ivx_iv() implements the instrumental-variable
predictability tests of Breitung & Demetrescu (2015). IVX is one
member of the family: any instrument \(z_{t-1}\) that is less persistent than
\(x_{t-1}\) but correlated with it
gives a 2SLS \(t\)-ratio with a
standard normal null limit whatever the local-to-unity parameter \(c\).
Instruments
Type-I instruments are functions of the predictor:
-
"frac": the truncated fractional difference \(\Delta_+^{d} x_{t-1}\), \(d = 1/2\) by default — \(I(1/2)\) when \(x\) is \(I(1)\), still informative when \(x\) is stationary; -
"diff": the long difference \(x_{t-1} - x_{t-1-k_T}\) with \(k_T = \lfloor 0.2\,T^{0.85} \rfloor\); - the IVX filter of Kostakis et al. (2015) is the paper’s “mild
integration” case (
ivx()).
Type-II instruments are deterministic and only correlate with a
near-integrated predictor: "sin" uses \(\sin(\pi t/T)\) (with \(K\) predictors, frequencies \(k = 1, \dots, K\)). They are weak when
\(x\) is stationary, so the paper
recommends the over-identified combination "comb" = sine +
fractional: the 2SLS statistic is asymptotically driven by whichever
instrument is informative (Theorem 3), so it keeps power in both
regimes.
The statistic
With \(Z\) the instrument matrix (intercept included in both stages) and \(\hat u_t\) the OLS residuals of the predictive regression, the squared \(t\)-ratio of eq. (12) is \[ t^2 = \frac{\big(x'Z(Z'Z)^{-1}Z'y\big)^2} {x'Z(Z'Z)^{-1}\big(\sum_t z_t z_t' \hat u_t^2\big)(Z'Z)^{-1}Z'x} \;\to\; \chi^2(1), \] and its \(K\)-predictor Wald analogue is \(\chi^2(K)\) (Theorem 7).
summary(ivx_iv(Ret ~ DP + TBL, data = kms))
#>
#> Call:
#> ivx_iv(formula = Ret ~ DP + TBL, data = kms)
#>
#> Coefficients:
#> Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP 0.0006284 0.0061730 0.102 0.010 0.919
#> TBL -0.0626958 0.0822806 -0.762 0.581 0.446
#> (Eicker-White standard errors)
#>
#> Joint Wald statistic: 0.9973 on 2 DF, p-value 0.6073
#> Multiple R-squared: 0.007404, Adjusted R-squared: 0.00451
sapply(c("comb", "sin", "frac", "diff"),
function(i) ivx_iv(Ret ~ DP, data = kms, instruments = i)$tstat)
#> comb.DP sin.DP frac.DP diff.DP
#> 0.4900820 0.4655899 -0.7089655 -1.6445824Replication of Table 1
The paper’s design: \(T = 250\), \(\mathrm{corr}(u, v) = 0.9\), \(\beta = b/T\), two-sided tests at 10%. Rejection rates in 1000 replications (paper values in brackets, 10 000 replications):
| \(\rho\) | \(b\) | comb | sin | frac | diff |
|---|---|---|---|---|---|
| 1.00 | 0 | 12.7 (11.2) | 11.5 (9.9) | 12.2 (11.1) | 11.2 (12.5) |
| 1.00 | 10 | 52.6 (65.7) | 38.5 (61.4) | 41.5 (33.4) | 30.4 (33.6) |
| 1.00 | 20 | 88.5 (91.3) | 59.6 (79.5) | 81.0 (66.9) | 68.9 (66.1) |
| 0.96 | 0 | 11.6 (10.4) | 9.7 (9.9) | 11.4 (9.4) | 10.7 (10.5) |
| 0.96 | 10 | 42.6 (47.3) | 29.4 (34.4) | 31.4 (30.1) | 33.2 (33.4) |
Sizes and the ranking (combination best, sine strong only near the unit root) reproduce; the split of power between the sine and fractional instruments differs from the paper, which does not report the exact demeaning and initialisation used for the deterministic instrument.
Caveats
- Short horizon only, no bootstrap.
- The sign (Cauchy) instrument of the paper needs recursive demeaning of the regressor and forward demeaning of the response and is not included.
- Several type-II instruments per predictor are not allowed (Assumption 5); the implementation uses one sine frequency per predictor.