Interrupted Time Series (ITS)

Motivating Example#

A public health researcher wants to know whether a comprehensive smoking ban introduced in January 2010 reduced hospital admissions for acute coronary events. She collects monthly hospital admission counts from January 2004 through December 2015 -- six years before and six years after the ban.

Here is the problem: she cannot randomly assign the smoking ban to some months and not others. The ban was a single policy change applied to an entire jurisdiction at a specific point in time. There is no control group -- every hospital in the region was affected simultaneously.

She cannot simply compare the average admission rate before and after the ban, either. Hospital admissions were already declining over time due to secular trends in cardiovascular health, improvements in emergency medicine, and other public health initiatives. A naive before-after comparison would attribute the entire pre-existing downward trend to the ban, vastly overstating its effect.

The interrupted time series design solves this problem (Wagner et al., 2002). It uses the pre-intervention trend as a counterfactual -- projecting what would have happened without the ban -- and then tests whether the post-intervention data deviates from that projection. The deviation, if any, is attributed to the intervention.

Specifically, the researcher fits a segmented regression model that allows both the level and the slope of the time series to change at the moment of the intervention. A sudden drop in admissions at the ban date indicates an immediate level change; a steeper post-ban decline indicates a gradual slope change. Both are policy-relevant: the level change captures the immediate effect, and the slope change captures whether the effect grows or diminishes over time.

A. Overview#

What the ITS Design Does#

The interrupted time series design estimates the causal effect of an intervention that occurs at a known point in time by modeling the outcome as a function of time, with a structural break at the intervention date. The standard segmented regression model is:

where:

• $Y_t$ is the outcome at time $t$ (e.g., monthly hospital admissions)
• $T_t$ is the time elapsed since the start of the series (1, 2, 3, ...)
• $D_t$ is a dummy variable equal to 1 after the intervention and 0 before
• $P_t$ is the time elapsed since the intervention (0 before, 1, 2, 3, ... after)

The four parameters have clear interpretations:

• $\beta_0$: baseline level at $T = 0$
• $\beta_1$: pre-intervention slope (the secular trend)
• $\beta_2$: immediate level change at the intervention -- the jump (or drop) in the outcome the moment the policy takes effect
• $\beta_3$: change in slope after the intervention -- the difference between the post-intervention trend and the pre-intervention trend

Level Change vs. Slope Change#

Different interventions produce different patterns:

• Level change only ($\beta_2 \neq 0$, $\beta_3 = 0$): the intervention causes an immediate, permanent shift. Example: a new billing code instantly changes how diagnoses are recorded.
• Slope change only ($\beta_2 = 0$, $\beta_3 \neq 0$): the intervention has no immediate effect but gradually changes the trajectory. Example: a new medical guideline slowly changes physician behavior.
• Both ($\beta_2 \neq 0$, $\beta_3 \neq 0$): the intervention causes an immediate shift and a change in trajectory. Example: a smoking ban immediately reduces exposure and also accelerates a downward trend as compliance increases.

How It Differs from Simple Before-After Comparison#

A simple before-after comparison estimates $E[Y_t | \text{post}] - E[Y_t | \text{pre}]$. This comparison conflates the intervention effect with any pre-existing trend. ITS explicitly models the pre-trend and asks whether the post-intervention data deviates from what the pre-trend would have predicted. This feature is why ITS is sometimes called "the strongest quasi-experimental design when randomization is not possible" (Shadish et al., 2002).

When to Use ITS#

• Your intervention occurs at a single known time point applied to a population
• You have a sufficient number of time points before and after the intervention (at least 8-12 per segment, ideally more) (Kontopantelis et al., 2015)
• The pre-intervention trend is reasonably stable and estimable
• No control group is available (though adding one strengthens the design -- see Section D)

When NOT to Use ITS#

• The intervention was phased in gradually with no clear start date
• The outcome is measured at only a few time points before or after (use DiD instead)
• Multiple major changes happened at the same time as the intervention
• The pre-intervention trend is highly volatile or nonlinear, making extrapolation unreliable

Common Confusions#

B. Identification#

For the ITS design to provide valid causal estimates, three key assumptions must hold (Lopez Bernal et al., 2017).

