PRICES OF THE LIBERTY STANDING QUARTER

Transcription

1 This document deals with the prices paid by collectors for quarters in the Liberty standing set, issued between 1916 and Year / Mint / Type Mintage Value ,000 14, Type 1 8,740, D Type 1 1,509, S Type 1 1,952, Type 2 13,880, D Type 2 6,224, S Type 2 5,552, ,240, D 7,380, S 11,072, ,324, D 1,944,000 1, S 1,836,000 1, ,860, D 3,586, S 6,380, ,916,000 1, ,716, S 1,360,000 2, ,920, D 3,112, S 2,860, ,280, ,316, D 1,716, S 2,700, , D 976, S 396,000 4, ,336, D S 2,644, ,140, D 1,358, S 1,764, ,632, S 1,556, The Mint identification is D for Denver and S for San Francisco. Coins without a mint identification were struck in Philadelphia. The 1917 coins were made in two types that differed according to the placement of the stars on the reverse. The prices here refer to 1

2 coins in collector grade MS60, which we can interpret crudely as shiny. Collectors are deeply concerned about quality. Prices change, of course, and printed and on-line prices do not always agree. These prices were found from in It s plausible that price should be inversely related to the quantity minted. Here is a scatterplot: Scatterplot of Price vs Mintage Price Mintage This plot was generated in Minitab through Graph Scatterplot. This reveals that the relatively less common, low-mintage coins are priced higher. However, we do not get to see detail on the other coins. 2

3 This situation is much improved by taking logarithms of both values. Here is that picture: 10 Scatterplot of logprice vs logmintage 9 logprice logmintage Several comments are in order here: (1) The logarithms are base e. We prefer base e to base 10 for nearly all purposes. There are many reasons for this, but for sure the present value formula P t = P 0 e rt gives a critical need for base e. (2) In recent years, about 250 million quarters have been struck at each of the Philadelphia and Denver mints. None of the Liberty standing quarters ever reached 28 million. (3) The coins seem to naturally divide into two groups. There are six valuable coins in the string at the upper left, and these separate themselves from all the others. For the remaining coins, the value seems relatively independent of the mintage. (4) The six coins in the string at the upper left need to be identified. These are 1916, 1919D, 1919S, 1921, 1923S, and 1927S. (5) It is worth identifying the coins from the less-valuable group that have mintage numbers below the six valuable coins. We will use the cutoff at 2,000,000. There are 1917D Type 1, 1917S Type 1, 1926D, 1927D, 1928D, 1929D, and 1929S. (6) Logarithms are also plausible from the model Price Mintage = Constant This model cannot be accurate, as price must level off at some point, regardless of mintage. Still it s a good start. It also suggests that the slope in a regression of logprice on logmintage ought to be close to -1. 3

4 Here is the output related to that regression. Regression Analysis: logprice versus logmintage The regression equation is logprice = logmintage Predictor Coef SE Coef T P Constant logmintage S = R-Sq = 52.2% R-Sq(adj) = 50.8% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations Obs logmintage logprice Fit SE Fit Residual St Resid X X denotes an observation whose X value gives it large leverage. This was done in Minitab through the command Stat Regression Regression. The regression produces numbers that suggest a plausible fit. The estimated slope is ; it s not exactly close to -1, but it s in the right spirit. The regression is certainly statistically significant, and R 2 is a respectable 52.2%. The residual versus fitted plot suggests that our model just doesn t work. 1.5 Versus Fits (response is logprice) Residual Fitted Value 8 9 In the command set for Stat Regression Regression, follow with Graphs Residuals versus fits. 4

5 The residual versus fitted plot should have no obvious pattern. In this case, there are six exceptional points and the remaining points fall along a line of negative slope. The six exceptional points are those with the highest values. While the residual versus fitted plot is an active window, the command Editor Brush will produce a pointing hand that allows individual points to be identified. See note (4) above. Perhaps the data should be separated into two groups, the six valuable coins and all others. For the six valuable coins, we don t have enough to make a serious analysis, but we ll look anyhow. This is the scatterplot: Scatterplot of logprice vs logmintage logprice logmintage Here is the regression for those six points: Results for: SixValuable Regression Analysis: logprice versus logmintage The regression equation is logprice = logmintage Predictor Coef SE Coef T P Constant logmintage S = R-Sq = 97.5% R-Sq(adj) = 96.9% Analysis of Variance Source DF SS MS F P Regression Residual Error Total

6 The fit is excellent. Here is the residual versus fitted plot: Versus Fits (response is logprice) Residual Fitted Value With six points, it is just not realistic to make decisions regarding the appropriateness of a model. Here is a scatterplot for the non-valuable coins. 6.2 Scatterplot of logprice vs logmintage logprice logmintage Here s the regression: Results for: NonValuable Regression Analysis: logprice versus logmintage The regression equation is logprice = logmintage Predictor Coef SE Coef T P Constant logmintage

7 S = R-Sq = 6.6% R-Sq(adj) = 3.4% Analysis of Variance Source DF SS MS F P Regression Residual Error Total Unusual Observations Obs logmintage logprice Fit SE Fit Residual St Resid R R denotes an observation with a large standardized residual. The residual versus fitted plot does not suggest any corrections Versus Fits (response is logprice) 0.50 Residual Fitted Value There is not much of a message in this regression. The R 2 value is very low at 6.6%, and the F statistic is not statistically significant. So... what really have we learned? There are six low-mintage coins for which price is approximately proportional to mintage. For the remaining coins, there appears to be no relation between price and mintage. 7

8 This glib analysis leaves a few things unresolved. (1) Why are there some coins with low mintages (below 2,000,000) that do not command high prices? (2) The decision to break the data into two subsets seems totally subjective and ad-hoc. Is this justified? Does it make sense? (3) This coin is not going to be issued again. What exactly is the valuable insight that we get here? 8