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Copyright © 2011, SAS Institute Inc. All rights reserved. #analytics2011 Copyright © 2011, SAS Institute Inc. All rights reserved. #analytics2011
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Media Mix Modeling
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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.
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1 3 5 7 9 111315171921232527293133353739414345474951 1 3 5 7 9 111315171921232527293133353739414345474951Year 1 2
$0
$500,000
$1,000,000
$1,500,000
$2,000,000
$2,500,000
$3,000,000
$3,500,000
Simulated Media Spend Data
Television_Spend Radio_Spend Newspaper_Spend Direct_Mail_Spend Digital_Spend
A Number Of Data Concerns Exist
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Data Concerns
Stationarity ! A simple “working definition” of stationarity is a process whose mean, variance and autocorrelation structures do not vary over time.
! 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.
Cointegration ! 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
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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).
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Data Concerns
GARCH ! 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.
Multicollinearity ! 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.
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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.
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Multicollinearity
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Multicollinearity
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.
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Multicollinearity
Parameter Estimates
Variable DF t Value Pr > |t|Intercept 1 -1.33934 163.23107 -0.01 0.9935 0holiday 1 3.61428 10.18642 0.35 0.7245 81880
Log_DM 1 0.00484 0.00279 1.73 0.0904 1.45394Log_TV 1 1.11802 11.78288 0.09 0.9249 51846
Log_Radio 1 0.20417 0.14413 1.42 0.1642 13.56417Log_Paper 1 -1.48058 13.72226 -0.11 0.9146 68041Log_Digital 1 2.73806 5.83569 0.47 0.6414 22776LogTVPaper 1 0.09406 0.9969 0.09 0.9253 253746LogTVDigital 1 -0.21443 0.43017 -0.5 0.6208 55934LogTVHoldy 1 -0.53949 0.6456 -0.84 0.4082 65908
LogRadioHoldy 1 0.36448 0.38295 0.95 0.3468 19923
ParameterEstimate
StandardError
VarianceInflation
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.
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Multicollinearity
proc reg data=mme.simulated_base outvif
outest=b ridge=0 to 0.40 by 0.02;
Title1 'Main and Interaction Effects -- Multicollinearity Demonstration';
Title2 'Ridge Regression';
where year = 1;
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 Regression Code.
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Multicollinearity
Ridge Plots
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Multicollinearity
Regression Plots
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Multicollinearity
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;
Ridge Regression Code Continued.
A variable selection mechanism is missing.
Each regressor is included (“considered statistically significant”).
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Multicollinearity
GLM Select (Lasso).
proc glmselect data=mme.simulated_base plots=all;
Title1 'Main and Interaction Effects -- Multicollinearity Demonstration';
Title2 'GLM Select -- Lasso';
where year = 1;
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;
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Multicollinearity
Step SBC ASE F Value Pr > F0 Intercept 1 0 0 -144.3294 0.0578 0 11 LogRadioHoldy 2 0.5747 0.5662 -184.8367 0.0246 67.57 <.00012 Log_Radio 3 0.8251 0.8179 -227.0783 0.0101 70.12 <.00013 Log_DM 4 0.8569 0.8479 -233.5702 0.0083 10.68 0.0024 Log_Digital 5 0.8695 0.8584* -234.4167* 0.0075 4.54 0.0383
EffectEntered
EffectRemoved
NumberEffects In
ModelR-Square
Adjusted R-Square
Lasso Variable Selection Summary
Analysis of Variance
Source DF F ValueModel 4 2.61132 0.65283 78.29Error 47 0.39192 0.00834
51 3.00324
Parameter EstimatesParameter DF EstimateIntercept 1 10.424565Log_DM 1 0.003916
Log_Radio 1 0.228549Log_Digital 1 -0.126448
LogRadioHoldy 1 0.037262
Sum ofSquares
MeanSquare
Corrected Total
Lasso Anova and parameter estimates
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Multicollinearity
Lasso Variable Selection Summary
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Multicollinearity
Lasso Variable Selection Summary
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Multicollinearity
Lasso Variable Selection Summary
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Stationarity
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Stationarity
proc varmax data=mme.simulated_base plots=(impulse) outest=est outstat=stat;
where year = 1;
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;
Vector Autoregression Code (model assumed to be ARMAX(1,1,0))
This model is testing for the need to difference the data.
