhibition of the 32 Campbell's Soup Cans was in an art gallery in Los Angeles in 1962 when he was 34 years old. From then on, most of Andy Warhol's best work was done over a span of about eight years, such as Death and Disaster series, as well as the photo silk screen-printing portraits of Marilyn Monroe, Elizabeth Taylor, and so forth. However, Warhol also produced some of his lowest-priced works at the same age. What is really striking from Fig. 3 is that his best work from 36 to 40 years old sold for more than 148 times his average work, while his worst sold for 0.67% of the average. The difference is 22,090 times! Thus, we can observe that while some innovations were highly valued by the market, others produced at the same time were poorly received. His success seems to be always accompanied by failure, but his failures early in life were seldom accompanied by successes. However, his innovation had a large price dispersion throughout his life.
Comparing the residuals from ages 36 to 40 with those from ages 56 to 60, we can see that Warhol had more successes as well as failures from ages 36 to 40, as there is more mass at both sides of the residual distribution. We may try to describe this mass on both sides of the residual distribution by computing the average squared residuals by age. In some sense, this is a measure of the artist's intensity of innovation at a certain age. The result is presented in the top left panel of Fig. 4. For Warhol, we can see this intensity peaks around 35, rises again in his early 40s, and then declines in his 50s.
For comparison, Fig. 4 also plots the age residual variance profile for other topranked artists. It presents several interesting results. First, there is clearly a large variation in the pattern of residual variance across artists. While most artists tend to peak early in their careers, Francis Bacon actually peaked later in life. While William De Kooning's residual variance was concentrated in his 40s, Jasper Johns had multiple peaks in his 20s, 30s, and 40s. The residual variance of artists fluctuated substantially over age and differently across artists 2 .
To better understand the residual variance, Panel A of Table 4 shows the summary statistics of the squared residuals from Eq. 1. We split the sample of artworks by each artist into four different life periods: the whole lifetime, ages between 25 where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. The time dummy (t) controls the market return of the art index, and the artist dummy (i) denotes the fixed effect of the artist. X represents the vector of the hedonic variables, such as the height, width, medium (oil or watercolor, etc.), shape (rectangular, or other shapes), whether or not the painting is signed, or whether the transaction took place in one of the three major auctions houses-Christie's, Sotheby's and Phillips. We use nominal prices, as the time dummies would automatically pick up inflation. We assume heteroscedasticity for residual variances across artists and we estimate the model using the GLS method. We separate the sample into different age groups: all the ages, from 25 to 35, from 36 to 65 and after 65 years old  May-June 1973(1973, Study for Self-Portrait (1982) and Study for a Self Portrait-Triptych (1985-86). and 35, ages after 36, and ages after 65. The mean for the residual variance measure is 0.937 for the whole sample, and the standard deviation is about 0.60, with a minimum of 0.248 and a maximum of 4.850. The mean of the residual variance decreases as artists age, and the standard deviation of residual variance increases. Thus, in general, the residual variance of contemporary artists tends to peak around their middle ages and then falls when artists grow older. However, the increasing standard deviation in the older age group indicates that a large number of older artists remain having high residual variances. Panel B of Table 4 presents the correlation matrix of the four residual variance measures. We can see there is a great deal of persistence in residual variance over the lifetime of an artist. As a further study of this lifetime persistence, we employ the GARCH (1,1) model of Engle and Bollerslev (1986) to investigate the persistence of residual variances. After estimating the model for each artist, we discover that the average for the sample artists is 0.16 and the average for the sample artists is 1.23. Again, the result is consistent with lifetime residual variance persistence.
Since the artist fixed effect in Eq. 1 measures the average achievement by an artist over his life as perceived by the market, in Table 5, we will examine the relationship between the fixed effect and residual variance. Thus, we regress the artists' fixed effects on the four residual variance measures derived from the squared residual from Eq. 1. It shows that the fixed effect is positively related to the residual variance of the whole lifetime, the residual variance between 25 and 35, the residual variance between 36 and 65, and the residual variance after 65, respectively. The coefficients are 0.734, 0.528, 0.785, and 0.705, while the t-statistics are 6.27, 3.76, 6.32, and 4.31, respectively. Thus, the residual variance measures are significantly and positively related to the average lifetime achievement by an artist. One may wonder if our result is a mechanical one, as high variance is often as associated with high mean in prices. However, as we have taken logs in art prices, this result is not determined a priori, just like the returns to large stocks (see Model 2). It could be higher or lower than small stocks. Contrary to large stocks, we find works by highly successful artists actually tend to have higher squared residuals. Table 6 reports the rankings of the artists based on the fixed effects and residual variance estimated from Eq. 1. It shows that a vast majority of the artists overlap in two categories. Examples are: Jeff Koons, Jasper Johns, Roy Lichtenstein, Andy Warhol, Yves Klein, Robert Rauschenberg, Wayne Thiebaud, Jackson Pollock, Willem De Kooning, Gerhard Richter, and Francis Bacon. This ranking table confirms our regression results that there is a high correlation between the residual variance and the average market performance of each artist.