Assumption 1: Stable Pre-Intervention Trend#

Plain language: The pre-intervention trend must be well-characterized and would have continued unchanged in the absence of the intervention. The counterfactual is the projection of the pre-trend into the post-period.

Formally: $E[Y_t^{(0)} | t > T^*] = \beta_0 + \beta_1 t$ for all $t > T^*$, where $Y_t^{(0)}$ is the potential outcome without the intervention and $T^*$ is the intervention time.

This assumption is violated if the pre-intervention trend was nonlinear (e.g., admissions were already accelerating downward before the ban), if it was driven by a transient shock, or if there was a "regression to the mean" effect from a temporary spike just before the intervention.

Assumption 2: No Concurrent Events (History Threat)#

Plain language: Nothing else that could affect the outcome happened at the same time as the intervention. If a new cardiac treatment was introduced in the same month as the smoking ban, the estimated effect of the ban is confounded.

Concurrent events are the most common threat to ITS validity (Shadish et al., 2002). The researcher must carefully document the policy landscape and argue that no other plausible cause of the observed change coincided with the intervention.

Assumption 3: No Anticipation Effects#

Plain language: Individuals, firms, or institutions did not change their behavior before the intervention in anticipation of it. If hospitals reduced admissions or smokers quit in the months leading up to the ban (because the ban was announced in advance), the pre-trend is contaminated and the level change at $T^*$ is attenuated.

If anticipation is plausible, the researcher can:

1. Move the intervention date earlier to the announcement date
2. Exclude a "transition window" around the intervention
3. Test for a structural break before the official date

When to Use#

1. A policy or event occurs at a single known date. Smoking bans, speed limit changes, new regulations, product launches, organizational restructurings -- any clearly dated intervention that applies to an entire population.

2. You have a long time series before and after the intervention. At least 8 observations per segment, ideally 24+ for seasonal data (Kontopantelis et al., 2015).

3. No suitable control group exists. ITS does not require a control group (though one helps). This flexibility makes it ideal for nationwide policies where everyone is treated.

4. You want to separate immediate from gradual effects. The level and slope change parameters distinguish immediate shifts from long-term trend changes.

Do NOT Use ITS When:#

1. The intervention timing is ambiguous. If the policy was phased in over months or years, the sharp break assumed by segmented regression is inappropriate.

2. You have very few time points. With 3-4 observations per segment, you cannot reliably estimate the pre-trend or the slope change. Consider DiD with panel data instead.

3. The pre-trend is chaotic or nonlinear. If the outcome fluctuates wildly before the intervention, the linear pre-trend extrapolation is unreliable and the counterfactual is poorly identified.

4. Multiple interventions overlap. If several policies changed simultaneously, ITS cannot disentangle their individual effects without strong additional assumptions.

Connection to Other Methods#

The ITS design relates to several other causal inference methods:

• Difference-in-Differences (DiD): DiD uses a control group to net out common time trends; ITS uses the pre-trend of the treated group as the counterfactual. When you add a control group to ITS, you get a controlled ITS (CITS), which is DiD with more flexible time trends. DiD is preferred when you have a good control group but few time points; ITS is preferred when you have many time points but no control group.

• Regression Discontinuity (RDD): Both ITS and RDD exploit a discontinuity, but the running variable differs. In RDD, the running variable is a score that determines treatment (e.g., test scores above a cutoff). In ITS, the running variable is time. ITS can be thought of as "RDD in time" (Lopez Bernal et al., 2017).

• Synthetic Control: When no single control group is available, synthetic control constructs a weighted combination of untreated units that matches the treated unit's pre-trend. ITS uses the treated unit's own pre-trend as the counterfactual. Synthetic control is preferred when you have a panel of potential control units; ITS is preferred when you have a single treated unit with a long time series.

• Event Studies: Event study designs estimate dynamic treatment effects at multiple leads and lags around the intervention. ITS can be viewed as a parametric event study that constrains the pre- and post-effects to follow linear trends. Event studies are more flexible but require more data.

C. Visual Intuition#

Compare three approaches to estimating the intervention effect. The naive pre-post difference ignores the pre-existing trend, the OLS trend model misses the slope change, and the segmented regression correctly captures both the level shift and the change in trajectory.

