Materials for Lecture 12• Chapter 7 – Study this closely• Chapter 16 Sections 3.9.1-3.9.7 and

4.3• Lecture 12 Multivariate Empirical

Dist.xls• Lecture 12 Multivariate Normal

Dist.xls

Multivariate Probability Distributions

• Definition: Multivariate (MV) Distribution --Two or more random variables that are correlated

• MV you have 1 distribution with 2 or more random variables

• Univariate distribution we have many distributions (one for each random variable)

Parameter Estimation for MV Dist.

• Data were generated contemporaneously – Output observed each year or month, – Prices observed each year for related

commodities• Corn 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

Different MV Distributions• Multivariate Normal distribution –

MVN• Multivariate Empirical – MVE • Multivariate Mixed where each

variable is distributed differently, such as – X ~ Uniform– Y ~ Normal– Z ~ Empirical– R ~ Beta– S ~ Gamma

Sim MV Distribution as Independent

• If correlation is ignored when random variables are correlated, results are biased:

• If Z = Ỹ1 + Ỹ2 OR Z = Ỹ1 * Ỹ2 and the model is simulated without correlation– But the true ρ1,2 > 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 * Ỹ2

– The Mean of Z is biased, as well

Parameters for a MVN Distribution

• Deterministic component– Ŷij -- a vector of means or predicted values for the period

i to simulate all of the j variables, for example:

Ŷij = ĉ0 + ĉ1 X1 + ĉ2 X2

• Stochastic component– êji -- 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

σêj

• Multivariate 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

3 Variable MVN Distribution

• Deterministic component for three random variables– Ĉi = a + b1Ci-1

– Ŵi = a + b1Ti + b2 Wi-1

– Ŝi = a + b1Ti

• Stochastic component – êCi = Ci – Ĉi

– êWi = Wi – Ŵi

– êSi = Si – Ŝi

• Multivariate componentσ2

cc σcw σcs

Σ = σ2ww σws

σ2ss

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

Enterwhere: 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

Example of Mean vs. Y-Hat Problem for Validation

0

20

40

60

80

100

120

140

160

180

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

X X-Bar Y-Hat

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 Ỹ1 = NORM( Mean1 , σ1 , CUSD1 )For Ỹ2 = NORM( Mean2 , σ2 , CUSD2 )For Ỹ3 = NORM( Mean3 , σ3 , CUSD3 )For Ỹ4 = NORM( Mean4 , σ4 , CUSD4 )

• Use Two Step if you want more control of the process

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

Two Step MVN Distribution

Review Steps for MVN• Develop parameters

– Calculate averages (and standard deviations used for two step procedure)

– Calculate Covariance matrix– Calculate 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

Parameters for MV Empirical

• Step I Deterministic component for three random variables– Ĉi = a + b1Ci-1

– Ŵi = a + b1Ti + b2 Wi-1

– Ŝi = a + b1Ti

• Step II Stochastic component will be calculated from residuals– êCi = Ci – Ĉi

– êWi = Wi – Ŵi

– êSi = Si – Ŝi

• Step III Calculate the stochastic empirical distributions parameter– SCi = 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 II1

e e e e

e e

c w c s

w s

, ,

1 ,

1

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 dataOption = 1 use Percent deviations from MeanOption = 2 use Percent deviations from TrendOption = 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

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 Ỹ1 = Mean1 + Mean1 * Empirical(S1, F(Si) , CUSD1)

For Ỹ2 = Mean2 + 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

Parameter Estimation for MVE

Simulate a MVE Distribution

Validation of MV Distributions• Simulate the model and specify the random

variables as the KOVs then test the simulated random values

• Perform the following tests– Use 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 correlationij = Historical correlation coefficientij

• 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

Validation of MV Distributions

Validation of MV Distributions

Test Correlation for MV Distributions

• Test simulated values for MVE and MVN distribution to insure the historical correlation matrix is reproduced in simulation– Data Series is the simulated values for all random

variables in the MV distribution– The original correlation matrix used to simulate

the MVE or MVN distribution

Validation Tests in Simetar• Student t Test is used to calculate statistical

significance of simulated correlation coefficient to the historical correlation coefficient– You want the test coefficient to be less than the

Critical Value– If the calculated t statistic is larger than the

Critical value it is bold

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

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 distribution

• Use the CUSDi values in the appropriate Simetar functions for each random variable

=NORM(Mean, Std Dev, CUSD1)

=BETAINV(CUSD2, Alpha, Beta)

=GAMMAINV(CUSD3, P1, P2)

=Mean*(1+EMP(Si, F(Si), CUSD4))