One interesting issue here is how the residual variance measure relates to the importance of artworks as perceived by art critics and historians. There are two conventional ways of measuring the importance of an artwork: First, by how many times, it is cited in important art history books; second, by how many times, it was exhibited in major museums. As noted in the data section, we measure the success of an artist by compiling the total number of citations of his sample works that were sold by the two major auction houses and by compiling the total number of exhibitions of these works displayed in major museums. Tables 7 and 8 show the regression results of the citation and the exhibition counts on the artists' fixed effects estimated from Eq. 1 and the four residual variances over the four age periods. Both tables show that the citation and the exhibition counts are significantly and positively related to the residual variance of the whole lifetime, the residual variance between 25 and 35, the residual variance between 36 and 65, and the residual variance after 65, respectively, controlling for the artists' individual fixed effects 3 . These results imply that the residual variance measures have additional explanatory power in terms of how often the artist's works are cited and exhibited, even after we control for artist fixed (reputation) effects.
A third way of ranking the importance of artists is to ask art historians to rank artists directly. We collect the ranking information from several Ph.D. students of art history at Florida State University and we take the average of all the rankings. Table 9 shows the regression results of the average relative rankings of the artists where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. The time dummy (t) controls the market return of the art index, and the artist dummy (i) denotes the fixed effect of the artist. X represents the vector of the hedonic variables, such as the height, width, medium (oil or watercolor, etc.), shape (rectangular, or other shapes), whether or not the painting is signed, or whether the transaction took place in one of the three major auctions houses-Christie's, Sotheby's and Phillips. We use nominal prices, as the time dummies would automatically pick up inflation. We assume heteroscedasticity for residual variances across artists and we estimate the model using the GLS method. We separate the sample into different age groups: all the ages, from 25 to 35, from 36 to 65 and after 65 years old. The artists' fixed effects are estimated from all the age groups. The numbers in the brackets are t-statistics. * denotes significant at 10%; * * denotes significant at 5%; * * * denotes significant at 1% Artist fixed effect (All) . whereas i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. The time dummy (t) controls the market return of the art index, and the artist dummy (i) denotes the fixed effect of the artist. X represents the vector of the hedonic variables. We use nominal prices, as the time dummies would automatically pick up inflation. We assume heteroscedasticity for residual variances across artists and we estimate the model using the GLS method. The numbers in the brackets are t-statistics. * denotes significant at 10%; * * denotes significant at 5%; * * * denotes significant at 1% ARTIST FULLNAME Fixed effects ARTIST FULLNAME Res. var.  on the artists' fixed effects and the four residual variance measures. The table shows that the rankings of art historians are significantly and positively related to the residual variance of the whole lifetime of the artist, the residual variance between 25 and 35, the residual variance between 36 and 65, and the residual variance after 65, respectively, controlling for the artists' individual fixed effects. These results imply that the residual variance measures have additional explanatory power in terms of the rankings of artists, even after we control for artist fixed (reputation) effects.

Asset pricing analysis
In this section, we employ an asset pricing analysis to examine whether the residual variance measure we developed in the above section was highly valued (or priced) by the art market.
To do this, we first mark-to-market, using a repeated sale index, all artworks sold during the sample period to 2012, essentially computing the present value of all artworks sold. We then compute the mean of the top five works' market value (as of 2012) for each artist. Table 10 shows the regression results of the mean and median of the present market value (as of 2012) for an artist on the artists' residual variance. It shows that the mean and median of the present market value are significantly and positively related to the various measures of residual variance, which indicate that the artist's residual variance helps determine the mean and median of the present market value. Table 11 shows the regression results of the mean of the top five works' market value (as of 2012) for an artist on the artists' residual variance. It shows that the top five works' market values are significantly and positively related to the various measures of residual variance. These results imply that the artist's residual variance helps determine the top five works' market value.
