equality in (7.2.0). This bias can be substantial: Will) p = 0.997. for example, it equals 3b/ 7 when ф = 0.9, 73/ 7 when ф = 0.95, and 171/7 \vhcii0 = 0.98.

л Л second problem is that the asymptotic theory given in the Appendix nay be misleading in (mite samples when the hori/.оп К is large relative t ) the sample size. Hodrick (1992) and Nelson and Kim (1993) use Monte (jarlo methods lo illustrate this for the case where stock returns are regressed dn dividend-price ratios. Richardson and Stock (1989) show that the linile-sjmiplc properties of regressions with large A. can be accounted for using ait alternative asymptotic theory in which К increases asymptotically at the same i jtlc as the sample size. Section 2.5.1 of Chapter 2 discusses the application ojf their theory to univariate regressions of returns on past returns.

j One way to avoid ibis problem is lo transform ihe basic regression so that il no longer has overlapping residuals. This has been proposed by Jfgadccsh (1991) for the Fama and French (1988b) regression of returns oln lagged returns, and it has been advocated more generally by Cochrane

(jl99l). For example, we might estimate

I V+t = y(A)(.v, + + а-,+ 1-л-)4-1 +.а-. (7.2.7)

wjherc ihe error term к,и.к is 1U)W serially uncorrelated. The numerator of the regression coefficient y(K) in (7.2.7) is the same as the numerator of the regression coefficient fi(K) in (7.2.2), because the covariance of x measured at one dale and / measured at another date depends only on the difference between the two dales. Hence у (A.-) = 0 in (7.2.7) if and only if fi(K) = 0 in (7.2.2). However it does not necessarily follow dial lesls of у (A ) = 0 and/)(&) = 0 have the same asymptotic properties .under the null or general alternative hypotheses. Hodrick (1992) presents Monte Carlo evidence on the distributions of both kinds of test statistics; he finds that they both tend to reject die null loo often if asymptotic critical values arc used, so lhat the long-horizon /-statistics reported in Tables 7.1 and 7.2 should be treated with caution. However he also finds that these biases are not strong enough to account for the evidence of return predictability reported in the tables.

An important unresolved question is whether there are circumstances under which long-horizon regressions have greater power to delect deviations from the null hypothesis than do short-horizon regressions. Hodrick (1992) and Mark (1995) present some suggestive Monte Carlo evidence thai this may he the case, and Campbell (1993b) also studies the issue, but the literature has not reached any firm conclusion at this stage.

Similar biases alllirl the rcgicssiun when lite horizon is greater titan one period. See I lixltit k (l!HI2) ami Matk (l!l.l.r>) fur M.....e Cat In evidence <m litis point.

7.2.2 Volatility Tests

In the previous section we have explored regressions whose dependent variables are returns measured over long horizons. One motivation for such regressions is that asset prices arc influenced by expectations of returns into the distant future, so long-horizon procedures are necessary if we arc lo understand price behavior. Wc now turn lo empirical work that looks at price variability more directly.

I.eRoyand Porter (1981) and Shillci (1981) stalled a heated debate in the early 1980s by arguing that slock prices are too volatile to he rational forecasts of future dividends discounted al a constant rate. This controversy has since died down, partly because it is now mote clearly understood that a rejection of constant-discount-rale models is not the same as a rejection of the Efficient Markets Hypothesis, and partly because regression tests have convinced many financial economists thai expected slock returns are time-varying rather than constant. Nonetheless the volatility literature has introduced some important ideas thai are closely connected with the work on multiperiod return regressions discussed in the previous section. Useful surveys of this literature include Gilles and I.cRoy (1991), l.cRoy (1989), Shillcr (1989, Chapter 4), and West (1988a).

The early papers in the volatility literature used levels of slock prices and dividends, but here we restate the ideas in logarithmic form. This is consistent with the more recent literature and with the exposition in the rest of this chapter. Wc begin by defining a log peiject-joresiglit stock price,

I, = YLp[{[ -/>) V<-M, + k~ г]. (7.2.8)

The perfect-foresight price /;* is so named because from the ex post slock price identity (7.1.21) it is the price that would prevail if realized returns were constant al some level r, dial is, if there were no revisions in expectations driving unexpected returns. Equivalemly, from the ex ante stock price identity (7.1.22) it is the price that would prevail if expected returns were constant and investors had perfect knowledge of future dividends. Substituting (7.2.8) into (7.1.21), we find that

P* - Pi = ХУ - ) (7.2.9)

>=

The difference between p* and p, is just a discounted .sum of future demeaned slock returns.

