Welcome to the Seminar in Asset Pricing Theory Consumption...

Welcome to the Seminar in Asset Pricing Theory

Consumption, Aggregate Wealth, and ExpectedStock Returns

Lettau and Ludvigson, 2001

September 25, 2007

Lettau and Ludvigson, 2001 Consumption, Aggregate Wealth, and Expected Stock Returns

Introduction

I Although macroeconomic variables have found to be poorpredictors of excess returns, the deviation from a long runtrend of a combination of macroeconomic aggregates forecastasset returns.

I The macroeconomic variables in question are aggregateconsumption, asset holdings, and labor income.

I The reason deviations in the common long-term trend ofthese variables predict asset returns is because the logconsumption-aggregate wealth ratio is a function of expectedfuture returns on the market portfolio in a range of economicmodels.


Introduction

I The economic intuition behind the result that deviations fromthe long-run trend predict excess returns is due toconsumption smoothing over time.

I With expected returns time-varying, asset wealth exhibitstransitory variations. If excess returns are expected to behigher in the future, the individual increases currentconsumption allowing it to rise above its common trend withlabor income and asset wealth. Thus, a positive deviation ofcurrent consumption indicates higher expected returns.


SetupThe Consumption-Wealth Ratio

I Representative agent economy in which all wealth, humancapital and assets, are tradable.

I The law of motion for the economy’s aggregate wealth isgiven by

Wt+1 = Rwt+1(Wt − Ct) (1)

and we can approximate (1) with a log-linear Taylor expansiongiven by

∆log(Wt+1) ≈ k +rwt+1 +(1−1/ρw )(log(Ct)− log(Wt)). (2)



I A solution to the difference equation (2) can be found bysolving it forward and assuming that the consumption-wealthratio is stationary, ignoring constants, we get the expression

log(Ct)− log(Wt) = Et

∞∑i=1

ρiw (rw

t+i −∆log(Ct+i )). (3)

I From (3), the log-consumption-wealth ratio is a forecast ofexpected returns on aggregate wealth and consumptiongrowth and thus consumption behavior reveals the agent’sexpectations about future returns and consumption growth.



I Although (3) summarizes information about investorsexpectations which are inherently unobservable, human capitalis also unobservable which makes log-wealth unobservable.

I Assume that human capital is the sum of a stochastic trendand a stationary component and assume that the stochastictrend of human capital is captured by labor income and z is amean zero stationary variable

log(Ht) = w + log(Yt) + zt . (4)



I How can this be motivated? Let human capital have a priceH and let the dividends paid from it, be labor income, Y . Thereturn to human capital is given by

Rht+1 =

Ht+1 + Yt+1

Ht. (5)

I Solve (5) forward, to get

log(Ht) ≈ log(Yt) + Et

∞∑j=0

ρjh(∆log(Yt+1+i )− rh

t+1+j). (6)

In (6) the first term on the right side is the stochastic trendand the second term is the stationary component,corresponding to (5) respectively.



I Further, approximate aggregate wealth Wt = At + Ht by alog-Taylor approximation to get

wt ≈ vat + (1− v)ht (7)

where v = ¯A/W .

I Since wealth is a portfolio consisting of asset wealth andhuman wealth weighted by its respective shares, it follows thatthe return to this portfolio is given by

Rwt+1 = vtR

at+1 + (1− vt)Rh

t+1

or to make the weights constant log-linearize the expression toget

rwt ≈ vra

t + (1− v)rht . (8)



I Substituting (8) and (7) into (3) yields

ct−vat−(1−v)ht = Et

∞∑i=1

ρiw

{[vra

t+i + (1− v)rht+i

]−∆ct+i )

}(9)

I Using (4) we get an expression with only observables on theleft side of the equation

cay ≡ ct − vat − (1− v)yt =

Et

∞∑i=1

ρiw

{[vra

t+i + (1− v)rht+i

]−∆ct+i )

}+ (1− v)zt . (10)

Thus, cay gives the deviation in the common trend, it is thecointegrating residual.



I Depending on what the stochastic properties of returns tohuman capital and consumption growth are, cay revealsexpectations of future asset returns to differing degrees.

I For example, if cay is relatively large then agents must expectlarge expected asset returns given that consumption growthand human capital return are unforecastable.



I Note, that consumption plays the role of dividends, when thedividend-price ratio is used as a forecasting variable ofexpected future returns.

