Data Errors in Small Data Sets Can Determine ...

Data Errors in Small Data Sets Can Determine Empirical Findings LING T. HE AND JOSEPH P. MCGARRITY∗

Abstract This paper provides an example of a model that yields widely divergent estimates when different stock market indexes are used to calculate two independent variables in Romer’s [1990] model. Her model sought to explain consumer durable good production before the Great Crash (31 observations). She used the Cowles Commissions Series P Stock Price Index to calculate two independent variables. However, when this paper uses the S&P Index to calculate these variables, its estimates completely contradict Romer’s Þndings. It discovered that one incorrect monthly observation in the S&P Index is responsible for this difference. It also found that robustness techniques serve to limit the impact of the errant observation, illustrating the importance of using robustness techniques in small c data sets.(JEL N11, N21); Atlantic Econ. J., 32(2): pp. 89-99, June 04. °All Rights Reserved

Introduction When estimating models with small data sets, empirical research runs the risk that errors may be responsible for the conclusions scholars draw. This paper provides an example of a model that yields widely divergent estimates when different stock market indexes are used to calculate two independent variables in the model. The difference is caused by a data error in one of the indexes. In particular, Romer’s [1990] model was examined. It sought to explain consumer durable good production.1 She was primarily interested in how inßuential the wealth effect and uncertainty effect were in explaining changes in this durable goods market. The wealth effect claims that an increase in stock prices represents an increase in expected future wealth, which causes consumers to increase their consumption in the current period, as suggested by the Permanent Income Hypothesis. Conversely, a decrease in stock prices implies a reduction in consumers’ expected future purchasing power, which will decrease current consumption.2 On the other hand, the uncertainty effect claims that stock price volatility, not necessarily an increase or decrease in stock prices, fundamentally alters consumers’ spending.3 Romer calculated a holding period return to capture the wealth effect and a measure of stock price volatility to capture the uncertainty effect, and included these independent variables in an equation explaining consumer durable production. She used the Cowles Commissions Series P Stock Price Index [Cowles, 1939] to calculate these variables in a pre-Great Crash sample period (31 observations). In a later sample (post World War II), she used the Standard and Poor’s (S&P) 500 Stock Index. Her early sample constructing the two stock market variables with the Cowles index is re-estimated here, as well as with the S&P Index. When using the Cowles Index to construct the stock market variables, Romer’s Þndings are replicated. She discovered strong evidence for an uncertainty effect, but none for the wealth ∗

University of Central Arkansas–U.S.A.

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effect. However, when using the S&P Index to calculate the variables, the uncertainty effect had a coefficient one-fourth the size of Romer’s original estimate and became insigniÞcant at conventional levels in the process. Furthermore, one incorrect monthly observation can be identiÞed in the S&P Index that is responsible for the conßicting results. When this inßuential row of data is deleted, the two estimates of Romer’s model become nearly identical across speciÞcations that use the Cowles and S&P Indexes. In addition, when a Bounded Inßuence Estimation is performed that limits the impact of outliers, both estimates are similar. Further, correcting for the error makes the divergent results disappear. With each method of accounting for the inßuential observation, the uncertainty effect is an important determinant of consumer durable production. Stock price variability becomes signiÞcant at higher levels when the problematic observation is deleted, corrected, or when robust estimation techniques are employed. Therefore, the results suggest that the choice of seemingly very similar data sets (such as different broad based stock market indexes) can alter the results of a study when the data sets are small and errors exist in the data. The results also highlight how useful robustness techniques are in small data sets. The Þndings are especially relevant in economic history where data limitations often require that scholars analyze small data sets. Methodology and Data In order to examine the inßuence of the choice of data sets on the uncertainty and the wealth effects, both the Cowles and S&P Indexes were applied to the volatile economic period deÞned in Romer’s study [1990]. The Bounced Inßuential Estimation was also used to measure alterations in regression coefficients of the uncertainty and wealth effects caused by outliers. The analysis is based on Romer’s model for the periods 1891-1913 and 1921-28 (a total of 31 observations). Her model appears below: Yit = f (Yi(t−1) , Y(t−1) , Vt , Wt )

