I employ descriptive regressions of the price of each commodity over time to examine the long-and short-run variation in prices as a whole. The results of estimating equations in the text show that contemporaries could not have predicted future prices; descriptive regressions can tell later observers what actually happened. The first such regression evaluates the long-run trend of prices. It models the relationship between each commodity’s log price and the year, year squared, year cubed, and three dummy variables for different intervals:
Log Pricei = α + β1year + β2year2 + β3year3 + β4Dum1 + β5Dum2
+ β6Dum3 + ε,
where the subscript of the log price refers to the commodity being observed. The first dummy variable, Dum1, controls for years between 323 BCE (-322) and 314 BCE (-313). This ten-year window is isolated to see if the death of Alexander had an effect on prices. The length of the window to account for any market disruptions in the wake of Alexander’s death is arbitrary. To allow for a possible longer disruption to markets, two other dummy variables, Dum2 and Dum3, control for ten-year windows from 313 BCE (-312) to 304 BCE (-303) and from 303 BCE (-302) to 294 BCE (-293). These added dummy variables are included in the regression to discover whether prices returned to their level before 323 BCE or if prices continued to be higher than normal ten years and twenty years after the initial shock. The year-squared and year-cubed variables allow the path of prices to curve over the years. Any polynomial is an approximation to the arbitrary path of prices over time; a third-order polynomial allows a reasonably good characterization of the time path. Slotsky also used a cubic equation to allow for an accurate representation of the prices’ patterns over four centuries.
It is hard to correct this equation for autocorrelation, and I have not done so. The problem is that the data come at irregular intervals while time is measured uniformly. Standard errors are incorrect as a result, and any significance tests need to be regarded as only approximate. I use a 1 percent confidence limit throughout to minimize the possibility of accepting an erroneous hypothesis. Results are shown in table 3.2.
In order to see if there was relative variation, that is, if the trends for the prices of different commodities differed from each other, I pooled the regressions for individual commodities to provide tests of significance for the trends of individual commodities. I expected market prices for agricultural commodities to have moved together since changes in supply and demand would have been similar for each crop. Yet wool might have moved differently from the agricultural commodities because the production of wool is quite distinct from raising a crop. To determine empirically whether the change of each price was significantly different from the change or other prices, I used a regression model of pooled commodity log prices that examines simultaneously the path of the six log prices over the years from 473 BCE to 72 BCE. The regression used is:
Log Price = α1 + β11 year + β12 year2 + β13 year3 + β4 Dum1 + β5Dum2 + β6Dum3+∑iδi (αi + βi1 year + βi2 year2 + βi3 year3) + ε,
where δi represents the dummy variable for the i-th commodity. Dates are the omitted commodity. Dum1, Dum2, and Dum3 are the dummy variables for the three ten-year windows from 323 BCE to 294 BCE. This regression determines the effect of time on each price. The extent to which trends of individual commodities differed is shown by the magnitude and significance of the α and β coefficients.
To capture seasonal variations in commodity log prices, it is necessary to model the relationship of seasons and years with log prices. Market prices could have shown consistent variation in seasons, while government determined prices would not have a clear pattern of variation. I incorporated dummy variables for winter, summer, and fall into the regressions for individual commodities. The dummy variable for spring (months I–III) is omitted. The seasonal regression is:
Log Pricei = α + β1 year + β2 year2 + β3 year3 + β4 Dum1 + β5Dum2
+ β6Dum3+ β7 winter + β8 summer + b9 fall + ε.
The year and commodity dummies are defined before. The dummy variables for seasons show the relative effect that each season has on prices. The scribes collected information on the height of the Euphrates River, and it can be added to or substituted for seasonal dummies. I also tested for correlation of the errors in this equation to see if there were good and bad years for agriculture as a whole.
TABLE 3.2.
Regressions of log prices on time
Although the seasonal dummies revealed a mixed pattern and there are not enough observations of the Euphrates River’s height to yield significant results, there is a way to identify good and bad years. If these are market prices, then we expect that a good year would produce bountiful harvests of most crops and therefore lower prices. A bad year, by the same reasoning, would result in high prices. As a result the errors in the individual price regressions shown in table 3.2 would be correlated. Zellner’s technique of “seemingly unrelated regressions” tests for any correlations and uses the additional information in the correlations to improve the estimates.
There are some costs to this procedure. The regression can only be done for years in which there are observations of all prices in order to calculate correlations among the residuals. The results of seemingly unrelated regressions will be more efficient, but using far fewer observations. The Breusch-Pagan test of independence of the regressions for different crops yields a chi-square with 15 degrees of freedom of 406. This is significantly different from zero at any conceivable level of significance. The errors of the equations in table 3.2 clearly are correlated; the prices behave just the way market prices of agricultural commodities in a local area are expected to act.
The resulting coefficients are almost the same as those in table 3.2. The standard errors are much larger, however, and not all coefficients are significant. This is due partly to the better estimation technique and partly to the reduced number of observations. Since the number of observations in each regression falls from over 500 to 90, the results in table 3.2 are more reliable. Of course, the trends shown in table 3.2 should be taken as purely descriptive rather than as evidence of an underlying price-formation process (Temin, 2002).