CS代考 Lecture Note 2 - PowCoder代写

Lecture Note 2
Stationarity, sample means, and robust regressions
. Lochstoer
UCLA Anderson School of Management

Winter 2022
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Overview of Lecture Note 2
Prices vs. returns, and methods for robust inference
E¢ cient markets, i.i.d. returns, and the Random Walk hypothesis
Covariance stationarity: returns vs. prices
The standard error of the mean return revisited: the central limit theorem Time-varying volatility and Generalized Least Squares
White (robust) standard errors
What drives stock market returns?
. Lochstoer UCLA Anderson School of Management () Winter 2022 2 / 29

A useful benchmark model of returns
Write log returns as:
where the error term, εt , has the following properties
1 Independent across time: f εt,εt+j = f (εt)f εt+j for any t,j
2 Has mean zero: Et1 [εt] = 0 for all t
3 Has unit variance: Vart 1 [εt ] = 1
4 Has Önite skewness and kurtosis (so that typical Central Limit and Law of Large Numbers theorems hold)
Notice: Returns have constant conditional mean and variance, but are not necessarily Normally distributed
rt=μ+σεt, forallt
. Lochstoer UCLA Anderson School of Management () Winter 2022

The Random Walk hypothesis
Given this model, consider the log value of a portfolio, pt , that earns this return each period and that has no distributions (all wealth is reinvested)
pt = pt1+rt
= pt1+μ+σεt.
This value process is said to follow a Random Walk with Drift
A Random Walk with Drift is a process with unforecasteable increments,
except for a constant drift term (μ)
I In particular, ∆pt pt pt1 = μ+σεt, so Et1 [∆pt] = μ, and
Et 1 [pt ] = pt 1 + μ
This is the original E¢ cient Markets model of (1970)
If markets are e¢ cient, you cannot forecast returns (other than the constant risk premium component)
We recognize now that the risk premium (μt rf ,t ) could be time-varying. More on this later in the class.
. Lochstoer UCLA Anderson School of Management () Winter 2022 4 / 29

The Random Walk hypothesis
α in the plot is our μ 2
1.8 1.6 1.4 1.2
1 0.8 0.6 0.4 0.2 0
Random Walk with Drift
α=0.2 per annum;σ=0.2 per annum
0 10 20 30 40 50 60 70 80 90 100
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Covariance Stationarity
In this model, prices are nonstationary while returns are stationary Technically, we will operate with a notion of stationarity that is called
covariance stationarity
Such stationarity is an important condition for most of the econometric techniques you will encounter
DeÖnition
A process fxt g∞ is covariance stationary if E [xt ] = μ and t=∞
Cov xt , xt +j = γj for all t and j . That is, the unconditional mean and covariances exist and are not a function of time t.
A corollary of this is, using the Law of Large Numbers, that the sample mean and covariances are consistent estimates of the true mean and covariances.
. Lochstoer UCLA Anderson School of Management () Winter 2022 6 / 29

Prices and ́s consider the Random Walk model of prices
We get the unconditional expectation by conditioning on the initial
observation, p0, and taking the limit as t ! ∞8:
limE[ptjp0]= limp0+μt= p0 ifμ=0 t!∞ t!∞ :∞ ifμ>0
Thus, if μ 6= 0, the unconditional mean does not exist and it is clear that for any Önite t the expectation is a function of t.
For μ = 0, it looks like weíre Öne. But, we need to check the covariances as well. Letís check for j = 0, i.e. the variance:
lim Var [ptjp0] = lim tσ2 = ∞ t!∞ t!∞
Thus, the unconditional variance of a Random Walk does not exist
=) The wealth process is nonstationary!
. Lochstoer UCLA Anderson School of Management () Winter 2022 7 / 29

Returns and ́s consider the return process:
E[rt] = E[μ+σεt]=μforallt
Var (rt) = Var (μ+σεt) = σ2 for all t =) The return process is stationary!
This is what holds in the data, as well. See next slide.
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Stationary of returns in a picture
Note that itís the unconditional mean and variance that needs to be constant The conditional mean and variance can move around
Monthly log Returns on S&P 500
1916 1930 1943 1957 1971 1984 1998 2012 2026
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

Nonstationarity in a picture
Aggregate output (GDP) and other macroeconomic series are nonstationary At least, thatís the consensus (more on this later)
1971 1984 1998 2012 2026
Real US GDP in 2009 dollars
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

What about valuation ratios?
Market price over cash dividends a stationary variable? S&P500 (Shiller data)
Hard to say using eyeball econometrics…!
100 90 80 70 60 50 40 30 20 10 0
Price/Dividend ratio for S&P500
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
1860 1880 1900 1920
1940 1960 Y ear
1980 2000 2020 2040