,

where Yit is the Þrst difference of natural logarithms of commodity output group i from year t − 1 to year t. The one commodity group is consumer durable goods. The years t and t − 1 are calendar years. Yi(t−1) is the one year lagged value of the dependent variable. Y(t−1) is the Þrst difference of natural logarithms of total commodity output from year t − 2 to year t − 1. Vt is a measure of the variability of real stock prices. The difference of the deßated natural logarithms of the monthly stock market index is squared and then averaged over 12 months from October of year t−1 to September of year t. Romer uses the Cowles Commissions Series P Stock Price Index. Also provided is a speciÞcation of this variable, calculated with the monthly Standard and Poor’s 500 (S&P 500) Index. Wt is the one year holding period return of the deßated stock market index calculated from September of year t − 1 to September of year t. Romer uses Vt in the model above to test the uncertainty hypothesis, which claims future uncertainty can be captured by stock market volatility. When people become more uncertain about their level of future income, they are more likely to postpone durable good purchases. The theory predicts a negative coefficient for Vt , which suggests that stock price variability and durable good purchases are negatively correlated. The model also can test the wealth effect, which suggests that an increase in stock prices represents more favorable expectations of future business conditions and thus, of future

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wealth. The permanent income hypothesis predicts that consumers with optimistic expectations of future wealth may buy more durable goods in the current period. Therefore, the wealth effect predicts Wt will be positive. The same data sources are used that Romer used in her study. The output indexes used to create Yit and Y(t−1) come from Shaw [1947, Table I-3, pp. 70-77], and are all annual data. The monthly S&P 500 Composite Index is from the Security Price Index Record [Standard & Poor’s, 1998]. The monthly Cowles Stock Index is from Common Stock Indexes [Cowles, 1939]. The stock price index is deßated with Warren and Pearson’s Wholesale Price Index [Warren and Pearson, 1933, pp. 10-13]. The Cowles Series P, which does not include cash dividends, covers the period 1871-1938. The Cowles Commission Index used the weekly Standard Statistics Index (the predecessor of the Standard and Poor 500 Index) to construct their indexes after 1917. Also, the Standard and Poor’s company used the Cowles index to construct their index before 1918. Basically, both the Series P of Cowles and the Standard Statistics Index use the same method of construction and the same companies as much as possible [Standard and Poor’s, 1998, p. 1] This is probably the reason why Cowles adopted the Standard Statistics Indexes into the Cowles Commission Indexes since 1918, in most cases without any adjustments [Cowles, 1939]. Starting in 1918, the Series P is simply the monthly averages of the weekly Standard Statistics Indexes. The only adjustment made by Cowles is the number of shares outstanding [Cowles, 1939, p. 3]. Results Summary Statistics for the Cowles and S&P Indexes In order to gauge the similarity of these two monthly stock indexes, the deßated monthly index numbers are calculated, which provide 374 observations over the periods of September 1890-September 1913 and September 1920-September 1928 and are consistent with Romer’s sample period. These two indexes are very similar, as suggested by their correlation coefficient of 0.9969. Statistical comparison of these two indexes is complicated by the potential that these series are not stationary. Indeed, Augmented Dickey Fuller tests are unable to reject the null hypothesis that the series has a unit root. The test statistics are -0.85 for the Cowles Index and -0.44 for the Standard and Poor’s Index, which are not close to the -2.57 critical value at the 10 percent level. To avoid non-stationary problems, the percentage change of the monthly indexes, calculated with the logged differences of the monthly deßated index, is considered. This procedure costs two observations, the Þrst for differencing and the second for the gap in the data that would have considered the change between September 1913 and September 1920. This leaves 372 observations which have a correlation coefficient of 0.93. The percentage changes in the S&P are somewhat more volatile than the percentage changes in the Cowles Index. The former has a standard deviation of 0.036, while the latter has a standard deviation of 0.034. An ordinary least squares estimation, with a dependent variable of the percentage change of the deßated Cowles Index and using a constant and the percentage change in the deßated S&P Index on the right hand side, can be used to spot differences between the two indexes. The residuals from the regression that are greater than the absolute value of two standard deviations (|0.023|) occur only twice, in November (0.156) and December (-0.138) of 1900. The Deßated Cowles Indexes for October, November, and December of 1900 are 59.75, 64.44, and 69.13, respectively. The index increases with each new month. In contrast, the deßated Standard and Poor’s Indexes for these three months are 7.42, 6.77, and 8.59. The

