money illusion in the stock market: the …personal.lse.ac.uk/polk/research/csinffinal.pdffrom money...

MONEY ILLUSION IN THE STOCK MARKETTHE MODIGLIANI-COHN HYPOTHESIS

RANDOLPH B COHEN

CHRISTOPHER POLK

TUOMO VUOLTEENAHO

Modigliani and Cohn hypothesize that the stock market suffers from moneyillusion discounting real cash flows at nominal discount rates While previousresearch has focused on the pricing of the aggregate stock market relative toTreasury bills the money-illusion hypothesis also has implications for the pricingof risky stocks relative to safe stocks Simultaneously examining the pricing ofTreasury bills safe stocks and risky stocks allows us to distinguish moneyillusion from any change in the attitudes of investors toward risk Our empiricalresults support the hypothesis that the stock market suffers from money illusion

I INTRODUCTION

Do people suffer from money illusion confusing nominaldollar values with real purchasing power When the differencebetween real and nominal quantities is small and stakes arerelatively low equating the nominal dollar amounts with realvalues provides a convenient and effective rule of thumb There-fore it seems plausible that people often ignore the rate of infla-tion in processing information for relatively small decisions1

Modigliani and Cohn [1979] hypothesize that stock marketinvestors may also suffer from a particular form of money illu-sion incorrectly discounting real cash flows with nominal dis-count rates An implication of such an error is that time variationin the level of inflation causes the marketrsquos subjective expectationof the future equity premium to deviate systematically from the

An earlier draft of the paper was circulated under the title ldquoHow InflationIllusion Killed the CAPMrdquo We would like to thank Clifford Asness John Camp-bell Edward Glaeser Jussi Keppo Stefan Nagel Andrei Shleifer Jeremy Steinand three anonymous referees for helpful comments

1 The term ldquomoney illusionrdquo was coined by John Maynard Keynes early inthe twentieth century In 1928 Irving Fisher gave the subject a thorough treat-ment in his book The Money Illusion Since then numerous papers have describedimplications of money illusion to test for its existence The most widely discussedof these implications is stickiness in wages and prices (see Gordon [1983] for areview of the evidence on this topic) Although money illusion can exist even in theabsence of inflation inflation is central to most money illusion stories Fisher andModigliani [1978] catalog the ways in which inflation could affect the real econ-omy with money illusion as one important source of real effects Shafir Diamondand Tversky [1997] examine in detail potential effects of money illusion andpresent evidence on these effects along with a theory of the psychological under-pinnings of the illusion

copy 2005 by the President and Fellows of Harvard College and the Massachusetts Institute ofTechnologyThe Quarterly Journal of Economics May 2005

639

rational expectation Thus when inflation is high (low) the ra-tional equity-premium expectation is higher (lower) than themarketrsquos subjective expectation and the stock market is under-valued (overvalued) The claim that stock market investors sufferfrom money illusion is a particularly intriguing and controversialproposition as the stakes in the stock market are obviously veryhigh

Nevertheless recent time-series evidence suggests that thestock market does suffer from money illusion of Modigliani andCohnrsquos variety Sharpe [2002] and Asness [2000] find that stockdividend and earnings yields are highly correlated with nominalbond yields Since stocks are claims to cash flows from real capitaland inflation is the main driver of nominal interest rates thiscorrelation makes little sense a point made recently by Ritterand Warr [2002] Asness [2003] and Campbell and Vuolteenaho[2004] These aggregate studies suffer from one serious weak-ness however Inflation may be correlated with investorsrsquo atti-tudes toward risk which directly influence stock prices even ifinvestors do not suffer from money illusion To the extent thatthese aggregate studies fail to fully control for risk the resultsmay confound the impact of risk attitudes and money illusion

Our novel tests explore the cross-sectional asset-pricing im-plications of the Modigliani-Cohn money-illusion hypothesis Si-multaneously examining the pricing of Treasury bills safe stocksand risky stocks allows us to distinguish money illusion fromchanging attitudes of investors toward risk The key insight un-derlying our tests is that money illusion will have a symmetriceffect on all stocksrsquo yields regardless of their exposure to system-atic risk In contrast the impact of investor risk attitudes on astockrsquos yield will be proportional to the stockrsquos risk as riskystocksrsquo yields will be affected much more than safe stocksrsquo yieldswill be This insight allows us to cleanly separate the two com-peting effects

Specifically we assume that investors use the logic of theSharpe-Lintner capital asset pricing model (CAPM) [Sharpe1964 Lintner 1965] to measure the riskiness of a stock and todetermine its required risk premium According to the CAPM astockrsquos beta with the market is its sole relevant risk measure Inthe absence of money illusion (and other investor irrationalities)the Sharpe-Lintner CAPM predicts that the risk compensationfor one unit of beta among stocks which is also called the slope ofthe security market line is always equal to the rationally ex-

640 QUARTERLY JOURNAL OF ECONOMICS

pected premium of the market portfolio of stocks over short-termbills For example if a risky stock has a beta of 15 and therationally expected equity premium is 4 percent then that stockshould have a rationally expected return of the Treasury-bill yieldplus 6 percent Conversely a safe stock with a beta of 05 shouldonly earn a 2 percent premium over Treasury bills and the riskystock will therefore return a premium of 4 percent over the safestock

The joint hypothesis of money illusion and the CAPM offersa sharp quantitative prediction We show that money illusionimplies that when inflation is low or negative the compensationfor one unit of beta among stocks is larger (and the securitymarket line steeper) than the rationally expected equity pre-mium Conversely when inflation is high the compensation forone unit of beta among stocks is lower (and the security marketline shallower) than what the overall pricing of stocks relative tobills would suggest Suppose that in our above example highinflation leads money-illusioned investors who still demand a 4percent equity premium to undervalue the stock market to theextent that the rational expectation of the equity premium be-comes 7 percent Then these investors will price the risky stock toyield only a 4 percent return premium over the safe stock Con-sequently when inflation is high the average realized equitypremium (7 percent) will be higher than the average returnpremium of the risky stock over the safe stock (4 percent)

Our empirical tests support this hypothesis First as anillustration we sort the months in our 1927ndash2001 sample intoquartiles based on lagged inflation and examine the pricing ofbeta-sorted portfolios in these quartiles The slope of the solid linein Figure I denotes the price of risk implied by the pricing of theoverall stock market relative to that of short-term bills ie theequity premium that a rational investor should have expectedThe dashed line is the security market line the slope of which isthe price of risk implied by the pricing of high-risk stocks relativeto that of low-risk stocks As predicted by the money-illusionhypothesis the figure shows that during months that are pre-ceded by inflation in the lowest quartile of our sample the rela-tion between average returns and CAPM betas is steeper thanthe slope predicted by the Sharpe-Lintner CAPM and no moneyillusion Conversely during months that are preceded by inflationin the highest quartile of our sample the security market line

641MONEY ILLUSION IN THE STOCK MARKET

estimated from the cross section of beta-sorted portfolios is muchshallower than the expected equity premium

Second we introduce a new method for estimating the excessslope and excess intercept of the security market line amongstocks relative to the predictions of the Sharpe-Lintner CAPMOur statistical test combines Fama-MacBeth [1973] cross-sec-tional and Black-Jensen-Scholes [1972] time-series regressions tosolve for the excess slope and excess intercept as a function of thebetas and conditional alphas from the time-series regressionrsquosparameters The idea behind this statistical test is exactly the

FIGURE IAverage Excess Returns and Beta in Different Inflation Environments

We first create ten portfolios by sorting stocks on their past estimated betas Wethen record the excess returns on these portfolios Next we sort months in our192706ndash200112 sample into four groups based on lagged inflation (defined as thesmoothed change in the producer price index) For each group we then estimatethe postformation betas and average excess returns The average annualizedexcess returns ( y-axis) and betas ( x-axis) of these portfolios form the graphs Thesolid line (drawn from the [00] to [1 average marketrsquos excess return in thissubsample]) is the relation predicted by the Sharpe-Lintner CAPM The dashedline is the fitted line computed by regressing the average returns on betas in eachsubsample


same as the one illustrated in Figure I but allows for a conve-nient and powerful statistical hypothesis test Our tests indicatethat the excess intercept of the security market line comovespositively and the excess slope negatively with inflation as pre-dicted by the Modigliani-Cohn money-illusion hypothesis