D. Mathematical Derivation#

E. Implementation#

Segmented Regression with Autocorrelation-Robust SEs#

1library(lmtest)
2library(sandwich)
3
4# ---- Step 1: Construct ITS variables ----
5# Assume df has columns: month (1..N), admissions, ban (0/1)
6df$time       <- 1:nrow(df)
7df$post       <- as.integer(df$month >= as.Date("2010-01-01"))
8df$time_since <- ifelse(df$post == 1,
9                      df$time - min(df$time[df$post == 1]) + 1, 0)
10
11# ---- Step 2: Fit segmented regression (OLS) ----
12its_ols <- lm(admissions ~ time + post + time_since, data = df)
13summary(its_ols)
14
15# ---- Step 3: Newey-West HAC standard errors ----
16# Bandwidth = floor(0.75 * N^(1/3)) is a common rule of thumb
17bw <- floor(0.75 * nrow(df)^(1/3))
18coeftest(its_ols, vcov = NeweyWest(its_ols, lag = bw,
19                                  prewhite = FALSE))
20
21# ---- Step 4: Check for autocorrelation ----
22dwtest(its_ols)                  # Durbin-Watson test
23bgtest(its_ols, order = 12)     # Breusch-Godfrey (up to lag 12)
24acf(resid(its_ols), main = "ACF of Residuals")
25
26# ---- Step 5: GLS with AR(1) errors (alternative) ----
27library(nlme)
28its_gls <- gls(admissions ~ time + post + time_since,
29             data = df,
30             correlation = corARMA(p = 1, q = 0))
31summary(its_gls)
32
33# ---- Step 6: Plot the ITS ----
34plot(df$time, df$admissions, pch = 19, cex = 0.6,
35   xlab = "Month", ylab = "Hospital Admissions",
36   main = "Interrupted Time Series: Smoking Ban")
37abline(v = min(df$time[df$post == 1]) - 0.5,
38     lty = 2, col = "red", lwd = 2)
39
40# Pre-intervention fitted line
41pre <- df[df$post == 0, ]
42lines(pre$time, predict(its_ols, pre), col = "blue", lwd = 2)
43
44# Post-intervention fitted line
45pst <- df[df$post == 1, ]
46lines(pst$time, predict(its_ols, pst), col = "darkgreen", lwd = 2)
47
48# Counterfactual projection
49cf <- data.frame(time = pst$time, post = 0, time_since = 0)
50lines(pst$time, predict(its_ols, cf),
51    col = "blue", lwd = 2, lty = 3)
52legend("topright",
53     c("Pre-trend", "Post-trend", "Counterfactual"),
54     col = c("blue", "darkgreen", "blue"),
55     lty = c(1, 1, 3), lwd = 2)

F. Diagnostics#

F.1 Durbin-Watson Test#

The Durbin-Watson test checks for first-order autocorrelation in OLS residuals. The test statistic ranges from 0 to 4: values near 2 indicate no autocorrelation, values significantly below 2 indicate positive autocorrelation (common in time series), and values above 2 indicate negative autocorrelation.

• In R: dwtest(model) from the lmtest package
• In Stata: estat dwatson after regress
• In Python: durbin_watson(results.resid) from statsmodels

If the Durbin-Watson statistic is far from 2, OLS standard errors are invalid and you must use Newey-West SEs, GLS, or an ARIMA-based approach.

F.2 Ljung-Box Test#

The Ljung-Box test checks for autocorrelation at multiple lags simultaneously. Unlike Durbin-Watson (which only tests lag 1), Ljung-Box tests whether the first $k$ autocorrelations are jointly zero. Multi-lag testing is especially important for seasonal data where autocorrelation may be present at lag 12 (monthly) or lag 4 (quarterly) even if lag 1 autocorrelation is mild.

• In R: Box.test(resid(model), lag = 12, type = "Ljung-Box")
• In Stata: estat bgodfrey, lags(1/12) (Breusch-Godfrey, similar purpose)
• In Python: acorr_ljungbox(results.resid, lags=12)

F.3 Residual Autocorrelation Plots#

Plot the autocorrelation function (ACF) and partial autocorrelation function (PACF) of the model residuals. Under correct specification:

• ACF should show no significant spikes beyond lag 0
• PACF should show no significant spikes beyond lag 0

Significant spikes at lag 1 suggest AR(1) errors. Spikes at seasonal lags (12 for monthly data) suggest seasonality not captured by the model. Either add seasonal dummies or use seasonal ARIMA.

