CausalImpact {CausalImpact}

Inferring causal impact using structural time-series models

Description

CausalImpact() performs causal inference through counterfactual predictions using a Bayesian structural time-series model.

See the package documentation (http://google.github.io/CausalImpact/) to understand the underlying assumptions. In particular, the model assumes that the time series of the treated unit can be explained in terms of a set of covariates which were themselves not affected by the intervention whose causal effect we are interested in.

The easiest way of running a causal analysis is to call CausalImpact() with data, pre.period, post.period, model.args (optional), and alpha (optional). In this case, a time-series model is automatically constructed and estimated. The argument model.args offers some control over the model. See Example 1 below.

An alternative is to supply a custom model. In this case, the function is called with bsts.model, post.period.response, and alpha (optional). See Example 3 below.

Usage

  CausalImpact(data = NULL, pre.period = NULL,
    post.period = NULL, model.args = NULL,
    bsts.model = NULL, post.period.response = NULL,
    alpha = 0.05)

Arguments

data

Time series of response variable and any covariates. This can be a zoo object, a vector, a matrix, or a data.frame. In any of these cases, the response variable must be in the first column, and any covariates in subsequent columns. A zoo object is recommended, as its time indices will be used to format the x-axis in plot().

pre.period

A vector specifying the first and the last time point of the pre-intervention period in the response vector y. This period can be thought of as a training period, used to determine the relationship between the response variable and the covariates. If data is a zoo object with a time attribute, pre.period must be indicated using the same time scale (i.e. using the same class as time(data), see Example 2 below). If data doesn't have a time attribute, post.period is indicated with indices.

post.period

A vector specifying the first and the last day of the post-intervention period we wish to study. This is the period after the intervention has begun whose effect we are interested in. The relationship between response variable and covariates, as determined during the pre-period, will be used to predict how the response variable should have evolved during the post-period had no intervention taken place. If data is a zoo object with a time attribute, post.period must be indicated using the same time scale. If data doesn't have a time attribute, post.period is indicated with indices.

model.args

Further arguments to adjust the default construction of the state-space model used for inference. One particularly important parameter is prior.level.sd, which specifies our a priori knowledge about the volatility of the data. For even more control over the model, you can construct your own model using the bsts package and feed the fitted model into CausalImpact(), as shown in Example 3.

bsts.model

Instead of passing in data and having CausalImpact() construct a model, it is possible to create a custom model using the bsts package. In this case, omit data, pre.period, and post.period. Instead only pass in bsts.model, post.period.response, and alpha (optional). The model must have been fitted on data where the response variable was set to NA during the post-treatment period. The actual observed data during this period must then be passed to the function in post.period.response.

post.period.response

Actual observed data during the post-intervention period. This is required if and only if a fitted bsts.model is provided instead of data.

alpha

Desired tail-area probability for posterior intervals. Defaults to 0.05, which will produce central 95% intervals.

Value

CausalImpact() returns a CausalImpact object containing the original observed response, its counterfactual predictions, as well as pointwise and cumulative impact estimates along with posterior credible intervals. Results can summarised using summary() and visualized using plot(). The object is a list with the following fields:

• series. Time-series object (zoo) containing the original response response as well as the computed inferences. The added columns are listed in the table below.

• summary. Summary statistics for the post-intervention period. This includes the posterior expectation of the overall effect, the corresponding posterior credible interval, and the posterior probability that the intervention had any effect, expressed in terms of a one-sided p-value. Note that checking whether the posterior interval includes zero corresponds to a two-sided hypothesis test. In contrast, checking whether the p-value is below alpha corresponds to a one-sided hypothesis test.

• report. A suggested verbal interpretation of the results.

• model. A list with four elements pre.period, post.period, bsts.model and alpha. pre.period and post.period indicate the first and last time point of the time series in the respective period, bsts.model is the fitted model returned by bsts(), and alpha is the user-specified tail-area probability.

