financial econometrics) econ 40357 topic 2: exploratory ...nmark/financialeconometrics/lecture...
TRANSCRIPT
![Page 1: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/1.jpg)
Financial Econometrics) Econ 40357Topic 2: Exploratory data analysis . . .
N.C. Mark
University of Notre Dame and NBER
Thursday 29 August 2019
1 / 18
![Page 2: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/2.jpg)
Concepts to cover
2 / 18
![Page 3: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/3.jpg)
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)
3 / 18
![Page 4: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/4.jpg)
20 replications
4 / 18
![Page 5: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/5.jpg)
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 .
5 / 18
![Page 6: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/6.jpg)
Ergodic Theorem
limT→∞1T
T
∑t=1
xt = E (xt )
limT→∞1T
T
∑t=1
x2t = E
(x2
t
)and so on
6 / 18
![Page 7: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/7.jpg)
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?
7 / 18
![Page 8: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/8.jpg)
Let’s look at plots of DJIA price and retunrs
8 / 18
![Page 9: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/9.jpg)
Quarterly returns
9 / 18
![Page 10: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/10.jpg)
Monthy returns
10 / 18
![Page 11: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/11.jpg)
Daily returns
11 / 18
![Page 12: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/12.jpg)
Questions
1 Do monthly DJIA returns look stationary?2 Do daily DJIA returns look stationary?
12 / 18
![Page 13: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/13.jpg)
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
13 / 18
![Page 14: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/14.jpg)
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
14 / 18
![Page 15: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/15.jpg)
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.
15 / 18
![Page 16: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/16.jpg)
Varying kurtosis
Figure: Distributions with differing kurtosis
16 / 18
![Page 17: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/17.jpg)
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.
17 / 18
![Page 18: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/18.jpg)
Visual
Figure: Testing a hypothesis
18 / 18
![Page 19: Financial Econometrics) Econ 40357 Topic 2: Exploratory ...nmark/FinancialEconometrics/Lecture Slides/Topic02.pdfFinancial Econometrics) Econ 40357 Topic 2: Exploratory data analysis](https://reader030.vdocuments.net/reader030/viewer/2022040105/5e9282acad9bf3220f7bfe21/html5/thumbnails/19.jpg)
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.
19 / 18