Transcript WCS Academy

Welcome to the course
Financial Econometrics II
Course objectives
Financial data analysis, based on
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Asset pricing models
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Econometric methods
Active knowledge
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Try it yourself!
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Share personal vision
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Avoid common pitfalls
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What you already know: Asset pricing models
Risk-return trade-off
Only systematic risk is priced!
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Idiosyncratic risk is reduced by diversification
What are the risk factors?
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CAPM: market
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APT: many other
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What you already know: Econometric methods
Regression
Coefficients
Standard errors
R-squared
Potential problems
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Outliers, heteroscedasticity, endogeneneity, etc.
Packages
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Eviews, Excel
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Plan for today
Specifics of financial data
The efficient market hypothesis
Tests for return predictability
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Asset prices: S&P500
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Asset prices
Convenient for graphical analysis, but..
Non-stationary
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Properties of the stochastic process change over time
Must be corrected for dividends, stock splits
Should be normalized to compare dynamics over time
and across securities
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Asset returns: S&P500 (monthly)
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Asset returns: discrete
Net return:
Rt = (Pt-Pt-1)/Pt-1
Gross return:
1+ Rt
Adjusting for dividends (total returns):
Rt = (Pt+Dt-Pt-1)/Pt-1
Adjusting for inflation (real returns):
1+Rt(real) = (Pt/Pt-1)*(CPIt-1/CPIt)
Excess returns (risk premia):
Zt = Rt – RF,t
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Asset returns: discrete
k-period return:
1+ Rt(k) = (1+ Rt)*…*( 1+ Rt-k+1)
Annualized return
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Geometric average (effective return): (1+ RA)k = 1+ Rt(k)
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Arithmetic average: RA = (1/k) j=0:k-1 Rt-j
Portfolio return: RP,t = i wi Ri,t
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wi: portfolio weights summing up to 1
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Asset returns: continuously compounded
Log return:
rt = ln(1+ Rt) = ln(Pt/Pt-1) = ln(Pt) – ln(Pt-1)
k-period return:
rt(k) = ln(Pt/Pt-k) = j=0:k-1 rt-j
Annualized return: rA = j=0:k-1 rt-j
Portfolio return:
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rP,t = ln(1+ RP,t) ≠ i wi ri,t
But rP,t ≈ i wi ri,t for small RP,t
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Asset returns: simple vs. log
Simple returns:
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Exact
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Easy to aggregate stocks into portfolio
Log returns:
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Easy to aggregate over time
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Closer to normal distribution
– Don't violate limited liability
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Market microstructure effects
Which prices to use to measure returns?
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Average vs. close
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Bid vs. ask
Can you make real profit out of paper returns?
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Transaction costs, including bid-ask spread
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Liquidity
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Price impact
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What do we know about stock returns?
Characteristics of the distribution
Mean (center):
μ = E[R]
Variance (spread): 2 = E[(R-μ)2]
Skewness (symmetry):
S = E[(R-μ)3/3]
Kurtosis (tail thickness): K = E[(R-μ)4/4]
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Normal distribution
X ~ N(μ, 2)
S=0
K=3
QQ-plot: standardized empirical quantiles vs.
theoretical quantiles from specified distribution
 2 ( K  3) 2
n
Jarque-Berra test for normality: JB   S 
6 
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n : # observations
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JB has an asymptotic chi-square distribution with 2 degrees of
freedom
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
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Histogram of S&P500 monthly returns
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Normal QQ-plot for S&P500 monthly returns
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S&P500: daily returns
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Normal QQ-plot for S&P500 daily returns
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Stylized facts about stock returns
Monthly returns
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Appoximately normal
Daily returns
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Non-normality
– Asymmetry: usually, S < 0 for the indices
– Thick tails: K > 3
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Volatility сlustering
– Esp. at daily frequency
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How can we model stock returns?
Traditional approach:
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Explain asset prices by rational models
Only if they fail, resort to irrational investor behavior
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Behavioral finance models
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What is an efficient market?
Informational efficiency (Fama, 1969):
asset prices accommodate all relevant information
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History of prices / all public variables / all private information
Price movements must be random!
