analytics2011 - Predictive Analytics World

Media Mix Modeling 

Predictive Analytics World 2011 

Copyright © 2011, SAS Institute Inc. All rights reserved. 

#analytics2011

Media Mix Modeling 

Objectives 

Demonstrate some of the commonly used techniques and methodologies used to 

estimate the impacts of media spend. 

Illustrate some of the most frequently encountered problems. 

Reference some of the newer Econometric techniques incorporated into SAS/ETS 

and Base Stat. 

Caveats 

It is not possible to provide an extensive catalog in the time provided. 

There are far more techniques and challenges than those listed here. 




A Number Of Data Concerns Exist 

Simulated Media Spend Data 

$3,500,000 

$3,000,000 

$2,500,000 

$2,000,000 

$1,500,000 

$1,000,000 

$500,000 

$0 

1 3 5 7 9 111315171921232527293133353739414345474951 1 3 5 7 9 111315171921232527293133353739414345474951 

Year 1 2 

Television_Spend Radio_Spend Newspaper_Spend Direct_Mail_Spend Digital_Spend 




Data Concerns 

Stationarity 

! A simple “working definition” of stationarity is a 

process whose mean, variance and autocorrelation 

structures do not vary over time. 

Cointegration 


! If variables in a regression model are not 

stationary, standard asymptotic assumptions are not 

valid (e.g. t-statistics will not follow a t distribution). 

! Generally, regression models with non-stationary 

predictors (that are not differenced) yield spurious 

(even nonsensical) results. 

! If a stationary linear combination of non-stationary 

regressors exists, these regressors are said to be 

cointegrated. 

! “Long run” and “short run” dynamic relationships 

exist amongst cointegrated regressors. 

! Generally, regression models that properly 

account for cointegrated predictors will not yield 

spurious results 



Data Concerns 

Exogeneity 

! Currently exogeneity is defined in terms of weak, 

strong and super exogeneity. 

! A regressor is said to be weakly exogenous if 

inference on the regression parameter estimates 

conditional upon the regressor involves no loss of 

information. If weak exogeneity does not hold the 

model's dynamic parameter estimates are 

inefficient. 

! A regressor is said to be super exogenous if it is 

weakly exogenous and the regression parameter 

estimates do not change when changes in the 

regressor's distribution occur. 

! A regressor is said to be strongly exogenous if it is 

weakly exogenous and the regressor is not 

preceded by an endogenous variable (in the model 

formulation). 




Data Concerns 

GARCH 

Multicollinearity 

! GARCH – Generalized Autoregressive 

Conditional Heteroscedasticity. 

! The variance of the current error term (or 

innovation) is a function of the size of the previous 

period's error term (or innovation). 

! Primarily used in variance modeling and may not 

necessarily improve forecasts. 

! Largely a question of degree or severity. 

! If severe multicollinearity exist, the variance 

estimates are inflated and the following may be 

observed: imprecise (or implausible) and unstable 

parameter estimates, a very high r-squared with 

statistically insignificant predictors, incorrect 

coefficient signs. 




Data Concerns 

Autocorrelation 

! When model residuals are correlated, parameter 

estimates are inefficient, t-statistics and r-squared 

values are upwardly biased. 

! Autocorrelation can be positive or negative. 

! First order autocorrelation is the most common 

variant. 

! Common causes for autocorrelation include 

observations being present in multiple time periods 

and omitted variables. 









proc reg data=mme.simulated_base; 

Title1 'Main and Interaction Effects -- Multicollinearity Demonstration'; 

where year = 1; 

model Log_Sales = Holiday Log_DM Log_TV Log_Radio Log_Paper Log_Digital 

LogTVPaper LogTVDigital LogTVHoldy LogRadioHoldy/vif; 

output out=p1 p=py r=residual; 

run; 

quit; 

Year 1 sales are regressed against Direct Mail, Television, Radio 

Newspaper and Digital Spend levels. 

A log-log functional form was assumed to enable easy elasticity 

estimates. 





Parameter Estimates 

Variable DF 

Parameter 

Es timate 

Standard 

Error t Value Pr > |t| 

Variance 

Inflation 

Intercept 1 -1.33934 163.23107 -0.01 0.9935 0 

holiday 1 3.61428 10.18642 0.35 0.7245 81880 

Log_DM 1 0.00484 0.00279 1.73 0.0904 1.45394 

Log_TV 1 1.11802 11.78288 0.09 0.9249 51846 

Log_Radio 1 0.20417 0.14413 1.42 0.1642 13.56417 

Log_Paper 1 -1.48058 13.72226 -0.11 0.9146 68041 

Log_Digital 1 2.73806 5.83569 0.47 0.6414 22776 

LogTVPaper 1 0.09406 0.9969 0.09 0.9253 253746 

LogTVDigital 1 -0.21443 0.43017 -0.5 0.6208 55934 

LogTVHoldy 1 -0.53949 0.6456 -0.84 0.4082 65908 

LogRadioHoldy 1 0.36448 0.38295 0.95 0.3468 19923 

Even though none of the regressors are statistically significant at the 

5% confidence level, the Adjusted R-square is .8453. 

