Parameter instability is a change in a regression model's coefficients or error variance at some point within the sample. A trading model fitted on decades of data may carry a coefficient that held in one regime and not the next. In "Testing for parameter instability in linear models" (1992), Hansen applied Lagrange multiplier stability tests to Romer's corrected U.S. GNP data for 1889-1987. The tests rejected a constant error variance (L = 0.45) but not a constant autoregressive parameter (L = 0.19).
What the Study Found
The individual stability test has an asymptotic 5 percent critical value of 0.470, so Hansen proposes treating values above one-half as significant. The persistence parameter swung from -0.12 in 1889-1929 to 0.60 in 1930-1947 and 0.05 in 1948-1987. The error variance moved from 0.0013 to 0.0054 to 0.0006 across the same three subperiods. The joint statistic for all three parameters was 0.72, with an R-squared of 0.13. After a GLS correction for variance shifts in 1930 and 1948, the autoregressive parameter's statistic was 0.18.
"We conclude that the data are not sufficiently informative to determine whether or not the autoregressive parameter is stable, given that the error variance is unstable."
Hansen (1992), Testing for parameter instability in linear models, p. 528.
Methodology
The first example uses Romer's corrected annual per capita real GNP, excluding pre-1888 data as unreliable. The second uses quarterly U.S. consumption and disposable income from Blinder and Deaton (1985), with inflation from the Citibase GNP deflator. The samples run 1889-1987 and 1953:2-1984:4. Standard errors use White's heteroskedasticity-robust estimator, and finite-sample size is checked with 2,000 simulated samples.
Key Statistics
| Metric | Finding | Context |
|---|---|---|
| Individual test, 5% critical value | 0.470 | One degree of freedom; informal rule: significant above one-half |
| Individual test, 10% critical value | 0.353 | One degree of freedom (Table 1) |
| Joint test, 5% critical value | 1.01 | Three degrees of freedom (Table 1) |
| Error variance stability statistic | 0.45* | GNP AR(1), 1889-1987; rejects at asymptotic 10% level |
| AR parameter stability statistic | 0.19 | GNP AR(1), 1889-1987; fails to reject |
| Size under the null, AR test | 0.11 | Nominal 10% test, 2,000 simulated samples |
| Consumption ECM error variance statistic | Levels: 1.14***, 1.09***; logs: 0.21, 0.16 | Models A, B (levels) and C, D (logs); *** rejects at 1% |
| Individual statistic | L_i = (1 / (n V_i)) Σ S_it² | S_it = cumulative first-order conditions; V_i = Σ f_it² |
| Joint statistic | L_c = (1/n) Σ S_t′ V⁻¹ S_t | Tests all regression coefficients and the error variance together |
Variance and AR Parameter Shifting vs AR Parameter Shifting
| Measure | α₁ and σ² varying | α₁ varying |
|---|---|---|
| AR parameter test rejection rate | 0.11 | 0.30 |
| Error variance test rejection rate | 0.61 | 0.10 |
| Joint test rejection rate | 0.25 | 0.17 |
Why This Matters
Splitting a sample at a date chosen after looking at the data biases conventional tests toward finding breaks that never happened. Hansen's tests need only one full-sample estimate and no chosen breakpoint, so they avoid that selection problem. For anyone checking whether a strategy or factor model has decayed, a noisy error variance can hide real coefficient drift. A significant statistic points to misspecification, not to a ready-made fix.