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major difference is that the November Index drops by 9.2 percent for the S&P index but increases by 7.6 percent for the Cowles Index. Because the November S&P Index dropped, the December monthly return started from a much lower November base and resulted in a very large increase (23.8 percent) in December. This is much greater than the percentage increase of 7.0 percent when the Cowles Index is used to calculate the December return. It is very likely that the November S&P Index is incorrect since the holding period return from October to December of 1900 (two month holding period) is 14.62 percent based on the Cowles data and 14.57 percent from the S&P data. They are virtually the same. Using the S&P Indexes, the December monthly return of 23.8 percent is twice as large as the next largest positive monthly return of 10.6 percent. Interestingly, when considering the Cowles Indexes, the same month produced a 10.6 percent return, which was also the largest positive return for this series. It may be helpful to consider the Dow Jones Industrial Average (DJIA) [Pierce, 1982] as a benchmark to compare the S&P and Cowles monthly holding period returns. The deßated monthly return from the DJIA index was 13.1 percent in November 1900. This is much closer to the Cowles monthly return of 7.6 percent than the S&P monthly return of -9.2 percent. The same holds true in December. The DJIA monthly return of 8.2 percent is much closer to the Cowles monthly return of 7.0 percent than it is to the S&P monthly return of 23.8 percent. The DJIA monthly returns give the same information about the market that the Cowles monthly returns do. There were positive returns in both November and December of 1900 and the December monthly return was in the 7 to 8 percent range, not the incredible 23.8 percent from the S&P Index. A comparison of S&P entries for November 1900 across different years of the Standard and Poor’s Index Record suggests that the data are indeed an error. The three most recent additions (1998, 2000, and 2002) all list 5.48 for the November 1900 S&P index. The 1988 edition lists this observation as 6.48. In order to explore the differences between monthly holding period returns based on the two indexes, the paper regresses returns of the S&P Index against returns of Cowles Index. When the monthly returns for November and December are retained in the regression model, the coefficient of the independent variable is 0.87 and the R-square is 0.88. Removing these two observations causes the coefficient to jump up to 0.9868 and the R-square to increase to 0.995. The Þrst estimate suggests that a 1 percent increase in the S&P Index results in a 0.88 percent increase in the Cowles Index. The Þrst estimate suggests that these two indexes clearly do not move in lock step. In the second estimation, which excludes the two observations, a 1 percent increase in the S&P corresponds with just under a 1 percent increase in the Cowles Index. That is, without November and December of 1900, the percentage changes in the monthly indexes have a Beta of nearly one. Further, the drastic increase in the R2 for the estimation without these two observations suggests that these two indexes convey the same information in 370 out of 372 months. The two indexes suggest very different market performance in November 1900. This can affect Vt , stock price variability, in Romer’s model. The low November index value will be counted twice because her measure takes the logged difference of each monthly observation. The November index number will be considered once when the October index number is subtracted from it and then when the November index number is subtracted from the December index. The problem will be exacerbated because the logged differences are squared before they are averaged. This makes the November index very inßuential. In fact, the measure of stock market volatility for October 1900 to September 1901 (Vt ) with the S&P index, which contains the problematic November 1900 index entry, is 0.008, by far the largest value in the sample. The next highest score is 0.0028 in 1918. The mean is quite low at 0.001. The November S&P Index, suggesting a large drop in the stock market, seems to have a large

