Materials for Lecture 12 - PowerPoint Presentation

Slide1

Materials for Lecture 12

Chapter 7 – Study this closelyChapter 16 Sections 3.9.1-3.9.7 and 4.3Lecture 12 Multivariate Empirical Dist.xlsLecture 12 Multivariate Normal Dist.xls

Slide2

Multivariate Probability Distributions

Definition: Multivariate (MV) Distribution --Two or more random variables that are correlatedMV you have 1 distribution with 2 or more random variables Univariate distribution we have many distributions (one for each random variable)

Slide3

Parameter Estimation for MV Dist.

Data were generated contemporaneously Output observed each year or month, Prices observed each year for related commoditiesCorn and sorghum used interchangeably for animal feed

Steer and heifer prices related

Fed steer price and Feeder steer prices related

Supply and demand forces affect prices similarly, bear market or bull market; prices move together

Prices for tech stocks move together

Prices for an industry or sector’s stocks move together

Slide4

Different MV Distributions

Multivariate Normal distribution – MVNMultivariate Empirical – MVE Multivariate Mixed where each variable is distributed differently, such as X ~ Uniform

Y ~ Normal

Z ~ Empirical

R ~ Beta

S ~ Gamma

Slide5

Sim MV Distribution as Independent

If correlation is ignored when random variables are correlated, results are biased:If Z = Ỹ

1

+ Ỹ

2

OR

Z =

Ỹ1

* Ỹ

2

and the model is simulated without correlation

But the true

ρ

1,2 > 0 then the model will understate the risk for Z But the true ρ1,2 < 0 then the model will overstate the risk for Z If Z = Ỹ1 * Ỹ2The Mean of Z is biased, as well

Slide6

Parameters for a MVN Distribution

Deterministic componentŶij -- a vector of means or predicted values for the period i to simulate all of the j variables, for example:

Ŷ

ij

= ĉ

0

+ ĉ1 X

1

+ ĉ

2

X

2

Stochastic componentêji -- a matrix of residuals from the predicted or mean values for each (j) of the M random variables êji = Yij – Ŷij and the Std Dev of the residuals σêjMultivariate component

Covariance matrix (

Σ

) for all M random variables in the distribution MxM covariance matrix (in the general case use correlation matrix) Estimate the covariance (or correlation) matrix using residuals about the forecast (or the deterministic component) σ211 σ12 σ13 σ14 1 ρ12 ρ13 ρ14 Σ = σ222 σ23 σ24 OR Ρ = 1 ρ23 ρ24 σ233 σ34 1 ρ34 σ244 1

13

Slide7

3 Variable MVN Distribution

Deterministic component for three random variablesĈi = a + b1

C

i-1

Ŵ

i

= a + b1Ti

+ b

2

W

i-1

Ŝi = a + b1TiStochastic component êCi = Ci – ĈiêWi

= W

i

– ŴiêSi = Si – ŜiMultivariate component σ2cc σcw σcs Σ = σ2ww σws σ2ss

Slide8

Simulating MVN in Simetar

One Step procedure for a 4 variable Highlight 4 cells if the distribution is for 4 variables, type

=MVNORM( 4x1Means Vector, 4x4 Covariance Matrix)

=MVNORM( A1:A4 , B1:E4)

Control Shift Enter

where:

the 4 means or forecasted values are in column A rows 1-4,

covariance matrix is in columns B-E and rows 1-4

If you use the historical means, the MVN will validate perfectly, but only forecasts (simulates) the future if the data are stationary.

If you use forecasts rather than means, the validation test fails for the mean vector.

The CV will differ inversely from the historical CV as the means increase or decrease relative to history

Slide9

Example of Mean vs. Y-Hat Problem for Validation

Slide10

Simulating MVN in Simetar

Two Step procedure for a 4 variable MVN Highlight 4 cells if the distribution is for 4 variables, and type

=CUSD (Location of Correlation Matrix)

Control Shift Enter

=CUSD (B1:E4) for a 4x4 correlation matrix in cells B1:E4

Next use the individual CSNDs to calculate the random values, using Simetar NORM function:

For

Ỹ

1

= NORM( Mean

1

,

σ

1 , CUSD1 )For Ỹ2 = NORM( Mean2 , σ2 , CUSD2 )For

Ỹ

3

= NORM( Mean3 , σ3 , CUSD3 )For Ỹ4 = NORM( Mean4 , σ4 , CUSD4 )Use Two Step if you want more control of the process

Slide11

Example of MVN Distribution

Demonstrate MVN for a distribution with 3 variables

One step procedure in line 63

Means in row 55 and covariance matrix in B58:D60

Validation test shows the random variables maintained historical covariance

Slide12

Two Step MVN Distribution

Slide13

Review Steps for MVN

Develop parameters Calculate averages (and standard deviations used for two step procedure)Calculate Covariance matrixCalculate Correlation matrix (Used for Two Step procedure and for validation of One Step procedure)