I In a Lucas-tree economy aggregate consumption is exactlyequal to dividends. This is not exactly true in economies withlabor income or storage technology.


EstimationCointegrating Residual

I Lettau and Ludvigson find that the variables consumption,asset wealth, and labor income have unit roots and so thereexists at least one linear combination which is stationary. Theyfind evidence in favor of a single cointegrating relationship.

I Consumption is measured as expenditures on nondurables andservices and is thus only a component of total consumption.To deal with this assume that log total consumption is amultiple of nondurable and service consumption, ct = λcn,t .



I Then, the estimated cointegrating vector for (cn,t , at , yt) isgiven by

[1,−(1/λ)v ,−(1/λ)(1− v)].

I Define the parameters to be estimated βa = (1/λ)v andβy = (1/λ)(1− v) and, note that, 1/λ is found by βa + βb.

I The estimated trend deviation is given by

ˆcay t ≡ cn,t − β̂aat − β̂yyt ,

where the coefficients can be estimated superconsistently withdynamic least squares.



I Using quarterly data from 1952 to 1998 they estimate theshared trend of the variables as

cn,t = 0.31at + 0.59yt , (11)

where the coefficients are all highly significant.

I As a point of interest, λ ≈ 1.10, so that almost 1/3 of wealthis held in assets and almost 2/3 in human capital.

I In order to understand the information content of cay one hasto know whether deviations in the shared trend is due totransitory movements in asset wealth, consumption, or laborincome.


EstimationTransitory Variation

I In order to address this question they estimate a cointegratedVAR of the log changes in consumption, asset wealth, andlabor income on lags of the dependent variables as well as onthe estimatd cay .

I Note that although the estimated cay has been estimated in afirst step it can be taken as given since it has been estimatedsuperconsistently.

I The estimates in the VAR reveal that ˆcay is a predictor onlyfor the growth rate in asset wealth.


EstimationTransitory Variation

I This implies that deviations in asset wealth from its sharedcommon trend uncovers transitory variation in asset holdings.

I Thus, they conclude that deviations from the shared trend arebetter described as transitory movements in asset wealth thanas transitory movements in consumption or labor income.

I The next step is to investigate the role of transitorymovements in asset wealth in forecasting asset returns.


ResultsQuarterly Forecasting Regressions

I Table III reports one-quarter ahead forecasts of the real returnon the S&P 500 and on the value-weighted CRSP as well asforecasts for excess returns.

I Regressions 1-8 use as regressors a constant as well as a lag ofthe dependent variable and/or current ˆcay on one step aheadforecasts. In all 8 regressions, current ˆcay is a statisticallysignificant predictor.

I Regressions 9-13 add control variables known to have forecastpower. Interestingly, current ˆcay is still statistically significant.



I In all regressions the adjusted R-squared is around 10% whencay is included. Other forecasting variables basically do notadd any additional forecasting power.

I The positive sign on cay adheres to the intuition outlinedbefore about consumption smoothing over time. If returns areexpected to increase and the investor wants to smoothconsumption, current consumption will increase temporarilyabove its long-term relationship with asset and labor income.

I The correlation of ˆcay with consumption growth and realGDP growth is slightly negative suggesting that ˆcay tends todecline during expansions and rise prior to a recession.



I This suggests that while expansions are characterized byincreasing consumption, assets increase at an even higher rate.

I These characteristics are consistent with Campbell andCochrane (1999). In an expansion, consumption increasesabove habit, leading to a decline in risk aversion, which leadsto greater demand of the risky asset and a decrease inexpected excess return.

I In their model booms are characterized by increasingconsumption but declining consumption-wealth ratios,consistent with what Lettau and Ludvigson find.


Address CriticismsOut-of-Sample tests

I The out-of-sample forecasting exercise starts as of 1968.

I Re-estimate cay each period with information available onlyup to the time of forecast. Compute the one step aheadforecast with a prediction equation using cay and abenchmark without and compare the resulting mean-squaredforecasting errors.

I Caveat: consistent estimation requires a sufficienlty largesample. Due to the nature of these tests, this procedureinduces a lot of sampling error, especially, in the early stages.



I Two benchmark models: (1) autoregressive benchmark, afirst-order autoregression of excess returns, and (2) constantexpected returns benchmark, a model of constant expectedreturns.