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effect on Vt . On the other hand, Wt , the wealth effect may not be affected by the November 1900 observation because this variable is calculated as holding period returns from the September Index numbers only. Indeed, the correlation between holding period returns based on Cowles and S&P Indexes is 0.9988. Next, consider the summary statistics of the annual data for the two stock market variables in Romer’s model. The one year holding period return is 5.29 percent when constructed with the Cowles Index and 5.56 percent based on the S&P Index (Table 1). The means of the stock price variability measure are also very similar (0.0011 and 0.0013). The standard deviations for the wealth effect variable are similar regardless of which index is used to construct the variables. However, stock price variability is greater when the S&P data is employed. The standard deviation for stock price variability is 0.0014 for the S&P data and 0.00079 for the Cowles data. This difference in volatility is largely caused by the observation for October 1900 to September 1901. When this observation is omitted, the indexes have nearly identical standard deviations (S&P 0.000655 and Cowles 0.00657).

Variables Durable Goods Lagged Final Goods Wt (Cowles) Wt (S&P) Vt (Cowles) Vt (S&P)

TABLE 1 Summary Statistics: 1891-1913 & 1921-28 Mean Std. Dev. Min. 0.0496 0.1439 -0.2829 0.0404 0.0614 -0.1126 0.0529 0.1720 -0.3730 0.0556 0.1746 -0.3723 0.00114 0.00079 0.00028 0.00131 0.00142 0.00026

Max. 0.4246 0.1505 0.3369 0.3378 0.0036 0.0082

Table 2 presents the correlations between the variables in the model. They seem to suggest that the index used to calculate the stock market variables matters for Vt but not for Wt . The coefficient of correlation for Cowles holding period returns and durables is 0.03. It is very close to the correlation between S&P holding period returns and durables (0.02). The correlation between the S&P variability and durables is only -0.19, compared with the correlation of -0.37 for the Cowles variability and durables.

Durable Goods Lagged Final Goods Wt (Cowles) Wt (S&P) Vt (Cowles) Vt (S&P)

Durable Goods 1 -0.316 0.034 0.024 -0.368 -0.186

TABLE 2 Correlations Wt Lagged Final Goods (Cowles) -0.316 0.034 1 -0.478 -0.466 0.044 0.007

-0.478 1 0.9988 0.052 0.209

Wt (S&P) 0.024

Vt (Cowles) -0.368

Vt (S&P) -0.186

-0.466 0.9988 1 0.043 0.203

0.044 0.052 0.043 1 0.882

0.007 0.209 0.203 0.882 1

Regression Results Based on the Cowles and S&P Indexes Table 3 contains Romer’s regression estimates for the periods 1891-1913 and 1921-28. Column 1 of the table reports the replication of her results, when the Cowles data is used to create the two stock market variables. The dependent variable is the growth rate in the consumer durable good production. The coefficient for stock price variability is -63.85, which is signiÞcant at the 5 percent level in a one-tailed test. The wealth effect coefficient is insigniÞcant at conventional levels.

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AEJ: JUNE 2004, VOL. 32, NO. 2 TABLE 3 OLS Results

Constant Lagged Durable Goods Lagged Final Goods Wt (Cowles) Wt (S&P) Vt (Cowles) Vt (S&P) R2

31 Observ. Cowles S&P 0.160 0.112 (3.36) (2.70) −0.035 0.110 (−0.09) (0.28) −0.766 −1.091 (−0.85) (−1.15) −0.100 − (−0.61) − − −0.096 − (−0.55) −63.85 − (−1.93) − − −14.43 − (−0.73) 0.24 0.15

30 Observ. Cowles S&P 0.210 0.213 (4.17) (4.22) −0.002 −0.011 (−0.006) (−0.03) −0.929 −0.881 (−1.10) (−1.05) −0.214 − (−1.32) − − −0.195 − (−1.24) −108.281 − (2.90) − − −109.54 − (−2.95) 0.35 0.36