One Step MVN procedure is easier

Use Two Step MVN procedure for more control of the process

Validate simulated MVN values vs. historical series

If you use different means than in history, the validation test for means vector WILL fail

Slide14

Parameters for MV Empirical

Step I Deterministic component for three random variablesĈi = a + b1

C

i-1

Ŵ

i

= a + b1T

i

+ b

2

W

i-1

Ŝi = a + b1TiStep II Stochastic component will be calculated from residualsêCi = Ci – ĈiêWi = W

i

– Ŵ

iêSi = Si – ŜiStep III Calculate the stochastic empirical distributions parameterSCi = Sorted (êCi / Ĉi)SWi = Sorted (êWi / Ŵi)SSi = Sorted (êSi / Ŝi)Step IV Multivariate component is a correlation matrix calculated using unsorted residuals in Step II

Slide15

Simulating MVE in Simetar

One Step procedure for a 4 variable MVE Highlight 4 cells if the distribution is for 4 variables, then type=MVEMP( Location Actual Data ,,,, Location Y-Hats, Option)

Option = 0 use actual data

Option = 1 use Percent deviations from Mean

Option = 2 use Percent deviations from Trend

Option = 3 use Differences from Mean

End this function with

Control Shift Enter

=MVEMP(B5:D14 ,,,, G7:I6, 2)

Where the 10 observations for the 3 random variables are in rows 5-14 of columns B-D and simulate as percent deviations from trend

Slide16

Two Step MVE

Two Step procedure for a 4 variable MVE Highlight 4 cells if the distribution is for 4 variables, type

=CUSD( Location of Correlation Matrix)

Control Shift Enter

=CUSD( A12:A15)

Next use the CUSDs to calculate the random values

(Mean here could also be

Ŷ)

For

Ỹ

1

= Mean

1 + Mean1 * Empirical(S1, F(Si) , CUSD1) For Ỹ

2

= Mean

2 + Mean2 * Empirical(S2, F(Si) , CUSD2) For Ỹ3 = Mean3 + Mean3 * Empirical(S3, F(Si) , CUSD3) For Ỹ4 = Mean4 + Mean4 * Empirical(S4, F(Si) , CUSD4)Use Two Step if you want more control of the process

Slide17

Parameter Estimation for MVE

Slide18

Simulate a MVE Distribution

Slide19

If Cannot Factor

Correl matrixWhen the Matrix is over defined then use “Always Calculate” Option

Slide20

If Cannot Get CUSD or CSDs

When the Matrix is over defined then you can not calculate CSNDs or CSNDs In that case use “Always Calculate” Option

Slide21

MV Mixed Distributions

What if you need to simulate a MV distribution made up of variables that are not all Normal or all Empirical? For example:

X is ~ Normal

Y is ~ Beta

T is ~ Gamma

Z is ~ Empirical

Develop parameters for each variable

Estimate the correlation matrix for the random variables in the distribution

Slide22

MV Mixed Distributions

Simulate a vector of Correlated Uniform Standard Deviates using =CUSD() function =CUSD( correlation matrix ) is an array function so highlight the number of cells that matches the number of variables in the distributionUse the CUSD

i

values in the appropriate Simetar functions for each random variable

=NORM(Mean, Std Dev, CUSD

1

)

=BETAINV(CUSD2, Alpha, Beta)

=GAMMAINV(CUSD

3

, P1, P2)

=Mean*(1+EMP(S

i

, F(Si), CUSD4))

Slide23

Validation of MV Distributions

Simulate the model and specify the random variables as the KOVs then test the simulated random valuesPerform the following testsUse the Compare Two Series Tab in HoHi to:

Test means for the historical series or the forecasted means vs. the simulated means

Test means and covariance for historical series vs. simulated

Use the Check Correlation Tab to test the correlation matrix used as input for the MV model vs. the implied correlation in the simulated random variables

Null hypothesis (Ho) is:

Simulated correlation

ij

= Historical correlation coefficient

ij

Critical t statistic is 1.98 for 100 iterations; if Null hypothesis is true the calculated t statistics will exceed 1.98

Use caution on means tests if your forecasted

Ŷ is different from the historical Ῡ

Slide24

Validation of MV Distributions

Slide25

Validation of MV Distributions

Slide26

Test Correlation for MV Distributions

Test simulated values for MVE and MVN distribution to insure the historical correlation matrix is reproduced in simulationData Series is the simulated values for all random variables in the MV distributionThe original correlation matrix used to simulate the MVE or MVN distribution

Slide27

Validation Tests in Simetar

Student t Test is used to calculate statistical significance of simulated correlation coefficient to the historical correlation coefficientYou want the test coefficient to be less than the Critical ValueIf the calculated t statistic is larger than the Critical value it is

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