Only Direct Mail had a variance inflation value less than 10. 

Many of the coefficient signs are reversed. 





Ridge Regression Code. 

proc reg data=mme.simulated_base outvif 

outest=b ridge=0 to 0.40 by 0.02; 


Title2 'Ridge Regression'; 


model Log_Sales = Holiday Log_DM Log_TV Log_Radio Log_Paper Log_Digital LogTVPaper LogTVDigital 

LogTVHoldy LogRadioHoldy/vif noprint; 

plot /ridgeplot; 

output out=p1 p=py r=residual; 

run; 

quit; 

proc print data=b;run; 





Ridge Plots 





Regression Plots 





Ridge Regression Code Continued. 

proc score data=mme.simulated_base score=b(where=(_RIDGE_=0.04)) out=p2 

type=RIDGE; 

var Holiday Log_DM Log_TV Log_Radio Log_Paper Log_Digital LogTVPaper LogTVDigital LogTVHoldy 

LogRadioHoldy; 

run; 

proc print data=p1;run; 

proc print data=p2;run; 

A variable selection mechanism is missing. 

Each regressor is included (“considered statistically significant”). 





GLM Select (Lasso). 

proc glmselect data=mme.simulated_base plots=all; 


Title2 'GLM Select -- Lasso'; 


model Log_Sales = Holiday Log_DM Log_TV Log_Radio Log_Paper Log_Digital 

LogTVPaper LogTVDigital LogTVHoldy LogRadioHoldy 

/details=all stats=all 

selection=lasso; 

*modelAverage nsamples=1000 subset(best=1); 

run; 





Lasso Variable Selection Summary 

Step 

Effect 

Entered 

Effect 

Removed 

Number 

Effects In 

Model 

R-Square 

Adjusted 

R-Square SBC ASE F Value Pr > F 

0 Intercept 1 0 0 -144.3294 0.0578 0 1 

1 LogRadioHoldy 2 0.5747 0.5662 -184.8367 0.0246 67.57
















Stationarity 




Stationarity 

Vector Autoregression Code (model assumed to be ARMAX(1,1,0)) 

proc varmax data=mme.simulated_base plots=(impulse) outest=est outstat=stat; 


nloptions tech=newrap maxiter=5000000000 maxfunc=5000000000; 

model Log_Sales = Log_TV Log_Digital Log_DM L1_Radio L3_Paper 

/print=(all) lagmax = 10 cointtest=(sw) /*dify=(1) difx=(1)*/ p=1 q=1; 

output out=out lead=5; 

causal group1=(Log_Sales) group2=(Log_TV L1_Radio L3_Paper Log_Digital Log_DM); 

causal group1=(Log_TV L1_Radio L3_Paper Log_Digital Log_DM) group2=(Log_Sales); 

run; 

This model is testing for the need to difference the data. 




Stationarity 


Dickey-Fuller Unit Root Tests 

Variable Type Rho Pr < Rho Tau Pr < Tau 

Log_Sales Zero Mean 0.04 0.6873 0.76 0.8742 

Single Mean -2.55 0.7028 -0.88 0.7866 

Trend -6.49 0.6818 -1.74 0.7176 

Dickey-Fuller Tests indicated model should be differenced 

Model Parameter Estimates 

Standard 

Equation Parameter Es timate Error t Value Pr > |t| Variable 

Log_Sales CONST1 -0.93697 0.90437 -1.04 0.3054 1 

XL0_1_1 0.40894 0.08564 4.78 0.0001 Log_TV(t) 

XL0_1_2 0.0546 0.05846 0.93 0.355 Log_Digital(t) 

XL0_1_3 0.00444 0.00217 2.05 0.0462 Log_DM(t) 

XL0_1_4 0.10212 0.1133 0.9 0.3719 L1_Radio(t) 

XL0_1_5 0.35952 0.14147 2.54 0.0143 L3_Paper(t) 

AR1_1_1 0.06687 0.24856 0.27 0.7891 Log_Sales(t-1) 

MA1_1_1 0.14939 0.33238 0.45 0.6551 e1(t-1) 

The model has an R-square value of .9027 




Stationarity 





model Log_Sales = Log_TV Log_Digital Log_DM L1_Radio L3_Paper 

/print=(all) lagmax = 10 cointtest=(sw) dify=(1) difx=(1) p=1 q=1; 




run; 

This model is estimated in first differences 




Stationarity 




Log_Sales Zero Mean -75.28

Model Diagnostics 

Durbin Watson does not indicate autocorrelations 

ARCH test statistic does not indicate heteroscedasticity 

Model residuals appear to be normally distributed 

Univariate Model White Noise Diagnostics 

Durbin 

Normality 

ARCH 

Variable Watson Chi-Square Pr > ChiSq F Value Pr > F 

Log_Sales 2.06684 3.35 0.1872 1.54 0.2212 

The model order does not appear to have an autoregressive error 

Univariate Model AR Diagnostics 

AR1 AR2 AR3 AR4 

Variable F Value Pr > F F Value Pr > F F Value Pr > F F Value Pr > F 

Log_Sales 0.25 0.6185 0.51 0.6043 0.59 0.6249 1.59 0.1962 




Stationarity 

Vector Autoregression Code (model assumed to be MAX(1,0)) 