1901 Observ. Corrected S&P 0.164 (3.42) −0.045 (−0.12) −0.736 (−0.82) − − −0.095 (−0.60) − − −66.15 (−1.99) 0.24

The next column presents the estimates when the same Wt and Vt are calculated with the S&P Index rather than the Cowles Index. The switch from the Cowles data to the S&P data reduces the importance of stock price variability. Its coefficient of -14.43 represents a drop to about one-fourth of the original estimate with the Cowles data. To illustrate the decreased effect of stock price volatility in the second estimate, consider the following. A one standard deviation increase in stock price volatility, measured with the Cowles data, results in a 5 percent decrease in consumer durable purchase. When measured with the S&P Index, it results in only a 2 percent decrease in consumer durable production. Stock price variability is insigniÞcant at standard levels (t-statistic = -0.73) in the second estimate. On the other hand, regardless of whether the Cowles or the Standard and Poor’s Index is used to calculate holding period returns, the wealth effect is statistically insigniÞcant (t-statistics of -0.61 and -0.55). In summary, the change to S&P data causes stock price variability to become statistically insigniÞcant at conventional levels and has no meaningful effect on the wealth effect. The earlier comparison of the two stock market indexes suggests that when the S&P Index is used, the observation of Vt that spans the period of October 1900 to September 1901 might be problematic. Column 3 of Table 1 contains estimates based on the Cowles Index without the 1901 observation. Results in Column 4 are from the S&P speciÞcation and with the omission of the 1901 observation. The two estimates are practically the same when the 1901 observation is excluded from the estimation model. The wealth effect coefficients of -0.195 and -0.214 both remain insigniÞcant. The stock price variability measures have larger coefficients and are very similar (-109.54 and -108.28). Indeed, a one standard deviation movement in either measure of Vt (which is now lower without the 1901 observation) decreases consumer durable production by 7 percent. This is even higher than the 5 percent increase reported using the Cowles data. Results without the 1901 data are also in line with the original Cowles estimate. This suggests that stock price variability is an important consideration.

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Year 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1921 1922 1923 1924 1925 1926 1927 1928 **Indicates

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TABLE 4 DF F IT S from the Model Estimated on the Full Sample DF F IT S from DF F IT S from Cowles SpeciÞcation S&P SpeciÞcation 0.072 0.026 -0.143 -0.055 -0.306 -0.268 -0.640** -0.571** 0.134 0.212 0.073 0.019 -0.107 -0.091 -0.061 -0.083 0.140 0.114 -0.215 -0.170 1.854** 6.106** 0.090 0.286 -0.204 -0.413 -0.145 -0.099 0.092 0.075 0.101 0.121 -0.104 -0.293 -0.368** -0.379** 0.304 0.318 0.066 0.091 -0.053 -0.108 0.032 0.102 -0.113 -0.046 -1.307** -1.129** 1.677** 1.719** 2.234** 2.100** -0.299 -0.193 0.169 0.203 0.211 0.240 -0.710** -0.597** -0.086 -0.002 value > |0.34|

The Þnal column estimates the model when the 1901 Standard and Poor’s observation is corrected. That is, for Vt , .008166 is replaced with .003607. The results are in line with the original estimates found with the Cowles data. Bounded Inßuence Estimation If one does not know that an outlier is an error, leaving out observations is not always the best way to deal with outliers. An observation may contain valuable information that is discarded in the process. Another approach is to limit the amount of inßuence of any outlier with Bounded Inßuence Estimation. Welsch [1980] suggests the following weighted least squares estimation to limit the amount of inßuence of outliers: min