The sample mean revisited
Sample means are tremendously important in econometrics
They make up the moments used for identiÖcation of parameters The mean and variance of a sample of returns fr1 , r2 , …, rT g are:
1T mT ET[rt]=T∑rt
E[mT] = T ∑E[rt]=μ
h 2i41T!5 E (mTE[mT]) = E T∑(rtμ)
241T 235σ2 =ET∑σεt =T
. Lochstoer UCLA Anderson School of Management () Winter 2022

Ergodicity
In order to do statistical inference on the sample mean, we need its distribution
Enter the magic of the Central Limit Theorem!
There are lots of them, with di§erent assumptions. We will assume ergocidity, which is a condition that ensures that the variance of the sample mean is Önite.
In the scalar case we are operating in, it is su¢ cient to assume the inÖnite sum of the autocovariances is Önite:
This is trivially the case in our example, where returns are i.i.d. (so all γj = 0 for j > 0) with Önite variance, σ2
. Lochstoer UCLA Anderson School of Management () Winter 2022 13 / 29

The Central Limit Theorem Theorem
If the sample mean has Önite variance and as T ! ∞, the sample mean estimate y ̄T convergesindistributiontoaNormallydistributedvariablewithmeanequalto the true mean and variance equal to the inÖnite sum of autocovariances, S:
p T ( y ̄ T μ ) N ( 0 , S )
Thus, the sample mean in our example is distributed as follows:
mT Nμ,σ2 T
and we can use the usual Normal 95% conÖdence band.
. Lochstoer UCLA Anderson School of Management ()
Winter 2022

