Financial Econometrics) Econ 40357Topic 2: Exploratory data analysis . . .

N.C. Mark

University of Notre Dame and NBER

Thursday 29 August 2019

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Concepts to cover

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Stochastic process, time series

Stochastic means randomA time-series is a sequence of observations over time. We think ofthem as stochastic processes.

The simplist stochastic process. xtiid∼ N(µ, σ2)

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20 replications

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Stationarity

Strict stationarity says the distribution of Xt is the same for all t .So the distribution of Xt is the same as for Xt+1, etc.Covariance stationarity A less restrictive form says thecovariance between Xt and Xt−s is the same, for all t .

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Ergodic Theorem

limT→∞1T

T

∑t=1

xt = E (xt )

limT→∞1T

T

∑t=1

x2t = E

(x2

t

)and so on

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First thing you do

Plot your data.Stare at it.What are we looking for?

Are there mistakes?Are there trends?If so, trends need to be eliminated before doing econometricanalysis.Structrual breaks?Volatility clustering?

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Let’s look at plots of DJIA price and retunrs

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Quarterly returns

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Monthy returns

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Daily returns

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Questions

1 Do monthly DJIA returns look stationary?2 Do daily DJIA returns look stationary?

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The Normal (Gaussian) benchmark

We usually take the normal distribution as a benchmark. Why?B/C properties are well understood, the normal is a good modelfor many natural phenomena, it has good mathematicalproperties, especially on the asymptotics.Focus on symmetry and tail thickness. We know what these arefor the normal. Is the data well described thusly, or are theresignificant deviations?First, we must be familiar with the following concepts

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Moments of a Distribution

The k -th theoretical moment of a distribution or of the randomvariable X )

E(

X k)

The k -th central moment

E (X − µ)k

where the first moment is µ = E (X ) .

Sample moments are the sample counterparts. Let {xt}Tt=1 be a

sequence of time-series observations (e.g., returns).Mean and variance. First two moments

µ = E (Xt ) ; X̄T =1T

T

∑t=1

xt

E (Xt − µ)2 ; σ̂2T =

1T − 1

T

∑t=1

(xt − X̄T

)2

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Moments of a distribution

Third moment for symmetry/asymmetry. Skewness measure

E (Xt − µ)3

σ3 ; skT =1

T−1 ∑Tt=1(xt − X̄T

)σ̂3

T

For normal distribution, skewness measure is 0. It is 0 for all symmetricdistributions.

Fourth moment measures tail thickness. The theoretical measure iskurtosis

E (Xt − µ)4

σ4 ; kurtT =1

T−1 ∑Tt=1(xt − X̄T

)4

σ̂4T

For normal distribution, kurtosis is 3.Distribution has excess kurtosis if the measure exceeds 3. These arefat-tailed distributions and peaked. There is a higher probability ofextreme events than predicted by the normal. Called leptokurtotic. Payattention to whether the software computes kurtosis or excess kurtosis.

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Varying kurtosis

Figure: Distributions with differing kurtosis

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Statistical Testing MethodologyThe classical hypothesis testing methodology is due to R.A. Fisher.

1 Assume the null hypothesis is true. (e.g., µ = 0)2 Determine the sampling distribution of your test statistic under the

null hypothesis. (e.g., the t-statistic, follows a student-t for smallsamples, and N (0,1) for larger samples).

t =1T ∑T

t=1 xt√1T ∑T

t=1(xt − µ̂)2=

µ̂

σ̂

3 Ask if the observed test statistic, computed using data, couldreasonably be drawn from the null distribution.

If answer is yes, data are consistent with the null. You cannot rejectthe null hypothesisIf answer is no, then you can reject the null.

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Visual

Figure: Testing a hypothesis

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Jarque-Bera test for normality

1 The Jarque-Bera statistic measures the difference betweenskewness and kurtosis in the data and the normal distribution.

Jarque-Bera =T6

(sk2

T +(kurtT − 3)2

4

)where skT is sample skewness, and kurtT is sample kurtosis.Under the null, statistic is χ2

2.2 Eviews produces JB test and p-values when asking for descriptive

statistics.

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