X

wi (yi − βxi )2

,

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where yi is the ith observation of the dependent variable, xi are the ith observations of a vector of explanatory variables and a constant, and wi is the weight which takes the following form: ½ ¾ 1 if |DF F IT Si | ≤ 0.34 wi = 0.34 if |DF F IT Si | > 0.34 |DF F IT Si | For each observation, DF F IT S represents the change in the Þtted value of the ordinary least squares regression due to deletion of that observation and is divided by a scaling factor hi Si , where Si2 is an estimate of the variance of the regression without the ith observation. That 1 is, DF F IT S = (hi/(1 − hi )) 2 Ui , where hi is the ith diagonal element of the hat matrix. Ui is the studentized residual for observation i. Table 4 lists the DF F IT S for each observation when the model is estimated on all 31 observations. The Þrst column shows the DF F IT S for the Cowles speciÞcation of the stock market variables, and the second column shows the DF F IT S for the S&P speciÞcation of these variables. In both the Cowles and S&P speciÞcations, there are seven observations with DF F IT S over 0.34. These occur in the same seven observations in the two speciÞcations. Interestingly, the largest DF F IT S value occurs for the 1901 observation in the estimation with the S&P data. Recall that this row of data includes the November 1900 stock market data that appeared so problematic.

Constant Lagged Durable Goods Lagged Final Goods Wt (Cowles) Wt (S&P) Vt (Cowles) Vt (S&P) R2

TABLE 5 Bounded Inßuence Estimation Cowles 0.166 (4.87) 0.032 (0.12) −1.027 (−1.75) −0.099 (−0.88) − − −72.15 (−2.80) − − 0.45

S&P 0.157 (4.58) 0.067 (0.24) −1.094 (−1.78) − − −0.101 (−0.86) − − −62.72 (−2.51) 0.42

The seven observations with DF F IT S over the absolute value of 0.34 are weighted in the weighted least squares estimation of the model as shown previously. The results appear in Table 5. This robust technique that limits the inßuence of outliers provides similar estimates, regardless of whether the Cowles Indexes or the S&P Indexes are used to calculate Wt and Vt . The estimated coefficient for stock price variability is -62.72 for the S&P speciÞcation. This is almost identical to the ordinary least squares estimate with the Cowles data in Table 3. The estimated coefficient for this variable is -72.15 in the Cowles speciÞcation. A one standard deviation movement in the Cowles constructed Vt results in a decrease in consumer durable production of 5.7 percent. If the same change in volatility (0.00079) is considered with the estimates using the S&P data, the model suggests a 5 percent decline in consumer durable good production. These estimates are fairly close to each other and in line with the estimated inßuence of the OLS estimates based on the Cowles data. The t-statistics for

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stock price variability in the two estimates are both over 2.5 and signiÞcant at the 1 percent level in a one-tailed test. Recall that only the Cowles data produced signiÞcant estimates of stock price variability and even then the t-statistic was only 1.93. The robust technique increases the signiÞcance of Vt . The estimates for the wealth effect variable are not signiÞcant at conventional levels, which is the same as the previous OLS results. Robustness techniques provide estimates that are in line with the estimates that occur when the indexes have no known errors (Cowles data) or when the indexes are created with the 1901 observation corrected (S&P index). Robustness techniques make it much less likely that data errors that create outliers are driving the results. Conclusions The analysis provides evidence that choice of seemingly interchangable stock indexes can alter the estimates and thus, conclusions of a paper. This paper estimates Romer’s consumer durable model over her sample periods 1891-1913 and 1921-28. When two independent variables are calculated using the Cowles Index, as she did, then her results are replicated. When the S&P Index is used to calculate these variables, the estimate of the coefficient of the uncertainty effect is one-fourth the size of Romer’s estimate and becomes insigniÞcant in the process. Furthermore, a single monthly observation of the S&P Index is identiÞed, which is driving this result. The S&P Index for November 1900 suggested a large drop in the stock market during this month that neither the Cowles Index or the Dow Jones Industrial Average was able to pick up. The inßuential observation was due to a data error, which, when corrected, gave results in line with Romer’s original estimates. Several methods were also employed to deal with inßuential observations, which served to mitigate the inßuence of the error. When the row of data with the problematic observation were deleted and the Bounded Inßuence Estimation method was used, the model produced very similar results. This study found that the uncertainty effect is highly signiÞcant and that stock market volatility has a negative impact on consumer durable production. In addition, the results point to the dangers that inßuential observations can have in small data sets. The one monthly S&P observation is enough to produce estimates that are completely counter to what Romer found. The results send up a warning ßag to those researchers who estimate models on small data sets. In these cases, careful examination of the data and robustness estimation techniques can clearly improve the analysis. APPENDIX Dependent Variable: Durable Goods ( Yit ) Yit = [log(Yij ) − log(Yi(j−1) )]

.

