This vignette covers the baseline method of the package: the IVX
estimator and Wald test of Kostakis, Magdalinos & Stamatogiannis
(2015, KMS), its long-horizon version (Kostakis, Magdalinos
& Stamatogiannis 2023), and the tuning and standard-error options
exposed by ivx().
The problem
The predictive regression
\[ y_t = \mu + \beta' x_{t-1} + u_t, \qquad x_t = \mu_x + R_n x_{t-1} + v_t, \qquad t = 1, \dots, n, \]
relates a stationary variable (a return) to lagged predictors that
are often highly persistent, \(R_n = I + C /
n^{\alpha}\) with \(\alpha \in (0,
1]\), and whose innovations \(v_t\) are correlated with \(u_t\). In that setting the OLS t-statistic
is not asymptotically normal: its distribution depends on the unknown
local-to-unity parameter \(C\) and on
the endogeneity correlation \(\delta =
\mathrm{corr}(u_t, v_t)\), and it over-rejects. The package
reports \(\delta\) through
delta().
The IVX instrument
KMS instrument \(x_{t-1}\) with a self-generated variable of controlled, mildly integrated persistence,
\[ z_t = \sum_{j=0}^{t-1} \rho_n^{\,j} \Delta x_{t-j}, \qquad \rho_n = 1 - \frac{c_z}{n^{\beta}}, \quad \beta \in (0, 1), \; c_z > 0, \]
i.e. \(z_t = \rho_n z_{t-1} + \Delta x_t\) with \(z_0 = 0\). Whatever the persistence of \(x_t\), \(z_t\) is mildly integrated, and the IV estimator
\[ \hat\beta_{ivx} = \Big( \sum_t z_{t-1} \tilde x_{t-1}' \Big)^{-1} \sum_t z_{t-1} \tilde y_t, \qquad \tilde x_{t-1} = x_{t-1} - \bar x, \; \tilde y_t = y_t - \bar y, \]
is asymptotically mixed normal. The Wald statistic for \(H_0: H\beta = h\),
\[ W = (H\hat\beta_{ivx} - h)' \big[ H \hat Q H' \big]^{-1} (H\hat\beta_{ivx} - h) \;\to\; \chi^2_{\mathrm{rank}(H)}, \]
uses the covariance \(\hat Q = (Z'\tilde X)^{-1} M (\tilde X' Z)^{-1}\) with, in KMS’s finite-sample corrected form,
\[ M = \hat\sigma_u^2 \sum_t z_{t-1} z_{t-1}' \;-\; n \, \bar z \bar z' \, \hat\Sigma_{FM}, \qquad \hat\Sigma_{FM} = \hat\sigma_u^2 - \hat\Omega_{uv}' \hat\Omega_{vv}^{-1} \hat\Omega_{uv}, \]
where \(\hat\Omega\) are Newey–West
long-run (co)variances of the OLS residuals \(\hat u_t\) and of the AR(1) residuals of
each predictor, with Bartlett bandwidth \(\lfloor n^{1/3} \rfloor\). The subtracted
term corrects the finite-sample effect of estimating the intercept (KMS,
p. 1516) and is what the package reports; summary() prints
the individual Wald statistics (equal to the squared t-ratios) and the
joint statistic.
mod <- ivx(Ret ~ DP + TBL, data = kms)
summary(mod)
#>
#> Call:
#> ivx(formula = Ret ~ DP + TBL, data = kms, horizon = 1)
#>
#> Coefficients:
#> Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP 0.006145 0.004557 1.349 1.819 0.177
#> TBL -0.080717 0.057701 -1.399 1.957 0.162
#>
#> Joint Wald statistic: 3.644 on 2 DF, p-value 0.1617
#> Multiple R-squared: 0.004968, Adjusted R-squared: 0.003036
delta(mod)
#> [1] -0.9755750 -0.0610432Tuning
KMS recommend \(\beta = 0.95\) and
\(c_z = 1\), the package defaults.
Both, and the Newey–West bandwidth, are arguments of
ivx():
coef(ivx(Ret ~ DP + TBL, data = kms, beta = 0.9, cz = 5, bandwidth = 10))
#> DP TBL
#> 0.00501996 -0.12896546Smaller \(\beta\) (or larger \(c_z\)) makes the instrument less persistent: better size under strong endogeneity at some cost in power. Hosseinkouchack & Demetrescu (2021) show the convergence to the \(\chi^2\) limit is slower the closer \(\rho_n\) is to one, and Lee (2016) uses \(c_z = 5\) for quantile regressions.
Long horizons
For a horizon \(K > 1\) the regression is between the \(K\)-period sum \(y_t(K) = \sum_{i=0}^{K-1} y_{t+i}\) and \(x_{t-1}\). Overlapping sums make the usual long-horizon OLS statistics oversized. The package implements the IVX-Wald statistic of Kostakis, Magdalinos & Stamatogiannis (2023), eqs (15) and (23): the estimator regresses \(y_t(K)\) on the \(K\)-period sum \(x_{t-1}(K)\) using the single-lag instrument \(z_{t-1}\),
\[ \tilde A_K = Y(K)' Z \,\big[ X(K)' Z \big]^{-1}, \]
while the covariance uses the \(K\)-period sum of the instrument, \(z_{t-1}(K)\), in place of \(z_{t-1}\) in \(M\) above, with \(\hat\sigma^2_u\) still from the one-step regression. The statistic keeps its \(\chi^2\) limit.
summary(ivx(Ret ~ DP + TBL, data = kms, horizon = 12))
#>
#> Call:
#> ivx(formula = Ret ~ DP + TBL, data = kms, horizon = 12)
#>
#> Coefficients:
#> Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP 0.008202 0.004760 1.723 2.970 0.0848 .