One interesting question arising from our study is whether residual variance estimated from an earlier sample helps forecast artist fixed effects and residual variance in later samples. Table 12 shows that the artist fixed effects and residual variance estimated from 1996 to 2012 are positively and significantly related to the artist fixed effect and residual variance estimated from an earlier sample of 1980 to 1995. The result again confirms the results of Panel B of Table 4 that residual variance is not only quite persistent over the life of an artist but also persistent over the sample period.
Another way of measuring whether residual variance is valued by the market is to ask if artworks by artists with high residual variance outperform the art market. We will address the question from both within and out of sample. To measure outperformance, we search our data for repeated sales and we define outperformance as the annualized excess returns of the artwork over the market return during the holding period. We use the time fixed effects in Eq. 1 as well as the market index computed from the repeated sales model (2) to compute the market return.
One important variable affecting art price could be liquidity. Collectors often need to sell their artworks for various reasons, and artworks differ in transaction cost and the level of difficulty in matching buyers with sellers. In the asset pricing Table 10 Residual variance and the mean and median of the present market value This table reports the regression results of the mean and median of the present market value of the artists'works on residual variance defined as the residual variance in the hedonic regression of the logarithm of price on various variables: where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. We separate the sample into different age groups: all the ages, from 25 to 35, from 36 to 65 and after 65 years old. For each age groups, we obtain residual variance estimated from the above method.
We first obtain the prices of all the works ever sold on the auction market for an artist as of 2012 and we take the mean and median of this series of prices. The numbers in the brackets are t-statistics. * denotes significant at 10%; * * denotes significant at 5%; * * * where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. We separate the sample into different age groups: all the ages, from 25 to 35, from 36 to 65 and after 65 years old. For each age groups, we obtain residual variance estimated from the above method. We first obtain the prices of the top works ever sold on the auction market for an artist as of 2012 and we take the mean of this series of prices. The numbers in the brackets are t-statistics. * denotes significant at 10%; * * denotes significant at 5%; * * * denotes significant at 1% Mean of the Top 5 works at the price of 2012 where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. We separate the sample into two periods-from 1980 to 1995 and from 1996 to 2012. For each period, we obtain the residual variance and the artists' fixed effects estimated from the above method. The numbers in the brackets are t-statistics. * denotes significant at 10%; * * denotes significant at 5%; * * * denotes significant at 1% Fixed effect (1996)(1997)(1998)(1999)(2000)(2001)(2002)(2003)(2004)(2005)(2006)(2007)(2008)(2009)(2010)(2011)(2012) Variance ( literature, Duffie et al. (2003), Vayanos and Wang (2007) and Weill (2007) provide theoretical models and empirical evidence to analyze the effects of liquidity on asset prices and trading volume, based on a search process between buyers and sellers. Pastor and Stambaugh (2003) also find the presence of a liquidity factor in equity pricing. It is quite intuitive that liquid assets tend to have higher prices and thus may under-or outperform the market. To study the effects of liquidity, we compute the total number of auctioned paintings for each artist and use that as a dependent variable.