Yi refers to durable goods. J refers to a calender year (January-December). The data are annual data and can be found in Shaw, Table I3 (pp. 70-7). Log refers to natural log. Independent Variables Stock Price Variability ( Vt ) Vt = {sum[log(S&Pj ) − log(S&P(j−1) )]2 }/12

.

S&Pj is the deßated monthly Standard and Poors 500 Index in month j. S&P(j−1) is the previous month’s value of the Standard and Poor’s Index. Log refers to natural log.

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The monthly difference is squared. These squared differences are summed and then averaged over 12 months. The j represents 12 months over a period of October of the previous year to September of the current year. The stock price index is deßated with the Warren and Pearson’s Wholesale Price Index, pp. 10-13. (In an alternate speciÞcation, this variable is also calculated with the Cowles Series P Index) Holding Period Return Wt = [log(S&Pj ) − log(S&P(j−1) )]

.

The one year holding period returns. S&Pj is the deßated stock market index in September of year t, while S&P(j−1) is the deßated stock market index in September of year t − 1. The stock price index is deßated with the Warren and Pearsons Wholesale Price Index, pp. 10-13. (In an alternate speciÞcation, this variable is also calculated with the Cowles Series P Index). Total Commodity Output ( Yt ) Yt = [log(Yt ) − log(Y(t−1) )]

.

Yt refers to the total Þnished commodities from Shaw, pp. 76-7. This series is annual data. Log refers to natural log. The deÞnition given here is for Yt . Y(t−1) is used in the estimated model. Footnotes 1 The following people would like to be thanked for their comments and suggestions: Kevin Grier, Chris Hanes, Bill Shughart, Dan Sutter, Robert Tollison, and the seminar series participants at the University of Mississippi. Marc Poitras provided exceptionally useful comments. The authors would especially like to thank Christina Romer for furnishing her data to them. 2 A number of studies on this issue report empirical evidence supporting the wealth effect. Examples include Fama [1981, 1990], Geske and Roll [1983], Huang and Kracaw [1984], Kaul [1987], Barro [1989, 1990], and Schwert [1990]. 3 This literature includes Romer [1990], Pindyck [1991], and Bittlingmayer [1998], who all build on Bernanke [1983].

References Barro, Robert J. “The Stock Market and the Macroeconomy: Implication of the October 1987 Crash,” in R. W. Kamphius, R. C. Kormendi, and J. W. H. Watson, eds: Black Moday and the Future of Financial Markets. Dow Jones-Irwin: Homewood, IL, 1989. –. “The Stock Market and Investment,” Review of Financial Studies, 3, 1990, pp. 115-31. Bernanke, Ben S. “Irreversibility, Uncertainty, and Cyclical Investment,” Quarterly Journal of Economics, 98, 1983, pp. 85-106. Bittlingmayer, George. “Output, Stock Volatility, and Political Uncertainty in a Natural Experiment: Germany, 1880-1940,” Journal of Finance, 43, 1998, pp. 2243-57. Coppock, Lee; Poitras, Marc. “Evaluating the Fisher Effect in Long-Term Cross-Country Averages,” International Review of Economics and Finance, 9(2), 2000, pp. 181-92. Cowles, Alfred, and Associates. Common-Stock Indexes, 2nd ed., Principia Press: Bloomington, IN, 1939. Fama, Eugene F. “Stock Returns, Real Activity, Inßation, and Money,” American Economic Review, 71, 1981, pp. 545-65. –. “Stock Returns, Expected Returns, and Real Activity,” Journal of Finance, 45, 1990, pp. 1089108.

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