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Otherwise one can forecast future price and make arbitrage profit
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Prices should immediately respond to new information
Why is it important?
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Stock markets: portfolio management
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Corporate finance: choice of the capital structure
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The efficient market hypothesis
EMH: stock prices fully and correctly reflect all relevant
information
Pt+1 = E[Pt+1 |It] + εt+1
Rt+1 = E[Rt+1 |It] + et+1
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The error has zero expectation and is orthogonal to It
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E[Rt+1 |It] is normal return or opportunity cost implied by some model
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Different forms of ME
Weak (WFE):
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I includes past prices
Semi-strong (SSFE):
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I includes all public info
Strong (SFE):
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I includes all (also private) info
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What if the EMH is rejected?
The joint hypothesis problem: we simultaneously test
market efficiency and the model
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Either the investors behave irrationally,
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or the model is wrong
One may not necessarily earn on this
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Ex-ante expected profit is within information acquisition and
transaction costs
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If the EMH is not rejected, then…
the underlying model is a good description of the
market,
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fluctuations around the expected price are unforecastable,
due to randomly arriving news
there is no place for active ptf management…
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technical analysis (WFE), fundamental analysis (SSFE),
or insider trading (SFE) are useless
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the role of analysts limited to diversification, minimizing taxes and
transaction costs
or corporate policy:
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the choice of capital structure has no impact on the firm’s value
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still need to correct market imperfections (agency problem, taxes)
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How realistic is it?
The Grossman-Stiglitz paradox:
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there must be some strong-form inefficiency left
to provide incentives for information acquisition
Operational efficiency:
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one cannot make profit on the basis of info,
accounting for transaction costs
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Different types of tests
Tests of informational efficiency:
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Finding variables predicting future returns
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Statistical significance
Tests of operational efficiency:
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Finding trading rules earning positive profit taking into account
transaction costs and risks
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Economic significance
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Different types of models
Constant expected return:
Et[Rt+1] = μ
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Tests for return predictability
CAPM:
Et[Ri,t+1] – RF = βi(Et[RM,t+1] – RF)
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Tests for mean-variance efficiency
Multi-factor models
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Topics covered in this course
Time series analysis of asset returns
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Predictability at different horizons
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Event study analysis
– Speed of stock price adjustment in response to news announcements
Cross-sectional analysis of asset returns
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CAPM and market efficiency
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Return anomalies and multi-factor asset pricing models
Performance evaluation of mutual funds
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Performance persistence, survivorship bias, dynamic strategies,
gaming behavior, etc.
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How can we model stock returns?
Is this market efficient / is price predictable?
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Martingale and MDS
Let {Yt} be a sequence of random variables
Let {It} be a sequence of info sets, It  I (universal info set)
(Yt, It) is a martingale if
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It  It+1 (filtration)
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Yt  It (Yt is adapted to It)
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E[|Yt|]<
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E[Yt|It-1] = Yt-1 (martingale property)
Example: random walk
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Yt = Yt-1 + εt, εt ~ IID(0, σ2)
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It = {εt, εt-1, …}
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Martingale and MDS
Law of iterated expectations:
E[Yt|It-2] = E[E[Yt|It-1] |It-2] = E[Yt-1|It-2] = Yt-2
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In general, E[Yt|It-k] = Yt-k
(Xt, It) is a martingale difference sequence (MDS) if
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E[Xt|It-1] = 0
Construct MDS from a martingale: Xt = Yt - E[Yt|It-1]
Can we apply it to stock prices/returns?