F.4 Seasonal Decomposition#

For monthly or quarterly data, decompose the outcome into trend, seasonal, and residual components before fitting the ITS model. If strong seasonality is present, add seasonal dummies (month indicators) to the segmented regression:

where $M_{st}$ are month dummies. Omitting seasonal controls when seasonality is present will bias the level-change estimate if the intervention happens to coincide with a seasonal peak or trough (Lopez Bernal et al., 2017).

F.5 Visual Inspection of the Fit#

Always plot three elements:

1. Raw data points -- the observed time series
2. Fitted segments -- the pre- and post-intervention regression lines
3. Counterfactual projection -- the dotted extension of the pre-trend into the post-period

The gap between the post-trend line and the counterfactual is the estimated intervention effect. If the data points do not track the fitted lines reasonably well, the linear specification may be wrong.

1library(lmtest)
2
3its_fit <- lm(admissions ~ time + post + time_since, data = df)
4
5# F.1 Durbin-Watson
6dwtest(its_fit)
7
8# F.2 Breusch-Godfrey for higher-order autocorrelation
9bgtest(its_fit, order = 12)
10
11# F.3 ACF / PACF plots
12par(mfrow = c(1, 2))
13acf(resid(its_fit), lag.max = 24, main = "ACF of Residuals")
14pacf(resid(its_fit), lag.max = 24, main = "PACF of Residuals")
15
16# F.4 Seasonal decomposition (pre-intervention only)
17pre_ts <- ts(df$admissions[df$post == 0], frequency = 12)
18decomp <- decompose(pre_ts)
19plot(decomp)
20
21# F.5 Add seasonal dummies if needed
22df$month_of_year <- factor(format(df$month, "%m"))
23its_seasonal <- lm(admissions ~ time + post + time_since +
24                   month_of_year, data = df)
25bgtest(its_seasonal, order = 12)

Reading the Coefficients#

The four coefficients from the segmented regression have direct policy interpretations:

The total effect at time $k$ after the intervention is:

At 12 months post-ban: $\hat{\tau}_{12} = -11.2 - 4.8 = -16.0$ admissions per month relative to counterfactual.

What to Report#

A well-reported ITS analysis should include:

1. Number of time points before and after the intervention
2. Level change ($\hat{\beta}_2$) with confidence interval and p-value
3. Slope change ($\hat{\beta}_3$) with confidence interval and p-value
4. Pre-intervention trend ($\hat{\beta}_1$) to show what the counterfactual trajectory looks like
5. Standard error type (Newey-West, GLS, etc.) and bandwidth if applicable
6. Autocorrelation diagnostics (Durbin-Watson, ACF)
7. A plot showing the raw data, fitted segments, and counterfactual projection

G. What Can Go Wrong#

H. Practice#

H.1 Concept Checks#

H.2 Guided Exercise#

H.3 Error Detective#

H.4 You Are the Referee#

I. Swap-In: When to Use Something Else#

• Difference-in-Differences (DiD) with a control group: when you have a suitable comparison group not affected by the intervention but with a similar pre-trend. DiD-with-control is more credible than single-group ITS because it nets out time-varying confounders that affect both groups. Preferred when control groups are available, even if you have only a few time periods.

• Synthetic Control: when you have one treated unit and a panel of untreated units that can be combined to match the treated unit's pre-trend. Synthetic control is the method of choice when a single jurisdiction adopts a policy and you have data on many non-adopting jurisdictions. More flexible than ITS because it does not assume a linear counterfactual.

• ARIMA-based ITS: when the time series has strong autocorrelation, seasonality, or nonlinear trends that the simple segmented regression cannot capture. ARIMA models (e.g., intervention analysis via transfer functions) fit the autocorrelation structure directly rather than relying on post-hoc corrections. More complex to specify but handles autocorrelation better. See Shadish et al. (2002) for discussion.

J. Reviewer Checklist#