At first it may seem incredible that stock market investorswith trillions of dollars at stake make such a pedestrian mistakeFortunately or perhaps unfortunately we need not look anyfurther than to the leading practitioner model of equity valuationthe so-called ldquoFed modelrdquo2 to find corroborating evidence of stockmarket investors falling prey to money illusion The Fed modelrelates the yield on stocks to the yield on nominal bonds Practi-tioners argue that the bond yield plus a risk premium defines aldquonormalrdquo yield on stocks and that the actual stock yield tends torevert to this normal yield Consistent with this practitionerargument Sharpe [2002] Asness [2000] and Campbell and Vuol-teenaho [2004] find that the Fed model is quite successful as anempirical description of aggregate stock pricesmdashprices are set asif the market used the Fed model to price stocks Logicallyhowever the Fed model is on weak grounds as it is based onprecisely the money-illusion error noted by Modigliani and Cohn

Even if most stock-market investors confuse nominal andreal quantities could a small number of wealthy and rationalarbitrageurs still eliminate any potential mispricing We believethat rational arbitrageurs would be very conservative in accom-modating supply and demand due to money illusion The Sharperatio (the expected excess return divided by the standard devia-tion of excess return) of a bet against the money-illusion crowd islikely to be relatively low because one can only make a single betat a time and because the mispricing may be corrected veryslowly This potential slow correction of mispricing is a particu-larly important limiting factor of arbitrage as any attempt tocorrect the inflation-related mispricing exposes the arbitrageur tothe uncertain development of the stock marketrsquos fundamentalsMispricing that corrects slowly necessarily requires long holdingperiods for arbitrage positions along with significant exposure tovolatility as the variance of fundamental risk grows linearly intime In fact if a rational arbitrageur had bet against moneyillusion by buying stocks on margin in the early 1970s his profits

2 Despite this name the model has absolutely no official or special statuswithin the Federal Reserve system


would have been negative for more than a decade As Modiglianiand Cohn noted in 1979 ldquoOn the other hand those experts ofrational valuation who could correctly assess the extent of theundervaluation of equities had they acted on their assessment inthe hope of acquiring riches would have more than likely endedup with substantial lossesrdquo In summary mispricing caused bymoney illusion has precisely those characteristics that Shleiferand Vishny [1997] suggest effectively prevent arbitrage activity

II MONEY ILLUSION AND ITS IMPLICATIONS

IIA Modigliani and Cohnrsquos Money-Illusion Hypothesis

The correct application of the present-value formula dis-counts nominal cash flows at nominal discount rates or real cashflows at real discount rates Modigliani and Cohn [1979] proposethat stock market investors but not bond market investors suf-fer from money illusion effectively discounting real cash flows atnominal rates

What mechanism could cause the bond market to correctlyreflect inflation while the stock market suffers from money illu-sion According to the Modigliani-Cohn hypothesis money illu-sion is due to the difficulty of estimating long-term future growthrates of cash flows Consider an investor who thinks in nominalterms Since nominal bonds have cash flows that are constant inthose terms estimating a growth rate for bonds is not difficult Incontrast the task of estimating the long-term expected cash-flowgrowth for stocks is far from trivial

For example suppose that this investor erroneously assumesthat long-term earnings and dividend growth are constant innominal terms and uses all past historical data to estimate along-term growth rate for a stock Of course a more reasonableassumption would be that expected long-term growth is constantin real terms If expected long-term growth is constant in realterms yet the investor expects it to be constant in nominal termsthen in equilibrium stocks will be undervalued when inflation ishigh and overvalued when inflation is low

The basic intuition of the Modigliani-Cohn hypothesis caneasily be captured by examining a money-illusioned investorrsquosapproach to stock valuation Consider the classic ldquoGordon growthmodelrdquo [Williams 1938 Gordon 1962] that equates the dividend-


price ratio with the difference between the discount rate andexpected growth

(1) DtPt1 R G

where R is the long-term discount rate and G is the long-termgrowth rate of dividends R and G can be either both in nominalterms or both in real terms but the Gordon growth model doesnot allow mixing and matching nominal and real variables If theexpected returns are constant the discount rate is exactly equalto the expected return on the asset If conditional expected re-turns vary over time however the discount rate is only approxi-mately equal to the long-horizon expected holding period returnon the asset

The Gordon growth model can also be thought of in terms ofthe investorrsquos first-order condition If an investor is at the opti-mum portfolio allocation then the discount rate or expected re-turn R on stocks must equal the yield on bonds plus a premiumdue to the higher covariance of stock returns with the investorrsquosconsumption If an otherwise optimizing investor suffers frommoney illusion of Modigliani and Cohnrsquos variety then he thinks ofR in nominal terms and expects G to be constant in nominalterms If the inflation is time varying however the assumption ofconstant nominal G does not make any sense as it would imply awildly variable real G In real terms there is no obvious reasonwhy either R or G should change mechanically with expectedinflation if the consumer is rational3

If stock market investors suffer from money illusion andexpect constant long-term growth in nominal terms what willhappen when inflation rises Higher nominal interest rates re-sulting from inflation are then used by stock market participantsto discount unchanged expectations of future nominal dividendsThe dividend-price ratio moves with the nominal bond yield be-cause stock market investors irrationally fail to adjust the nomi-nal growth rate G to match the nominal discount rate R Fromthe perspective of a rational investor stocks are thus underval-ued when inflation is high and overvalued when inflation is lowA single small rational investor facing a market populated bymoney-illusioned investors would then tilt his portfolio toward

3 Some business-cycle dynamic (such as Famarsquos [1981] proxy hypothesis)might create a correlation between inflation and either near-term discount ratesor near-term growth rates However such movements are a priori unlikely tomove long-term discount rates or growth rates much


stocks when inflation is high and away from stocks when inflationis low so that the equilibrium risk premium of stocks would bejustified by stock returnsrsquo covariance with his consumption

To adapt the notation to conform with our subsequent em-pirical tests first subtract the riskless interest rate from both thediscount rate and the growth rate of dividends We define theexcess discount rate as Re R Rf and the excess dividendgrowth rate as Ge G Rf where all quantities should again beeither nominal or real As we are considering the possibility thatsome investors are irrational we follow Campbell and Vuol-teenaho [2004] and distinguish between the subjective expecta-tions of irrational investors (superscript SUBJ) and the objectiveexpectations of rational investors (superscript OBJ)

As long as irrational investors simply use the present valueformula with an erroneous expected growth rate or discount rateboth sets of expectations must obey the Gordon growth model

(2) DP ReOBJ GeOBJ ReSUBJ GeSUBJ

GeOBJ ReSUBJ GeOBJ GeSUBJ

In words the dividend yield has three components (1) the nega-tive of objectively expected excess dividend growth (2) the sub-jective risk premium expected by irrational investors and (3) amispricing term due to a divergence between the objective (ierational) and subjective (ie irrational) growth forecast ε GeOBJ GeSUBJ Notice that mispricing ε is specified in termsof excess yield with ε 0 indicating overpricing and ε 0underpricing Notice also that the Gordon growth model requiresthat the expectational error in long-term growth rates GeOBJ GeSUBJ be equal to the expectational error in long-term expectedreturns ReOBJ ReSUBJ

Campbell and Vuolteenaho [2004] formalize the Modigliani-Cohn money-illusion story by specifying that mispricing or expec-tational error is a linear function of past smoothed inflation

(3) ε GeOBJ GeSUBJ ReOBJ ReSUBJ 0 1

where is the expected inflation and 1 0 If one takes theModigliani-Cohn hypothesis literally one could argue that 1 1ie inflation is (irrationally) fully priced into stock yields Thecase in which 1 1 is consistent with the simple form of moneyillusion in which investors assume that future expected cash-flowgrowth is constant in nominal terms


IIB Cross-Sectional Implications of Money Illusion

While previous research has tested the aggregate time-seriespredictions of the Modigliani-Cohn money-illusion hypothesisthe cross-sectional implications of this hypothesis have beenlargely unexplored in either the literature on behavioral financetheory or the empirical literature in general (The main exceptionis Ritter and Warrrsquos [2002] study which examines the differentialimpact of inflation on a firmrsquos stock price as a function of itsfinancial leverage) We fill this gap in the literature by developingand testing cross-sectional predictions resulting from the originalModigliani-Cohn hypothesis