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Under EMH, stock prices are martingales after detrending
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the unexpected stock returns are MDS: E[Rt+1 - E[Rt+1 |It] |It] = 0
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The random walk hypotheses
Random walk with drift:
Δln(Pt) = μ + εt or rt = μ + εt
RW1: independent and identically distributed
increments, εt~IID(0, σ2)
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Any functions of the increments are uncorrelated
Unrealistic
RW2: independent increments
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Allows for heteroskedasticity
Test using filter rules, technical analysis
RW3: uncorrelated increments, cov(εt, εt-k) = 0, k>0
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Allows for dependence on higher moments (e.g., GARCH)
Test using autocorrelations, variance ratios, regressions
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Autocorrelation tests
Assume that rt is covariance stationary
Test that autocorrelation is 0
For a given lag: Fuller’s test
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Asymptotically, T k ~ N(0,1)
For m lags:
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Box-Pierce statistic: Q ≡ T Σk=1:mρ2(k) ~ 2(m)
Ljung-Box statistic (finite-sample correction):
Q’ ≡ T(T+2) Σk=1:mρ2(k)/(T-k) ~ 2(m)
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Autocorrelation tests: results
CLM, Table 2.4: US, 1962-1994
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The CRSP stock index has significantly positive first-order
autocorrelation at D, W, and M frequency
Economic significance: 12% (R2 = square of 1) of the variation in
daily value-wtd CRSP index is predictable from the last-day return
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The equal-wtd index has higher autocorrelation
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Predictability declines over time
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Variance ratios
Intuition: how to aggregate volatility over time?
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Usually, T rule for standard deviation
Example: compute the variance of a 2-day return:
var(rt + rt-1) = var(rt) + var(rt-1) + 2 cov(rt, rt-1) = {assuming covariance
stationarity and no autocorrelation}= 2 var(rt)
Variance ratio: VR(2) ≡ var(rt(2)) / 2var(rt) = 1 + 1
In general,
VR(q)≡Var[rt(q)]/(qVar[rt]) = 1 + 2k=1:q-1 (1-k/q) k
Under RW1, VR=1
Test statistic:
3Tq
 (q)  (VR (q)  1)
~ asy N (0,1)
2(2q  1)(q  1)
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Variance ratios: results
CLM, Tables 2.5, 2.6, 2.7: US, 1962-1994, weekly data
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Indices: VR(q) rises with time interval (positive autocorrelation),
predictability declines over time, is larger for small-caps
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Individual stocks: weak negative autocorrelation
How to reconcile this contradiction?
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Table 2.8: size-sorted portfolios
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Large positive cross-autocorrelations,
large-cap stocks lead small-caps
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Regression analysis: ARMA models
Testing for long-horizon predictability:, Rt+h(h)=a+bRt(h)+ut+h,
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Non-overlapping horizons: too few observations
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Overlapping horizons: autocorrelation ρ(k)=h-k => use HAC s.e.
Results from Fama&French (1988): US, 1926-1985
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Negative autocorrelation (mean reversion) for horizons from 2 to 7 years,
peak b=-0.5 for 5y
Poterba&Summers (1988): similar results based on VR
Critique:
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Small-sample and bias adjustments lower the significance
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Results are sensitive to the sample period, largely due to 1926-1936 (the
Great Depression)
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Interpretation: Behavioral finance
Investor overreaction
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Assume RW with drift, Et[Rt+1] = μ
There is a positive shock at time τ
The positive feedback (irrational)
traders buying for t=[τ+1:τ+h] after
observing Rτ>μ
SR (up to τ+h): positive
autocorrelation, prices overreact
LR (after τ+h): negative
autocorrelation, prices get back to
normal level
Volatility increases
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Interpretation: Market microstructure
Non-synchronous trading
Low liquidity of some stocks (assuming zero returns for
days with no trades) induces
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negative autocorrelation (and higher volatility) for them
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positive autocorrelation (and lower volatility) for indices
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lead-lag cross-autocorrelations
Consistent with the observed picture (small stocks are
less liquid), but cannot fully explain the magnitude of
the autocorrelations
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Interpretation: More complicated model
Time-varying expected returns:
Et[Rt+1] = Et[RF,t+1] + Et[RiskPremiumt+1]
Changing preferences / risk-free rate / risk premium
Decline in interest rate => increase in prices
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If temporary, then positive autocorrelation in SR, mean reversion in
LR
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Conclusions
Reliable evidence of return predictability at short
horizon
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Mostly among small stocks, which are characterized by low liquidity
and high trading costs
Weak evidence of return predictability at long horizon
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May be related to business cycles (i.e., time-varying returns and
variances)
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