Lecture 3 Autocorrelation
. Lochstoer
UCLA Anderson School of Management
Winter 2022
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. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Overview of Lecture 3
Autocorrelation
1 Introduction to autocorrelations
I The autocorrelation function 2 The Ljung-Box Q-test
. Lochstoer UCLA Anderson School of Management () Winter 2022
Correlation DeÖnition
The correlation between two random variables X and Y is deÖned as ρx,y = p Cov(X,Y)
Var(X)Var(Y)
ρx,y is known as Pearsonís correlation
measures linear dependence
bounded between 1 and 1
two variables are uncorrelated if ρx,y = 0, perfectly (negatively) correlated if ρx,y =1(ρx,y = 1)
if X and Y are random normal variables, then ρx,y = 0 if and only if X and Y are independent
. Lochstoer UCLA Anderson School of Management () Winter 2022 3 / 32
Sample Correlation DeÖnition
The sample correlation between two random variables X and Y is: bρx,y = q ∑Tt=1(xt x)(yt y)
∑Tt=1(xt x)2 ∑Tt=1(yt y)2 where x and y are the sample means.
this is not a regression coe¢ cient
bρx,y consistently estimates ρx,y
bρx,y is built from method of moments estimators
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
The autocorrelation for a series frt g is deÖned as:
ρj = q Cov(rt,rt j) = Cov(rt,rt j) = γj .
Var(rt)Var(rt j) Var(rt) γ0
a covariance-stationary series rt is not serially correlated if ρj = 0 for all j
autocorrelations are a key signature of the dynamics of the time series youíre interested in modeling
líth autocorrelation is a regression coe¢ cient in univariate regression of rt and rt j
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Autocorrelation: why important?
First, from covariance-stationarity
Cov(rt,rt j) = Cov(rt+j,rt)
Any time Cov(rt +j , rt ) 6= 0, we have that current value of the series, rt , can
predict future realizations.
Patterns in Cov(rt +j , rt ) vs j tell you a lot about the nature of predictability within the series you are looking at.
Note: autocorrelation, as opposed to autocovariance, is convenient for intuition as the scale is easy to understand
In fact, autocorrelations tell you about which model you need for capturing the predictability of the series at any horizon
. Lochstoer UCLA Anderson School of Management () Winter 2022
1st Order Önition
The sample autocorrelation for a series fxt g is:
bρ 1 = ∑ Tt = 2 ( x t x ) ( x t 1 x ) , 0 j T 1
∑Tt=1(xt x)2 where x are the sample means.
under some conditions, bρ1 is a consistent estimator of ρ1
bρ1 is asymptotically normal with mean zero and variance (1/T ) if fxt g are
independently and identically distributed over time. totestH0 :ρ1 =0,uset-stat:
t = p T bρ 1
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Autocorrelation in stock returns?
-0.4 -0.3 -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 Returns?
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
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Log Housing Returns at t+1
Higher-order Önition
The sample autocorrelation for a series fxt g at lag j is:
bρ j = ∑ Tt = j + 1 ( x t x ) ( x t j x ) , 0 j T 1
∑Tt=1(xt x)2 where x are the sample means.
. Lochstoer UCLA Anderson School of Management ()
Winter 2022
Autocorrelation
1 Önancial time series: e.g. stock returns and housing returns
I Önancial returns tend to be only very weakly autocorrelated [if markets are fairly e¢ cient and liquid]
I strong autocorrelations in returns would create huge proÖt opportunities! 2 macroeconomic time series: e.g. GDP growth rates
I macroeconomic time series have growth rates that are highly autocorrelated I macroeconomic shocks tend to have very persistent e§ects (e.g. think about
the e§ect of the subprime crisis on GDP growth rates)
. Lochstoer UCLA Anderson School of Management () Winter 2022 11 / 32
Autocorrelation vs. “Persistence”
Persistence (or persistent) is not a precisely deÖned term, but refers to how long-lasting shocks are in terms of their impact on the series at hand
A “persistent” time series refers to a time series with high autocorrelation at some lag (in absolute value, really, but in economics typically high and positive)
Persistence is typically used in qualitative description/discussion
Autocorrelations and the autocorrelation function (to be deÖned) are a little more technical
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The Autocorrelation Function (ACF)
DeÖnition
The sample autocorrelation function (ACF) of a time series is deÖned as bρ1,bρ2,…,bρk,…,
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Autocorrelation in Quarterly GDP Growth
Sample Autocorrelation Function
Autocorrelation Function for Quarterly U.S. GDP growth. Two standard error bands around zero. 1947.I-2012.IV
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Autocorrelation in Monthly Ináation
Monthly Autocorrelation — U.S. Inf lation
Autocorrelation Function for Monthly U.S. Ináation. 1950-2007. ρˆ1 = 0.52. . Lochstoer UCLA Anderson School of Management () Winter 2022
0 10 20 30 40 50 60 70
Autocorrelation of Annual Log Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Annual log Returns on VW-CRSP Index. Two standard error bands around zero. 1926-2012 . ρˆ1 = 0.05.
