Class Review and Final Preparations
. Lochstoer
UCLA Anderson School of Management
Winter 2022
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. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 1 / 28
1 Empirical Methods in Finance vs. the rest of the curriculum
2 Review of main topics
I What are main lessons?
I What will be on the Önal?
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 2 / 28
Empirical Methods in Finance vs.
Other Classes
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 3 / 28
Empirical Methods in Finance
This class is a continuation of Econometrics and Investments Goals are to
1 provide you with the baseline econometric tools needed for subsequent MFE classes and to conduct empirical quantitative research
2 enable you to more easily understand and implement other, potentially more advanced, time-series models for use in quantitative analysis
The class is hard, but necessarily so as we need to get you quickly up to speed in order to focus the rest of your time here on tackling a wide variety of interesting, realistic problems
Other topic of interest: ̈lter, cointegration and non-stationary time series analysis, see chapters 8.5, 8.6, and 11 in Tsayís book
I regime switching (and other nonlinear) models (see Ch. 4 in Tsay).
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 4 / 28
The Link to Future Classes
Quantitative Asset Management
Factor models, Fama-MacBeth regressions, portfolio sorts, íalphaí, information and appraisal ratios
Fixed Income Markets
Principal Components Analysis, forecasting regressions, VARs
Advanced Stochastic Calculus
Multifactor models and no-arbitrage pricing, models of heteroscedasticity, jumps, non-normalities
Financial Risk Management
Models of heteroscedasticity, Value-at-Risk, factor models and hedging, non-normalities
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022
The Link to Future Classes (contíd)
Data Analytics
ARMA, VARs, Principal Components Analysis, cross-sectional regressions, learning and shrinkage, factor models
Behavioral Finance
Portfolio sorts, forecasting regressions, cross-sectional regressions, factor models
Statistical Arbitrage
íAlphaí, Principal Components Analysis, shrinkage, factor models
Credit Markets
VARs, Principal Components Analysis, jumps, non-normalities
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 6 / 28
Class Review
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 7 / 28
Basics of asymptotics (Central Limit Theorem and Law of Large Numbers)
Stationarity important for Central Limit Theorem, important for any moment condition that is a sample average
White standard errors, later Hansen-Hodrick and Newey-West standard errors Note on Asymptotic Theory posted on CCLE
I will not ask detailed questions about aymptotic theory on the Önal. But:
You have to understand what stationarity means and how to achieve a stationary time-series
Asset returns typically highly non-normal (non-zero skewness, excess kurtosis): how does this a§ect regression inference?
You have to understand when to apply White standard errors
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 8 / 28
Lectures 3 and 4
Autocorrelations and, in particular, the autocorrelation function describe the time-dependencies in a time series.
Stock returns exhibit interesting autocorrelation patterns
1 Short-term reversal (negative autocorrelation)
2 Momentum (positive autocorrelation)
3 Long-term reversal (negative autocorrelation)
Di§erent patterns are more dominant at di§erent frequencies
Understand what the autocorrelation function is Know the Ljung-Box test
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 9 / 28
ARMA models
ARMA models can capture any linear function of past data, any autocorrelation pattern
Optimal linear forecast (based on univariate data)
Stationarity requirements
Multi-period forecasting
Autocorrelation functions (how to derive), partial autocorrelation function AIC, BIC
Conditional and unconditional variances
Dynamic multipliers
Exact vs. conditional likelihood functions for AR
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 10 / 28
Return predictability Forecasting regressions
I Overlapping observations
Newey-West standard errors (when do we use them, what is lag length?)
Campbell-Shiller approximation and expression for price-dividend ratio Variance decomposition of prices: discount rates or cash áows
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 11 / 28
Vector Autoregressions Multi-variate AR model
Get multi-horizon forecasts based on set of predictors Now how to forecast using VAR(1), dynamic multiplier
Donít worry about:
Vec and kronecker products
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 12 / 28
GARCH models
Stylized facts on stock market conditional variance
ARCH and GARCH models in general
I In particular, know GARCH(1,1) well
I What additional stylized fact can EGARCH and GJR-GARCH account for? I Forecasting with GARCH models
Realized variance
I How to construct, what are beneÖts?
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 13 / 28
Multivariate volatility models will not be on Önal
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 14 / 28
Lecture 10
Please understand portfolio construction through factors RPt = Rft +w1RFe1,t +…+wkRFek,t
Note: no restriction that wís need to sum to 1!
Example 1:
Clearly, 0.8 6= 1. But, not a problem.
R P t = R f t + 0 . 8 R Me k t , t
I Net weight in risk-free asset is 1 0.8 = 0.2. Net weight in market is 0.8.
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 15 / 28
Lecture 10 – contíd
Example 2:
RPt = Rft + 0.8RMe kt,t 0.3RHML,t Clearly, 0.8 0.3 6= 1. But, not a problem.
I Net weight in risk-free asset is 1 0.8 = 0.2. Net weight in market is ..?
I You have an initial position in market of 0.8. But, you are then overweighting growth stocks and underweighting value stocks relative to the market portfolio since RHML,t = RV ,t RG ,t
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 16 / 28
Lecture 10 – contíd
Example 3:
You are evaluating a fund with historical returns RPt.
