If you ever spent time in the field of marketing analytics, chances are that you have analyzed the existence of a causal impact from a new local TV campaign, a major PR event, or the emergence of a new local competitor. From an analytical standpoint these type of events all have one thing in common: the impact cannot be tracked at the individual customer level and hence we have to analyze the impact from a bird’s eye view using time series analysis at the market level (e.g., DMA, state, etc.). Data science may be changing at a fast pace but this is an old school use-case that is still very relevant no matter what industry you’re in.

Intervention analyses require more judgment than evaluation of
randomized test/control studies. When analyzing interventions through
time series analysis we typically go through two steps, each of which
can involve multiple analytical decisions: 0. Identify the *test*
market(s) where the intervention will happen. This can be based on data
(optimal splitting) as well as business reasons and limitations. 1. Find
matching *control* markets for the test market(s) where the event
took place using time series matching based on historical data prior to
the event (the “pre period”). If the intervention has not happened,
check the accuracy of the test/control design via a fake intervention
analysis.. 2. Analyze the causal impact of the event by comparing the
observed data for the test and control markets following the event (the
“post period”), while factoring in differences between the markets prior
to the event.

The purpose of this document is to describe a robust approach to
intervention analysis based on two key `R`

packages: the
`CausalImpact`

package written by Kay Brodersen at Google and
the `dtw`

package available in CRAN. In addition, we will
introduce an `R`

package called `MarketMatching`

which implements a simple intervention analysis workflow based on these
two packages.

For the time series matching step the most straightforward approach is to use the Euclidian distance. However, this approach implicitly over-penalizes instances where relationships between markets are temporarily shifted. Although it is preferable for test and control markets to be aligned consistently, occasional historical shifts should not eliminate viable control market candidates. Or another option is to match based on correlation, but this does not factor in size.

For the inference step, the traditional approach is a “diff in diff” analysis. This is typically a static regression model that evaluates the post-event change in the difference between the test and control markets. However, this assumes that observations are i.i.d. and that the differences between the test and control markets are constant. Both assumptions rarely hold true for time series data.

A better approach is to use *dynamic time warping* to do the
time series matching (see [2]) . This technique finds the distance along
the *warping curve* – instead of the raw data – where the warping
curve represents the best alignment between two time series within some
user-defined constraints. Note that the Euclidian distance is a special
case of the warped distance.

For the intervention analysis the `CausalImpact`

package
provides an approach that is more flexible and robust than the “diff in
diff” model (see [1]). The `CausalImpact`

package constructs
a synthetic baseline for the post-intervention period based on a
Bayesian structural time series model that incorporates
*multiple* matching control markets as predictors, as well as
other features of the time series.

We can summarize this workflow as follows:

- Pre-screening step: find the best control markets for each market in the dataset using dynamic time warping. The user can define how many matches should be retained. Note that this step merely creates a list of candidates markets; the final markets used for the post-event inference will be decided in the next step.

Note: If you don’t have a set of test markets to match, the
`MarketMatching`

can provide suggested test/control market
pairs using the `suggest_market_splits`

option in the
`best_matches()`

function. Also, the
`test_fake_lift()`

function provides pseudo prospective power
analysis if you’re using the `MarketMatching`

package to
create your test design (i.e., not just doing the post inference).

- Inference step: fit a Bayesian structural time series model that utilizes the control markets identified in step 1 as predictors. Based on this model, create a synthetic control series by producing a counterfactual prediction for the post period assuming that the event did not take place. We can then calculate the difference between the synthetic control and the test market for the post-intervention period – which is the estimated impact of the event – and compare to the posterior interval to gauge uncertainty.

As mentioned above, the purpose of the dynamic time warping step is
to create a list of viable control market candidates. This is not a
strictly necessary step as we can select markets directly while building
the time series model during step 2. In fact, the
`CausalImpact`

package selects the most predictive markets
for the structural time series model using spike-and-slab priors (for
more information, see the technical details below).

However, when dealing with a large set of candidate control markets
it is often prudent to trim the list of markets in advance as opposed to
relying solely on the variable selection process. Creating a synthetic
control based on markets that have small *distances* to the test
market tends to boost the face-validity of the analysis as similar-sized
markets are easily recognized as strong controls through simple line
plots.

Ultimately, however, this is a matter of preference and the good news
is that the `MarketMatching`

package allows users to decide
how many control markets should be included in the pre-screen. The user
can also choose whether the pre-screening should be correlation-based or
based on time-warped distances, or some mix of the two.

