1 A Motivating Example

In January 2012, a US state passes a law requiring prescribers to check a prescription drug monitoring database before dispensing opioids. Monthly opioid prescription rates are recorded for five years before and five years after the law. Did the law reduce prescriptions?   

This question sounds simple, but answering it causally is harder than it appears. Opioid prescription rates were already declining nationally before the law (due to changing medical guidelines and media coverage). A concurrent federal crackdown may have started at the same time. Any naive before-after comparison conflates the law's effect with these contemporaneous trends.

The interrupted time series (ITS) design is tailor-made for settings like this. When a policy is implemented at a known, sharp point in time and time-series data are available before and after, ITS can recover the causal effect by modelling the pre-policy trend and asking how the series deviated from that trend after the policy.  

2 The Fundamental Problem

The causal question is: what would prescription rates have been in the post-policy period had the law not been passed? This counterfactual is never observed. ITS approximates it by extrapolating the pre-policy trend forward in time.

This approximation is valid under one key assumption: in the absence of the policy, the time series would have continued on its pre-policy trajectory. This is the ITS analogue of the parallel trends assumption in DiD. It fails when:  

• A concurrent event happened to occur at the same time as the policy (e.g., a national media campaign).  

• The series was on a non-linear trajectory that would have "naturally" changed direction.  

• The policy was implemented precisely because the trend was changing (selection into treatment based on outcome dynamics).  

3 The Method Step by Step

3.1 The Segmented Regression Model

The standard ITS model is a segmented regression with three components: a pre-intervention trend, a level change at the intervention, and a post-intervention change in trend (Wagner et al., 2002):  

Where:

• t = 1, 2, ..., T indexes time periods.  

• Dₜ = 1[t ≥ T₀] is an indicator equal to 1 in the post-intervention period.  

• (t - T₀)Dₜ is time elapsed since the intervention (zero in the pre-period).  

The parameters have clear interpretations:

• β₀: the baseline level of Y at time t = 0.  

• β₁: the pre-intervention trend (slope per period).  

• β₂: the immediate level change at the interruption—a step up or down at T₀.  

• β₃: the change in slope after the intervention—how much the trend accelerates or decelerates.  

3.2 Visualising the ITS Design

Figure 1 illustrates the design for a series that shows both an immediate level drop (β₂ < 0) and an accelerated declining slope (β₃ < 0) after the policy.  

3.3 A Numerical Example

Suppose monthly prescription rates in the five years before the policy (January 2007-December 2011, T = 60 months) average 85 per 1,000 people and show a mild decline of β̂₁ = -0.3 per month. After the policy (January 2012 onwards), estimation of (1) gives:  

• β̂₂ = -12 (immediate drop of 12 per 1,000 at the policy date; p < 0.001).  

• β̂₃ = -0.8 (the trend becomes -0.3 + (-0.8) = -1.1 per month after the policy; p = 0.02).  

After 24 months, the cumulative effect attributable to the policy is approximately β̂₂ + 24β̂₃ = -12 + 24(-0.8) = -31.2 prescriptions per 1,000 relative to the counterfactual trend.  

4 The Controlled ITS

The key limitation of single-series ITS is vulnerability to co-interventions—events occurring simultaneously with the policy. The controlled ITS adds a control group (e.g., a state that did not pass the law) and estimates:  

Where λᵢ are unit fixed effects and δₜ are time fixed effects. This is precisely a difference-in-differences regression with an interaction term for the change in trend. Parallel trends in the pre-period is required but it can now be tested empirically.  

5 Dealing with Autocorrelation

A critical practical issue: ITS residuals are almost always serially correlated. Prescription rates in month t depend on rates in month t-1; OLS standard errors that assume i.i.d. errors will be anti-conservative (too small). Solutions include:  

1. Newey-West HAC standard errors: correct for both heteroskedasticity and autocorrelation of up to L lags.  ‍
2. ARIMA errors: explicitly model the autocorrelation structure via a pre-whitening ARIMA filter (Prais-Winsten or Cochrane-Orcutt).  ‍
3. Block bootstrap: resample entire time-series blocks to preserve autocorrelation structure.  

Check for residual autocorrelation using the Durbin-Watson test or an ACF/PACF plot of residuals before reporting results.  

6 Common Mistakes

1. Too few pre-intervention observations. Estimating β₁ reliably requires at least 20 pre-intervention time points (Bernal et al., 2017). With fewer, the extrapolated trend is unreliable.  ‍
2. Ignoring autocorrelation. Failure to correct for serial correlation inflates t-statistics and can turn a null result into a "significant" one.  ‍
3. Confounding concurrent events. If another policy or shock occurred at the same time as T₀, ITS attributes all effects to the policy. Always discuss co-interventions.  ‍
4. Confusing ITS with DiD. ITS is a single-series design; DiD compares two groups. A controlled ITS is a DiD with a slope change interaction—it is more credible but requires a valid control group.  

7 Where to Learn More

The most accessible treatment is Bernal et al. (2017), who provide a practical tutorial with code. Kontopantelis et al. (2015) address the small-sample case. For Bayesian ITS with uncertainty about the intervention date, Brodersen et al. (2015) develop the CausalImpact framework (R package CausalImpact) based on structural time series models.  

References

1. Bernal, J. L., Cummins, S., and Gasparrini, A. (2017). Interrupted time series regression for the evaluation of public health interventions: A tutorial. International Journal of Epidemiology, 46(1):348-355.  
2. Brodersen, K. H., Gallusser, F., Koehler, J., Remy, N., and Scott, S. L. (2015). Inferring causal impact using Bayesian structural time-series models. Annals of Applied Statistics, 9(1):247-274.  
3. Campbell, D. T. (1969). Reforms as experiments. American Psychologist, 24(4):409-429.  
4. Kontopantelis, E., Doran, T., Springate, D. A., Buchan, I., and Reeves, D. (2015). Regression based quasi-experimental approach when randomisation is not an option: Interrupted time series analysis. BMJ, 350:h2750.  
5. Wagner, A. K., Soumerai, S. B., Zhang, F., and Ross-Degnan, D. (2002). Segmented regression analysis of interrupted time series studies in medication use research. Journal of Clinical Pharmacy and Therapeutics, 27(4):299-309.