The fund claims it follows a stock-picking strategy, but is actually a closet-indexer. In particular, it invests all itís money each period in a market index fund and goes short an amount equal to half of that in a growth stock portfolio. It then invests the proceeds of the short position in a long value portfolio. Assume the growth and value portfolios are the same as those underlying the HML factor.
You run the regression
RPt Rft =α+βMktRMe kt,t +βHMLRHML,t +εt What are your estimated βˆMkt and βˆHML? βˆMkt = 1, βˆHML = 0.5
What is your estimate of α? αˆ = 0.
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 17 / 28
Lectures 10 and 11
Please know factor portfolio math:
Re =α +β0F +ε , foralli
it i it it
where E ε2it = σ2i , and where E εitεjt = 0 for all i 6= j.
Know how to calculate expected returns, variances, and covariances in this setup
Please note: the factor model does not necessarily imply that αi = 0. Need additional assumptions for this.
I Typically: assume εís can be diversiÖed away (Arbitrage Pricing Theory)
Principal Components Analysis: natural way to Önd factors Run PCA using excess returns
Eigenvectors are then the wís for the corresponding factor in the previous slides and give each assetís weight in the zero-investment PCA factors
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 18 / 28
Lecture 12
The assumption of no-arbitrage makes the αís in the factor model zero if the factors are traded
In general, the factor model implies that:
E (Re ) = β0λ it i
If factor j is traded and expressed as an excess return, we have: E Fjt = λ(j)
as the factor has a beta of one with respect to itself and zero to all other factors.
In other words, the price of risk of a traded factor is its risk premium
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 19 / 28
Lecture 12
Empirically, a lot of factors that drive the covariance matrix of stock returns arenít priced
I.e., λ(j) = 0 if factor j is not priced.
I Thus, the risk premium of a traded factor that is priced is signiÖcantly
di§erent from zero.
Example: industry factors are not priced, typically, while the HML factor of Fama and French is.
Thus, while PCA o§ers an intuitive way of getting at the most important factors (e.g., industry factors), it is an empirical stylized fact that priced factors in stock returns are not well-identiÖed by PCA analysis
PCA is useful, however, for Önding factors that add variance We may want to hedge out such factors
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 20 / 28
Lecture 12
A linear beta-pricing model (our factor models) with traded factors prices all assets if and only if…
..the factors span the mean variance e¢ cient portfolio
That means, the mean-variance e¢ cient portfolio can be constructed by
trading the factors only:
μ0V 1μ = λ0Σ 1λ F
where the left-hand side is the maximum squared Sharpe ratio of all assets and the right-hand side is the maximum squared Sharpe ratio of the factors
Please, know mean-variance math!
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 21 / 28
Lecture 12
Continuing from the previous slide. In general, it is true that
μ0V 1μ = λ0Σ 1λ+α0Σ 1α. Fε
Thus, under the null that a given factor model prices all assets, we have that α0Σ 1α = 0.
This is not a function of investorsípreferences, mean-variance risk criteria, etc. Itís just math. An implication of the linear factor model framework.
Another mechanical implication:
If you uncover αís for a particular factor model, it implies that you can achieve a higher Sharpe ratio than one can using the factors alone
Understand how to implement this
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 22 / 28
Lecture 13
Fama-MacBeth runs this cross-sectional regression each period.
Mostly useful is betaís or other Örm characteristics vary over time:
1T1T1T λ0=T ∑λ0,t, λ1=T ∑λ1,t, α ̃i=T ∑α ̃i,t
Re=λ+λbm +α ̃ i,t 0,t 1,t i,t 1
t=1 t=1 t=1
Note that this is simply a panel forecasting regression, where the regression
coe¢ cient is the same across Örms. From this regression, we have:
Re =λ +λ bm
t 1 i,t 0 1 i,t 1
Note, this regression does not have λ0 = 0 as a null hypothesis (bmi,t 1 is
not a beta)
What is λ1 in this case?
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 23 / 28
FM Interpretation: Example
In this case, OLS implies that:
bm N 1 bmi,t Eti [bmi,t] e λt+1=∑N vari(bm ) Ri,t+1
i=1| t{zi,t} =wi,t
where Eti [] and varti () denote the cross-sectional mean and variance of bmi,t at time t.
Note that λbm is a long-short portfolio return t+1
The (excess) return to a portfolio that is long high book-to-market stocks and short low book-to-market stocks.
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 24 / 28
FM Interpretation: Example (contíd)
The Önal estimate of the Fama-MacBeth regression is ˆ1Tˆ
λ=T ∑λt+1 t=1
Thus, the estimated coe¢ cient λˆbm is the mean return to the portfolio that goes long high bm stocks and short low bm stocks.
The expected excess return to a factor-mimicking portfolio
Note that the intercept is capturing the average return to all stocks and is what makes the mimicking portfolio a long-short portfolio
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 25 / 28
How Should I Study for the Final?
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 26 / 28
How to Study
1 Read lecture notes and handout on factor models and asymptotics I The Tsay text book is a useful reference
2 Understand the homeworks, in particular the parts that were analytical.
3 You do not need to know any of the coding for the Önal.
4 Finals from preceding years are posted
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 27 / 28
Good luck and thanks for being great students!!
I hope to see many of you next quarter for Data Analytics and Machine Learning
. Lochstoer UCLA Anderson School of Management () Class Review and Final Preparations Winter 2022 28 / 28
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