Another variable related to art investment return is the "masterpiece effect." Studies such as Pesando (1993), Moses ((2002, 2005)) show that masterpieces (defined by their high purchase prices) tend to underperform the market. Thus, we include the purchase price to control for the "masterpiece effect." Table 13 shows the regression of outperformance for 3,361 paintings with repeated sales against residual variance and several control variables over the whole sample period. The outperformance is defined as the (annualized) difference of the two adjacent log sale prices adjusted, respectively, by the corresponding change in the Repeated Sales (RS) Contemporary Art Index (Model 1) or the year fixed effects estimated from Model 2. We find that even controlling for reputation, liquidity, and where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. We separate the sample into different age groups: all the ages, from 25 to 35, from 36 to 65 and after 65 years old. For each age groups, we obtain residual variance and the artists' fixed effects estimated from the above method. The outperformance is defined as the difference of the two adjacent log sale prices from t to t + 1 adjusted, respectively, by the corresponding change in the Repeated Sales (RS) Contemporary Art Index (Eq. 1) and the year fixed effects estimated from the above model (Eq. 2). Log (Auction Number) is the logarithm of the total number of auctions for an artist in each age group. Log (Sale Price) is the logarithm of the auction price at time t. We cluster the standard errors by painting. The numbers in the brackets are t-statistics. * denotes significant at 10%; * * denotes significant at 5%; * * * denotes significant at 1% where i denotes the artist i; j denotes the jth painting, s is the age of the artist at the time of creating the art (the date signed) and t is the time of the transaction. We separate the sample into two periods, respectively-the first pair is from 1980 to 1995 and from 1996 to 2012, and the second pair is from 1980 to 1999 and from 2000 to 2010. For the first period of each pair, we obtain residual variance estimated from the above method. For the second period of each pair, we obtain the outperformance estimated as the difference of the two adjacent log sale prices adjusted, respectively, by the corresponding change in the Repeated Sales (RS) Contemporary Art Index and the year fixed effects estimated from the above model. For Eqs. 1 and 2, the outperformance of a painting is from year t to t + 1 and the residual variance is over the period from t − 15 to t − 1 . For Eqs. 3 and 4, the outperformance of a painting is at year t and the residual variance is over the period from t − 20 to t − 1 . Log (Auction Number) is the logarithm of the total number of auctions for an artist in each period. Log (Sale Price) is the logarithm of the auction price at time t. We cluster the standard errors by painting. The numbers in the brackets are t-statistics. * denotes significant at 10%; * * denotes significant at 5%; * * * denotes significant at 1% the masterpiece effect, the adjusted outperformance is significantly and positively related to the residual variance of the artist measured over his lifetime. We discover that reputation (the artist fixed effect) is significantly and positively related to outperformance. This is not surprising as these artists on average have high prices estimated over the sample period. We also discover that the liquidity (the total number of auctioned paintings by the artist) is positively related to outperformance, but the effect is statistically not significant. We further confirm the presence of a very strong and negative "masterpiece effect." Table 14 provides an out-of-sample outperformance study by splitting the sample in two ways : 1980-1995 and 1996-2010 vs. 1980-1999 and 2000-2010. The earlier sample is used for computing residual variance and liquidity, while the latter sample is used for analyzing outperformance. Due to the smaller sample, we drop reputation as a control variable. Again, our results show that even controlling for liquidity and the masterpiece effect, residual variance of the artist can significantly predict outperformance. The result is robust over different sample splits and models used for controlling market returns. We also discover the presence of a very strong and negative "masterpiece effect" though the liquidity effect is not significant. It is worth noting, This graph shows the residual estimated from Eq. 1 sorted by age group for Andy Warhol. The X-axis is the nth painting created in that period however, that residual variance becomes statistically insignificant once we introduce reputation as a controlling variable.
In summary, we find that there is some evidence that the market seems to reward those artworks created by artists with high residual variance as defined in this The residual variance profiles for top-ranked artists. This graph shows the residual variance profiles for some top-ranked artists. The residual variance measure is the average residual variances by the age of an artist. The residual is estimated from Eq. 1 paper-but collectors should avoid paying excessive amounts. We also note that "liquid" and well-known artists seem to enjoy small excess returns. Given the fact that the residual variance is highly related to artistic creativity as perceived by art historians and curators and that there is preliminary evidence that it is also valued by collectors, we may use it as an alternative measure of creativity in addition to those based on first moments used by Galenson and Weinberg ((2000) and (2001)).

Sample selection bias
One possible critique of our empirical work is that it might be subject to a serious sample selection bias. Just imagine that auction houses would not accept any painting whose price was below a certain level in a given year. As a result, we would not be able to observe a large section of the price (and return) distribution of low-priced artists, but this is not the case for high priced artists, such as Warhol. As a result, the residual variance of low-priced artists could be underestimated, thus creating a spurious correlation between residual variance and art pricing.
To address this potential bias, we will use a bootstrap approach to create a data set similar to ours under the assumption of constant residual variance across artists. Using Eq. 1 and estimated fixed effects, we will then create the same number of observation for each artist i as in our sample by randomly drawing ijt from N(0, 2 ) , where 2 is assumed to be 0.937, the mean residual variance from Panel A of Table 4. Whenever the created price is below the lowest price observed in data for a given year, we will throw away that data point and generate another one. We will then re-estimate the relevant tables and compare them with actual data.