We base our cross-sectional predictions on three substantiveassumptions First we assume that the market suffers frommoney illusion of the type described by equation (3) Second weassume that the market makes no other type of systematic mis-take in valuing stocks Together these two assumptions implythat equation (3) holds not only for the market but also for eachindividual stock

(4) εi GieOBJ Gi

eSUBJ RieOBJ Ri

eSUBJ 0 1

An important result of these assumptions is that money illu-sionrsquos influence on mispricing is equal across stocks ie εi εM 0 1

Our final assumption is that investors behave according tomodern portfolio theory in evaluating risks that is they use theSharpe-Lintner CAPM to set required risk premiums This im-plies that the slope of the relation between the subjective returnexpectation on an asset and that assetrsquos CAPM beta is equal tothe subjective market premium

(5) RieSUBJ iRM

eSUBJ

This is in contrast with the usual rational-expectations specifi-cation of the CAPM Ri

eOBJ iRMeOBJ Note that we implicitly

assume that betas are known constants so that subjective andobjective expectations of betas are thus equal

These assumptions allow us to derive the cross-sectionalimplication of the Modigliani-Cohn [1979] money-illusion hypothe-sis Substituting the subjective Sharpe-Lintner CAPM into (4)yields

(6) εi RieOBJ iRM

eSUBJ


Recognizing that market mispricing εM equals the wedge be-tween objective and subjective market premiums results in

(7) εi RieOBJ iRM

eOBJ εM13

N iOBJ Ri

eOBJ iRMeOBJ εi iεM

Above iOBJ is an objective measure of relative mispricing called

Jensenrsquos [1968] alpha in the finance literature Since mispricingfor both the market and stock i is equal to the same linearfunction of expected inflation 0 1 we can write

(8) iOBJ 0 1 i0 1

Equation (8) predicts that the (conditional) Jensenrsquos alpha of astock is a linear function of inflation the stockrsquos beta and theinteraction between inflation and the stockrsquos beta If the marketsuffers from money illusion then when inflation is high a rationalinvestor would perceive a positive alpha for low-beta stocks and anegative alpha for high-beta stocks Conversely when inflation islow (or negative) a rational expectation of a stockrsquos alpha isnegative for low-beta stocks and positive for high-beta stocks

Recall that the security market line is the linear relationbetween a stockrsquos average return and its beta Equivalentlyequation (8) states that both the intercept and the slope of theobserved security market line deviate systematically from therational-expectation Sharpe-Lintner CAPMrsquos prediction More-over this deviation is a function of inflation Define the excessslope of the security market line as the cross-sectional slope of(objective) alpha on beta Define the excess intercept of the secu-rity market line as the (objective) alpha of a unit-investmentstock portfolio that has a zero beta Equation (8) predicts that theexcess intercept of the security market line equals 0 1 andthe excess slope equals (0 1) under the joint hypothesis ofmoney illusion and the Sharpe-Lintner CAPM

The above reasoning assumes that prices are exclusively setby investors who suffer from money illusion What happens ifsome investors suffer from money illusion while other investorsdo not and the two groups interact in the market In the Appen-dix we describe a very stylized equilibrium model in which afraction of the risk-bearing capacity in the market suffers frommoney illusion This stylized model gives an intuitive predictionthe excess slope of the security market line is determined by the


product of inflation and the fraction of the marketrsquos risk-bearingcapacity controlled by money-illusioned investors

The above hypotheses tie in closely with recent research onequity-premium predictability and inflation A paper by PolkThompson and Vuolteenaho [forthcoming] assumes that theCAPM holds in terms of investorsrsquo subjective expectations anduses the relative prices of high and low beta stocks to derive anestimate of the subjective equity premium Polk Thompson andVuolteenaho find that this estimate correlates well with proxiesfor the objective equity premium such as the dividend yield andalso has predictive power for the future equity premium Themajor exception to their finding occurs in the early 1980s whentheir subjective equity premium measure is low but the dividendyield as well as the subsequent aggregate stock market return ishigh It is noteworthy that this period was also the peak of U Sinflation

Campbell and Vuolteenaho [2004] assume the validity ofPolk Thompson and Vuolteenahorsquos [2004] measure of the sub-jective equity premium Campbell and Vuolteenaho combine thismeasure with the Gordon growth model for the aggregate marketto estimate the subjectively expected growth rate of aggregatecash flows It appears that inflation drives a wedge between thesubjective and objective estimates of aggregate growth just aspredicted by the Modigliani-Cohn hypothesis

In contrast we essentially circle back to ask how moneyillusion affects the objective validity of the CAPM Even if inves-tors subjectively use the CAPM does the CAPM describe thepattern of objective returns in the cross section The answer isthat there should be an objective security market line but it canbe steeper or flatter than the prediction of the Sharpe-LintnerCAPM ie the rational expectation of the equity premium

III EMPIRICAL METHODOLOGY AND RESULTS

Our main tests examine time variation in the excess inter-cept and slope of the security market line and the relation of thistime variation to inflation Our estimation strategy is the follow-ing First we construct dynamic stock portfolios that are likely toshow a large and consistent cross-sectional spread in their CAPMbetas The natural way to construct such portfolios is to sortstocks into portfolios each month on their past estimated stock-


level betas We record the returns on these value-weight portfo-lios which become our basis assets

Specifically we generate our basis asset returns from theCenter for Research in Securities Prices (CRSP) monthly stockfile which provides monthly prices shares outstanding divi-dends and returns for available NYSE AMEX and NASDAQstocks We measure betas it for individual stocks using at leastone and up to three years of monthly returns in a market-modelOLS regression on a constant and the contemporaneous return onthe value-weight NYSE-AMEX-NASDAQ portfolio4 As we some-times estimate beta using only twelve returns we censor eachfirmrsquos individual monthly return to the range (50 percent 100percent) in order to limit the influence of extreme firm-specificoutliers We use these stock-level estimates to form beta-sortedportfolios The portfolios are value-weight and re-formed eachmonth using the most recent available betas We consider sortsinto 10 20 and 40 portfolios The results are not sensitive to thenumber of portfolios and we thus concentrate on the twenty-portfolio data set for most tests These portfolio-return seriesspan the 895-month period 192706ndash200112

Second we estimate rolling betas on these 20 beta-sortedportfolios using a trailing window of 36 months (We have repli-cated our results using 24- and 48-month beta-estimation win-dows and the results are robust to variation in window length)We denote the time series of these rolling betas as the ldquopostfor-mation betasrdquo of the basis assets These postformation beta seriesspan the 860-month period 193005ndash200112

At this stage of the analysis it is important to verify that ourstock-level beta estimates are actually useful and result in cross-sectional spread in the average postformation betas We find thatthey are as the average postformation beta of the lowest betaportfolio is 063 while the postformation beta of the highest betaportfolio is 177 However the estimated postformation betas fora particular portfolio are not constant through time For thelowest beta portfolio the postformation beta varies from 035 to192 while the highest beta portfoliorsquos postformation beta variesfrom 059 to 363 Of course most of this time-series variation inthe postformation betas is simply due to sampling variation

Third we form two portfolios from these 20 basis assets using

4 We skip those months in which a firm is missing returns However werequire all observations to occur within a four-year window


Fama and Macbethrsquos [1973] cross-sectional regression techniqueThe purpose of this step is to directly control for the time varia-tion in postformation betas documented above Specifically foreach cross section we regress the future excess return on the 20basis assets on a constant and the portfoliosrsquo trailing-windowpostformation beta As shown by Fama and Macbeth the timeseries of these cross-sectional regression coefficients are excessreturns on portfolios as well

