Chapter 4 Linear Econometrics for Finance
Market efficiency and the predictability of stock returns
Time series models
• Consider a time series {yt}Tt=1 . Let us think of yt as a stochastic process. • Two features to discuss:
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1. Stationarity vs. nonstationarity
2. Dependence (on the past, particularly on the recent past)
Stationarity
1. Strong: The joint distribution of yt, …, yt−k and ys, …, ys−k is the same for all t, s, k.
2. Weak: yt, …, yt−k and ys, …, ys−k have the same first two moments. The mean vector and the covariance matrix are the same for all s, t, k.
Clearly, you could have a time series which is weakly stationary but not strongly sta- tionary. You could also have a time series which is strongly stationary but not weakly stationary if its first two moments do not exist.
Dependence
Look at the covariance (or correlation, of course) between yt and yt−k for different values of k.
1.1 A workhorse model: The autoregressive model of order 1 (AR(1))
yt = c + φyt−1 + εt, 1
Chapter 4 Linear Econometrics for Finance where εt is a white noise process (uncorrelated, mean zero, with variance σε2).
Recall how we modeled the divided-to-price ratio in Chapter 3? As an AR(1) process. We will return to data below.
• The parameter φ is key:
1. Ifφ=0,thenyt =c+εt (whitenoisewithmeanc)
2. If |φ| ≤ 1, the process is stationary. Expected value:
= c + φE(yt−1) ⇒ E(yt)= c
= φ2V ar(yt−1) + V ar(εt) σε2
⇒ Var(yt)=1−φ2.
Autocovariance function (i.e., the covariance of the process as a function
of k, the time lag):
= Cov(yt, yt−1) = Cov(c + φyt−1 + εt, yt−1)
= φV ar(yt−1)
= φV ar(yt)
Linear Econometrics for Finance
Cov(yt, yt−2) = Cov(c + φyt−1 + εt, yt−2) Cov(c + φ(c + φyt−2 + εt−1) + εt, yt−2) Cov(c + φc + φ2yt−2 + φεt−1 + εt, yt−2) φ2V ar(yt−2)
φ2V ar(yt) φ2γ(0).
= = = = = =
Using the same method, we can show that
= Cov(yt,yt−k) = φkV ar(yt)
Hence, naturally, the autocorrelation function would be
ρ(k) γ(k) γ(0)
= Corr(yt, yt−k)
Cov(yt,yt−k)
= V ar(yt)V ar(yt−1)
Cov(yt,yt−k)
= V ar(yt)V ar(yt)
Note: the mean value of the process c can be interpreted as some sort of
“steady state” value.
Linear Econometrics for Finance
yt− c = c+φyt−1+εt− c 1−φ 1−φ
Let us show it …
c = c − cφ − c + φyt−1 + εt
c = −cφ + φyt−1 + εt 1−φ 1−φ
cc yt−1−φ = φ yt−1−1−φ +εt.
cc E yt−1−φ =φE yt−1−1−φ .
These expressions are important.
Assume 0 < φ < 1. If yt−1 is above its steady state value c , it will be above
the steady state level even the next period (on average, i.e., in expectation)
but less so (since φ < 1). In other words, the process tends to go back to its steady state (on average, i.e., in expectation). Of course, in the short term the shocks ε may divert the process from its convergence to the “steady state”. Technically, we say that the process “mean-reverts” to its steady state.
Assume −1 < φ < 0. Again the process has a steady state level c . If yt−1 1−φ
is above its steady state value c , it will be below the steady state level next 1−φ
period (on average) but less so (since −1 < φ < 0). In other words, the process tends to go back to its steady state (on average). Of course, again, in the short term the shocks ε could divert the process from its convergence to the “steady state”. The process “mean-reverts” to its steady state. Contrary to the case 0 < φ < 1, though, the process has a tendency to flip back and forth. If it is above the steady state at t − 1, it tends to be below the steady state at t.
Note: it is of course not coincidental that (while decreasing) the autocovari- ance function of the AR(1) process with −1 < φ < 0 goes from positive to
Linear Econometrics for Finance negative values, whereas the autocovariance function of the AR(1) process
with 0 < φ < 1 is always positive.
