Bayesian Structural Time-Series

Introducing a Bayesian Structural Time-Series Model

• A BSTS model is a state-space mdoel used for estimating and identifying causal effects

• It's also used for:

  • Feature selection
  • Time series nowcasting/forecasting
• The model is designed to be used with time series data

• In contrast to diff-in-diff, state-space models make it possible to do the following:

  • Infer the temporal evolution of attributable impact
  • Incorporate empirical priors on the parameters in a fully Bayesian treatment

  • Flexibly accommodate multiple sources of variation, including the time-varying influence of contemporaneous covariates

    • I.e. synthetic controls

Addressing the Limitations of Diff-in-Diff

1. DD is based on a static regression model that assumes iid data, despite the fact the design has a temporal component

   • Thus, when fit to serially correlated data, static models yield overoptimistic inferences
   • Meaning, uncertainty windows become too narrow

2. Most DD analyses only consider two time points: before and after the intervention

   • In practice, we're usually interested in more than just these two time points
   • Specifically, we may be interested in how the treatment effects evolve over time (after the treatment)
   • Especially, we're interested in its onset or decay structure

3. When DD analyses are based on time series, they sometimes impose restrictions on the way a synthetic control is constructed from a set of predictor variables

   • This is something we'd likely like to avoid

Defining Advantages of Bayesian Structural Time-Series

• The limitations of DD schemes can be addressed by using state-space models, coupled with highly flexible regression components, to explain the temporal evolution of an observed outcome:

  1. We can flexibly accommodate different kinds of assumptions about the latent state and emission processes underlying the observed data, including local trends and seasonality

  2. We use a fully Bayesian approach to infer the temporal evolution of counterfactual activity and incremental impact

     • One advantage of this is the flexibility with which posterior inferences can be summarised

  3. We use a regression component that precludes a rigid commitment to a particular set of controls by:

     • Integrating out our posterior uncertainty about the influence of each predictor
     • Integrating our uncertainty about which predictors to include in the first place, which avoids overfitting

Defining Structural Time-Series Models

• BSTS models are state-space models for time-series data
• They can be defined in terms of a pair of equations:

$$Y_{t} = \mu_{t} + x_{t} \beta + S_{t} + \epsilon_{t}$$ $$\mu_{t+1} = \mu_{t} + \eta_{t}$$

• Here, we assume $\epsilon_{t} \sim \mathcal{N}(0, \sigma_{\epsilon}^{2})$ are independent
• And, we assume $\eta_{t} \sim \mathcal{N}(0, \sigma_{\eta}^{2})$ are independent

• Essentially, the random variables in the above equations represent:

  • $x_{t}$ represents a set of regressors at a point in time $t$
  • $S_{t}$ represents a seasonality effect at a point in time $t$
  • $\mu_{t}$ represents a localized trend around a point in time $t$

• Note, regressor coefficients, seasonality and trend are estimated simultaneously

  • This helps avoid strange coefficient estimates due to spurious relationships

• Since the model is bayesian, we can shrink the elements of $\beta$ to promote sparsity or specify outside priors for the means

  • In case, we’re not able to get meaningful estimates from the historical data

Assembling the State-Space Model

• A structural time-series model allows us to flexibly choose appropriate components for the following terms:

  • Trend terms
  • Seasonality terms
  • Static/dynamic regression terms for the controls

• Static coefficients are a good option when the relationship between control and treated units has been stable in the past

  • This is because a spike-and-slab prior can be implemented efficiently within a forward-filtering, backward-sampling framework
  • This makes it possible to quickly identify a sparse set of covariates (even from tens or hundreds of variables)

References

• Post about Bayesian Structural Time-Series
• Paper proposing Bayesian Structural Time-Series
• Presentation about Causal Impact by Google