II wc now take expectations and use the definition given in (7.1.22) and (7.1.23) of the price component / , we find that

li[/,*] - pi = p - j- = p - F.l/i ]. (7.2.10)

I - p

/. lleseiit-Vulue Relations

Recall dial \i can be interpreted as thai component oilhe slock price which is associated with changing expectations of future slock relurns. Thus the conditional expectation of p\ - measures the effect of changing expected slock relurns on the ( uncut stock price. In the AR(l) example developed earlier, the conditional expectation of/,; - is just .v,/( 1 -рф) from (7.1.29).

It expected slock returns arc constant through time, then die right-hand side ol (7.2.1()) is /его. The conslanl-expected-relurn hypothesis implies lhat /;* - p, is a forecasl error uncorrelated with information known al time I. Equivalent, ii implies that the stock price is a rational expectation of the perfect-foresight stock price:

l>i = V-i\l>\. (7.2.11)

I low can these ideas be used to test the hypothesis that expected stock returns are constant? For simplicity of exposition, we begin by making two unrealistic assumptions: fust, that log slock prices and dividends follow stationary stochastic processes, so that they have well-defined fust and second moments; and second, that log dividends are observable into the infinite Inline, so dial (he perfect-foresight price p* is observable lo the econome-trician. Below we discuss how these assumptions are relaxed.

Orthogonality tin/l Variaiice-Boiiiiih les(\

Equation (7.2.1 I) implies thai /> - p, is orthogonal lo information variables known at time /. Ли orthogonality test of (7.2.11) regresses / * - p, onlo information variables and tests for /его coefficients. If the inforinalio.i variables include the slock price p, itself, this is equivalent to a regression of />, onto p, and oilier variables, where die hypothesis to be tested is now that />, has a unit coefficient and the other variables have zero coefficients. These regressions are variants of the long-horizon return regressions discussed in the previous section. Equation (7.2.9) shows lhat p* - p, is just a discounted sum ol future demeaned stock returns, so an orthogonality test of (7.2.11) is a return regression with an inliniie horizon, where more distant relurns are geometrically dowiiweighted.1-

Instead of testing orthogonality directly, much of the literature tests the implications of orthogonality for the volatility of stock prices. The most famous such implication, derived by I.eRoy and Porter (1981) and Shiller (1981), is the variance inequality for the slock price:

Var/i,* = Var/;,+Var/,*-/>,J > Var]/ ]. (7.2.12)

- I In- ilowuwcigliiiiie, allows ilie It- slaiisiii in iln- regression lo lie positive, wlirreas we

showed in Sn lion 7.LM llial tin- /,- sialism in a......weighted linite-h.iiwim return recession

converges ю /em as ilie lioii/on им .eases. Ihirlanlanil Hall (I.iwi), Seoll (I.iki), anil Shiller (I.WI, ( :llapler I I ) liave Hill regressions ol litis soil.

7.2. lresent-Value Illations and US Stock l*rice Behavior 277 j

The equality in (7.2.12) holds because under the null hypothesis (7.2.11) p* - pi must be uncorrelated with p, so no covariance term appears in the variance of/;*; the variance inequality follows directly. Equation (7.2.12) can also be understood by noting that an optimal forecast cannot be more variable than the quantity it is forecasting. With constant expected returns ihe stock price forecasts only the present value of future dividends, so it cannot be more variable than the realized present value of future dividends. Tests of this and related propositions are known as variance-bounds tests.

As Duiiauf and Phillips (1988) point out, variance-bounds tests can be restated as orthogonality tests. To see this, consider a regression of pt on pi - pi. This is the reverse of the regression considered above, but it too should have a zero coefficient under the null hypothesis. The reverse regression coefficient is always t9 = Cov[p - p p,]/Var[p* - />,]. It is straightforward to show that

Var-Vart ] Var[/>, - p,}

so the variance inequality (7.2.12) will be satisfied whenever the reverse regression coefficient 9 > -1/2. This is a weaker restriction than the orthogonality condition в = 0, so the orthogonality test clearly has power in some situations where the variance-bounds test has none. The justification for using a variance-bounds test is not increased power; rather it is that a variance-bounds test helps one to describe the way in which the null hypothesis fails.

Unit Roots

Our analysis so far has assumed that the population variances of log prices and dividends exist. This will not be the case if log dividends follow a unit-root process; then, as Kleidon (1986) points out, the sample variances of prices and dividends can be very misleading. Marsh and Merton (1986) provide a particularly neat example. Suppose that expected stock returns are constant, so the null hypothesis is true. Suppose also that a firms managers use its stock price as an indicator of permanent earnings, setting the firms dividend equal to a constant fraction of its stock price last period. In log form, wc have

4+t = &+p (7.2.14)

where there is a unique constants that satisfies the null hypothesis (7.2.11). It can be shown that both log dividends and log prices follow unit-root processes in this example. Substituting (7.2.14) into (7.2.8), we find that the perfect-foresight stock price is related to the actual stock price by

p, = (l-P)£p/>,+;. (7.2.15)

This is just a smoothed version of the actual stock price p so its variance depends on the variance and autocorrelations of/;,. Since autocorrelations can never be greater than one, p] must have a lower variance than />,. The importance of this result is not that it applies to population variances (which are not welt defined in this example because both log prices and log div- idends have unit roots), but that it applies to sample variances in every simple. Thus the variance inequality (7.2.12) will always be violated in the (Varsh-Mcrton example.

