1 Today’s Agenda

1. Market Efficiency2. Variance Ratio Tests3. Forecastability of Returns: Univariate and BivariateSystems

4. Long-Horizon Forecasts of Returns5. Multivariate Systems (VAR): Reduced forms6. A Basic Structural model: Consumption-basedmodel

7. Risk Return Trade-Off (ICAPM)8. ARCH9. GMM

Warm-up question: Is FED policy risk priced?Can you answer this question within the framework ofthe APT?

2 Market Efficiency (Chapter 2)

• Earlier Models of Asset Pricing:– Fair Game: Cardano’s (c.1565) LiberdeLudoAleae(CLM, 1997)

The most fundamental principle of all in gamblingis simply equal conditions, e.g. of opponents, ofbystanders, of money, of situation, of the dice box, andof the die itself. To the extent to which you depart fromthat equality, if it is in your opponent’s favor, you are afool, and if in your own, you are unjust.

• We have already seen the market efficiencycondition in class 1, but now we will go deeper andprovide a test based on the price process.

E (Pt+1|It) = Ptwhich implies that

Pt+1 = Pt + εt+1with E (εt+1|It) = 0

• A martingale is a special case of the above. IfIt = {Pt, Pt−1,....}, so that

E (Pt+1|Pt, Pt−1,....) = Pt

• Note: This is an AR(1) model with φ = 1 (non-stationary).

• Themartingale condition (also known as a “RandomWalk”) places restrictions only on the first momentson the process. However, it is silent on second,third, fourth, etc. moments.– Specifically, we know that there has got to be atrade-off between risk and return.

• LeRoy (1973), Lucas (1978): The Martingalecondition is neither a necessary nor a sufficientcondition for rational expectations models of assetprices.

• However, the martingale is an important startingpoint.

• There are three versions of the Random Walkmodel.

• The variations place various restrictions on theinnovations (surprise) process, εt in

Pt = µ + Pt−1 + εt

– In its weakest form, εt are i.i.d E (εt) = 0,E¡ε2t¢= σ2.

– If we place some restrictions, we can drop the“identically distributed” condition, but keep the“independent” condition. I.e. εt are independent.

– An even weaker form states that εt are uncorre-lated (but they might be dependent).

• Note that the above process has a “stochastictrend”. To see why, notice that, if we solverecursively the above process, we have

Pt = µt +tXi=1

εi + P0

– E (Pt) = µt– V ar (Pt) = σ2t

– Note that all three forms have the same twomoments.

• Review Q: Do you see why Pt is not (covariance)stationary?

0 10 20 30 40 50 60 70 80 90 100-10

0

10

20

30

40

50

60

70

80

90

time

Pt

1.

• Is the process Pt = µ+Pt−1+ εt adequate to modelstock prices? Here is a hint:

We will use pt = logPt (or Pt = ept) andpt = µ + pt−1 + εt

so that pt = µt +Pt

i=1 εt + p0, or to get back toprices, P = eP0eµte

Pti=1 εt.

• Note that pt − pt−1 = µ + εt = rt (in our previousnotation), is the continuously compounded return.

• If we assume that εt (and hence rt) is normallydistributed, then we are back to the case where rt isnormally distributed and 1 +Rt = ert is lognormallydistributed.

• The process pt = µ + pt−1 + εt is a discrete(arithmetic) Brownian motion

• The process Pt = ept is a discrete geometricBrownian motion.

• Note: An immediate observation of the abovespecification for prices is that returns are notforecastable by past returns or by any otherconditioning variable, since

rt = µ+ εtand

• εt are either uncorrelated or independent.• µ is the mean and it does not vary with time.

3 Variance Ratio Tests

• The variance ratio test is a neat and “natural” wayto test market efficiency.

• We cannot work with prices pt or Pt, because thoseseries are non-stationary. So far, we know how towork only with stationary time-series.

• Here is a good rule for empirical work: DO NOTwork with non-stationary time series, unless youknow what you are doing.

• The non-stationary processes of interest to us (pt)can be stationarized....How?

• Therefore, we will have a preference to work with....

• As we saw above, an implication of the RandomWalk model of prices is that:– Continuously compounded (or log) returns areunforecastable

– The increments (log returns) are uncorrelated (orindependent, or iid)

• Here is a useful observation about the variance ofa two period (log) return rt,t+2

V ar (rt,t+2) = V ar (rt+1 + rt+2)

= V ar (rt+1) + V ar (rt+2)

= 2σ2r

• In general, for the k-th period (long-horizon) returnrt,t+q:

V ar (rt,t+q) = V ar (rt+1 + ... + rt+q)

= qσ2r

• Those implications were derived under the hypoth-esis (the null) that returns are unforecastable.

