Question
A council installs a new appointment-booking system in East district at the start of period 1. West district will receive it later, at the start of period 3. The table gives average waiting times (minutes) in three observation periods; West has not yet received the system in any shown period:
Period −1: East 22, West 14.
Period 0: East 23, West 16.
Period 2: East 18, West 20.
An announcement about East's installation may already affect East in period 0. Suppose the untreated mean in each district is a district constant plus a time shock common to both districts, and allow East additionally to have a district-specific linear trend \(\delta\) per period. The system changes East's waiting time by a constant \(\tau\) from period 1 onward, while an anticipation effect a applies in period 0 only. West has neither effect before period 3.
Calculate the usual difference-in-differences (DiD) contrast using periods 0 and 2. Express that contrast and the earlier −1-to-0 placebo contrast in terms of \(\tau\), \(\delta\) and a. What treatment effect would you infer if \(\delta=0\) and \(a=-1\)? Give a different \((\delta,a,\tau)\) triple consistent with the same data, and explain what the pre-period check can and cannot establish.
Get hint
Hint 1
For each contrast, subtract West's change from East's change. The period-0 observation is not necessarily an untreated East baseline because anticipation may already operate.
Check your answer
Solution
The ordinary DiD for periods 0 to 2 is
\(D=(E_{2}-E_{0})-(W_{2}-W_{0})=(18-23)-(20-16)=-5-4=-9\) minutes.
The earlier placebo contrast is
\(D_{\mathrm{pre}}=(E_{0}-E_{-1})-(W_{0}-W_{-1})=(23-22)-(16-14)=1-2=-1\) minute.
To see what these quantities identify, parameterize untreated East by \(\mu_E+\lambda_t+\delta t\) and untreated West by \(\mu_W+\lambda_t\), where \(\lambda_t\) is an arbitrary shared time shock. Add a only to \(E_0\) (advance response to publicity) and \(\tau\) to \(E_t\) for \(t\ge1\). In the period-0-to-2 DiD the district constants and shared time shocks cancel, leaving
\(D=[(2\delta+\tau)-(0\delta+a)]-0=\tau+2\delta-a\).
In the −1-to-0 contrast there is no installed treatment, so
\(D_{\mathrm{pre}}=[(0\delta+a)-(-\delta)]-0=\delta+a\).
Thus the observed data give the two equations \(\tau+2\delta-a=-9\) and \(\delta+a=-1\), which do not determine all three unknowns. Under the additional restrictions \(\delta=0\) and \(a=-1\), \(\tau=D+a=-10\) minutes.
But another coherent explanation is \(a=0\) and \(\delta=-1\), which forces \(\tau=-7\) minutes because \(-7+2(-1)-0=-9\). It also gives \(\delta+a=-1\) as observed. The underlying district intercept can be adjusted to fit the recorded levels under either account, without changing the common shocks. This demonstrates the confounding, not just a verbal warning.
If one instead imposed no anticipation and parallel untreated trends (\(a=\delta=0\)), DiD would identify \(\tau\) and equal −9, but that joint model would predict \(D_{\mathrm{pre}}=0\), contrary to this pre-period comparison. The observed −1 warns that at least one of the assumed features may fail. More pre-periods can help diagnose past trend differences, yet even a perfectly flat pre-period gap cannot prove that a post-period common-trend assumption will hold. West's later rollout is irrelevant to the control contrast here precisely because it has not yet received the system in periods 0 or 2.
Assumptions
Period numbers −1,0,1,2,3 represent equally spaced intervals, so East's differential untreated trend contributes \(\delta\) per period.
Anticipation enters East only in period 0; the installed system's effect \(\tau\) starts in period 1 and is constant through period 2.
West is untreated and has no anticipation throughout the three observed periods, before its period-3 installation.
Common time shocks are unrestricted but affect both districts equally; any East-specific linear drift is represented by \(\delta\).
Final Answer
The period-0-to-2 DiD is −9 minutes, with \(D=\tau+2\delta-a\). The earlier DiD is −1 minute, equal to \(\delta+a\). If \(\delta=0\) and \(a=-1\), \(\tau=-10\) minutes. The alternative \(\delta=-1\), \(a=0\), \(\tau=-7\) minutes also matches both contrasts; the effect is not identified without restrictions.