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Stationarity
Vector Autoregression Code (model assumed to be ARMAX(1,1,0))
Dickey-Fuller Unit Root TestsVariable Type Rho Pr < Rho Tau Pr < Tau
Log_Sales Zero Mean 0.04 0.6873 0.76 0.8742Single Mean -2.55 0.7028 -0.88 0.7866
Trend -6.49 0.6818 -1.74 0.7176
Model Parameter Estimates
Equation Parameter Estimate t Value Pr > |t| VariableLog_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)
StandardError
Dickey-Fuller Tests indicated model should be differenced
The model has an R-square value of .9027
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Stationarity
proc varmax data=mme.simulated_base plots=(impulse) outest=est outstat=stat;
where year = 1;
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;
Vector Autoregression Code (model assumed to be ARMAX(1,1,0))
This model is estimated in first differences
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Stationarity
Vector Autoregression Code (model assumed to be ARMAX(1,1,0))
Dickey-Fuller Tests indicated the model is “fully differenced
The model has an R-square value of .6882
Dickey-Fuller Unit Root TestsVariable Type Rho Pr < Rho Tau Pr < Tau
Log_Sales Zero Mean -75.28 <.0001 -5.44 <.0001Single Mean -81.28 0.0004 -5.51 0.0001
Trend -89.13 <.0001 -5.46 0.0003
Model Parameter Estimates
Equation Parameter Estimate t Value Pr > |t| VariableLog_Sales CONST1 0.00224 0.00096 2.33 0.0241 1
XL0_1_1 0.35504 0.08953 3.97 0.0002 Log_TV(t)XL0_1_2 0.07366 0.05414 1.36 0.1802 Log_Digital(t)XL0_1_3 0.00527 0.00198 2.66 0.0107 Log_DM(t)XL0_1_4 0.15012 0.08453 1.78 0.0822 L1_Radio(t)XL0_1_5 0.40703 0.09935 4.1 0.0002 L3_Paper(t)AR1_1_1 -0.10757 0.1264 -0.85 0.3991 Log_Sales(t-1)MA1_1_1 0.98452 0.05045 19.52 0.0001 e1(t-1)
StandardError
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Model Diagnostics Durbin Watson does not indicate autocorrelations
The model order does not appear to have an autoregressive error
Univariate Model White Noise Diagnostics
VariableNormality ARCH
Chi-Square Pr > ChiSq F Value Pr > FLog_Sales 2.06684 3.35 0.1872 1.54 0.2212
DurbinWatson
ARCH test statistic does not indicate heteroscedasticity
Univariate Model AR Diagnostics
VariableAR1 AR2 AR3 AR4
F Value Pr > F F Value Pr > F F Value Pr > F F Value Pr > FLog_Sales 0.25 0.6185 0.51 0.6043 0.59 0.6249 1.59 0.1962
Model residuals appear to be normally distributed
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Stationarity
Vector Autoregression Code (model assumed to be MAX(1,0))
Dickey-Fuller Tests results are unchanged
The model has an R-square value of .6917
Dickey-Fuller Unit Root TestsVariable Type Rho Pr < Rho Tau Pr < Tau
Log_Sales Zero Mean -75.28 <.0001 -5.44 <.0001Single Mean -81.28 0.0004 -5.51 0.0001
Trend -89.13 <.0001 -5.46 0.0003
Model Parameter Estimates
Equation Parameter Estimate t Value Pr > |t| VariableLog_Sales CONST1 0.00188 0.00082 2.29 0.0267 1
XL0_1_1 0.36226 0.09098 3.98 0.0002 Log_TV(t)XL0_1_2 0.06322 0.05588 1.13 0.2636 Log_Digital(t)XL0_1_3 0.00491 0.00199 2.47 0.0173 Log_DM(t)XL0_1_4 0.11312 0.07101 1.59 0.1179 L1_Radio(t)XL0_1_5 0.36666 0.08721 4.2 0.0001 L3_Paper(t)MA1_1_1 0.99856 0.05334 18.72 0.0001 e1(t-1)
StandardError
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Model Diagnostics Durbin Watson does not indicate autocorrelations
The model order does not appear to have an autoregressive error
ARCH test statistic does not indicate heteroscedasticity
Model residuals appear to be normally distributed
Univariate Model White Noise Diagnostics
VariableNormality ARCH
Chi-Square Pr > ChiSq F Value Pr > FLog_Sales 2.27232 1.6 0.4499 1.03 0.3152
DurbinWatson
Univariate Model AR Diagnostics
VariableAR1 AR2 AR3 AR4
F Value Pr > F F Value Pr > F F Value Pr > F F Value Pr > FLog_Sales 1.08 0.3035 0.99 0.3784 0.76 0.5207 1.58 0.1983
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Forecasts
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Forecasts
Error=(1-(Forecast/Sales))
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Cointegration
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Cointegration
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 p=4 cointtest=(johansen=(normalize=Log_Sales));
cointeg rank=4 normalize=Log_TV exogeneity;
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;
Vector Error Correction Model (VECM) Code
This model is estimated in levels.
