# Identification Strategy This document is the place where causal language is earned or refused. Nothing in `reports/` may carry tier `quasi_experimental` unless the corresponding argument here is written, and the corresponding diagnostic is reported. --- ## 1. The estimand Let $g$ index geographies and $t$ months. Let $E_g$ be predetermined exposure to low-coupon mortgages, measured at a pre-shock date $t_0$. Let $R_t$ be the national market mortgage rate. The target is the coefficient path $$ y_{gt} \;=\; \alpha_g + \gamma_t + \sum_{k \neq k_0} \beta_k\,\bigl(E_g \times \mathbf 1\{t=k\}\bigr) + X_{gt}'\theta + \varepsilon_{gt}. $$ $\beta_k$ is the **differential** response of outcome $y$ in a high-exposure geography relative to a low-exposure geography, at date $k$, relative to reference date $k_0$. **What is not identified.** The level effect of the national rate increase. $R_t$ is common to all geographies and is absorbed by $\gamma_t$. Any statement of the form "the rate increase reduced national transactions by X%" is *not* available from this design. Only cross-sectional differences in the response are. **Interpretation of $\beta_k$.** Under the assumptions in §2, $\beta_k$ is the causal effect of an additional unit of predetermined lock-in exposure on the outcome, at date $k$, holding common national shocks fixed. It is a *relative* effect and it embeds general-equilibrium spillovers between geographies (a locked-in household in one metro who does not move also does not buy in another metro). Spillovers bias $\beta_k$ toward zero if high- and low-exposure markets are linked by migration. --- ## 2. Assumptions required, stated explicitly **A1 (Parallel trends in exposure).** Absent the national rate increase, outcomes in high- and low-exposure geographies would have evolved in parallel, conditional on $\alpha_g$, $\gamma_t$, and $X_{gt}$. *Diagnostic:* joint test that $\beta_k = 0$ for all $k < k_0$. Reported, with the F/Wald statistic and p-value, in `outputs/eventstudy/*.json` under `pretrend_test`. A pre-trend failure demotes the result to `descriptive`. **A2 (No anticipation).** Geographies did not adjust before $t_0$ in anticipation of the rate increase. Plausible: the speed of the 2022 rate increase was widely unforecast, and exposure was built up by the 2020–21 refinance wave, whose motivation was the *low* rate then prevailing. *Diagnostic:* leads in the event study; a placebo shock date in a low-rate-volatility period (2018-01, 2019-06). **A3 (Exposure is predetermined).** $E_g$ is computed only from information available at $t_0$. Enforced in code: the exposure builder takes a `as_of` date and asserts that no input row has a date after it. **A4 (Shock–share exogeneity, for the shift-share variant).** Predetermined is **not** exogenous. The shift-share form $E_g = \sum_k \omega_{gk}^{\text{pre}} \cdot s_k$ requires either (i) the shares $\omega_{gk}$ are conditionally uncorrelated with unobserved determinants of the outcome trend, or (ii) the shocks $s_k$ are as-good-as-random across coupon bins. *Neither is credible here without argument.* The coupon distribution at $t_0$ is a function of when a geography's housing stock last turned over, which correlates with pandemic in-migration, price growth, and construction. The exposure is therefore **not** an instrument. We report: - the concentration of exposure across coupon bins (Herfindahl of $\omega_{gk}$), - the correlation of $E_g$ with pre-period covariates (a balance table), - results with and without controls for pre-period price growth and refi intensity. **We do not use IV language.** The design is a *conditional* difference-in-differences with a continuous, predetermined treatment. If a future version wants IV, the exclusion restriction must be stated here first: "$E_g$ affects post-2022 purchase originations only through the lock-in channel", and the obvious violation — that the same 2020–21 refinance wave also reflects a local demand boom that independently predicts 2023 outcomes — must be defended, not asserted. **A5 (SUTVA / limited spillovers).** Treated above. Direction of bias: toward zero. **A6 (Measurement).** $E_g$ is measured on the Freddie-acquired population, not all mortgages. If Freddie's share of a geography's mortgages varies systematically with the outcome, $E_g$ is measured with non-classical error. We report Freddie loan counts per geography as a coverage variable and test sensitivity to dropping low-coverage geographies. --- ## 3. Threats, one by one ### 3.1 Pandemic housing-demand reallocation 2020–21 saw large, geographically uneven demand shifts. Markets with the biggest price booms also had the most refinancing (equity + rate incentive), hence the lowest coupons at $t_0$, hence the highest $E_g$. Those same markets then mean-reverted in 2022–23 for reasons unrelated to lock-in. *Response:* control for 2019-01→2021-12 log price growth; exclude top-decile boom markets as a robustness cell; report both. ### 3.2 