• So, we have a “natural” test for the Random Walkmodel (or equivalently, for unforecastable returns):V ar (rt,t+q)

Under the Null of a Random Walk= qV ar (rt+1)

orV R(q) =

V ar (rt,t+q)

qV ar (rt+1)= 1

• This is a simple test.• Compute the variance of the q-period returns.• Compute the variance of the 1-period returns.• It must be the case that the ratio of V ar(rt,t+q)qV ar(rt+1)

mustbe close to 1.

• Q: How close? We should not forget that the testV R(q) is a random variable with a correspondingdensity, etc.

• Q: For what values of V R(q) is the model a RW andfor what values it is not?

• ASIDE: We can relate the V R(q) test to autocorre-lations (see CLM, 1997).

• So, under the null hypothesis that rt are iid, we canuse the CLT to show thatp

Tq³\V R (q)− 1

´∼a N (0, 2 (q − 1))

(recall last lecture and the results about X and β)or ³

\V R (q)− 1´∼a N

µ0,2 (q − 1)Tq

¶• Therefore, we can conduct testing in the usual way:Form a statistic

Z =\V R (q)− 1q

2(q−1)Tq

∼a N (0, 1)

• So, if Z is greater than 1.96, we reject the null of IIDreturns at the 5% level.

• Note: We must be careful to compute the long-horizon returns rt,t+q from overlapping observations.

• Q: Why?• A: Because, under the null hypothesis, rt,t+q are iid.If we use overlapping observations, they will NOTbe iid by construction.

• Some people advocate the use of overlappingobservations and correcting for correlation from theoverlap as:pTq³

\V Roverlap (q)− 1´∼a N

µ0,2 (2q − 1) (q − 1)

3q

¶• The correction of the variance is supposed tocorrect for the overlap. At the same time we havemore observations, so the test might be “better”, ormore powerful.

• People are deluding themselves!• If we correct exactly for the overlap, whether we useoverlap or not would make no difference. But theuncertainty introduced around how to deal with theoverlap makes the second test less desirable.

• At the end of the day, we have the same amount ofinformation (returns), not more.

• NOTE: As part of your homework, you will have toprogram the V R test using non-overlapping andoverlapping returns and for different portfolios.

• Compare your results to those in CLM (chapter 2).• Answer the question: Do you believe the RWhypothesis?

4 Forecastability of Returns: Univariate andBivariate Systems (Chapters 2 and 7)

• Thus fare, we defined returns rt = log (1 +Rt) asrt = pt − pt−1= log

PtPt−1

• Of course, if rt = µ + εt is true, we can also runsimple regression tests:– We can write rt = µ+φrt−1+ εt , where under thenull hypothesis, φ = 0.

– Run: rt on rt−1 (this is an AR(1) model ).– Test that the parameter φ is equal to zero.– Those tests are perfectly valid and can be relatedto the V R test.

• ASIDE: There is some evidence that, a shorthorizons, rt is positively correlated for some stocks.That implies that if returns of certain stocks arehigher than average at t− 1, they would tend to behigher than average at t. (Jagadeesh and Titman)

• This is called MOMENTUM and it is not uncontro-vercial.

• Momentum disappears at longer horizon and even“reverses” itself.– Q: If agents are rational and markets are efficient,why don’t people take advantange of momentum?

– A: Well, markets are not efficient and argentsare not rational. This is precisely the source ofmomentum.

• There is no agreed upon concensus on momentum.

• Ifrt = pt − pt−1= log

PtPt−1

• Thenert = (1 +Rt) =

PtPt−1

Rt =Pt − Pt−1Pt−1

• Note: This definition has no place for dividends. Butdividends must carry important information aboutthe viability of a company.

• Can it be the case that, if we do the calculationscorrectly and incorporate dividends, then thatinformation might forecast returns?

• The exact definition of a return isRt+1 =

Pt+1 +Dt+1 − PtPtor

1 +Rt+1 =Pt+1Pt

+Dt+1Pt

• Therefore, the increase in returns can come eitherfrom an increase in the stock price or from adividend payment.

• Here, we will use a trick that has proven useful.• Take logs

log (1 +Rt+1) = log (Pt+1 +Dt+1)− log (Pt)• Now, I will do a trick using the Taylor’s expansion(see CLM, p. 261)

•rt+1 ≈ k + ρpt+1 − pt + (1− ρ) dt+1

where– dt+1 = log (Dt+1) is the log dividend– k is a constant term– ρ is a constant of linearization that we will treat asa parameter.

• The above transformation is “cool” because nowreturns, prices, and dividends are related by a linearrelation.

• But there is a problem: This is not exactly aregression, because prices and dividends are notstationary....Remember.