If φ = 1, the process is nonstationary. It is called a “random walk.” Why is it
nonstationary? No “steady state.” The quantity E(yt)= c ,
Var(yT) = Var(yT−1)+Var(εT)
= V ar(yT−2) + V ar(εT−1) + V ar(εT )
= V ar(y0) + ΣTt=1V ar(εt).
The variance increases with the horizon.
The process starts from a certain level and then drifts away from it without returning to the steady state. If c > 0, we call it a “random walk” with positive drift. If c < 0, we call it a “random walk” with negative drift. If c = 0, we have a “random walk” with no drift.
Note: “Differenced” random walks are “white noise,” i.e.,
yt −yt−1 =c+εt
So, if the original data looks like realizations from a random walk, the “differ-
enced data” should look like realizations from a white noise process.
If φ > 1 or φ < −1, the process is nonstationary. It simply tends to grow larger and larger, or smaller and smaller.
cannot be defined when φ = 1. What about the variance:
The dividend-to-price ratio. We use dyny from the file “predictability.xls.” Here is what the data looks like. This is clearly stationary, autoregressive material. There is persistent fluctuation around some mean level.
Chapter 4 Linear Econometrics for Finance
.08 .07 .06 .05 .04 .03 .02 .01
50 55 60 65 70 75 80 85 90 95 00
Figure 1: The dividend-to-price ratio.
Let us look at the autocorrelations (from the 1st to the 10th). We will see that the estimated φ is very close to 1, thereby demonstrating (as discussed in Chapter 3) that the dividend-to-price ratio is stationary, but very highly persistent.
Date: Time:
Sample: 1946M01 2001M12 Included observations: 672
Linear Econometrics for Finance
Autocorrelation .|*******
.|******* .|******* .|******* .|******* .|******* .|******* .|******| .|******| .|******|
Partial Correlation .|*******
AC PAC 0.987 0.987
0.975 −0.002 0.964 0.036 0.951 −0.048 0.938 −0.043 0.923 −0.075 0.908 0.013 0.894 −0.008 0.881 0.073 0.868 −0.014
Q-Stat Prob 658.13 0.000
1300.7 0.000 1929.2 0.000 2542.4 0.000 3139.3 0.000 3718.1 0.000 4279.7 0.000 4824.5 0.000 5354.7 0.000 5870.2 0.000
1 2 3 4 5 6 7 8 9
Estimation: One can easily run a regression of the yt values on their lagged values yt−1. The dividend-to-price ratio: continued.
Chapter 4 Linear Econometrics for Finance
Dependent Variable: DYNY
Method: Least Squares (Gauss-Newton / Marquardt steps) Date: Time:
Sample (adjusted): 1946M02 2001M12
Included observations: 671 after adjustments
DYNY= C(1)+C(2)*DYNY(-1)
Coefficient C(1) 0.000281
C(2) 0.991834
Std. Error 0.000220
t-Statistic 1.280866
Prob. 0.2007
0.0000 0.037631
0.012236 −9.850981 −9.837542 −9.845776
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
0.979483 0.979453 0.001754 0.002058 3307.004 31938.83 0.000000
0.005550 178.7144 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
• We note that the autoregressive parameter is very accurately estimated.
• We also note that its magnitude is consistent with the first-order autocorrelation
found earlier.
Specification: If the model is correctly specified, the residuals should be roughly un- correlated. If the errors are not roughly uncorrelated, then the structure of the model should be made more complex. One can easily go from an AR(1) to an AR(p) model
yt =c+φ1yt−1 +φ2yt−2 +...+φpyt−p +εt,
where p is selected so that the residuals are “as uncorrelated as possible.” This model
can of course be estimated by running a multiple regression of yt on lagged values.
The dividend-to-price ratio: continued. What do the residuals from the previous autoregression look like? Let us see.
Chapter 4 Linear Econometrics for Finance
.012 .008 .004 .000
.08 .06 .04 .02 .00
50 55 60 65 70 75 80 85 90 95 00
Residual Actual Fitted
Figure 2: The residuals from a dividend-to-price ratio autoregression of order 1.