(9) rinterceptte

rslopete 1

t1131

t113

11

t113r te

Above 1 is a vector of constants and

t1 a vector of postforma-tion betas of beta-sorted portfolios estimated using a trailingwindow that ends at t 1 r t

e is the vector of excess returns on thebeta-sorted portfolios

We present the regression coefficients in matrix notation inequation (9) to highlight the fact that the cross-sectional regres-sion coefficients are portfolios As long as the trailing postforma-tion betas are accurate forecasts of future postformation betasthe intercept portfolio return will be the excess return on aunit-investment zero-beta stock portfolio and the slope portfolioreturn will be the excess return on a unit-beta zero-investmentportfolio Furthermore these portfolio strategies are implement-able as long as the explanatory variables (ie the betas) areknown in advance of the dependent variables (ie the basis-assetexcess returns) The intercept and slope portfolio have averagereturns of 44 and 19 basis points per month respectively thoughonly the intercept portfoliorsquos mean return is statistically signifi-cantly different from zero These two excess-return series spanthe 859-month period 193006ndash200112

Though the steps taken so far are complicated these compli-cations are justified as they will produce two portfolio returnseries with relatively constant precisely measured betas of zeroand one for the intercept and slope portfolios respectively This isdesirable as the time-series regressions in the next stage criti-cally require that the portfolios we use have constant betas

Fourth we regress the intercept and slope portfoliorsquos excessreturns on a constant the contemporaneous market excess re-turn and lagged inflation As above we use the value-weightNYSE-AMEX-NASDAQ portfolio as our proxy of the market port-folio The excess return is computed by subtracting the three-


month Treasury-bill rate from CRSP Our measure of inflation isthe series used by Campbell and Vuolteenaho [2004] in theirstudy investigating aggregate market valuations and inflationWe first compute log growth rates on the producer price index Asthese growth rates are very noisy especially in the first part of oursample we smooth these log growth rates by taking an exponen-tially weighted moving average with a half-life of 36 months (iemonthly decay to the power of 09806) Note that the exponen-tially weighted moving averages use trailing inflation data sothere is no look-ahead bias in our smoothing We also demean thisinflation series using its full sample mean in order that thesubsequent regression parameters are easier for the reader tointerpret

The two time-series regressions (10) are analogous to BlackJensen and Scholes [1972] and Gibbons Ross and Shanken[1989] time-series regressions with time-varying Jensenrsquos [1968]alphas

(10) rinterceptte a1 b1rMt

e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t

The empirical estimates of the two regression equations in (10)show that both portfolios have very precisely measured betasTable I shows that for our preferred specification (20 beta-sortedportfolios where postformation betas are estimated using a 36-month trailing window) the intercept portfolio has a beta of00041 with a t-statistic of 014 while the slope portfolio has abeta of 10205 with a t-statistic of 3438 We also find that theconditional alpha of the intercept portfolio varies positively withlagged inflation as the estimate of c1 is 150 with a t-statistic of241 Our estimate of c2 is reliably negative (value of 148t-statistic of 235) indicating that inflation tracks the condi-tional alpha of the slope portfolio in an opposite fashion

Because of our novel methodology we now have identifiedtwo portfolios with relatively stable betas If we could be confidentthat the trailing-window postformation beta estimates are perfectforecasts of the future basis-asset betas the excess intercept andexcess slope of the security market line would be given by a1 c1t1 and a2 c2t1 In that hypothetical case the time-series regression coefficients b1 and b2 would be exactly equal tozero and one Despite the usefulness of our new approach real-istically speaking the trailing-window betas we use as inputs of


the Fama-MacBeth stage will never be perfect forecasts of futurebetas there is no guarantee that b1 0 and b2 1 exactly Sincethe point estimates are always close for the basis assets weconsider our method is informative enough to allow us to simplymodify the formulas for the conditional excess intercept andexcess slope of the security market line to take these small de-viations into account

As we can confidently reject the hypotheses that b2 0 andb2 b1 for all sets of basis assets straightforward algebraprovides the alphas of a zero-beta and a unit-beta stock portfolioimplied by the estimates of equation (10) The functions that mapestimates of the parameters in regression (10) into the parame-ters of equation (8) are as follows The excess slope of the securitymarket line is

g0 g1t1

(11) g0 a2b2

g1 c2b2

The excess intercept of the security market line is given by thefunction

TABLE ITIME-SERIES REGRESSIONS OF INTERCEPT AND SLOPE PORTFOLIOS

K N a1 a2 b1 b2 c1 c2 Rintercept2 Rslope

2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)

The table shows OLS regressions of the intercept portfoliorsquos (rinterceptte ) and the slope portfoliorsquos (rslopet

e )excess return on a constant contemporaneous excess market return (rMt

e ) and demeaned lagged inflation(t1)

rinterceptte a1 b1rMt

e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t

The intercept and slope portfolios are constructed using Fama-Macbeth [1973] regressions of excessreturns on N beta sorted portfolios on a constant and the portfoliosrsquo lagged K-month postformation betast-statistics are in parentheses R2 is adjusted for degrees of freedom The regressions are estimated fromthe sample period 193006 ndash200112 859 monthly observations


h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2

To summarize these two formulas are the result of solving for theconditional alpha of a zero-beta and a unit-beta portfolio impliedby estimates of system (10)

It is important to note that equations (11) and (12) alsoprovide a correction for any potential measurement error problemcaused by the use of estimated betas at the Fama-Macbeth stageEven if betas are estimated with error in earlier stages our finalestimates of the excess slope and the excess intercept of thesecurity market line are consistent

Table II reports the point estimates of the excess slope of thesecurity market line We focus on the specification using 20 port-folios and a 36-month beta-estimation window in the Fama-Mac-Beth stage but as the table shows the results are robust to small

TABLE IIEXCESS INTERCEPT AND SLOPE OF THE SECURITY MARKET LINE

K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]

The table shows the estimated function that maps inflation into the excess slope and intercept of thesecurity market line First we regress the intercept portfoliorsquos (rinterceptt

e ) and the slope portfoliorsquos (rslopete )

excess return on a constant contemporaneous excess market return (rMte ) and lagged inflation (t1)


e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t

The intercept and slope portfolios are constructed using Fama-Macbeth [1973] regressions of excessreturns on N beta sorted portfolios on a constant and the portfoliosrsquo lagged K-month postformation betasSecond we compute the functions that map the regression parameters to the excess slope and interceptof the security market line The excess slope is defined as g0 g1t1 where g0 a2b2 and g1 c2b2 Theexcess intercept is computed as h0 h1t1 where h0 a1 a2b1b2 and h1 c1 c2b1b2 t-statisticscomputed using the delta method are in parentheses We also report the test statistic and the two-sidedp-values [in brackets] for the hypotheses that [g1h1] [00] and g1 h1 0 The regressions areestimated from the sample period 193006 ndash200112 859 monthly observations


variations in these choices Increasing the number of basis-assetportfolios in the tests typically strengthens our results

We estimate g0 as 00009 with a t-statistic of 040 and h0as 00008 with a t-statistic of 036 The interpretation of thesenear-zero intercept estimates is that when inflation is at itsmean the empirical beta slope and the zero-beta rate amongstocks are consistent with the prediction of the Sharpe-LintnerCAPM In other words when inflation is at its time-series aver-age the Sharpe-Lintner CAPM ldquoworksrdquo This is consistent with aform of money illusion in which people use historical averagenominal growth rates to value the stock market ignoring thecurrent level of inflation which may be very different from infla-tionrsquos historical average

Our estimate for g1 is 14487 with a t-statistic of 235 Aspredicted by the Modigliani-Cohn hypothesis the excess slope ofthe security market line comoves negatively with inflation Ourpoint estimates for the excess-intercept function are also consis-tent with the predictions of the theory the estimate of h1 is15108 with a t-statistic of 240 which is statistically significantlydifferent from zero but not from one In words we can reject thehypothesis that the market does not suffer from money illusionbut we cannot reject the hypothesis that inflation is (irrationally)fully priced into real stock yields Furthermore g1 is economicallyand statistically very close to h1 as predicted Finally we canreject the joint hypothesis that both g1 0 and h1 0 againstthe two-sided alternative at the 5 percent level of significance

IIIA Additional Robustness Checks

Our results are not sensitive to small variations in the infla-tion measure For example all of our conclusions remain valid ifwe use as our measure of inflation the fitted value from a regres-sion of monthly (unsmoothed) inflation on its lagged value thethree-month Treasury-bill yield and the ten-year Treasury-bondyield