• But, we can writept ≈ k + ρpt+1 − rt+1 + (1− ρ) dt+1

• Solve FOREWARD (not backward as before)pt+1 ≈ k + ρpt+2 − rt+2 + (1− ρ) dt+2

orpt ≈ k + ρ [k + ρpt+2 − rt+2 + (1− ρ) dt+2]− rt+1 + (1− ρ) dt+≈ K + ρ2pt+2 − ρrt+2 − rt+1 + ρ (1− ρ) dt+2 + (1− ρ) dt+1≈ ....

≈ K + ρjpt+j −jXi=0

ρirt+1+i +

jXi=0

ρi (1− ρ) dt+1+i

• Now, we have to assume a no-bubbles condition(interpretation):

limj→∞

ρjpt+j = 0

Therefore:

pt ≈ K −∞Xi=0

ρirt+1+i +∞Xi=0

ρi (1− ρ) dt+1+i

• This is quite intuitive: If price is “high”, it means thateither– dividends in the future would be high– returns in the future would be low– This presumes a no-bubbles condition. Note adifference between fundamentals and the pricecan exist, but it cannot last indefinitely.

• The last step is to subtract the above expressionfrom dt to obtain:

dt − pt = K −∞Xi=0

ρi (1− ρ)∆dt+1+i +∞Xi=0

ρirt+1+i

• This is a very, very useful relation, because– The log dividend price ratio (dt − p) is stationary– ∆dt+1+i is stationary– rt+1+i is stationary.– The log dividend price ratio must forecast eitherfuture dividend growth or future returns

– This is the relationship we had obtained a fewslides ago, but now it is in a “regression” form.

– We can use regressions to test this relation.• The last observation is that we are dealing withfuture values. But this has never stopped us.

dt−pt = K−Et( ∞Xi=0

ρi (1− ρ)∆dt+1+i +∞Xi=0

ρirt+1+i

)

Before going further, let’s take a step back andthink of what we have done so far:• In the absence of dividends, we argued that returnsare unforecastable

• Now, we argue that dividends can help us forecastreturns.

• Isn’t that inconsistent?

Testing the above relationship:• Regress: rt+1 on dt − pt• Regress: ∆dt+1 on dt − pt• Which relationship is stronger?• Note: This is a forecasting relationship: the dividendprice ratio today helps us forecast tomorrow’s(expected) returns

• It is different from the CAPM, APT, where we weretrying to explain ex-post variations in returns.

• Here, we are doing an ex-ante regression.

5 Long-Horizon Regressions:

People have noticed that the relationship

dt − pt = K −∞Xi=0

ρi (1− ρ)∆dt+1+i +∞Xi=0

ρirt+1+i

might not hold in the short run, but it has got to holdin the long run.• The above observation has prompted people to runregressions:– rt+1 + ... + rt+k = α1 + β1 (dt − pt) + ut+k– ∆dt+1 + ... +∆dt+k = α2 + β2 (dt − pt) + et+k

• But then, it must be the case thatβ1 + β2 = 1

by definition. We have not used anything, butthe definition of returns, no-bubbles condition, and alog-linearization.• The above restriction is never used in practice. Howcan it be implemented?

• Remark: As in the variance ratio tests, it isimportant to run the long horizon regressions usingnon-overlapping returns.

• If we use overlapping returns, we get spuriousresults.

• Correcting for the overlap is difficult and introducesanother layer of complication.

Vector Autoregressions (VAR)• First, we should not confuse VAR (vector autore-gression) with VaR (Value at Risk).

• The VAR is a natural generalization of the autore-gressive process, for multivariate series.

• Suppose we are interested in the joint, dynamicinteraction between a few series, say returns andvolatility.– We suppose that higher volatility must lead tohigher returns

– We also think that there might be some feedbackeffect from returns to volatility Campbell andHentschel (1992).

– Ultimately, we are not sure about the dynamicrelationship between those two series

Then, we write

Yt =

·rtσt

¸where rt is the return and σt is its volatility

(standard deviation) at time t. We want to write thefollowing dynamic relationship

rt = α1 + β11rt−1 + β12σt−1 + ε1,tσt = α2 + β21rt−1 + β22σt−1 + ε2,t

• Both series depend on their own lagged values (asin the AR process)

• Both series depend on the lagged realization of theother process

• The residuals [ε1t ε2t] have a covariance matrix Σ.• This system can be written in a more elegant formas:

Yt = α + ΦYt−1 + εt

where Φ =

·β11 β12β21 β22

¸and α =

·α1α2

¸.

The beauty of VARs• Very easy to estimate: Run regressions of variableson their lagged values and on the lagged values ofall other variables

• Run the regressions, equation by equation, andthen stack them together.

• We don’t need a priori theory to know what variablecauses what variable, etc.

• The VARs help us make the notion of statistical“causality” very precise.

Statistical (Granger) causality:• Example: Martian observes that when people bringumbrellas, it starts raining. She concludes thatpeople bringing umbrellas causes rain.

• But it is the other way around...The anticipation ofrain makes people bring their umbrellas.