1.2 Prices and returns
We will show that:
1. Prices look like “random walks” (AR(1) processes with φ = 1). Prices are non- stationary.
2. Returns look like a “white noise” process. Returns are stationary.
Since prices and returns depend on each other, are these empirical statements compatible with each other? Let us see:
log Pt+1 − log Pt
1 + Rt,t+1 = ⇒ log(1 + Rt,t+1) = ⇒
Chapter 4 Linear Econometrics for Finance log Pt+1 = log Pt + εt+1,
where εt+1 = log(1 + Rt,t+1) = rt,t+1 is a continuously-compounded return. Roughly speaking, (continuously-compounded) returns are (log) price differences. If they are fairly uncorrelated (and they are in the short-run), then the price process has to behave like a “random walk.”
Let us look at market data. The data comes from the file “S&P500daily-level.xls”. The file contains daily closing values of the S&P500 index.
1,600 1,400 1,200 1,000
800 600 400 200
sp5 00le vel
80 82 84 86 88 90 92 94 96 98 00 02 04 06
Now, in logs.
Figure 3: The S&P 500 index.
Chapter 4 Linear Econometrics for Finance
LOGSP500LEVEL
80 82 84 86 88 90 92 94 96 98 00 02 04 06
Figure 4: The log S&P 500 index. Finally, in log differences (or continuously-compounded returns).
.10 .05 .00
80 82 84 86 88 90 92 94 96 98 00 02 04 06
Figure 5: First differences in the log S&P 500 index. 11
Chapter 4 Linear Econometrics for Finance
Let us now run a couple of regressions. First, we will look at autoregressions of prices on prices.
Dependent Variable: LOGSP500LEVEL
Method: Least Squares (Gauss-Newton / Marquardt steps) Date: Time:
Sample (adjusted): 1/03/1980 12/29/2006
Included observations: 6814 after adjustments LOGSP500LEVEL=C(1)+C(2)*LOGSP500LEVEL(-1)
Coefficient C(1) 0.001439
C(2) 0.999827
Std. Error t-Statistic 0.000942 1.528347
0.000152 6561.962 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
As is clear from the output, the slope is
close to 1. The market index behaves as a random walk.
Prob. 0.1265
0.0000 6.125694
0.822429 −6.304476 −6.302473 −6.303785
Adjusted R-squared
S.E. of regression
Sum squared resid
Log likelihood
F-statistic 43059343 Prob(F-statistic) 0.000000
0.999842 0.999842 0.010344 0.728909 21481.35
very accurately estimated. It is economically
Chapter 4 Linear Econometrics for Finance
7.5 7.0 6.5 6.0 5.5 5.0 4.5
80 82 84 86 88 90 92 94 96 98 00 02 04 06
Residual Actual Fitted
Figure 6: First differences in the log S&P 500 index.
The residuals (which are returns, since the slope is effectively one) show no obvious persistence, only a few jumps and time-varying variance (to which we will return in the next chapter.) Let us now look at autoregressions of returns on returns.
Chapter 4 Linear Econometrics for Finance
Dependent Variable: DIFF
Method: Least Squares (Gauss-Newton / Marquardt steps) Date: Time:
Sample (adjusted): 1/04/1980 12/29/2006
Included observations: 6813 after adjustments DIFF=C(1)+C(2)*DIFF(-1)
Coefficient C(1) 0.000374
C(2) 0.019901
Std. Error 0.000125
t-Statistic 2.983989
Prob. 0.0029
0.1005 0.000382
0.010345 −6.304578 −6.302574 −6.303887
Adjusted R-squared S.E. of regression Sum squared resid Log likelihood F-statistic Prob(F-statistic)
0.000396 0.000249 0.010344 0.728728 21478.55 2.698593 0.100483
0.012115 1.642740 Mean dependent var
S.D. dependent var Akaike info criterion Schwarz criterion Hannan-Quinn criter. Durbin-Watson stat
Returns “look” white noise. We saw, however, that their conditional mean is likely predictable by slow-moving variables. Hence, they cannot be white noise. Also, their conditional volatility structure will make them a more complicated object than a simple white noise process but, to this, we will return in the next chapter.