The `MarketMatching`

package implements the workflow
described above by essentially providing an easy-to-use “wrapper” for
the `dtw`

and `CausalImpact`

. The function
`best_matches()`

finds the best control markets for each
market by looping through all viable candidates in a parallel fashion
and then ranking by distance and/or correlation. The resulting output
object can then be passed to the `inference()`

function which
then analyzes the causal impact of an event using the pre-screened
control markets.

Hence, the package does *not* provide any new core
functionality but it simplifies the workflow of using `dtw`

and `CausalImpact`

together *and* provides charts and
data that are easy to manipulate. `R`

packages are a great
way of implementing and documenting workflows.

- Minimal inputs required. The only strictly necessary inputs are the name of the test market (for inference), the dates of the pre-period and post-period and, of course, the data.
- Provides a data.frame with the best matches for all markets in the input dataset. The number of matches can be defined by the user.
- Outputs all inference results as objects with intuitive names (e.g., “AbsoluteEffect” and “RelativeEffect”).
- Checks the quality of the input data and eliminates “bad” markets.
- Calculates MAPE and Durbin-Watson for the pre-period. Shows how these statistics change when you alter the prior standard error of the local level term.
- Plots and outputs the actual data for the markets selected during the initial market matching.
- Plots and outputs actual versus predicted values.
- Plots the final local level term.
- Shows the average estimated coefficients for all the markets used in the linear regression component of the structural time series model.
- Allows the user to choose how many markets are sent to the slab-and-prior model.
- All plots are done in
`ggplot2`

and can easily be extracted and manipulated. - Facilitates pseudo prospective power analysis for geo-based experiments (measuring causal impact of fake interventions).
- Provides suggested optimal test/control market pairs for future studies (in case the test markets have not been identified).

```
## THIS PACKAGE IS IN CRAN.
## If you want to install from Github, use devtools version 1.11.1
## packageurl <- "http://cran.r-project.org/src/contrib/Archive/devtools/devtools_1.11.1.tar.gz"
## install.packages(packageurl, repos=NULL, type="source")
## library(devtools)
## install_github("klarsen1/MarketMatching", build_vignettes=TRUE)
library(MarketMatching)
```

The dataset supplied with the package has daily temperature readings for 20 areas (airports) for 2014. The dataset is a stacked time series (panel data) where each row represents a unique combination of date and area. It has three columns: area, date, and the average temperature reading for the day.

This is *not* the most appropriate dataset to demonstrate
intervention inference, as humans cannot affect the weather in the short
term (long term impact is a different blog post). We’ll merely use the
data to demonstrate the features.

```
##-----------------------------------------------------------------------
## Find the best matches (default is 5) for each airport time series
##-----------------------------------------------------------------------
library(MarketMatching)
data(weather, package="MarketMatching")
<- best_matches(data=weather,
mm id_variable="Area",
date_variable="Date",
matching_variable="Mean_TemperatureF",
parallel=FALSE,
warping_limit=1, # warping limit=1
dtw_emphasis=1, # rely only on dtw for pre-screening
matches=5, # request 5 matches
start_match_period="2014-01-01",
end_match_period="2014-10-01")
##-----------------------------------------------------------------------
## Or just search for 5 control markets for CPH and SFO
##-----------------------------------------------------------------------
<- best_matches(data=weather,
mm_only_cph id_variable="Area",
date_variable="Date",
markets_to_be_matched=c"CPH", "SFO"),
="Mean_TemperatureF",
matching_variable=FALSE,
parallel=1, # warping limit=1
warping_limit=1, # rely only on dtw for pre-screening
dtw_emphasis=5, # request 5 matches
matches="2014-01-01",
start_match_period="2014-10-01")
end_match_period
##-----------------------------------------------------------------------
## Analyze causal impact of a made-up weather intervention in Copenhagen
## Since this is weather data it is a not a very meaningful example.
## This is merely to demonstrate the functionality.
##-----------------------------------------------------------------------
<- MarketMatching::inference(matched_markets = mm,
results test_market = "CPH",
end_post_period = "2015-10-01")
##-----------------------------------------------------------------------
## You can also pass specific bsts model arguments (see bsts documentation)
##-----------------------------------------------------------------------
<- MarketMatching::inference(matched_markets = mm,
results test_market = "CPH",
analyze_betas=TRUE,
bsts_modelargs = list(niter=2000, prior.level.sd=0.001),
end_post_period = "2015-10-01")
```

A view of the best matches data.frame generated by the best_matches() function:

`::kable(head(mm$BestMatches)) knitr`

Plot actual observations for test market (CPH) versus the expectation. It looks like CPH deviated from its expectation during the winter:

`$PlotActualVersusExpected results`

Plot the cumulative impact. The posterior interval includes zero as expected, which means that the cumulative deviation is likely noise:

`$PlotCumulativeEffect results`

Although it looks like some of the dips in the *point-wise*
effects toward the end of the post period seem to be truly negative:

`$PlotPointEffect results`

Store the actual versus predicted values in a data.frame:

```
<- results$Predictions
pred ::kable(head(pred)) knitr
```

Plot the actual data for the test and control markets:

`$PlotActuals results`

Check the Durbin-Watson statistic (DW), MAPE and largest market coefficient for different values of the local level SE. It looks like it will be hard to get a DW statistic close to 2, although our model may benefit from a higher local level standard error than the default of 0.01:

`$PlotPriorLevelSdAnalysis results`

Store the average posterior coefficients in a data.frame. STR (Stuttgart) receives the highest weight when predicting the weather in Copenhagen:

```
<- results$Coefficients
coeff ::kable(head(coeff)) knitr
```

In this example, the probability of a causal impact at different levels of fake interventions starting after 2014-10-01 and ending at 2015-10-01. We’re analyzing fake lifts from 0 to 5 percent in 5 steps (default is 10). This will help you evaluate if your choice of test and control markets creates a sufficient model to measure a realistic lift from a future intervention.

```
##-----------------------------------------------------------------------
## Find the best 5 matches for each airport time series. Matching will
## rely entirely on dynamic time warping (dtw) with a limit of 1
##-----------------------------------------------------------------------
library(MarketMatching)
data(weather, package="MarketMatching")
<- best_matches(data=weather,
mm id_variable="Area",
date_variable="Date",
matching_variable="Mean_TemperatureF",
parallel=FALSE,
warping_limit=1, # warping limit=1
dtw_emphasis=1, # rely only on dtw for pre-screening
matches=5, # request 5 matches
start_match_period="2014-01-01",
end_match_period="2014-10-01")
#' ##-----------------------------------------------------------------------
#' ## Power analysis for a fake intervention ending at 2015-10-01
#' ## The maximum lift analyzed is 10 percent, the minimum is 0 percent
#' ## Since this is weather data it is a not a very meaningful example.
#' ## This is merely to demonstrate the functionality.
#' ##-----------------------------------------------------------------------
<- MarketMatching::prospective_power(matched_markets = mm,
power test_market = "CPH",
end_fake_post_period = "2015-10-01",
prior_level_sd = 0.002,
steps=5,
max_fake_lift=0.05)
```

Inspecting the power curve:

`$ResultsGraph power`

This example shows how to get test/control market pair suggestions from the distances. The package stratifies the markets by size (sum of Y) and the creates pairs based on the correlation of logged values. To invoke this markets_to_matched must be NULL.

Once the optimized pairs have been generated they are passed to the
pseudo power function for evaluation. The `synthetic`

parameter in the roll_up_optimal_pairs function determines if the
control markets will be aggregated (equal weights in `bsts`

and `CausalImpact`

) or if they’ll be left as individual
markets and get separate weights (synthetic control).

```
##-----------------------------------------------------------------------
## Find all matches for each airport (market) time series.
##-----------------------------------------------------------------------
library(MarketMatching)
data(weather, package="MarketMatching")
<- MarketMatching::best_matches(data=weather,
mm id_variable="Area",
date_variable="Date",
matching_variable="Mean_TemperatureF",
suggest_market_splits=TRUE,
parallel=FALSE,
warping_limit=1, # warping limit=1
dtw_emphasis=0, # rely only on correlation
start_match_period="2014-01-01",
end_match_period="2014-10-01")
##-----------------------------------------------------------------------
## The file that contains the suggested test/control splits
## The file is sorted from the strongest market pair to the weakest pair.
##-----------------------------------------------------------------------
head(mm$SuggestedTestControlSplits)
##-----------------------------------------------------------------------
## Pass the results to test_fake_lift to get pseudo power curves for the splits
## Not a meaningful example for this data. Just to illustrate.
## Note that the rollup() function will label the test markets "TEST"
##-----------------------------------------------------------------------
<- MarketMatching::roll_up_optimal_pairs(matched_markets = mm,
rollup synthetic=FALSE)
<- MarketMatching::test_fake_lift(matched_markets = rollup,
power test_market = "TEST",
end_fake_post_period = "2015-10-01",
lift_pattern_type = "constant",
steps=20,
max_fake_lift = 0.1)
```

[1] CausalImpact version 1.0.3, Brodersen et al., Annals of Applied Statistics (2015).

[2] The vignette for the `dtw`

package
(browseVignettes(“dtw”))

[3] Predicting the Present with Bayesian Structural Time Series, Steven L. Scott and Hal Varian