• In this example, the statistical causality runs from“umbrellas” to “rain”

• The structural causality runs from “rain” to “umbrel-las”.

• Problem: In empirical work, if we don’t have amodel (or more information about the system), wecannot distinguish between the two alternatives.

Causality in a bivariate VAR:rt = α1 + β11rt−1 + β12σt−1 + ε1,tσt = α2 + β21rt−1 + β22σt−1 + ε2,t

• Suppose that β12 = 0. Then rt = α1 + β11rt−1 + ε1,t,or rt is not affected by σt−1. In such a case, we saythat σt−1 does not Granger-cause (or just cause) rt.

• However, since β21 6= 0, rt−1 does Granger-causeσt.

• It is important to understand that Granger-causalitygives us the timing (umbrella, then rain) but not theeconomic story.

• This is a very common mistake in academia, inpractice, and in everyday life.

Test for Granger-causality:• We run the two regressions.• For the hypothesis: “σt−1 Granger-causes rt”, wetest β12 = 0

• For the hypothesis: “rt−1 Granger-causes σt”, wetest β21 = 0

• It is always a good idea to start an empirical workwith some background Granger-causality tests, beit only to get a feel for the data.

• We will come back to VARs.

Here is a real research problem (Goto andValkanov (2001)):• Fact: The effect of the FED actions on the stockmarket is negative. In other words, a contractionarymonetary policy (aimed at curbing inflation) willresult in lower returns for some time in the future.

• Question: Why is this so? There are two possibili-ties– Fed has a better forecast of the state of theeconomy, but its policy has no real effect onstock fundamentals (the umbrella). This is onlyGranger-causation without structural effect.

– Fed has an impact on the economy and bycontracting the economy, cash flows go down,returns decrease (the cloud). This is Granger-causation and a structural effect.

• Q: Is there something about the stock marketthat would help us distinguish between the twoalternatives?

6 A Basic Structural model: Consumption-basedmodel (CCAPM)

• So far, we have estimated the APT and the CAPM• The CAPM and the APT capture risk and return, butare their related to our more fundamental needs:consumption of goods.

• Some of you have asked me to clarify: What do youmean by “The equity premium puzzle is too high”?

• We will work out a “simple” model where assetsare priced explicitly relative to our utility fromconsumption.

• This explicit model will generate a familiar stochasticdiscount factor pricing relation.

• First, we have to model the behavior of a represen-tative investor

• Think of this investor as the average person in theeconomy.

• The investor invests primarily so she consumegoods (break, cheese, or Ferraris) tomorrow.

• The utility function of this investor, today andtomorrow is:

U (ct, ct+1) = u (ct) + βu (ct+1)

where ct denotes consumption at date t and β is asubjective discount factor.

• At the beginning, we will work with u(ct), where– u(.) is concave, reflecting a decreasing marginalvalue of consumption

– u(.) is increasing, reflecting a insatiable desire formore consumption

– the curvature generates aversion to risk and tointer-temporal substitution: The investor prefers aconsumption stream that is steady over time andacross states of nature.

• But to mare the model operational (estimable), wehave to give a functional form to u(.).

• We will assume thatu (ct) =

1

1− γc1−γt

where γ captures the curvature.

• Here is the game:– We want to value an asset with an uncertainpayoff xt+1

– It is a two-period problem (today/tomorrow oryoung/old, etc.)

– The problem is:maxζu (ct) +Et [βu (ct+1)]

such that : ct = et − ptζct+1 = et+1 + xt+1ζ

– et is the original endowment of the individual(cash he inherited from his parents)

– At time t, he has an endowment et.– He decided to purchase ζ shares of the asset atprice pt

– Whatever is left over after the purchase of theasset et − ptζ, is used for consumption.

– At time t + 1, we has an endowment et+1 but alsothe payoff from the asset xt (think pt+1+dt+1) timethe number of shares.

– Since the individual “dies” at t + 1, he mustconsume everything, so ct+1 = et+1 + xt+1ζ.

• To recapitulate: We want to find the number ofshares that would be bought at price pt, given thatthe payoff from this investment is xt+1.

• Implicitly, we will find the price of the asset as afunction of the payoff and everything else.

• This is called an equilibrium model.• This is also a structural model.• Solving the maximization problem yields the FOC:

ptu0(ct) = Et [βu0 (ct+1)xt+1]

orpt = Et

·βu0 (ct+1)u0(ct)

xt+1

¸

• The equationpt = Et

·βu0 (ct+1)u0(ct)

xt+1

¸is quite interesting.