Log_Sales Zero Mean -75.28


Durbin Watson does not indicate autocorrelations 

ARCH test statistic does not indicate heteroscedasticity 

Model residuals appear to be normally distributed 


Durbin 

Normality 

ARCH 

Variable Watson Chi-Square Pr > ChiSq F Value Pr > F 

Log_Sales 2.27232 1.6 0.4499 1.03 0.3152 

The model order does not appear to have an autoregressive error 

Univariate Model AR Diagnostics 

AR1 AR2 AR3 AR4 

Variable F Value Pr > F F Value Pr > F F Value Pr > F F Value Pr > F 

Log_Sales 1.08 0.3035 0.99 0.3784 0.76 0.5207 1.58 0.1983 




Forecasts 




Forecasts 

Error=(1-(Forecast/Sales)) 




Cointegration 




Cointegration 

Vector Error Correction Model (VECM) Code 



model Log_Sales Log_TV Log_Digital Log_DM L1_Radio L3_Paper 

/print=(all) lagmax = 10 p=4 cointtest=(johansen=(normalize=Log_Sales)); 

cointeg rank=4 normalize=Log_TV exogeneity; 




run; 

This model is estimated in levels. 

Estimating a VECM with differenced data results in a lost of 

information. 




Cointegration 

Granger Causality 

Granger-Causality Wald Test 

Test DF 

Chi- 

Square 

Pr > 

ChiSq 

1 20 54.61

Cointegration 

Johansen Rank Test 

Cointegration Rank Test Using Trace 

5% 

H0: 

Rank=r 

H1: 

Rank>r 

Eigenvalu 

e Trace 

Critical 

Value 

Drift in 

ECM 

Drift in 

Process 

0 0 0.4891 188.8512 93.92 Constant Linear 

1 1 0.3978 123.7144 68.68 

2 2 0.271 74.5222 47.21 

3 3 0.2289 43.8567 29.38 

4 4 0.113 18.6465 15.34 

5 5 0.0697 7.012 3.84 

Cointegration Rank Test Using Trace Under Restriction 

5% 

H0: 

Rank=r 

H1: 

Rank>r 

Eigenvalu 

e Trace 

Critical 

Value 

Drift in 

ECM 

Drift in 

Process 

0 0 0.4892 189.2263 101.84 Constant Constant 

1 1 0.3984 124.0615 75.74 

2 2 0.2712 74.7639 53.42 

3 3 0.2293 44.0806 34.8 

4 4 0.1138 18.8118 19.99 

5 5 0.0705 7.0963 9.13 




Cointegration 

Johansen Rank Test 

Hypothesis of the Restriction 

Hypothesis 

Drift in 

ECM 

Drift in 

Process 

H0(Case 2) Constant Constant 

H1(Case 3) Constant Linear 

Hypothesis Test of the Restriction 

Rank Eigenvalue 

Restricted 

Eigenvalue DF 

Chi- 

Square 

Pr > 

ChiSq 

0 0.4891 0.4892 6 0.38 0.999 

1 0.3978 0.3984 5 0.35 0.9967 

2 0.271 0.2712 4 0.24 0.9933 

3 0.2289 0.2293 3 0.22 0.9736 

4 0.113 0.1138 2 0.17 0.9206 

5 0.0697 0.0705 1 0.08 0.7715 




Cointegration 



Normality 

ARCH 

Variable 

Durbin 

Watson Chi-Square 

Pr > 

ChiSq F Value Pr > F 

Log_Sales 2.02311 3.86 0.145 0.04 0.8432 

Log_TV 2.027 76.5

Cointegration 


Univariate Model ANOVA Diagnostics 

Standard 

Variable R-Square Deviation F Value Pr > F 

Log_Sales 0.3979 0.11948 1.98 0.0139 

Log_TV 0.2892 0.16767 1.22 0.2547 

Log_Digital 0.499 0.1971 2.99 0.0002 

Log_DM 0.4996 3.33832 3 0.0002 

L1_Radio 0.6721 0.10855 6.15

Cointegration 

Weak Exogeneity 

Testing Weak Exogeneity of Each Variables 

Variable DF Chi-Square 

Pr > 

ChiSq 

Log_Sales 4 11.01 0.0265 

Log_TV 4 5.72 0.2212 

Log_Digital 4 23.18 0.0001 

Log_DM 4 44.98

VECM Forecasts 




VECM Forecasts 

Error=(1-(Forecast/Sales)) 




Impulse Response Functions 
















































Thank You 

Feel free to contact me at dccozine@gmail.com.

analytics2011 - Predictive Analytics World

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