#> TBL -0.062903 0.059869 -1.051 1.104 0.2934
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> Joint Wald statistic: 3.998 on 2 DF, p-value 0.1355
#> Multiple R-squared: 0.05429, Adjusted R-squared: 0.06255Lag augmentation for power
The IVX instrument is less persistent than \(x_t\), so the test loses local power
against OLS exactly when \(x_t\) is
near-integrated and endogenous. Demetrescu (2014) shows that adding
\(y_{t-1}\) to the regression,
instrumented by itself, \[
y_t = \phi\, y_{t-1} + \beta' x_{t-1} + u_t, \qquad \phi = 0 \text{
under the null},
\] feeds the signal back into the instrument and can raise power
substantially when the instrument is weak (small \(\eta\)), while being asymptotically
equivalent to plain IVX otherwise. lag_y = TRUE does this;
the joint Wald statistic still tests only \(\beta\).
ivx(Ret ~ DP + TBL, data = kms, lag_y = TRUE)
#>
#> Call:
#> ivx(formula = Ret ~ DP + TBL, data = kms, lag_y = TRUE, horizon = 1)
#>
#> Coefficients:
#> DP TBL y_lag
#> 0.007248 -0.073288 0.091950Heteroskedasticity-robust standard errors
Demetrescu, Georgiev, Rodrigues & Taylor (2023) show that the IVX statistics keep their standard limits under much weaker assumptions on the innovations (unconditional and conditional heteroskedasticity) if Eicker–White standard errors are used, i.e. \(\hat\sigma_u^2 \sum z z'\) is replaced by \(\sum z_{t-1} z_{t-1}' \hat u_t^2\):
summary(ivx(Ret ~ DP + TBL, data = kms, robust = TRUE))
#>
#> Call:
#> ivx(formula = Ret ~ DP + TBL, data = kms, robust = TRUE, horizon = 1)
#>
#> Coefficients:
#> Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP 0.006145 0.004792 1.282 1.644 0.200
#> TBL -0.080717 0.057250 -1.410 1.988 0.159
#> (Eicker-White standard errors)
#>
#> Joint Wald statistic: 2.893 on 2 DF, p-value 0.2354
#> Multiple R-squared: 0.004968, Adjusted R-squared: 0.003036robust = TRUE is available for horizon = 1;
the long-horizon heteroskedasticity-robust form is not defined in the
literature. See vignette("robust-inference") for the
bootstrap alternative.
Replication of Kostakis et al. (2015)
The kms dataset is the authors’ monthly file. The
published estimates (Table 6, p. 1531 and Table 8, p. 1537) are
reproduced to the printed precision, and these values are asserted in
the package tests.
m6 <- ivx(Ret ~ DE, data = kms)
c(coef = round(coef(m6), 4), wald = round(m6$Wald_Ind, 3), delta = round(delta(m6), 3))
#> coef.DE wald.DE delta
#> -0.0033 0.3930 -0.0670
# paper: -0.0033, 0.393, -0.067
m8 <- ivx(Ret ~ DP + TBL, data = kms)
c(round(coef(m8), 4), joint = round(m8$Wald_Joint, 3))
#> DP TBL joint
#> 0.0061 -0.0807 3.6440
# paper: 0.0061, -0.0807, 3.644Long-horizon Wald statistics for Ret ~ EP + TBL (Table
13, p. 1547):
h <- c(4, 12, 24, 36, 48, 60)
t13 <- t(sapply(h, function(k) {
m <- ivx(Ret ~ EP + TBL, data = kms, horizon = k)
c(EP = m$Wald_Ind[["EP"]], TBL = m$Wald_Ind[["TBL"]], joint = m$Wald_Joint)
}))
round(cbind(horizon = h, t13), 3)
#> horizon EP TBL joint
#> [1,] 4 5.778 3.894 7.638
#> [2,] 12 6.383 3.166 7.614
#> [3,] 24 4.990 2.124 5.794
#> [4,] 36 4.599 1.915 5.383
#> [5,] 48 4.983 1.441 5.660
#> [6,] 60 4.321 1.039 4.822
# paper: EP 5.778 6.383 4.990 4.599 4.983 4.321 | TBL 3.894 3.166 2.124 1.915 1.441 1.039Caveats
- The regression must carry an intercept;
ivx()handles it internally and warns if the formula removes it. - The instrument uses the whole sample, so
ivx()on a subsample is not the same as a subsample statistic with the full-sample instrument; useivx_episodic()for the latter. - With strong endogeneity (\(|\delta|\) near 1) and near-unit-root
predictors the asymptotic test still over-rejects in samples of a few
hundred observations, especially one-sided.
ivx_boot()andivx_ra()address this. - Weighted fits (
weights) are supported but rarely appropriate for the theory.
References
Demetrescu, M. (2014). Enhancing the local power of IVX-based tests in predictive regressions. Economics Letters, 124(2), 269–273.
Kostakis, A., Magdalinos, T., & Stamatogiannis, M. P. (2015). Robust econometric inference for stock return predictability. Review of Financial Studies, 28(5), 1506–1553.
Kostakis, A., Magdalinos, T., & Stamatogiannis, M. P. (2023). Taking stock of long-horizon predictability tests: Are factor returns predictable? Journal of Econometrics, 237(2), 105380.
Magdalinos, T., & Phillips, P. C. B. (2009). Limit theory for cointegrated systems with moderately integrated and moderately explosive regressors. Econometric Theory, 25(2), 482–526.
Phillips, P. C. B., & Lee, J. H. (2013). Predictive regression under various degrees of persistence and robust long-horizon regression. Journal of Econometrics, 177(2), 250–264.
Hosseinkouchack, M., & Demetrescu, M. (2021). Finite-sample size control of IVX-based tests in predictive regressions. Econometric Theory, 37(4), 769–793.