1. Note that if we write mt+1 = βu0(ct+1)u0(ct)

and interpretmt+1 as the stochastic discount factor, then weget the familiar pricing equation, where mt+1 is afunction of consumption. Hence, this is called aconsumption-based pricing model.

pt = Et [mt+1xt+1]

2. Note that we cannot go beyond this point withoutspecifying a functional form for u(), and hence, foru0(). In the above parameterization, u0(c) = c−γ andthe FOC is:

pt = Et

"β

µct+1ct

¶−γxt+1

#3. Now, we can rewrite the model as:

1 = Et

"β

µct+1ct

¶−γxt+1pt

#= Et

"β

µct+1ct

¶−γ(1 +Rt+1)

#4. If we have data on Rt+1 and on ct, we can think ofa way to estimate the parameters γ and β.

5. Problem: The above relationship is nonlinear...weonly know how to run linear regressions....

• Here is a good insight: The equation1 = Et

"β

µct+1ct

¶−γ(1 +Rt+1)

#must hold for any asset. Indeed, we did notspecify a particular asset when deriving the aboveequations.– Let’s consider a particular asset, the risk freeasset with return Rf (and forget uncertainty andexpectations for a moment)

– The pricing equation can be rewritten as

(1 +Rf) =1

β

µct+1ct

¶γ

– Take logs:ln(1 +Rf) = rf = − lnβ + γ (ln ct+1 − ln ct)

– Note that ln ct+1− ln ct is nothing but the growth inconsumption between t + 1 and t.

– We can estimate the above equation if we havedata on rf and ct.

– What is the interpretation of γ?

• Now, we have to take care of the uncertainly. Forthat we will use log-normality

• If X is conditionally lognormally distributed, it hasthe convenient property

lnEt (X) = Et (ln(X)) +1

2V art (ln(X))

• Recall our discussion of: g(E(X)) 6= E(g(X)). Inthe above example, g() = ln().

• In addition, we will assume that V art (ln(X)) =V ar (ln(X)) = σ2x. (What is this assumption called?)

• Recall1 = Et

"β

µct+1ct

¶−γ(1 +Rt+1)

#• Taking logs0 = lnEt

"β

µct+1ct

¶−γ(1 +Rt+1)

#

= Et

"ln

Ãβ

µct+1ct

¶−γ(1 +Rt+1)

!#+

1

2V ar

Ãln

Ãβ

µct+1ct

¶−γ(1 +Rt+1)

!!= Et [ln (1 +Rt+1) + lnβ − γ (ln ct+1 − ln ct)] +1

2V ar (ln (1 +Rt+1) + lnβ − γ (ln ct+1 − ln ct))

= Etrt+1 + lnβ − γEt [(ln ct+1 − ln ct)] +1

2

£σ2r + γ2σ2∆c − 2γσr,∆c

¤

• The equation0 = Etrt+1 + lnβ − γEt [(ln ct+1 − ln ct)] +

1

2

£σ2r + γ2σ2∆c − 2γσr,∆c

¤is valid for any asset. It holds for the risk free

asset, where σ2rf = σrf,∆c = 0. Therefore, we canwrite

rf = − lnβ + γEt [(ln ct+1 − ln ct)]− 12γ2σ2∆c

• Comparing this formula to what we had before,the term −12γ2σ2∆c adjusts for the variance, oruncertainly in consumption, provided that allprocesses are lognormal.

• The advantage of this formula is that we can handleuncertainly and any other asset.

• FromEtrt+1 = − lnβ + γEt [(ln ct+1 − ln ct)]−

1

2

£σ2r + γ2σ2∆c − 2γσr,∆c

¤• And

rf = − lnβ + γEt [(ln ct+1 − ln ct)]− 12γ2σ2∆c

we obtain the cool expression

Errt+1 − rf = −γσr,∆c − σ2r2

= −γCov (rt+1, (ln ct+1 − ln ct))− V ar (rt+1)2

• In words, the excess return is equal to the co-variance of the asset return with consumptiongrowth.

• What is this result reminiscent of?• This is called the Consumption CAPM (or CCAPM)?• But then:– The CAPM does not hold. Does CCAPM hold?– Why is this model better? (before β, now γ)

• The CCAPM is as successful as the CAPM, or evenless, but– The coefficient γ has a very nice interpretation: Itmeasures our aversion to risk.

– We have a consumption variable in the pricingkernel.

– To test the CAPM we needed the market port-folio (Roll’s critique). Similarly, now we needconsumption.

– The CCAPM (with the added log-linearity restric-tions) is easy to test using regressions.

• Note that we have two regressions that we can runin order to estimate γ

• First, using the riskless raterf = − lnβ + γEt [(ln ct+1 − ln ct)]− 1

2γ2σ2∆c

• Second, using the risky rateErrt+1 − rf = −γσr,∆c − σ2r

2• Note that both equations must give us the sameresult (statistically speaking, at least).

• The trouble is that the estimates of γ in thoseregressions are in total disagreement.

• Either the risky rate has been “too high” or theriskless rate has been “too low” to reconcile themodel with the data.