2 Market efficiency: the classical approach
Suppose that a security price at time t can be written as the rational expectation of some “fundamental” value V ∗ conditional on information It available at time t:
Pt = E[V ∗|It]. The same equation has to hold tomorrow, i.e.,
Linear Econometrics for Finance
Pt+1 = E[V ∗|It+1].
Note that It ⊂ It+1. In other words, the information set at time t is contained in the information set at time t + 1 (people learn over time). Can we predict price changes, i.e., returns? No!
E[(Pt+1 − Pt) |It]
= E[E[V ∗|It+1] − Pt|It]
= E[E[V ∗|It+1]|It] − Pt
= E[V ∗|It] − Pt
= Pt−Pt=0.
In other words, in a frictionless market in which prices contain all available information, prices changes (returns) ought to be unforecastable given information (given the past).
What should we include in the information set? Assume the information set is ex- pressed as It = {Pt , Pt−1 , ...}. In other words, the information set only contains past prices. Write
Pt = Pt−1 + εt.
This model implies that Et[Pt+1] = Pt (this is a “martingale” condition). In other words, the best predictor for future prices are prices today. Prices are “martingales” or “random walks”.
To re-cap: it is not surprising, given classical notions of market efficiency, that prices roughly behave as random walks and returns are roughly uncorrelated (and unpredictable given their past and information).
Chapter 4 Linear Econometrics for Finance
2.1 Two important tests of the classical notion of market effi- ciency
1. Regression-based tests.
Consider continuously-compounded returns rt,t+1 t = 1, ..., T . Run:
rt,t+1 = μ + φrt−1,t + εt
and test the null φ = 0 (returns should be uncorrelated). We tested this hypothesis before, right?
Note: there is some evidence that at short-horizons rt is positively correlated (φ > 0) for some portfolios. This implies that if returns are higher than average at time t, they tend to be higher than average at time t + 1 as well. This phenomenom is typically called ”momentum” (Jegadeesh and Titman, 1993). Momentum reverses itself over long horizons and this is, of course, consistent with an autoregressive process for returns. Momentum is controversial. However, it seems to be pervasive (Asness, Moskowitz and Pedersen, 2013). If agents are rational and markets are efficient, why don’t people take advantage of momentum?
2. Variance ratio-test.
Long-run continuously-compounded returns are just sums of short-run continuously- compounded returns, i.e., rt,t+k = rt,t+1 + rt+1,t+2 + …. + rt+k−1,t+k.
If returns are uncorrelated, then
V ar(rt,t+k) = V ar(rt,t+1 + rt+1,t+2 + … + rt+k−1,t+k) = kV ar(rt,t+1).
The test simply compares the variance of the k period returns to k times the variance of the single period returns. Formally, H0 : V ar(rt,t+k) = kV ar(rt,t+1). Assume the returns
Chapter 4 Linear Econometrics for Finance are i.i.d. (more than just uncorrelated). One can show (try it for yourself…) that under
√ V ar(rt,t+k) T − 1
T Var(rt,t+k) −1 Yd N 0,2k−1 . k
kV ar(rt,t+1)
The 5% level test is therefore trivial. Reject H0 if
kV ar(rt,t+1)
2k−1 k
Note: you need to be careful when computing the multiperiod returns. You should not use overlapping returns. Under the null, these returns should be i.i.d. If you use overlapping returns, you would be inducing correlation by construction.
Poterba and Summers (1988), for example, find ratios smaller than one. This implies presence of negative correlations in short-term returns. Is this in contraddiction with momentum?
Critiques of classical efficiency tests
How can we test conditional restrictions if “information” can not be evaluated? (See, for example, Hansen and Richard, 1987.) Standard tests largely impose that “information” only comprises prices and/or returns. This is likely very restrictive. This is similar to Roll’s critique.
Is it really true that Pt = E[V ∗|It]? Or, should we write Pt = E[mt+1V ∗|It] where mt+1 is a stochastic discount factor?