We have also replicated our results with expanded sets ofbasis assets presented in Table III The first panel uses 20beta-sorted and 10 size-sorted portfolios as basis assets Thesecond panel uses 20 beta-sorted and 10 book-to-market-sortedportfolios as basis assets The third and final panel uses 20beta-sorted 10 size-sorted and 10 book-to-market-sorted portfo-lios as basis assets The size-sorted and book-to-market-sortedportfolios are provided by Kenneth French on his Web site Add-


ing these characteristics-sorted portfolios to the set of basis assetsdoes not alter our basic conclusions as the point estimates re-main close to those obtained in the earlier tests Thus we arguethat our main conclusions are not sensitive to small changes inthe set of basis assets

In unreported tests we also examine the Modigliani-Cohnhypothesis using long-horizon returns We use the same portfo-lios as in our previous tests except we hold the stocks for horizonsranging from 3 to 60 months Our market return is also com-pounded in the same way and then the compounded three-monthTreasury-bill interest rate is subtracted Smoothed inflation isscaled to the same time units as the returns Other than thechange in the holding period the test procedure is exactly thesame as in the previous tests We find point estimates consistent

TABLE IIIRESULTS FROM EXPANDED ASSET SETS

20 beta-sorted and 10 ME-sorted portfoliosK g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]

20 beta-sorted and 10 BEME-sorted portfoliosK g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]

20 beta-sorted 10 ME-sorted and 10 BEME-sorted portfoliosK g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]

The table shows the estimated function that maps inflation into the excess slope and intercept of the securitymarket line estimated from expanded asset sets First we regress the intercept portfoliorsquos (rinterceptt

e ) and the slopeportfoliorsquos (rslopet

e ) excess return on a constant contemporaneous excess market return (rMte ) and lagged inflation

(t1)rinterceptt

e a1 b1rMte c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t

The intercept and slope portfolios are constructed using Fama-Macbeth [1973] regressions of excessreturns on basis-asset portfolios on a constant and the portfoliosrsquo lagged K-month postformation betasSecond we compute the functions that map the regression parameters to the excess slope and interceptof the security market line The excess slope is defined as g0 g1t1 where g0 a2b2 and g1 c2b2 Theexcess intercept is computed as h0 h1t1 where h0 a1 a2b1b2 and h1 c1 c2b1b2 t-statisticscomputed using the delta method are in parentheses We also report the test statistic and the two-sidedp-values [in brackets] for the hypotheses that [g1h1] [00] and g1 h1 0 The regressions areestimated from the sample period 193006 ndash200112 859 monthly observations


with the joint hypothesis of money illusion and the CAPM at thequarterly horizons and at horizons of three years and five yearsHowever for intermediate horizons (12ndash24 months) any effect issmall with point estimates occasionally having the wrong signThough unfortunate the low power and large standard errors ofthese long-horizon tests are at least partially to blame as theModigliani-Cohn hypothesis is never rejected statistically

As part of our long-horizon tests we also check to seewhether our point estimates of the cross-sectional effect of moneyillusion are consistent with the aggregate mispricing of stocksversus bonds by Campbell and Vuolteenaho [2004] In particularwe estimate a regression forecasting the excess market returnwith smoothed inflation while controlling for the subjective risk-premium measure SRC of Polk Thompson and Vuolteenaho[2004] As predicted by the Modigliani-Cohn hypothesis the par-tial regression coefficient on inflation is positive significant andsimilar to our short-horizon cross-sectional estimate at allhorizons

Though we find evidence of Modigliani and Cohnrsquos moneyillusion our tests so far have only considered the Sharpe-Lintnerversion of the CAPM However it is theoretically possible thatour results are simply due to an incorrect restriction on theintercept of the security market line implicit in that version of theCAPM

Black [1972] considers the possibility that investors cannotborrow at the Treasury-bill rate If so the likely effect of suchinability to borrow is that the zero-beta rate among stocks devi-ates from the Treasury-bill rate In other words the Black CAPMallows the excess intercept and slope of the security market lineto be nonzero Therefore an alternative explanation for our find-ings is that the spread between the true borrowing rate facinginvestors and the Treasury-bill rate comoves with inflation

Fortunately for our conclusions data on actual borrowingrates indicate that the spread does not comove positively withinflation Our three empirical proxies for the true borrowing rateare car-loan rates from commercial banks personal-loan ratesfrom commercial banks and credit-card interest rates We obtainthese quarterly data from the Federal Reserversquos Web site Thedata from commercial banks begin 197202 while the credit-cardrate data begin 199411 We first compute the yield spread be-tween these loans and maturity-matched Treasury yields We


then regress these spreads on smoothed inflation (in the sameannualized units)

The regression results in Table IV show that the yield spreadbetween individualsrsquo borrowing rates and Treasury rates comovesnegatively not positively with lagged inflation This result is notsurprising as Ausubel [1991] finds that credit-card interest ratesappear ldquostickyrdquo in responding to changes in market interest ratesThus we reject the Black CAPM as an alternative explanation forthe observed time-variation in the excess slope of the securitymarket line

We also consider subjective risk premiums determined in aworld where multiple risk factors determine the cross section ofsubjective expected returns That is we assume a world in whichinvestors mistakenly misestimate real cash-flow growth of (andthus expected returns on) all stocks due to money illusion butotherwise price stocks correctly in accordance with a multifactormodel Furthermore we assume that measured betas are notmaterially affected by this mispricing

TABLE IVINFLATION AND THE SPREAD BETWEEN BORROWING AND TREASURY RATES

48-month car loans from commercial banks spread over the 48-month T-note yieldconstant (t-statistic) slope on (t-statistic) Adj R2 N

34327 (139) 00489 (09) 001 124

24-month personal loans from commercial banks spread over the 24-monthT-note yield

constant (t-statistic) slope on (t-statistic) Adj R2 N

92135 (294) 05596 (79) 058 124

Credit card accounts (interest rates) spread over the 90-day T-bill yieldconstant (t-statistic) slope on (t-statistic) Adj R2 N

114532 (228) 04270 (20) 015 33

Credit card accounts (assessed interest) spread over the 90-day T-bill yieldconstant (t-statistic) slope on (t-statistic) Adj R2 N

111719 (211) 04547 (20) 012 33

The table regresses proxies for the spread between borrowing rates that individuals face and Treasuryrates on lagged inflation The inflation series () is the smoothed inflation used in earlier tests annualizedby multiplying the series by twelve The t-statistics are based on Newey-West standard errors computedusing four lags and leads The heading of each panel specifies the spread measure being used as thedependent variable Data are quarterly


In our robustness checks below we employ the well-knownthree-factor model of Fama and French [1993] but the stepsbelow will easily generalize to any multifactor model for whichthe additional factors are expressed as long-short stock port-folios The equations given above for the Sharpe-LintnerCAPM case and therefore the regressions we will run to testthe model and the Modigliani-Cohn hypothesis easily gener-alize to this case We begin by replacing equation (5) with themultifactor beta representation of assetsrsquo subjective riskpremiums

(13) RieSUBJ iRM

eSUBJ i f

f is a column vector of factor realizations for the given period andi is a column vector of asset irsquos multiple-regression loadings onthose factors Here we assume that the factor-mimicking portfo-lios are long and short stocks in equal dollar amounts Underthese conditions there is no need for SUBJ superscripts as theinflation-related mispricing affects the yields of all stocks identi-cally so that the expected return of any long-short stock portfoliois unaffected Thus

(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri

eOBJ iRMeOBJ i f εi iεM

iOBJ denotes the Jensenrsquos alpha relative to the multifactor

model and is almost identical to the expression derived in theCAPM case except that is a multifactor sensitivity on themarket return

(16) iOBJ 0 1 i0 1

In the Fama-MacBeth regressions we now include as explana-tory variables the estimated loadings on all three factors includ-ing multifactor market betas Let the additional nonmarket fac-tor loadings be denoted by

t1 where

t1 has one row for each

asset and one column for each nonmarket factor The returns onthe intercept and (all) slope portfolios are then given by


(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e

rinterceptte represents the return (in excess of the riskless rate) on a

portfolio anticipated to have zero loadings on all factors (includingthe market) and a unit net investment in stocks rslopet

e which isdefined as the first element of rallslopest

e is the return on a portfolioanticipated to have a unit market loading and a zero loading on theother factors The remaining elements of rallslopest

e are returns onportfolios with unit loadings on the other factors they are not usedin our subsequent analysis