• What’s next: The assumption that the risk premiumErrt+1 − rf does not vary with time has been seen,lately, as being particularly bothersome.

• A few models have tried to relax this assumption,while keeping the economic story of the model.

7 Risk Return Trade-Off (ICAPM)

• One of the main tenets of modern finance is that wehave to be compensated with higher returns if weare exposed to higher risk.

• We measure risk with the variance.• So far, we have assumed that V ar (rt) = σ2.• This is clearly an untenable assumption.• There is no reason why the variance of return willstay constant.

• Why does the variance fluctuate (Schwert (1990)?– Macroeconomic factors– Microeconomic factors– Who knows why?

• Bottom line: V ar(rt) = σ2t (What does this imply forreturns?)

• Modelling σt will be a separate discussion.

• Suppose we have an estimate of σt.• We can run the regression

rt = α + βσt−1 + εt

• Under the null hypothesis that there is a risk-returntrade-off, we expect that β is positive.

• If the relationship does not hold, there are severalpossibilities:– The relationship is not well specified (perhaps seeVAR)

– The volatility is not well estimated– There are other variables that must enter into aVAR (investment opportunity set is time-varying).

– A combination of the above• Surprisingly, thus far, the evidence for β > 0 isnon-existent.

• Several papers find β < 0.

A naive way to estimate σt (it is unobservable)• Note that before we can estimate the conditionalsecond moment, we have to estimate the condi-tional first moment. A lot of people forget to specifythe conditional first moment and get garbage. Fromthe definition, V art (rt+1) = Et

³(rt+1 −Etrt+1)2

´=

Et¡r2t+¢− (Et (rt+1))2

• Here are the steps.– Run daily returns on lagged daily returns andlagged dividend yields. The residuals from thisregression will be

et = rt − Et−1(rt)where e denotes the residual from the OLSregression.

– Using the daily data, we can form an estimate forthe monthly volatility as in:

σ2t =

Ã1

n

nXi=1

e2i

!∗ 22

where i = 1, ..., n are the daily observationswithin month t.

– This is a particularly good way to estimate themonthly σ2t . It is non-parametric (no parametersto estimate, just σt).

• If we plot σ2t over time, it becomes immediatelyclear that σ2t is not white noise, but follows a certainprocess.

• Recall, how we characterized the levels Yt =φYt−1 + εt.

• We want to find a similar model for the volatility,something like σ2t = φσ2t−1 + εt.

• But a simple AR model would not work. Why?

8 Autoregressive Conditional Heteroskedasticity(ARCH)

• Suppose we have the returns process {rt}Tt=1 .• First, we model the conditional mean, for exampleas

rt = µ+ φrt−1 + ut• We know that the unconditional first moments areE (rt) =

µ1−φ and V ar(rt) =

σ2u1−φ2

• We also know that the conditional first momentEt−1 (rt) = µ + φrt is time varying, even though theunconditional moment is not!

• Q: Can we also have the same situation for thesecond conditional moment, i.e. to have a time-varying conditional second moment, although theunconditional second moment constant over time?

• A: Yes.• Note: The unconditional second moment of ut is σ2u.

• Intuitively, we want to model u2t to follow an ARprocess just as we had rt follow an AR process.

• We can write such a process asu2t = ζ + αu2t−1 + wt

where wt is a white noise process with E¡w2t¢=

σ2w.

• Note that Et−1¡u2t¢= ζ + αu2t−1, so that the condi-

tional moment is time-varying althoughE¡u2t¢= σ2u.

• The above process is called an autoregressiveconditional heteroskedasticity model of order 1, orARCH(1).

• We can generalize to an ARCH(p) model as:u2t = ζ + α1u

2t−1 + α2u

2t−2 + ... + αpu

2t−p + wt

• Note that a variance cannot be negative. We needto place certain restrictions on

u2t = ζ + αu2t−1 + wtin order to insure that u2t is always positive.

• We need:– wt to be bound from below by −ζ, where ζ > 0.

– α ≥ 0.– For covariance stationarity of u2t , we also need

α < 1 as in the other AR model.• With all those conditions, we can see that

V ar (ut) = E¡u2t¢= σ2u =

ζ

1− α

• This is the ARCH(1) model and its historical relationto what we have done before

• It is more convenient, but less intuitive, to presentthe ARCH(1) model as:

ut =phtvt

where vt is iid with mean 0, and E¡v2t¢= 1.

• Suppose thatht = ζ + au2t−1

then combining the above equations, we obtain:u2t = htv

2t

• Now, since vt is iid thenEt−1

¡u2t¢= Et−1

¡h2t v

2t

¢= Et−1

¡h2t¢Et−1

¡v2t¢

= ζ + au2t−1as before.