This unit-root problem is important, but it is also easy to circumvent. Tjhc variable p* - p, is always stationary provided that slock returns arc stationary, so any test that p* - p, is orthogonal lo stationary variables will be well-behaved. The problems pointed out by Klcidon (1986) and Marsh and Mcrton (1986) arise when p - p, is regressed on the stock price /; which has aunit root. These problems can be avoided by using unit-root regression theory or by choosing a stationary regressor, such as the log dividend-price ratio. Some other ways lo ileal with the unit-root problem are explored in Problem l.b.

Finite-Sample Considerations

So far wc have treated the perfect-foresight stock price as if it were an ol>-scrvable variable. But as defined in (7.2.8), the perfect-foresight price is unobservable in a finite sample because il is a discounted sum of dividends out to the infinite future. The definition of p* implies that

t~t~l

Р, = (1 - P) £ M+i+y + A - r) + р7--Уг. (7.2.16)

Given data up through lime 7 the first term on the right-hand side of (7.2.16) is observable but the second term is not.

Following Shillcr (1981), one standard response to this difficulty is to replace the unobservable p by an observable proxy pr thai uses only in-sample information:

ptT = (\-p) ]Г рми-, + Л~г) + р--/ , (7.2.17)

>=<>

1 lere the terminal value of the actual stock price, pr, is used in place of the terminal value of the perfect-foresight stock price, />*-.4 Several points are

Diiilatif and Hall (1.Ш) apply unit-root tegrexsion theory, while Campbell and Shillcr (1988a,b) replace the log slock price with the log dividend-price ratio. Problem 7.!> is based on the work of Mankiw, Romer, and Shapiro (1985) and West (1988b).

,4Shiller (1981) used the sample average price instead of the end-of-sample price in his terminal condition, but later work, including Shillcr (I9H9). (ollows the approach discussed her].

worth noting about ihe variable p r. First, if expected returns are constant, (7.2.11) continues lo hold when p* r is substituted for p*. Thus tests of the conslant-expecled-rclurn model can use p* r. Second, a rational bubble in the stock price will affect both pi and p*,. Thus lests using p , include bnl>-blcs in the null rather than the alternative hypothesis. Third, the difference P i ~ li Cim he written as a discounted sum of demeaned stock returns, with the sum terminating at the end of the sample period / rather than al some fixed horizon from the present date I. Thus orthogonality tests using Pi.t ~ li arc )llsl long-horizon return regressions, where future returns are geometrically discounted and the horizon is the end of the sample period.

As one might expect, the asymptotic, theory for statistical inference in orthogonality and variance-bounds tests is essentially the same as the theory used to conduct statistical inference in long-horizon return regressions.- As always, in finite samples it is important lo look at the effective order of overlap (that is, the number of periods in (7.2.9) during which discounted future returns make a nonnegligiblc contribution to todays value of/-/ -/),). If this is large relative lo the sample size, then asymptotic theory is unlikely to be a reliable guide for statistical inference.

Flavin (1983) gives a particularly clear intuition for why this might he a problem in ihe context of variance-bounds tests. She points out that whenever a sample variance around a sample mean is used lo estimate a population variance, there is some downward bias caused by the fact that the true mean of the process is unknown. When the process is white noise, it is well-known that this bias can be corrected by dividing the sum of squares by Г - 1 instead of 7 . Unfortunately, the downward bias is mote severe for serially correlated processes (intuitively, there is a smaller number of effective observations for these processes), so this correction becomes inadequate in the presence of serial correlation. Now ji , is more highly serially cot related than /; since p* T changes only as dividends (hop out of the present-value formula and discount factors are updated, while p, is affected by new information about dividends. Thus the ralio of the sample variance of p у to the .sample variance of p, is downward-biased, and this can cause the variance inequality in (7.2.12) to be violated too often in finite .samples. From the equivalence ol variance-bounds and orthogonality tests, die same problem arises in a regression context.

7.2. 3 Vector Autorrjiessive Methods

The methods discussed in the previous two sections have the common feature that they try U> look directly at long-horizon properties of the data. This

I.eRoy and Stcigci tvald (I!I!I2) use Mottle Carlo methods to study the power of orthogonality and variance-bounds tests.