Predictability and market efficiency
Consider the optimality condition of an agent allocating wealth between consumption and a risky asset (recall from your previous classes?):
Linear Econometrics for Finance
Ptu′(ct) = Et[βu′(ct+1)(Pt+1 + dt+1)],
βu′(ct+1) Pt = Et u′(ct) (Pt+1 + dt+1)
= Et[mt+1(Pt+1 + dt+1)], (1) = βu′(ct+1) is the stochastic discount factor. Consider a short horizon (i.e.,
where m t+1
dt+1 ≈ 0 and β ≈ 1). If investors are risk neutral and/or there is little variation in
consumption (which is expected over short horizons), then
Pt ≈ Et[Pt+1],
i.e., prices are roughly martingales (as implied by classical approaches to market effi-
ciency) and returns are not predictable.
More generally, prices should satisfy Eq. (1). Hence, prices are martingales only after adding dividends and after rescaling by the stochastic discount factor. Returns can be predictable, mainly in the medium and long-run! Let us see:
Pt = Et[mt+1(Pt+1 + dt+1)]
Pt+1 +dt+1 1=Et mt+1 Pt
1 = Et[mt+1(1 + Rt,t+1)]
Chapter 4 Linear Econometrics for Finance
1 = Covt(mt+1, (1 + Rt,t+1)) + E(mt+1)Et(1 + Rt,t+1) and,sinceE(m )= 1 ,
t t+1 1+Rf
Et(1 + Rt,t+1) − (1 + Rf ) = −Covt(mt+1, 1 + Rt,t+1).
Et (mt+1 )
Et(Rt,t+1 − Rf ) = −Covt(mt+1, 1 + Rt,t+1). Et (mt+1 )
This equation implies that conditional expected excess returns depend on the beta of asset returns with respect to the stochastic discount factor. Assets which give high returns when marginal utility is high are assets which payoff in bad times. These assets are relatively preferable and sell at a premium (their expected returns are lower).
If u(c ) = 1 c1−γt, one can derive t 1−γtt
Et(Rt,t+1 − Rf ) ≈ γtV art(∆ct)V art(Rt,t+1)Corrt(mt+1, Rt,t+1). (2)
Neglecting the correlation just for simplicity, whatever predicts changes in the variance of consumption growth, the variance of stock returns, and relative risk aversion should help us predict excess returns. These variables change with the business cycle. So, variables that are correlated with the business cycle should predict expected (and realized) returns precisely at business cycle frequencies.
Returns are hardly predictable in the short-run. They are predictable in the medium/long- run (does this ring a bell?). The variables that should predict stock returns are vari- ables that are correlated with the business cycle. The most well-known variable is the dividend-to-price ratio. Fama and French (1989) use the term spread between long- and short-term bonds, the default spread, and the t-bill rate. Cochrane (1991) uses the in- vestment/capital ratio. Lettau and Ludvingson (2003) advocate the consumption/wealth ratio. Some variables have a financial nature. Some variables have a macro nature. Most variables are correlated with the business cycle. Some operate at even lower frequencies (Bandi, Perron, Tamoni and Tebaldi, 2019).
Chapter 4 Linear Econometrics for Finance
In sum, stock return predictability is not inconsistent with market efficiency. It is coherent with general asset pricing models to observe (roughly) uncorrelated and unpre- dictable returns in the short-term and predictable returns in the medium/long-run.
Naturally, testing the implications of market efficiency in the medium/long run is difficult unless we take a stand on a specific model. To test Eq. (2), for example, one would have to take a stand on a specification for mt+1 (which depends on the investor’s utility) and, importantly, on “information”.
As discussed at length earlier, it is now standard to run forecasting regressions like:
Rt,t+k − Rf = β0 + β1 (d/p)t + ut,t+k,
where (d/p)t is the dividend-to-price ratio at t, for example, k is the adopted horizon
in years, and ut,t+k is a forecast error. As shown, it is typically found that the
slope estimates, i.e., the β s, their statistical significance, as well as the R2 of the 1
regressions increase with the horizon.
Important econometric issue: Recall,
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