• Reconciling the two definitions.• From one side, we have u2t = htv2t• From another side, we have u2t = ht +wt. Therefore

htv2t = ht + wt

orwt = ht

¡v2t − 1

¢• From here, we can see that Et−1

¡w2t¢is time

varying, whereas E¡w2t¢= σ2w

• Note: The unconditional second moment of wt(the unconditional fourth moment of ut) does notalways exist for an ARCH model. Not a big deal, butmight be annoying if we want to look at conditionalkurtosis.

• The ARCH process gives us conditional het-eroskedasticity, but it turns out that σ2u is a verypersistent process.

• We can capture such a process with an ARCH(p)process u2t = ζ + α1u

2t−1 + α2u

2t−2 + ...+ αpu

2t−p +wt

where p is very large.• This solution is inefficient. There are too manyparameters to estimate!

• What do do? GARCH• The GARCH, Generalized ARCH allows us tocapture the persistence of conditional volatility in aparsimonious way.

• GARCH: Recall that we could write:ut =

phtvt

where ht = ζ + au2t−1 for an ARCH process.• Suppose, we specify ht as

ht = ζ + δht−1 + au2t−1• The direct link between ht and ht−1 is exactly whatis needed to capture the dependence between σ2tand σ2t−1.

• A process with ht = ζ + δht−1 + au2t−1 is called aGARCH(1,1)

• Of course, we can generalize to a GARCH(p,q) as:u2t = ζ+δ1ht−1+δ2ht−2+...+δpht−p+α1u2t−1+α2u

2t−2+...+αqu

2t−q+

.

• For most practical purposes a GARCH(1,1) isGREAT.– There is a trade-off. You introduce more parame-ters to capture the accurate dynamics, but thereare more parameters to estimate

– Those parameters have restrictions. The estima-tion is tricky.

– Bottom line, for 99% of the applications,GARCH(1,1) does a great job.

• GARCH is successful, because it can capture thepersistence in σ2t , which is the most significantfeature that needs to be captured.

• Another useful model to estimate is the IGARCHmodel, or integrated GARCH

• The IGARCH(1,1) is a GARCH(1,1) whereδ + α = 1

• If this condition is satisfied, it can be shown that theconditional variance of ut is infinite.

• The processes ut and u2t are not covariancestationary.

• However, the process ut is stationary (i.e itsconditional density does not depend on t ).

• The IGARCH is important because it captures theimportant case of a strong dependence that leadsto non-stationarity.

The GARCH literature has gone crazy chasingafter the perfect conditional heteroskedasticity model.

Some of the models we have are:• ARCH in Means• Exponential GARCH• Nonlinear GARCH• Asymmetric GARCH• Fractionally Integrated GARCH (FIGARCH)• ABS.ASYMM.FIGARCH ????

• The interest in forecasting conditional volatilityusing past volatility has been greater in the appliedfield more so than in academia.

• The GARCH models are unsatisfactory, from aneconomic perspective, because– Explaining vol with past vol tells us nothing aboutthe underlying economic factors that cause thevolatility to move.

– If a structural break occurs, a recursive model willfail miserably...a structural model might not.

– In general, the GARCH models are difficult togeneralize to a multivariate setting.

Future Research:• The conditional volatility literature is huge.• There are very few papers on conditional correla-tion.

• But, understanding how and why correlations varywith time is crucial in finance.– Think of a portfolio choice problem (correlationsmatter a lot)

– Think of a CCAPM model. The correlationbetween consumption growth and the return isassumed constant (and, hence the risk premiumis constant), but it need not be.

• There are no good models that try to capture thecorrelation as function of macroeconomic factors(except SVV (2001)).

• This is great, but how do we estimate a GARCHmodel.

• OLS clearly does not work• We need a method that would allow us to donon-linear estimation

• We also need a general method that we can apply toany nonlinear problem, with minimal assumptions.

• We do not want to have assumptions on thedistribution of the residuals.

• Therefore, we need GMM (Generalized Method ofMoments)

9 Simple Introduction to GMM

• Recall that any variable xt has a distribution Fx(x).If x has moments E(xj), j = 1, ...then thosemoments can be used to retrieve Fx(x).

• Caution: Some variables do not have moments(Cauchy distributon case).

• Suppose we have random variables xt, yt, zt.– A population moment of those variables is

E [g (xt, yt, zt)]

– A sample moment of those variables is1

T

TXt=1

g (xt, yt, zt)

– By the ergoticity theorem (or the LLN in crosssection, we know that)

1

T

TXt=1

g (xt, yt, zt)→p E [g (xt, yt, zt)]

under some mild conditions on the function g(.).

• In other words, the distance between the sam-ple and the population moment goes to zero inprobability as T →∞ :(

1

T

TXt=1

g (xt, yt, zt)−E [g (xt, yt, zt)])→p 0

• Can we use this “insight” to estimate parameters.Suppose that the function g depends not only onthe data but also on the unknown parameters, θ.

