# Bayesian structural time series

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Statistical technique used for feature selection

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**Bayesian structural time series** (**BSTS**) model is a [statistical](/source/Statistical) technique used for [feature selection](/source/Feature_selection), time series forecasting, [nowcasting](/source/Nowcasting_(economics)), inferring causal impact and other applications. The model is designed to work with [time series](/source/Time_series) data.

The model has also promising application in the field of analytical [marketing](/source/Marketing). In particular, it can be used in order to assess how much different marketing campaigns have contributed to the change in web search volumes, product sales, brand popularity and other relevant indicators. [Difference-in-differences](/source/Difference_in_differences) models[1] and [interrupted time series](/source/Interrupted_time_series) designs[2] are alternatives to this approach. "In contrast to classical difference-in-differences schemes, state-space models make it possible to (i) infer the temporal evolution of attributable impact, (ii) incorporate empirical priors on the parameters in a fully Bayesian treatment, and (iii) flexibly accommodate multiple sources of variation, including the time-varying influence of contemporaneous covariates, i.e., synthetic controls."[1]

## General model description

The model consists of three main components:

1. **[Kalman filter](/source/Kalman_filter)**. The technique for time series decomposition. In this step, a researcher can add different state variables: trend, seasonality, regression, and others.

1. **[Spike-and-slab](/source/Spike-and-slab_variable_selection) method.** In this step, the most important regression predictors are selected.

1. **[Bayesian model averaging](/source/Ensemble_learning).** Combining the results and prediction calculation.

The model could be used to discover the causations with its counterfactual prediction and the observed data.[1]

A possible drawback of the model can be its relatively complicated mathematical underpinning and difficult implementation as a computer program. However, the programming language [R](/source/R_(programming_language)) has ready-to-use packages for calculating the BSTS model,[3][4] which do not require strong mathematical background from a researcher.

## See also

- [Bayesian inference using Gibbs sampling](/source/Bayesian_inference_using_Gibbs_sampling)

- [Correlation does not imply causation](/source/Correlation_does_not_imply_causation)

- [Spike-and-slab regression](/source/Spike-and-slab_regression)

## References

1. ^ [***a***](#cite_ref-:0_1-0) [***b***](#cite_ref-:0_1-1) [***c***](#cite_ref-:0_1-2) ["Inferring causal impact using Bayesian structural time-series models"](http://research.google.com/pubs/pub41854.html). *research.google.com*. Retrieved 2016-04-17.

1. **[^](#cite_ref-2)** ["Interrupted Time-Series Design"](https://web.archive.org/web/20190321192704/https://www.insightsassociation.org/issues-policies/glossary/interrupted-time-series-design). *Interrupted Time-Series Design*. Insights Association. Archived from [the original](https://www.insightsassociation.org/issues-policies/glossary/interrupted-time-series-design) on 21 March 2019. Retrieved 21 March 2019.

1. **[^](#cite_ref-3)** ["bsts"](https://cran.r-project.org/web/packages/bsts/bsts.pdf) (PDF).

1. **[^](#cite_ref-4)** ["CausalImpact"](https://google.github.io/CausalImpact/CausalImpact.html). *google.github.io*. Retrieved 2016-04-17.

## Further reading

- Scott, S. L., & Varian, H. R. 2014a. [Bayesian variable selection for nowcasting economic time series](http://people.ischool.berkeley.edu/~hal/Papers/2012/fat.pdf). *Economic Analysis of the Digital Economy.*

- Scott, S. L., & Varian, H. R. 2014b. [Predicting the present with bayesian structural time series](http://people.ischool.berkeley.edu/~hal/Papers/2013/pred-present-with-bsts.pdf). *International Journal of Mathematical Modelling and Numerical Optimisation.*

- Varian, H. R. 2014. [Big Data: New Tricks for Econometrics](http://people.ischool.berkeley.edu/~hal/Papers/2013/ml.pdf). *Journal of Economic Perspectives*

- Brodersen, K. H., Gallusser, F., Koehler, J., Remy, N., & Scott, S. L. 2015. [Inferring causal impact using Bayesian structural time-series models](http://research.google.com/pubs/pub41854.html). *The Annals of Applied Statistics.*

- R package ["bsts"](https://cran.r-project.org/web/packages/bsts/bsts.pdf).

- R package ["CausalImpact"](https://google.github.io/CausalImpact/CausalImpact.html).

- O’Hara, R. B., & Sillanpää, M. J. 2009. [A review of Bayesian variable selection methods: what, how and which](https://projecteuclid.org/euclid.ba/1340370391). *Bayesian analysis.*

- [Hoeting, J. A.](/source/Jennifer_A._Hoeting), Madigan, D., Raftery, A. E., & Volinsky, C. T. 1999. [Bayesian model averaging: a tutorial](https://projecteuclid.org/euclid.ss/1009212519). *Statistical science.*

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