The actual factor loadings of the rinterceptte and rslopet

e port-folios are again reasonably close to their hypothetical values Weobserve this by regressing the time series of returns on thefactors as well as on our inflation variable


e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t

ft is a vector of factor realizations at time t B1 and B2 areregression coefficients on the nonmarket factors As above inorder to estimate the slope of the security market line we need toadjust the intercept and slope portfolios slightly to get portfoliosthat (in sample) actually have the necessary loadings Again theprocess of cleaning out any extraneous loadings on other factorsconveniently leaves us with security market line equations thatare virtually identical to those in the CAPM case (except that theb1 and b2 now come from the regression that includes the otherfactors (ie they are multifactor betas) The excess slope andexcess intercept of the security market line are again given byequations (11) and (12)

Table V contains our estimates for the Fama and French[1993] multifactor model which contains two factors in additionto the market factor The factor series are provided by KennethFrench on his Web site The first is SMB the difference betweenthe return on small and big market-capitalization stocks Thesecond is HML the difference between the return on high andlow book-to-market ratio stocks In Table V we find that theestimated g1 is close to 1 the estimated h1 is close to 1 and thetwo are close to equal in absolute value but opposite in sign just


as predicted by the Modigliani and Cohn hypothesis (and just aswe found using the CAPM as the risk model) For our preferredspecification (36 months in postformation loading regressions 20test asset portfolios) we obtain point estimates of 128 for g1(t-statistic of 175) and 128 for h1 (t-statistic of 171) The testsusing the multifactor model have less power but we can stillreject at the 10 percent level the hypothesis that inflation playsno role in the determination of the cross-sectional beta premiumThe results for other specifications are qualitatively similar ascan be seen in Table V

TABLE VRESULTS FOR THE FAMA-FRENCH THREE-FACTOR MODEL

20 beta-sorted portfoliosK g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]

The table repeats the tests of Table II using the Fama-French [1993] three-factor model First we regressthe excess returns on the basis-asset portfolios on a constant and the portfoliosrsquo lagged K-month postforma-tion factor loadings The intercept portfoliorsquos (rinterceptt

e ) and the slope portfoliorsquos (rslopete ) excess returns are

the coefficient time series corresponding to the intercept and the three-factor modelrsquos market loadingrespectively Second we regress these returns on a constant contemporaneous factor returns and laggedinflation (t1)


e b12rSMBte b13rHMLt

e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t

The excess slope is defined as g0 g1t1 where g0 a2b21 and g1 c2b21 The excess intercept isdefined as h0 h1t1 where h0 a1 a2b11b21 and h1 c1 c2b11b21 t-statistics computed using thedelta method are in parentheses We also report the test statistic and the two-sided p-values [in brackets]for the hypotheses that [g1h1] [0 0] and g1h1 0 The regressions are estimated from the sampleperiod 193006 ndash200112 859 monthly observations


IV CONCLUSIONS

Do people suffer from money illusion While we may disagreeon the answer to this question its importance is indisputableMany important decisions such as choosing between buying andrenting a home or allocating onersquos portfolio between stocks andnominal bonds depend critically on the decision-makerrsquos abilityto distinguish between nominal and real quantities

Modigliani and Cohn [1979] suggest that stock market inves-tors suffer from money illusion Consistent with this hypothesisprevious time-series studies have found that high inflation coin-cides with low prices for stocks relative to bonds This observedrelation may be caused by money illusion however as noted byCampbell and Vuolteenaho [2004] and others it may also becaused by the real discount rates used by investors being posi-tively correlated with inflation

We present novel cross-sectional evidence supportingModigliani and Cohnrsquos hypothesis Simultaneously examining thefuture returns of Treasury bills safe stocks and risky stocksallows us to distinguish money illusion from any change in theattitudes of investors toward risk The key insight underlying ourtests is that money illusion will have a constant additive effect onall stocksrsquo future returns regardless of their exposure to system-atic risk This constant effect is in contrast to the impact ofinvestor risk attitudes on future stock returns which is propor-tional to the stockrsquos risk as risky stocksrsquo future returns will beaffected much more than safe stocksrsquo future returns Our empiri-cal tests indicate that when inflation is high (low) stock returnsare higher (lower) than justified by an amount that is constantacross stocks irrespective of the riskiness of the particular stock

A critical assumption in our tests is that investors use theSharpe-Lintner capital asset pricing model (CAPM) to evaluatethe risk of a stock Our cross-sectional tests leverage thisSharpe-Lintner CAPM assumption by noting that any effect ofrisk attitudes on a stockrsquos price would have to result in anexpected-return effect that is proportional to the stockrsquos CAPMbeta Consequently one may never completely rule out thepossibility of our results being due to a misspecified model ofrisk

In addition to suggesting that stock market investors sufferfrom money illusion our results offer a partial explanation for theSharpe-Lintner CAPMrsquos poor empirical performance in recent


samples In an influential paper Fama and French [1992] fail tofind support for ldquothe central prediction of [the CAPM] that aver-age returns are positively related to market [betas]rdquo Curiouslythis negative result is primarily driven by their 1951ndash1960 and1981ndash1990 subsamples both of which were preceded by highinflation The cross-sectional implication of the Modigliani-Cohnhypothesis is that the slope of average returns on beta should bemuch lower than the equity premium in precisely those sub-samples In a sense money illusion may have killed the Sharpe-Lintner CAPM

Although we do not explicitly consider money illusionrsquos effecton investor welfare we believe that our results may neverthelesshave some policy implications however speculative First if in-vestors suffer from money illusion stable and low inflation islikely to result in a less mispriced stock market than volatile andhigh inflation To the extent that real investment decisions areinfluenced by stock market (mis)valuations one would expect lowand stable inflation to be beneficial to society5 Second if govern-ment borrowing shifts from nominal bonds to inflation-indexed orreal bonds it is possible that the stock market will value stocksrelative to real (instead of nominal) bonds eliminating the effectof money illusion on stock prices Third and most importantly tothe extent that investors perceive a benefit from valuing stocksusing nominal quantities they should pay more attention toexpected inflation when forecasting future nominal cash flows

APPENDIX

In this Appendix we present a stylized model of marketequilibrium where some investors suffer from money illusion andothers do not The model gives a simple prediction the differencebetween the equity premium and the cross-sectional beta pre-mium (ie the slope of the security market line) is equal toc(1 ) where is the rate of inflation and c is a measure ofthe risk-bearing capacity of money-illusioned investors relative to

5 Dow and Gorton [1997] model the connection between stock market effi-ciency and economic efficiency Stein [1996] focuses on the link between marketinefficiency and firmsrsquo real investment policies modeling how an inefficient cap-ital market can result in managers catering to market mispricing Polk andSapienza [2004] document catering effects in firmsrsquo capital expenditures related tomispricing


that of rational investors On the one hand as expected if money-illusioned investors have all of the risk-bearing capacity in theeconomy inflation translates almost one-for-one into the excessslope of the security market line On the other hand also asexpected if rational investors have all of the risk-bearing capac-ity then the Sharpe-Lintner CAPM holds For intermediatecases the excess slope of the security market line depends lin-early on the share of risk-bearing capacity of the two investorgroups

The model has two periods denoted by 0 and 1 At time 0the investors trade but do not consume At time 1 investorsconsume the payoffs from their portfolios and there is notrading

There are three traded assets one nominal bond (asset in-dexed 1) in zero net supply and two stocks (assets indexed 2 and3) both with one share outstanding The prices of these assets attime 0 are p1 p2 and p3 Because we have no consumption attime 0 asset prices have meaning only in relative terms We setp1 1 as an arbitrary normalization interpreting all relative tothe nominal bondrsquos price

The asset payoffs are as follows The nominal debt has nodefault risk The rate of inflation is known and the nominaldebt is thus risk-free in real terms as well The real payoff to thedebt is X1 F(1 ) where F is the nominal face value of thebond Set F 1 without any loss of generality and thus the realdebt payoff is X1 1(1 )