The field series is a zoo time-series object with the following columns:

response	Observed response as supplied to CausalImpact().
cum.response	Cumulative response during the modeling period.
point.pred	Posterior mean of counterfactual predictions.
point.pred.lower	Lower limit of a (1 - alpha) posterior interval.
point.pred.upper	Upper limit of a (1 - alpha) posterior interval.
cum.pred	Posterior cumulative counterfactual predictions.
cum.pred.lower	Lower limit of a (1 - alpha) posterior interval.
cum.pred.upper	Upper limit of a (1 - alpha) posterior interval.
point.effect	Point-wise posterior causal effect.
point.effect.lower	Lower limit of the posterior interval (as above).
point.effect.lower	Upper limit of the posterior interval (as above).
cum.effect	Posterior cumulative effect.
cum.effect.lower	Lower limit of the posterior interval (as above).
cum.effect.upper	Upper limit of the posterior interval (as above).

Note

Optional arguments can be passed as a list in model.args, providing additional control over model construction:

• niter. Number of MCMC samples to draw. Higher numbers yield more accurate inferences. Defaults to 1000.

• standardize.data. Whether to standardize all columns of the data using moments estimated from the pre-intervention period before fitting the model. This is equivalent to an empirical Bayes approach to setting the priors. It ensures that results are invariant to linear transformations of the data. Defaults to TRUE.

• prior.level.sd. Prior standard deviation of the Gaussian random walk of the local level, expressed in terms of data standard deviations. Defaults to 0.01, a typical choice for well-behaved and stable datasets with low residual volatility after regressing out known predictors (e.g., web searches or sales in high quantities). When in doubt, a safer option is to use 0.1, as validated on synthetic data, although this may sometimes give rise to unrealistically wide prediction intervals.

• nseasons. Period of the seasonal components. In order to include a seasonal component, set this to a whole number greater than 1. For example, if the data represent daily observations, use 7 for a day-of-week component. This interface currently only supports up to one seasonal component. To specify multiple seasonal components, use bsts to specify the model directly, then pass the fitted model in as bsts.model. Defaults to 1, which means no seasonal component is used.

• season.duration. Duration of each season, i.e., number of data points each season spans. For example, to add a day-of-week component to data with daily granularity, supply the arguments model.args = list(nseasons = 7, season.duration = 1). Alternatively, use model.args = list(nseasons = 7, season.duration = 24) to add a day-of-week component to data with hourly granularity. Defaults to 1.

• dynamic.regression. Whether to include time-varying regression coefficients. In combination with a time-varying local trend or even a time-varying local level, this often leads to overspecification, in which case a static regression is safer. Defaults to FALSE.

• max.flips. By default, each iteration tries to flip each predictor in or out of the model. Setting max.flips tells the algorithm to randomly choose that many variables to explore flipping. Useful to set (e.g. 100) when you have a large number of controls (e.g. 10000); in such cases, a lower max.flips can speed up computation. Defaults to -1.

Author(s)

Kay H. Brodersen [e-mail]

Examples

# Example 1
#
# Example analysis on a simple artificial dataset
# consisting of a response variable y and a
# single covariate x1.
set.seed(1)
x1 <- 100 + arima.sim(model = list(ar = 0.999), n = 52)
y <- 1.2 * x1 + rnorm(52)
y[41:52] <- y[41:52] + 10
data <- cbind(y, x1)
pre.period <- c(1, 40)
post.period <- c(41, 52)
impact <- CausalImpact(data, pre.period, post.period)

# Print and plot results
summary(impact)
summary(impact, "report")
plot(impact)
plot(impact, "original")
plot(impact$model$bsts.model, "coefficients")

# For further output, type:
names(impact)

## Not run: 
# Example 2
#
# Weekly time series: same data as in example 1, annotated
# with dates.
times <- seq.Date(as.Date("2016-01-03"), by = 7, length.out = 52)
data <- zoo(cbind(y, x1), times)

impact <- CausalImpact(data, times[pre.period], times[post.period])

summary(impact)  # Same as in example 1.
plot(impact)  # The plot now shows dates on the x-axis.

# Example 3
#
# For full flexibility, specify a custom model and pass the
# fitted model to CausalImpact(). To run this example, run
# the code for Example 1 first.
post.period.response <- y[post.period[1] : post.period[2]]
y[post.period[1] : post.period[2]] <- NA
ss <- AddLocalLevel(list(), y)
bsts.model <- bsts(y ~ x1, ss, niter = 1000)
impact <- CausalImpact(bsts.model = bsts.model,
                       post.period.response = post.period.response)
plot(impact)

## End(Not run)