I Comparison of cay -augmented model and the autoregressivebenchmark indicates that the mean-squared forecasting erroris lower in the former.

I Due to sampling error the differences in MSE’s are not assignificant as if they would when using the entire sample toestimate cay .



I Addressing concerns of availability of macro data, they alsoshow results for using a lagged value of cay and still find animprovement relative to the benchmark models.

I The cay forecasting model also has superior forecastingperformance relative to the constant expected returnsbenchmark. As a note of interest, it has been shown that thedividend yield has no predictive ability relative to thisbenchmark.

I The differences in the models are all statistically significantusing the Harvey, Leyborne, and Newbold test statistic. Thenull is that the restricted model (model without cay)encompasses the unrestricted model.



I Also, using an out-of-sample F test, in which the test statisticis compared to the 95th percentile of the asymptoticdistribution as well as 95 percent bootstrapped critical values,suggesst that cay contains information over and above thebenchmarks.

I So far the out-of-sample test have involved an unrestrictedmodel which nests the benchmark model. An alternative is touse two competing models which are not nested.

I They investigate a variety of such competing models such asdividend-price ratio and relative t-bill rate forecasting modelsand find overwhelming statistical evidence in favor of the caymodel.


ResultsLong Horizon Forecasts

I Recall (4),

log(Ct)− log(Wt) = Et

∞∑i=1

ρiw (rw

t+i −∆log(Ct+i )).

I One would expect that cay tracks longer term tendencies inasset markets.

I Also, note that cay could also be a long-horizon forecaster ofconsumption growth.


ResultsLong-Horizon Forecasts

I In order to disentangle the implications, as well as consideringforecasting over longer horizons, Lettau and Ludvigson runtwo sets of long-horizon regressions, for H-period consumptiongrowth and H-period excess returns respectively, where Hspans 1 to 24 quarters.

I They find that cay has no forecasting power for futureconsumption growth at any horizon.

I From (4), this reinforces the notion that cay should predictasset returns.



I Indeed, using cay as the sole predictor shows that, inparticular, for 3 to 8 eight quarters ahead, it predicts excessreturns with adjusted R-squared of around 17%.

I In contrast, using the dividend yield as the sole predictor, itbecomes significant only at a horizon of 24 quarters, with anadjusted R-squared of 25%.

I Thus, the dividend yield seems to be a strong forecastor oflong-horizon returns, whereas cay performs well in the shortto medium term.



I The dividend-yield together with the dividend-earnigs ratioexplain about 40% in returns at a 6 year horizon.

I The relative t-bill rate used as the sole predictor is statisticallysignificant up to 4 quarters but has lower adjusted R-squaresthan cay in that horizon range.

I When using all 4 predictor variables cay , d − p, d − e, andrrel the results are similar in the sense that cay has its bestperformance in the short to medium term, rrel in the shortterm and d − p and d − e in the long term.



I The best overall performance is at a 6 year horizon with anadjusted R-squared of 42%.

I An alternative approach for long-horizon regressions, withoutactually using long-horizon data, is to use a VAR to imputelong-horizon statistics.

I Using this approach the results of the long-horizon regressionare confirmed.


DiscussionStrengths

I A contribution of the paper is the use of cointegrationtechniques in the predictability literature.

I Cointegration is a result of many economic models and thuspredictability is associated with them.

I Technique develops a clear link between the real economy andthe financial markets.


DiscussionStrengths

I Very careful empirical analysis of the implications of themodel using real data and a variety of robustness checks.

I Cay is less persistent than the dividend-price ratio and, thus,less subject to statistical criticism.

I Consumption and labor income are widely studied variables, soframework is relevant.


DiscussionWeaknesses

I Recall Lustig and Van Nieuwerburgh (2006), who demonstratethat the implications of constant expected returns on humancapital is inconsistent with reconciling the joint behavior ofconsumption and market return (their model IV underscrutiny).

I If returns to human capital are forecastable then implicationsof cay change and the documented predictability may bespurious (error-in-variables).

I Superconsistent estimation requires sufficient data toovercome endogeneity problems and it is not clear how muchdata is sufficient. For example, estimating the cointegratingvector now is different then from estimating it in 1998.


DiscussionWeaknesses

I Difficulty of macroeconomic data availability in terms oftiming and also, data is subject to revision.

I Problem for applied work not so much for an equilibriummodel.

I Log-linear framework may neglect important second-orderelements.


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