1Dr. Simon Sheather SAS Day 2016

Assessing the effects of recent events on Chipotle sales revenue

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InFebruary2005,Imovedfrom

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HeadoftheDepartmentofStatistics:March1,2005untilFebruary28,2014

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InFall2007,MS(Statistics)onlinebeganwith20students

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In2012,TexasA&MStatisticalServicesLPwasformed

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InFall2013,MS(Analytics)face‐to‐faceandonlinebegan

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PredictingquarterlyrevenueforChipotleMexicanGrill

• WewishtodevelopatimeseriesmodelforthequarterlyrevenueofChipotleMexicanGrill,inordertoprovideforecastsofquarterlyrevenueforthenexttwoquarters,namelyquarters3and4of2016.WeshallconsiderChipotle’squarterlyrevenuefromthefirstquarterof2004untilthelatestavailableresultsnamely,thesecondquarterof2016.Weshalldevelop2models,namely,

1. Atimeseriesmodel2. Aregressionmodel(withtimeserieserrors)usingthe

followingpredictors1) TotalnumberofexistingChipotlestoresatthebeginningof

eachquarter2) NumberofnewChipotlestoresopenedduringeachquarter

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Source: http://www.foodsafetynews.com/2015/12/a‐timeline‐of‐chipotles‐five‐outbreaks/#.WAz‐H3nrvbh

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Regressionmodel

Conclusions:• Evidence of non constant variance in the residuals, we shall consider a logarithmic 

transformation of revenue and the predictors.

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Regressionmodel

Conclusions:• Evidence of outliers in the residuals• Evidence of autocorrelation in the residuals

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Residualsfromregressionmodel

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Residualsfromregressionmodel

Conclusions:• We shall consider an AR(5) model for the residuals from the least squares regression 

model

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Residualsfromregressionmodel

Conclusions:• We shall begin by considering an AR(5) model for the residuals from the least squares 

regression model

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RegressionmodelwithAR(5)errors

Notice that the coefficient of Log[New Store Openings] is positive and statistically significant 

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RegressionmodelwithAR(5)errors

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Comparisonofregressionmodels

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RegressionmodelwithARgapmodelerrors

Conclusions: For every 1% increase in • Existing stores, Chipotle’s quarterly revenue is predicted to increase by 100*(1.011.26554 – 1) = 1.27%• New stores, Chipotle’s quarterly revenue is predicted to increase by 100*(1.010.03651 – 1) = 0.04%

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RegressionmodelswithARgapmodelerrors

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RegressionmodelwithARgapmodelerrors

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Timeseriesmodel

Conclusions:• Quarterly revenue increases over time, as does the variability in the quarterly revenue. • In order to stabilize the variability, we shall consider a logarithmic transformation of revenue.

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Timeseriesmodel

Conclusions:• A logarithmic transformation of revenue stabilizes the variability of the series• With an increasing trend in Log[Revenue ($M)] there is a need for differencing

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Timeseriesmodel

Conclusions:• Apart from 2013, Log[Revenue ($M)] is lower in the first quarter when compared to the other 

three quarters, for which Log[Revenue ($M)] is similar each year. • In other words, there is evidence of seasonality in the log transformed revenue data.

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Timeseriesmodel

Conclusions:• The Ljung‐Box Q statistic is statistically significant at lags 1, 7 and 8, with lag 1 being the most 

statistically significant. • With parsimony in mind, we begin by considering models just based on lag 1 autoregressive (AR) 

and/or moving average (MA) terms. 

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Timeseriesmodel

Conclusions:• The seasonal ARIMA(0,1,1)(0,1,0)4 model produces the lowest values of AIC (Akaike

Information Criterion) and SBC (Schwartz Bayesian Criterion) and hence is deemed to be the “best” of the four models considered. 

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Timeseriesmodel

Conclusions:• The seasonal ARIMA(0,1,0)(0,1,0)4 is a valid model.• A number of outliers are evident in the plot of the residuals 

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Timeseriesmodel

Time Type2014Q1 Level Shift (LS)2013Q2 Level Shift (LS)2016Q1 Additive Outlier (AO)2015Q4 Level Shift (LS)2006Q1 Level Shift (LS)

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Timeseriesmodelallowingfor1AOand4LS

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EstimatedpercentagechangeinChipotle’squarterlyrevenuedueto4levelshiftsandanadditiveoutlier

Time Type % change

2006Q1 Level Shift (LS) ‐5.7

2013Q2 Level Shift (LS) ‐14.0

2014Q1 Level Shift (LS) 22.9

2015Q4 Level Shift (LS) ‐15.5

2016Q1 Additive Outlier (AO) ‐13.1Chipotle (https://chipotle.com/2015incidents) reports on the following “food safety incidents of 2015” 

Norovirus in August and December, 2015 Salmonella in August, 2015 E. coli in October‐November, 2015

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Timeseriesmodelallowingfor1AOand4LS

Level shift

Level shift

Additive outlier

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Timeseriesmodelallowingfor1AOand4LS

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Timeseriesmodelpredictions

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Source: http://ir.chipotle.com/phoenix.zhtml?c=194775&p=irol‐newsArticle&ID=2206614

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Source: http://ir.chipotle.com/phoenix.zhtml?c=194775&p=irol‐newsArticle&ID=2215726

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Analyst’sexpectationsaboutrevenue