The second asset is the stock of a relatively safe companyand the third asset is a relatively risky companyrsquos stock The realpayoffs to the second asset and the third asset are X2 and X3 Forsimplicity we make the expected payoffs for both assets equalthat is E(X2) E(X3) X The uncertain future values of realassets are independent random variables with known variancesvar(X2) 2 and var(X3) k2 k 1 In particular these realpayoffs do not depend on the rate of inflation The real payoff tothe market portfolio of all assets is thus X2 X3 and the realreturn on the market portfolio is (X2 X3)(p2 p3) Thenominal return on the market portfolio is (1 )(X2 X3)(p2 p3)

Suppose that two investors (or groups of investors) de-noted by A and B have mean-variance preferences over time 1


consumption and behave competitively as price takers Ini-tially the assets are endowed evenly between the two groups ofinvestors Both investors maximize a mean-variance objectivefunction but in addition investor B suffers from moneyillusion

Investor A has an absolute risk-bearing capacity of (1 c)0 c 1 Investor A does not suffer from money illusionperceives the above real payoffs correctly and maximizes themean-variance preference of time 1 consumption

(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2

subject to the budget constraint 5( p2 p3) w1 p2w2 p3w3 by choosing his portfolio allocations w The first-order con-ditions yield demand curves for risky assets

(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2

The idea that investor B suffers from money illusion mani-fests itself in two ways First investor B maximizes mean-vari-ance preferences over nominal wealth Second investor B be-lieves that the nominal growth in the value of corporate assets isa random variable with a distribution that does not depend on therate of inflation In other words investor B perceives nominalpayoffs XB1 F 1 XB2 X2 and XB3 X3 irrespective of theinflation environment We set the risk-bearing capacity of inves-tor class B to c which has the advantage of keeping the risk-bearing capacity of the economy as a whole constant as c changesThus investor B maximizes

(21) wB1XB1 wB2EXB2 wB3EXB3 12c wB2

2 varXB2

12c wB3

2 varXB3 wB1 wB2 wB3X 12c wB2

2 2 12c wB3

2 k2


subject to the budget constraint 5( p2 p3) wA1 p2wA2 p3wA3 by choosing his portfolio allocations The first-order condi-tions yield demand curves for risky assets

(22) wB2 cX 2 cp22

wB3 cX 2 cp32

In the equilibrium the asset market clears and one share ofboth assets must be held by both investors

(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2

Solving for prices yields

(24) p2 1

1 c X 213

p3 1

1 c X k213

These prices makes sense if money-illusioned investors dominatethe market (ie c 1) then the price of a real asset in relationto the price of a nominal asset does not change with the rate ofinflation

The expected return premium of asset 3 over that of asset 2is

(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2

The return on the market portfolio is (X2 X3p2 p3) and theCAPM betas of the two stocks are

(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2

The difference between the betas of the two assets is


(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213

The expected equity premium or the expected return on themarket portfolio less that on a risk-free asset is

(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11

Finally the slope of the security market line (the premium for oneunit of beta exposure among stocks) is

(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213

The difference between the beta premium and the equity pre-mium simplifies to

(30) 1 c

1

32

11

c

1

In words the excess slope of the security market line (c(1 )) is determined by the product of inflation and the fraction ofthe marketrsquos risk-bearing capacity controlled by money-illusionedinvestors

HARVARD BUSINESS SCHOOL

KELLOGG SCHOOL OF MANAGEMENT NORTHWESTERN UNIVERSITY

HARVARD UNIVERSITY AND NATIONAL BUREAU OF ECONOMIC RESEARCH

REFERENCES

Asness Clifford ldquoStocks versus Bonds Explaining the Equity Risk PremiumrdquoFinancial Analysts Journal LVI (2000) 96ndash113

mdashmdash ldquoFight the Fed Model The Relationship between Future Returns and Stockand Bond Market Yieldsrdquo Journal of Portfolio Management XXX (2003)11ndash24

Ausubel Lawrence M ldquoThe Failure of Competition in the Credit Card MarketrdquoAmerican Economic Review LXXXI (1991) 50ndash81

Black Fischer ldquoCapital Market Equilibrium with Restricted Borrowingrdquo Journalof Business XLV (1972) 444ndash454

Black Fischer Michael C Jensen and Myron Scholes ldquoThe Capital Asset Pricing


Model Some Empirical Testsrdquo in Studies in the Theory of Capital MarketsMichael Jensen ed (New York NY Praeger 1972)

Campbell John Y and Tuomo Vuolteenaho ldquoInflation Illusion and Stock PricesrdquoAmerican Economic Review XCIV (2004) 19ndash23

Dow James and Gary Gorton ldquoStock Market Efficiency and Economic EfficiencyIs There a Connectionrdquo Journal of Finance LII (1997) 1087ndash1129

Fama Eugene F ldquoStock Returns Real Activity Inflation and Moneyrdquo AmericanEconomic Review LXXI (1981) 545ndash565

Fama Eugene F and Kenneth R French ldquoThe Cross-Section of Expected StockReturnsrdquo Journal of Finance XLVII (1992) 427ndash467

Fama Eugene F and James D MacBeth ldquoRisk Return and Equilibrium Em-pirical Testsrdquo Journal of Political Economy LXXXI (1973) 607ndash636

Fischer Stanley and Franco Modigliani ldquoTowards an Understanding of the RealEffects and Costs of Inflationrdquo Weltwirtschaftliches Archiv CXIV (1978)810ndash833

Fisher Irving The Money Illusion (New York NY Adelphi 1928)Gibbons Michael Steven A Ross and Jay Shanken ldquoA Test of the Efficiency of

a Given Portfoliordquo Econometrica XCIX (1989) 1121ndash1152Gordon Myron The Investment Financing and Valuation of the Corporation

(Homewood IL Irwin 1962)Gordon Robert J ldquoA Century of Evidence on Wage and Price Stickiness in the

United States the United Kingdom and Japanrdquo in Macroeconomics Pricesand Quantities James Tobin ed (Washington DC Brookings Institution1983)

Jensen Michael C ldquoThe Performance of Mutual Funds in the Period 1945ndash1964rdquoJournal of Finance XXIII (1968) 389ndash416

Lintner John ldquoThe Valuation of Risk Assets and the Selection of Risky Invest-ments in Stock Portfolios and Capital Budgetsrdquo Review of Economics andStatistics XLVII (1965) 13ndash37

Modigliani Franco and Richard Cohn ldquoInflation Rational Valuation and theMarketrdquo Financial Analysts Journal XXXV (1979) 24ndash44

Polk Christopher and Paola Sapienza ldquoThe Real Effects of Investor SentimentrdquoNorthwestern University October 2004 photocopy

Polk Christopher Samuel Thompson and Tuomo Vuolteenaho ldquoCross-SectionalForecasts of the Equity Premiumrdquo Journal of Financial Economicsforthcoming

Ritter Jay R and Richard S Warr ldquoThe Decline of Inflation and the Bull Marketof 1982ndash1999rdquo Journal of Financial and Quantitative Analysis XXXVII(2002) 29ndash61

Shafir Eldar Peter A Diamond and Amos Tversky ldquoOn Money Illusionrdquo Quar-terly Journal of Economics CXII (1997) 341ndash374

Sharpe Steven A ldquoReexamining Stock Valuation and Inflation The Implicationsof Analystsrsquo Earnings Forecastsrdquo Review of Economics and StatisticsLXXXIV (2002) 632ndash648

Sharpe William ldquoCapital Asset Prices A Theory of Market Equilibrium underConditions of Riskrdquo Journal of Finance XIX (1964) 425ndash442

Shleifer Andrei and Robert W Vishny ldquoThe Limits of Arbitragerdquo Journal ofFinance LII (1997) 35ndash55

Stein Jeremy ldquoRational Capital Budgeting in an Irrational Worldrdquo Journal ofBusiness LXIX (1996) 429ndash455

Williams John Burr The Theory of Investment Value (Cambridge MA HarvardUniversity Press 1938)


rational expectation Thus when inflation is high (low) the ra-tional equity-premium expectation is higher (lower) than themarketrsquos subjective expectation and the stock market is under-valued (overvalued) The claim that stock market investors sufferfrom money illusion is a particularly intriguing and controversialproposition as the stakes in the stock market are obviously veryhigh