For brevity reason, we summarize our results in this paragraph; all tables are available upon request. There is a weakly (at 10% significance level) positive relationship between artist fixed effect estimated with the created prices and residual variance from the simulated data for the whole sample. For each age group, there is no significant relationship between artist fixed effect estimated with the created prices and the simulated residual variance. Across the whole sample and all age groups, there is no significant relationship between the simulated residual variance and the literature citation, the museum exhibition or the average ranking by art scholars. Across the whole sample and all age groups, there is no significant relationship between the simulated residual variance and the mean or median of the present market value from the created prices, or the top five works' market value from the created prices. In summary, there is no statistic relationship between the residual variance from the simulated data generated from the created price which has the cutoff at the lowest price in our actual data for a given year, and the economic variables which are significantly related to the residual variance from the real data. Therefore, our results are not driven by the serious problem of the sample-section bias.

Discussion
So far, this paper is quite mute on the question why high priced artists tend to have high residual variance. By taking logs of art prices, we point out that this result is not due to data construction. After ruling out the possibility of sample selection bias, our conjecture is that our result may have something to do with the creative process. Art creation is a process full of uncertainties. Ex ante, the most creative artists need to take huge risks because their art could be misunderstood by the market. The most creative, relatively speaking, are also likely to have the most to lose. This may lead the log price distribution of their paintings to have fat tails on the left side. But they must continue to innovate despite the huge risk.
But our results could be affected by some important assumptions underlying the analysis. First of all, we assume that the market value provides an unbiased estimate of the creative value of art. As a result, we assume away all micro-factors that may affect the demand for and supply of art by collectors. This is certainly not always the case, as Beggs and Graddy ((2008) and (2009)) find direct evidence of anchoring and loss aversion in the art market. As a result, the conditional mean of the residuals in Model 1 may not be zero. Second, other macro-factors, such as market sentiment and speculative bubbles, may affect art pricing. 4 Third, the hedonic Model 1 might be miss-specified. For example, not all exhibitions and citations are similar, but we have no way of distinguishing them. They may not enter the model linearly either. Fourth, the market may not correctly price young artists who have yet to establish themselves in the market. For example, Vincent van Gogh created some of the most expensive paintings that have been sold on the modern market, but he nonetheless was unable to sell his artworks during his lifetime. In this study, we bypass this problem by focusing on artists whose works are sold by the world's top auction houses.
Our analysis is broadly consistent with a body of the literature in corporate finance, which studies firm innovation and idiosyncratic volatility of stock prices. The theory of firms by Myers and Majluf (1984) suggests that firms with more growth opportunities are more volatile and especially have more idiosyncratic volatility. Firms innovate through R&D, so firms that invest more in R&D are expected to have more volatile cashflows. That is to say innovation constantly creates winners and losers, so we should expect innovative firms to have more idiosyncratic risk. Pastor and Veronesi (2009) study technological revolutions and stock prices across countries, and they find that countries where technological revolutions originate are associated with higher idiosyncratic volatility. Bartram et al. (2012) also find that the higher idiosyncratic volatility of US firms is driven by a higher share of R&D in the sum of capital expenditures and more R&D than comparable firms in foreign countries.

Conclusion
In this paper, we compute residual variance of art prices to examine asset pricing in contemporary art market. Our empirical work has yielded a few interesting results. First, we discover that the residual variance is significantly and positively related to the average price level achieved by an artist. Second, the residual variance has additional explanatory power in terms of how often the artist's works are cited and exhibited, even after we control for artist fixed (reputation) effects. Third, collectors tend to value more those artworks with higher residual variance. Artworks by those artists with high residual variance tend to outperform the market within and out of sample. While our results are subject to different interpretations, it could be consistent with risk taking in the creative process.
In November of 2017, Leonardo da Vinci's famous work of art, Salvator Mundi, sold for close to $450.3 million at Christie's, New York, setting the world record for any work of art sold at an auction. In May of 2015, Pablo Picasso's famous work of art, Les Femmes d'Alger, sold for close to $179.4 million at Christie's, New York. The hundreds of millions dollars were almost the pure value of artistic creativity, as the material cost of the painting is negligible. Creativity is at the center of almost all economic innovations. It is the root source of total factor productivity as new technology, new business models, and new markets are developed by creativity. Yet, the economics profession scarcely studies the subject. Art historians have even resisted the introduction of economic analysis into art. We hope that the new measure of residual variance may help shed some light on the creative process.
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