0 1 2 3 4 5 6 7 8 9 10 Lag
. Lochstoer UCLA Anderson School of Management () Winter 2022
Sample Autocorrelation
Autocorrelation of Monthly Log Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Monthly log Returns on VW-CRSP Index. Two standard error bands around zero. 1926-2012 . ρˆ1 = 0.088.
0 2 4 6 8 10 12 14 16 18 20 Lag
. Lochstoer UCLA Anderson School of Management () Winter 2022
Sample Autocorrelation
Autocorrelation of Monthly Log Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Monthly log Returns on EW-CRSP Index (equal weighted). Two standard error bands around zero. 1926-2012 . ρˆ1 = 0.12.
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Autocorrelation of Monthly Log Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Monthly log Returns on VW-CRSP Index. Two standard error bands around zero. 1990-2012. ρˆ1 = 0.09.
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Autocorrelation of Monthly Log Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Monthly log Returns on EW-CRSP Index (equal weighted). Two standard error bands around zero. 1990-2012. ρˆ1 = 0.17.
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Monthly ACF for the Stock Market
some positive autocorrelation in stock returns (for the market) at the one-month horizon
autocorrelation is stronger for small stocks (see EW-CRSP)
I bρ1 varies between 0.09 (VW) and 0.17 (EW) on the short sample
I bρ1 varies between 0.08 (VW) and 0.12 (EW) on the long sample
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Autocorrelation of Daily Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Daily log Returns on VW-CRSP Index (value-weighted). Two standard error bands around zero. 1926-2012. ρˆ1 = 0.07.
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Autocorrelation of Daily Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Daily log Returns on EW-CRSP Index (equal-weighted). Two standard error bands around zero. 1926-2012. ρˆ1 = 0.21.
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Autocorrelation of Daily Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Daily log Returns on VW-CRSP Index. Two standard error bands around zero. 1990-2012 . ρˆ1 = 0.01.
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Autocorrelation of Daily Stock Returns
Sample Autocorrelation Function
Autocorrelation Function for Daily log Returns on EW-CRSP Index. Two standard error bands around zero. 1990-2012 . ρˆ1 = 0.13.
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Daily ACF for the Stock Market
some positive autocorrelation in stock returns (for the market) at the one-day horizon, but mainly for small stocks
I bρ1 varies between 0.01 (VW) and 0.13 (EW) on the short sample I bρ1 varies between 0.07 (VW) and 0.21 (EW) on the long sample
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Benchmark Model of Portfolio Theory
In the benchmark model of portfolio theory, returns are assumed to be independently and identically distributed (i.i.d.) over time.
I If returns are i.i.d., the variance grows linearly in the investment horizon I The investorís horizon turns out to be irrelevant for optimal portfolio
allocation.
Not quite true in the data
. Lochstoer UCLA Anderson School of Management () Winter 2022
Autocorrelation of Monthly Log House Price Changes
Sample Autocorrelation Function
Autocorrelation Function for Monthly log House Price Changes. Two standard error bands around zero. 1987-2013
0 2 4 6 8 10 12 14 16 18 20 Lag
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Sample Autocorrelation
Testing for autocorrelation: Ljung and Box (1978)
DeÖnition
The Ljung-Box statistic tests the null that H0 : ρ1 = … = ρm = 0 Q ( m ) = T ( T + 2 ) ∑m bρ 2i
i=1 T i Q(m) is asymptotically χ2 with m degrees of freedom.
reject the null if Q (m) > χ2 (α) where χ2 (α) denotes the (1 α) 100-th percentile
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Q-test on Monthly Returns
100 90 80 70 60 50 40 30 20 10
00 0 10 20 30 40 50
0.05 0.045 0.04 0.035 0.03 0.025 0.02 0.015 0.01 0.005
Q-test for Monthly log Returns on VW-CRSP Index. 1926-2012
. Lochstoer UCLA Anderson School of Management () Winter 2022
0 10 20 30 40 50 horizon
Monthly Stock Returns
choice of m matters: rule of thumb m = ln(T )
if we use this rule of thumb, m = 6 and Q(6) = 26 and p-value is 1.9e 4
in any case, we reject the null that there is no autocorrelation in monthly U.S. stock returns for all holding periods considered..
even though these autocorrelations are small, theyíre measured rather precisely, allowing us to reject the null.
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White Noise DeÖnition
A time series εt is said to be white noise if fεt g is a sequence of independent and identically distributed random variables.
Notation: εt WN 0, σ2ε
If εt is white noise + normally distributed with mean zero and variance σ2, then it
is called Gaussian white noise. Notation: εt GWN 0, σ2ε
There is no autocorrelation. ACFís are all zero
. Lochstoer UCLA Anderson School of Management () Winter 2022 32 / 32
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