The sample mean with heteroskedasticity
Letís extend our model to include time-varying volatility: rt =μ+σt1εt
where jσt1j < ∞ for all t and where V [rt] = σ2. Does this a§ect our test? Notice that the central limit theorem only asks for the unconditional moments. Thus, the test is the same In sum, despite the non-normalities found in the data, the Central Limit Theorem provides a robust testing framework as long as the sample is su¢ ciently large . Lochstoer UCLA Anderson School of Management () Winter 2022 15 / 29 OLS revisited Letís next consider how time-varying volatility and non-normalities a§ects regressions Recall OLS: Y=Xβ+ε. (T 1) (T K )(K 1) (T 1) εt N 0,σ forallt Thus, the residualsívariance-covariance matrix is: E εε0 = σ2IT The standard OLS assumption is that the error term is normally i.i.d. distributed: i.i.d. 2 . Lochstoer UCLA Anderson School of Management () Winter 2022 16 / 29 Heteroskedastic Error Terms What if, as is typically the case for Önancial data, error terms are heteroskedastic? Letís stick with Normal distribution for now: εt N 0, σ2t Also, let error terms be uncorrelated across time: E εt εt +j = 0 for all j 6= 0. Now the residual variance-covariance matrix is 26 σ 21 0 0 37 Σ=6 0 σ2 0 7 4 . . ... . 5 0 0 σ2T Intuitively, when estimating the regression coe¢ cients, you want to weight observations with lower residual variance (less noisy observations) more than observations with higher residual variance . Lochstoer UCLA Anderson School of Management () Winter 2022 17 / 29 Generalized Least Squares (GLS) Matrix inversion can be tricky, but not with diagonal matrices Consider the matrix Σ1/2: 2 σ1 0 0 3 611 7 Σ1/2 =6 0 σ2 0 7 4 . . ... . 5 0 0 σ1 RedeÖne the independent and dependent variables: Y ̃ = Σ1/2Y and X ̃ = Σ1/2X Consider the GLS regression: Y ̃ =X ̃β+ε ̃ What is the covariance matrix of the GLS residuals? I Simply, IT . Thus, OLS is optimal in this alternative regression! The regression coe¢ cients thus can be written: βˆGLS = X0Σ1X1 X0Σ1Y N βnull,X0Σ1X1 . Lochstoer UCLA Anderson School of Management () Winter 2022 Feasible GLS Issue: we need to know the variance-covariance matrix of the residuals before running the regresion Feasible GLS is a two-pass approach 1 First pass: Run OLS, estimate σ2j using σˆ2j = εˆ2OLS,j for j = 1,...,T 2 Second pass: Run GLS using σˆ2j instead of (the unknown) σ2j Issue: The σˆ2j are quite noisy estimates, can lead to very noisy βˆGLS estimates I Defeats the purpose, which was e¢ ciency gain Many researchers prefer to run OLS and instead adjust the standard errors for the heteroskedasticity So-called írobust standard errorsí An asymptotic adjustment that relies on the Central Limit Theorem is also robust to unconditionally non-normal residuals . Lochstoer UCLA Anderson School of Management () Winter 2022 19 / 29 Asymptotic OLS Consider the OLS regression: wherext andβareK1vectors yt =xt0β+εt, εt is a mean-zero error term with variance σ2t < ∞. It need not be Normally distributed. Assume as before that E εtεt+j = 0 for all j 6= 0 We still need the OLS identifying assumption: E x t0 ε t = 0 . Lochstoer UCLA Anderson School of Management () Winter 2022 20 / 29 Autocorrelation Is the assumption E εt εt +j = 0 reasonable? Since the E [εt ] = 0, this is the same as asking whether residuals are correlated across time Not a bad assumption for, say, monthly returns in a relatively e¢ cient market I E.g. stock market Can be a bad assumption in more ine¢ cient prices series I E.g., real estate market First-order autocorrelation is simply corr (εt , εt +1 ) Correlation of adjacent observations . Lochstoer UCLA Anderson School of Management () Winter 2022 Autocorrelation in stock market Not much with monthly data -0.2 -0.1 0 0.1 Returns at t 0.2 0.3 0.4 Scatter plot of monthly log returns (VW-CRSP) 1925-2013. . Lochstoer UCLA Anderson School of Management () Winter 2022 Returns at t+1 Autocorrelation in real estate market -0.03 -0.02 0 0.01 0.02 0.03 Log Housing Returns at t Monthly Case- Returns for Arizona Scatter plot for Monthly log House Price Changes in AZ. Case- . 1987.1-2013.10 . Lochstoer UCLA Anderson School of Management () Winter 2022 Log Housing Returns at t+1 The OLS Moment Condition DeÖne the OLS moment condition for the estimated βˆ: f t βˆ = x t y t x t0 βˆ Let the sample mean of the moment condition be: ˆ1Tˆ From the Central Limit Theorem: In standard OLS, the squared error term is uncorrelated with xt xt0 as the variance is constant. gT β = T ∑ ft β = 0 t=1 p T g T βˆ N ( 0 , S T ) 1T ˆˆ0 1T 02 ST=T∑ft βft β =T∑xtxtεˆt t=1 t=1 . Lochstoer UCLA Anderson School of Management () Winter 2022 White (robust) standard errors In the end, we want the distribution of βˆ Note that, asymptotically gT (β)=E[xtyt]Extxt0β βˆβN0,T1ET xtxt01STET xtxt01 With constant variance OLS, ST = ET [xt xt0 ] ET εˆ2t . In sum, OLS regressions in large samples Are unbiased Standard errors need to be adjusted for heteroskedasticity Do not require normally distributed errors We will deal with cases where E [εt εt +1 ] 6= 0 later . Lochstoer UCLA Anderson School of Management () Winter 2022 Take-aways 1 Make sure you are estimating your model using stationary data I Historical samples "representative," Central Limit Theorem applies I Possible to work with non-stationary data using cointegration analysis, but in practice not much used 2 OLS regressions are unbiased and yield correct inference if sample is large I Do not need Normally distributed errors or constant variance of residuals (homoscedasticity) I Adjust standard errors (robust; White (1980)), we will do Newey-West for autocorrelation later I Explicitly modeling non-normalities and heteroskedasticity in a small sample often entails estimation error that outweighs potential beneÖt . Lochstoer UCLA Anderson School of Management () Winter 2022 26 / 29 What drives stock returns? A big question is "What are the sources of stock returns?" Macroeconomic factors, e.g. consumption? Aggregate Örm earnings? Monetary policy, interest rate movements? Answer: Yes, to some extent, but large fraction of stock returns cannot easily be tied to the above factors . Lochstoer UCLA Anderson School of Management () Winter 2022 27 / 29 Contemporaneous regressions Using annual data from ́s webpage (1889 - 2015), we run: rt = β0 + β1∆const + β2∆earnt + β3∆intt + εt ∆const is the annual di§erence in log per capita real consumption ∆earnt is the annual di§erence in log real stock market earnings ∆intt is the annual di§erence in 1-year T-bill rate Result: White t-stat in parenthesis, Ra2dj = 13.4% rt = 0.06 0.49 ∆const + 0.21 ∆earnt 2.57 ∆intt + εt (3.49) (1.02) (3.28) (3.01) . Lochstoer UCLA Anderson School of Management () Winter 2022 Forward-looking regressions Markets are forward-looking Use 5-year moving average of regressors instead E.g., use ln(const+4)ln(const1) instead of Örst di§erence ∆const =ln(const)ln(const1) Keep one-year return, rt , on left hand side Result: White t-stat in parenthesis, Ra2dj = 4.1% rt = 0.04 + 0.16∆const1,t+4 + 0.07∆earnt1,t+4 0.83 ∆intt1,t+4 +εt (1.06) (0.53) (1.69) (1.73) So, what drives stock returns?? Speculation? Errors in expectations? Risk tolerance? . Lochstoer UCLA Anderson School of Management () Winter 2022 程序代写 CS代考加微信: powcoder QQ: 1823890830 Email: powcoder@163.com

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