• We want to choose the parameter θ in order tominimize the distance between the data and thepopulation moment.

• In a simpler example, let’s concenrate on a univari-ate case. Then g(x|θ) = µ, the population mean. Inother words, θ = µ.

• The problem becomes (trivially):(1

T

TXt=1

xt − µ)→p 0

• Here is a more interesting example: OLS as GMM• The FOC in the OLS case could be written as:

E (xtεt) = 0

• This is a moment condition that also depends onparameters. To see that, write

E (xt (yt − xtβ)) = 0

E (xtyt) = βE¡x2t¢

β =E (xtyt)

E (x2t )

• Therefore, approximating the population means bytheir sample analogues, we get

1

T

TXt=1

(xt (yt − xtβ)) = 0

1

T

TXt=1

xtyt = β1

T

TXt=1

x2t

β =1T

PTt=1 xtyt

1T

PTt=1 x

2t

• But we can also write another moment condition:E¡x2tεt

¢= 0

• Then, as aboveE¡x2t (yt − xtβ)

¢= 0

β =E¡x2t yt

¢E (x3t )

• Therefore, using sample moments to approximatepopulation moments, we get

β2 =1T

PTt=1 x

2t yt

1T

PTt=1 x

3t

• We can also useE (g(xt)εt) = 0

for some function g(.). Note: You should also beable to show that E (xtεt) = 0 implies E (g(xt)εt) =0. Then, for a known function g(.)

βg =1T

PTt=1 g(xt)yt

1T

PTt=1 g(xt)xt

• Oupss! Problem. We have one parameter, β, butthree possible estimators

β =1T

PTt=1 xtyt

1T

PTt=1 x

2t

→p β

β2 =1T

PTt=1 x

2t yt

1T

PTt=1 x

3t

→p β

βg =1T

PTt=1 g(xt)yt

1T

PTt=1 g(xt)xt

→p β

• Which one do we choose?• Result: Under some very restrictive assumptions(i.e. exogeneity of xt, homoskedasticity, uncorre-lated εt, etc), the OLS is the best linear unbiasedestimator (BLUE). In other words, in has thesmallest variance.

• However, who knows if those assumptions aresatisfied. In all likelihood, they are not.

• Q: Can we stack all the moments in a vector as

E(g (x|β))3x1

= E

xtεx2tεg(xt)εt

= 0and choose the value of β that satisfies the threesample moments?

• A: Off course, not! Three equations, potentiallynonlinear, with only one unknown....Who knowshow many solutions there are, if any.

• But, we can construct a quadratic function, as:E

µg (x|β)01x3

W3x3g (x|β)3x1

¶= 0

for some symmetric positive definite matrixW.• Now, we have the information into the threeequations, weighted by the elements of the matrixW.

• Problem: What matrixW to choose.• A: Any symmetric positive matrix will give usconsistent estimates (i.e. βW →p β), but we areconcerned with efficiency, or smallest possiblestandard errors around βW.

• Endogeneity, Instrumental Variables (IV) and GMM.• By construction, we had E (ε|x) = 0 implied thatE (εx) = 0. In other words, the residuals and theexplanatory variables are uncorrelated.

• However, in structural models, it is often thecase that we want to run regressions when thisrequirement is not satisfied. For example:

FirmV aluet = α + βDebtt + εt

• But it is not reasonable to assume that Debt is anexogenous variable. For example, new (relativelylow Firm Value) firms do not have access to debt.Indeed, we might try to run the opposite regression:

Debtt = δ + ζFirmV aluet + υt

• So, hereE (Debttεt) = E ((δ + ζFirmV aluet + υt) εt)

6= 0

• Q: If E (Debttεt) 6= 0, can we still have β →p β?

• Breaking the E (Debttεt) = 0 condition is thecardinal sin in empirical work!!!

• Q: What to do?• Note: We can argue that most equations in financesuffer from this endogeneity problem.

• Well, we can look for a variable, Zt which is:– Correlated with Debtt (it proxies for Debtt )– Is uncorrelated with εt.

• If such a variable exists, then we can write themoment condition:

E (Ztεt) = 0

• Or, if we look at the sample moments, then1

T

TXt=1

Ztεt =1

T

TXt=1

Zt (yt − βxt)

1

T

TXt=1

Ztyt = β1

T

TXt=1

Ztxt

βIV =1T

PTt=1Ztyt

1T

PTt=1Ztxt

• Then we can show that βIV →p β, whereas βolsdoes not.

• This estimator was motivated from GMM.

• GMM is a very powerful way of looking at anestimation problem.

• All we need is a moment condition that holds.• The problem does not have to be linear• No distributional assumptions are needed.• We will use GMM to estimate– The non-linearized version of the CCAPM– Nonlinear process, such as ARCH, GARCH,etc.– Joint estimation of linear models (CAPM).