Nevertheless recent time-series evidence suggests that thestock market does suffer from money illusion of Modigliani andCohnrsquos variety Sharpe [2002] and Asness [2000] find that stockdividend and earnings yields are highly correlated with nominalbond yields Since stocks are claims to cash flows from real capitaland inflation is the main driver of nominal interest rates thiscorrelation makes little sense a point made recently by Ritterand Warr [2002] Asness [2003] and Campbell and Vuolteenaho[2004] These aggregate studies suffer from one serious weak-ness however Inflation may be correlated with investorsrsquo atti-tudes toward risk which directly influence stock prices even ifinvestors do not suffer from money illusion To the extent thatthese aggregate studies fail to fully control for risk the resultsmay confound the impact of risk attitudes and money illusion

Our novel tests explore the cross-sectional asset-pricing im-plications of the Modigliani-Cohn money-illusion hypothesis Si-multaneously examining the pricing of Treasury bills safe stocksand risky stocks allows us to distinguish money illusion fromchanging attitudes of investors toward risk The key insight un-derlying our tests is that money illusion will have a symmetriceffect on all stocksrsquo yields regardless of their exposure to system-atic risk In contrast the impact of investor risk attitudes on astockrsquos yield will be proportional to the stockrsquos risk as riskystocksrsquo yields will be affected much more than safe stocksrsquo yieldswill be This insight allows us to cleanly separate the two com-peting effects

Specifically we assume that investors use the logic of theSharpe-Lintner capital asset pricing model (CAPM) [Sharpe1964 Lintner 1965] to measure the riskiness of a stock and todetermine its required risk premium According to the CAPM astockrsquos beta with the market is its sole relevant risk measure Inthe absence of money illusion (and other investor irrationalities)the Sharpe-Lintner CAPM predicts that the risk compensationfor one unit of beta among stocks which is also called the slope ofthe security market line is always equal to the rationally ex-

























(1) DtPt1 R G



















(4) εi GieOBJ Gi

eSUBJ RieOBJ Ri

eSUBJ 0 1



(5) RieSUBJ iRM

eSUBJ





(6) εi RieOBJ iRM

eSUBJ



(7) εi RieOBJ iRM

eOBJ εM13

N iOBJ Ri




(8) iOBJ 0 1 i0 1




















(9) rinterceptte

rslopete 1

t1131

t113

11

t113r te











e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t






g0 g1t1

(11) g0 a2b2

g1 c2b2




2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t



h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES






















































(1) DtPt1 R G



















(4) εi GieOBJ Gi

eSUBJ RieOBJ Ri

eSUBJ 0 1



(5) RieSUBJ iRM

eSUBJ





(6) εi RieOBJ iRM

eSUBJ



(7) εi RieOBJ iRM

eOBJ εM13

N iOBJ Ri




(8) iOBJ 0 1 i0 1




















(9) rinterceptte

rslopete 1

t1131

t113

11

t113r te











e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t






g0 g1t1

(11) g0 a2b2

g1 c2b2




2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t



h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES












































(4) εi GieOBJ Gi

eSUBJ RieOBJ Ri

eSUBJ 0 1



(5) RieSUBJ iRM

eSUBJ





(6) εi RieOBJ iRM

eSUBJ



(7) εi RieOBJ iRM

eOBJ εM13

N iOBJ Ri




(8) iOBJ 0 1 i0 1




















(9) rinterceptte

rslopete 1

t1131

t113

11

t113r te











e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t






g0 g1t1

(11) g0 a2b2

g1 c2b2




2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t



h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES
































(7) εi RieOBJ iRM

eOBJ εM13

N iOBJ Ri




(8) iOBJ 0 1 i0 1




















(9) rinterceptte

rslopete 1

t1131

t113

11

t113r te











e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t






g0 g1t1

(11) g0 a2b2

g1 c2b2




2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t



h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES














































(9) rinterceptte

rslopete 1

t1131

t113

11

t113r te











e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t






g0 g1t1

(11) g0 a2b2

g1 c2b2




2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t



h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES


































e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t






g0 g1t1

(11) g0 a2b2

g1 c2b2




2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t



h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES

































g0 g1t1

(11) g0 a2b2

g1 c2b2




2

36 20 00008 00009 00041 10205 15048 14784 044 5803(036) (040) (014) (3438) (241) (235)

36 10 00008 00009 00344 10558 14588 14151 049 5769(033) (040) (112) (3415) (223) (216)

36 40 00007 00004 00575 09732 15099 15722 102 5858(033) (021) (207) (3475) (257) (265)

24 20 00001 00001 00589 09641 16340 15561 112 5719(006) (003) (210) (3402) (273) (257)

48 20 00016 00017 00163 10394 12269 12224 022 5723(070) (072) (053) (3355) (186) (185)





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t



h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES































h0 h1t1

(12) h0 a1 a2b1b2

h1 c1 c2b1b2





K N g0 g1 h0 h1 [g1h1] 0 g1 h1 0

36 20 00009 14487 00008 15108 582 000(040) (235) (036) (240) [005] [096]

36 10 00009 13403 00007 14126 520 000(040) (216) (033) (223) [007] [095]

36 40 00004 16155 00007 16029 712 000(021) (264) (032) (257) [003] [099]

24 20 00001 16140 00001 17290 837 001(006) (256) (005) (271) [002] [093]

48 20 00016 11761 00016 12077 346 000(072) (185) (071) (186) [018] [098]





e c1t1 u1t

rslopete a2 b2rMt

e c2t1 u2t














36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES










































36 00005 16660 00002 15911 694 000(022) (255) (007) (241) [003] [095]


36 00001 19251 00000 20335 1090 001(003) (312) (002) (323) [000] [093]


36 00012 20503 00007 20644 986 000(052) (313) (030) (313) [001] [099]




(t1)rinterceptt


rslopete a2 b2rMt

e c2t1 u2t














34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES










































34327 (139) 00489 (09) 001 124



92135 (294) 05596 (79) 058 124


114532 (228) 04270 (20) 015 33


111719 (211) 04547 (20) 012 33




(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES
































(13) RieSUBJ iRM

eSUBJ i f


(14) εi RieOBJ iRM

eSUBJ i f

and therefore

(15) εi RieOBJ iRM

eOBJ i f εM13

N iOBJ Ri




(16) iOBJ 0 1 i0 1


t1 where




(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES































(17) rinterceptte

rallslopeste

1

t1

t1131

t1

t11311

t1

t113r t

e









e B1 ft c1t1 u1t

rslopete a2 b2rMt

e B2 ft c2t1 u2t







36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES


































36 00003 12788 00003 12787 313 000(011) (175) (013) (171) [021] [100]


36 00018 09980 00018 10099 230 000(076) (151) (075) (152) [032] [099]


36 00014 09099 00009 09615 228 000(040) (312) (035) (136) [032] [099]


36 00014 11630 00012 11950 354 000(062) (180) (060) (184) [017] [098]






e c1t1 u1t

rslopete a2 b21rMt


e c2t1 u2t



IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES































IV CONCLUSIONS









APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES

































APPENDIX













(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES








































(19) w1X1 w2EX2 w3EX3 1

21 cw2

2 varX2

1

21 cw3

2 varX3 1

1 w1 w2 w3X

1

21 cw2

22 1

21 cw3

2k2


(20) w2 1 cX

2 1 c

1

p2

2

w3 1 cX

k2 1 c

1

p3

k2



2 varXB2

12c wB3


2 2 12c wB3

2 k2



(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES
































(22) wB2 cX 2 cp22

wB3 cX 2 cp32


(23) 1 w2 wB2 X

2 1 c p2

1 2

1 w3 wB3 X

k2 1 c

1

p3

k2


(24) p2 1

1 c X 213

p3 1

1 c X k213



(25)EX3

p3

EX2

p2 1 c

1 X

X k2 X

X 2


(26) 2 21 kX 2

X 2

3 2k1 kX k2

X k2



(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES































(27) 32

2k1 kX k213X 213 X kX1321 kX 213

X k213X 213


(28)

EX2 EX3

p2 p3 1 c

1 2X

X 213 X k213

11


(29) 1 c

1

X 213X X k213X

2k1 kX k213X 213 X k21321 kX 213


(30) 1 c

1

32

11

c

1





REFERENCES































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