Estimating a VECM with differenced data results in a lost of information.
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Cointegration
Granger Causality
Sales drives (Granger causes) advertising.
Test 1: Group 1 Variables: Log_SalesGroup 2 Variables: Log_TV L1_Radio L3_Paper Log_Digital Log_DM
Test 2: Group 1 Variables: Log_TV L1_Radio L3_Paper Log_Digital Log_DMGroup 2 Variables: Log_Sales
Advertising drives (Granger causes) sales.
Granger-Causality Wald Test
Test DF1 20 54.61 <.00012 20 32.01 <.0432
Chi-Square
Pr > ChiSq
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Cointegration
Johansen Rank Test
Cointegration Rank Test Using Trace
Trace0 0 0.4891 188.8512 93.92 Constant Linear1 1 0.3978 123.7144 68.682 2 0.271 74.5222 47.213 3 0.2289 43.8567 29.384 4 0.113 18.6465 15.345 5 0.0697 7.012 3.84
H0: Rank=r
H1: Rank>r
Eigenvalue
5% Critical Value
Drift in ECM
Drift in Process
Cointegration Rank Test Using Trace Under Restriction
Trace0 0 0.4892 189.2263 101.84 Constant Constant1 1 0.3984 124.0615 75.742 2 0.2712 74.7639 53.423 3 0.2293 44.0806 34.84 4 0.1138 18.8118 19.995 5 0.0705 7.0963 9.13
H0: Rank=r
H1: Rank>r
Eigenvalue
5% Critical Value
Drift in ECM
Drift in Process
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Cointegration
Johansen Rank Test
Hypothesis of the Restriction
HypothesisH0(Case 2) Constant ConstantH1(Case 3) Constant Linear
Drift in ECM
Drift in Process
Hypothesis Test of the Restriction
Rank Eigenvalue DF0 0.4891 0.4892 6 0.38 0.9991 0.3978 0.3984 5 0.35 0.99672 0.271 0.2712 4 0.24 0.99333 0.2289 0.2293 3 0.22 0.97364 0.113 0.1138 2 0.17 0.92065 0.0697 0.0705 1 0.08 0.7715
RestrictedEigenvalue
Chi-Square
Pr > ChiSq
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Cointegration
Model Diagnostics
Univariate Model White Noise Diagnostics
Variable
Normality ARCH
Chi-Square F Value Pr > FLog_Sales 2.02311 3.86 0.145 0.04 0.8432Log_TV 2.027 76.5 <.0001 0.01 0.9315
Log_Digital 2.00815 13.3 0.0013 0.06 0.8026Log_DM 1.89245 1.9 0.387 0.6 0.441L1_Radio 2.05207 1.51 0.4708 0.02 0.8891L3_Paper 2.01872 0.74 0.6907 0.46 0.4999
DurbinWatson
Pr > ChiSq
There is no evidence of heteroscedasticity
Only the residuals for Television and Digital Spend are not normally distributed
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Cointegration
Model Diagnostics
Univariate Model ANOVA Diagnostics
Variable R-Square F Value Pr > FLog_Sales 0.3979 0.11948 1.98 0.0139Log_TV 0.2892 0.16767 1.22 0.2547
Log_Digital 0.499 0.1971 2.99 0.0002Log_DM 0.4996 3.33832 3 0.0002L1_Radio 0.6721 0.10855 6.15 <.0001L3_Paper 0.7234 0.06408 7.85 <.0001
StandardDeviation
Except for television spend, each of the models is statistically significant.
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Cointegration
Weak Exogeneity
Amongst the six regressors, only Television Spend appears to be weakly exogenous
Testing Weak Exogeneity of Each Variables
Variable DF Chi-SquareLog_Sales 4 11.01 0.0265Log_TV 4 5.72 0.2212
Log_Digital 4 23.18 0.0001Log_DM 4 44.98 <.0001L1_Radio 4 14.83 0.0051L3_Paper 4 20.18 0.0005
Pr > ChiSq
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VECM Forecasts
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VECM Forecasts
Error=(1-(Forecast/Sales))
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Impulse Response Functions
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Thank You
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