Remote-work exposure Teleworkable employment share drives both migration and construction, and correlates with the pandemic boom. *Response:* optional adapter for a teleworkable-share control; heterogeneity split. Documented as an *unresolved* threat if the control is unavailable in the slice. ### 3.3 Differential refinancing booms A market where nearly everyone refinanced in 2020–21 has both extreme exposure and an exhausted refinance pipeline, which mechanically depresses subsequent refi counts regardless of lock-in. *Response:* refi-origination outcomes are reported but treated as **mechanically contaminated**; the headline outcome is *purchase* originations. Exclude top-decile refi-intensity markets as a robustness cell. ### 3.4 Local labour-market shocks *Response:* state unemployment control (optional adapter); region × period fixed effects as a robustness cell. ### 3.5 Housing-supply constraints Supply elasticity determines whether a demand shift shows up in prices or quantities. This is not a nuisance — it is part of the mechanism. *Response:* predetermined supply-constraint proxy (historical permits per housing unit); interact with exposure rather than only controlling for it. ### 3.6 Composition change in the observed mortgage stock The active stock shrinks and its composition drifts. Contemporaneous exposure is endogenous to the outcome (markets with more transactions churn their stock faster). *Response:* exposure is fixed at $t_0$ and never recomputed. Contemporaneous exposure is reported *only* as a descriptive series. ### 3.7 National monetary-policy endogeneity The Fed raised rates in response to macro conditions that also affect housing. Because the rate path is national, this is absorbed by $\gamma_t$. The residual concern is that the *interaction* of the national shock with exposure picks up the interaction of macro conditions with whatever else exposure proxies for. *Response:* this is exactly A1/A4. Handled by the balance table and controls, and flagged as the deepest remaining threat. ### 3.8 Geography-specific mortgage-rate differences PMMS is national. Local offered rates differ by tens of basis points. *Response:* measurement error in the *level* of the gap, attenuating loan-level coefficients. Robustness: HMDA-reported interest rates (available 2018+) to build a local rate series; documented as future work if not in the slice. ### 3.9 Differential credit conditions Tightening credit standards in 2022–23 varied locally and reduce originations independent of lock-in. *Response:* HMDA denial rates as a control/placebo outcome. --- ## 4. Falsification tests | Test | Prediction if lock-in is the mechanism | Prediction if confounded | |---|---|---| | Placebo shock date 2018-01 or 2019-06 | $\beta_k \approx 0$ (rate move too small) | non-zero, similar sign | | Multifamily (5+) permits as outcome | Weak — multifamily demand is renter-driven, not lock-in-driven | similar magnitude to single-family | | HMDA denial rate as outcome | $\approx 0$ | non-zero | | Exposure among *investor* loans only | Weaker (investors are less locked-in behaviourally, and second-home/investment loans are a small share) | similar | | Reverse the sign of the shock (2019 rate *decline*) | Opposite-signed | same sign | Each writes a row to `outputs/robustness/grid.parquet` and a paragraph to `reports/failed_hypotheses.md` if it fails. --- ## 5. Loan-level vs local-level: the firewall The loan-level hazard results and the local-market results answer different questions and carry different tiers. They are reported in separate files (`reports/loan_hazard_analysis.md` vs `reports/local_market_event_study.md`) and the synthesis in `reports/technical_report.md` must state the tier of each sentence it combines. The loan-level rate-gap coefficient is **not** a causal elasticity of mobility. It is the conditional association between a point-in-time rate gap and the probability that a loan's balance goes to zero, in a selected population, where the gap is mechanically a function of the note rate the borrower chose and the national rate path. A borrower with a 2.8% note rate in 2023 is different from a borrower with a 6.8% note rate in 2023 in cohort, credit, equity, and tenure. The age dummies, cohort controls, and covariates reduce but do not eliminate that. --- ## 6. Decision rule for causal language A report sentence may use causal language ("reduced", "caused", "led to") only if **all** of the following hold for the underlying artifact: 1. `evidence_tier == "quasi_experimental"`. 2. `pretrend_test.pvalue >= 0.10` (or the failure is disclosed in the same paragraph). 3. At least one placebo specification is reported and does not itself produce a significant effect of the same sign. 4. Clustered standard errors are reported, with the cluster count. 5. The exposure definition and pre-shock date are stated in the sentence or its table. Otherwise the sentence must read "is associated with" / "predicts" / "under the model".