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... by construction pt will differ from and be much smoother than rational prices if discount rates are constant. Further, prices appear nonstationary, which can account for the previously reported gross violations of variance bounds. Conditional variance bounds that are valid under nonstationarity are not violated for Standard and Poor's data. The results are consistent with changes in expectations of future cash flows causing changes in stock prices. I am grateful for the assistance and encouragement of my dissertation committee, Merton Miller (chairman), Time FIcG. 1.-Standard and Poor's (real) annual composite stock price index 1926-79 augmented with Cowles Commission common stock index 1871-1925 (solid line) fnd corresponding perfect-foresight series, including terminal condition pl = po. 956 JOURNAL OF POLITICAL ECONOMY 700 -600 -500-* 400-0 300 200-100 0 20 40 60 50 100 Time FiG;. 2.-Nonista tionary (geometric random wa1k) price series (solid line) and corresponding perfect-foresight series, including terminal condition PI1 = P'r Again, it seems obvious that the bound (4) is violated and that consequently the valuation model (1) is empirically untenable. However, suc-h conclusions based on figure 2 are absolutely unfounded. This figure is based not on real data but on simulated data that by construction are generated by the rational valuation model (1). The variance bound (4) is not violated, and absolutely nothing can be inferred from the plots about the validity of the model (1). This seems startling at first glance. Much of the impact of the variance bounds literature has come from the apparent clear violation of the inequality (4) by plots such as figure 1.-Indeed, it has been claimed that an inspection of these plots provides such obvious evidence against the inequality (4) and the valuation model (1) that formal empirical tests of (4) need not be relied on (see Shiller 198 la, pp. 4, 7; 1984). Tirole (1985, p. 1085) also claims: "Simply by looking at Figures 1 and 2 in Shiller [ 198 lb], this inequality [i.e., (4)1 is not satisfied." This interpretation is clearly false if plots virtually identical to figure 1 can be readily created when (1) holds by construction. More important, the price process used in figure 2 is not an unusual or artificial construct, but rather is the (geometric) random walk traditionally regarded in finance as an excellent empirical description of the price process in actual data.3 This paper examines Standard and 3 For construction details, see Sec. 114 below, particularly n. 7. Note also that the primary characteristics of time-series plots such as figs. I and 2 do not depend on the nonstationarity assumption and are present even in stationary AR(1) processes for prices, as demonstrated in Sec. II below. See Kleidon (1986) for more detail on the stationary case. VARIANCE BOUNDS TFEsTS