Residual-augmented IVX

Idea

Write the innovations of the predictor’s autoregression as \(v_t\) and decompose the regression error as \(u_t = \gamma' v_t + \eta_t\) with \(\eta_t\) uncorrelated with \(v_t\). The finite-sample bias of any estimator of \(\beta\) comes from \(\gamma \ne 0\). If \(v_t\) were observed, adding it as a regressor,

\[ \tilde y_t = \beta' \tilde x_{t-1} + \gamma' v_t + \eta_t, \]

would remove the problem. Amihud & Hurvich estimate \(v_t\) from a bias-corrected autoregression; Demetrescu & Rodrigues show that augmenting the IVX regression with plain autoregressive residuals is enough to reduce the bias substantially while retaining the persistence-robust \(\chi^2\) inference.

Procedure

1. Fit an autoregression of order \(p\) in levels to the predictors (a VAR with several predictors), without intercept, \(p\) chosen by AIC (the paper’s recommendation); keep the residuals \(\hat\varepsilon_t\), \(t = p+1, \dots, n\), and demean them.
2. Regress the demeaned response on \(\hat\varepsilon_t\) by OLS and keep the residuals \(\tilde y_t = \tilde y_t^{\,0} - \hat\gamma' \hat\varepsilon_t\).
3. Estimate \(\beta\) by IVX of \(\tilde y_t\) on \(\tilde x_{t-1}\) with the KMS instrument (not demeaned).

Standard errors are the heteroskedasticity-robust form of the paper’s eq. (9)/(14), which stay valid whether \(x_t\) is stationary or near-integrated:

\[ \widehat{\mathrm{Cov}}(\tilde\beta_{ivx}) = B^{-1} M B^{-1\prime}, \quad B = \sum_t z_{t-1}\tilde x_{t-1}', \quad M = \sum_t z_{t-1} z_{t-1}' \tilde\varepsilon_t^2 + \hat Q_T - n \bar z \bar z' \hat\Sigma_{FM}, \]

where \(\tilde\varepsilon_t\) are the OLS residuals of the augmented regression, \(\hat Q_T = H_{zx} H_{xx}^{-1} \tilde H_{xx} H_{xx}^{-1} H_{zx}'\) accounts for the estimation of \(\hat\varepsilon_t\) (it matters only in the stationary case), and the last term is the Kostakis et al. (2015) intercept correction that the paper also uses in its simulations.

m <- ivx_ra(Ret ~ DP + TBL, data = kms)
m
#> 
#> Call:
#> ivx_ra(formula = Ret ~ DP + TBL, data = kms)
#> 
#> Residual-augmented IVX, AR order p = 4 (aic)
#> 
#> Coefficients:
#>        DP        TBL  
#> -0.001934  -0.053984
summary(m)
#> 
#> Call:
#> ivx_ra(formula = Ret ~ DP + TBL, data = kms)
#> 
#> Coefficients:
#>      Estimate Std. Error t value Wald Ind Pr(> chi)
#> DP  -0.001934   0.005010  -0.386    0.149     0.699
#> TBL -0.053984   0.055313  -0.976    0.953     0.329
#> (Eicker-White standard errors)
#> 
#> Joint Wald statistic:  1.423 on 2 DF, p-value 0.4908
#> Multiple R-squared:  0.01794,    Adjusted R-squared:  0.01602

Compared with plain IVX the point estimate on the strongly endogenous predictor (DP, \(\delta \approx -0.98\)) shrinks and the standard error is smaller — the efficiency gain from removing the \(\gamma' v_t\) component of the error.

rbind(ivx = coef(ivx(Ret ~ DP + TBL, data = kms)), ivx_ra = coef(m))
#>                  DP         TBL
#> ivx     0.006145163 -0.08071667
#> ivx_ra -0.001934204 -0.05398376

Replication: size in the paper’s Monte Carlo

The paper has no reproducible empirical application, so the check is against the size rows of its Table 3 (right-sided 5% tests, \(T = 200\), DGP (24)–(25) with \(\rho = 1 - c/T\), short-run AR parameter \(-0.5\) and innovation correlation \(-0.95\)). With 10 000 replications:

The over-rejection of plain IVX under strong endogeneity is largely removed; the residual offset of about 0.01 at small \(c\) is shared by the plain IVX column and so reflects a detail of the simulation design rather than the estimator.

Long horizons: the transformed regression

Demetrescu, Rodrigues & Taylor (2023) extend the estimator to the \(h\)-period regression without HAC estimation. Following Britten-Jones et al. (2011), the overlapping regression of \(\sum_{j=1}^h y_{t+j}\) on \(x_t\) is numerically the non-overlapping regression of \(y_{t+1}\) on the transformed regressor \(A_h' x\), and the same trick applies to the instrument: \[ z_t^{trf,(h)} = \sum_{i=\max(1,\,t-h+1)}^{\min(t,\,T-h)} z_i, \qquad \hat\beta_h = \Big(\sum_{t=1}^{T-h} z_t \bar x_t'\Big)^{-1} \sum_{t=p}^{T-1} z_t^{trf,(h)}\,(\bar y_{t+1} - \hat\gamma'\hat\varepsilon_{t+1}), \] with the sandwich covariance of eq. (5.7), i.e. the short-horizon one with \(z_t\) replaced by \(z_t^{trf,(h)}\). horizon = h does exactly this and reduces to the short-horizon estimator at h = 1.

sapply(c(1, 3, 12), function(h) ivx_ra(Ret ~ DP, data = kms, horizon = h)$tstat)
#>         DP         DP         DP 
#> -0.4642328 -0.3057244 -0.3139751

Under the null with a near-integrated, strongly endogenous predictor (\(c = 0\), \(\delta = -0.95\), \(T = 500\)) the two-sided 5% test rejects at 0.034 (\(h = 10\)) and 0.040 (\(h = 20\)) in 2000 replications, inside the [0.023, 0.058] range the paper reports for \(T = 500\).

Caveats and implementation notes

• Two details the paper leaves implicit are decisive: the autoregression must be fitted without an intercept (the paper’s \(\mu_0\) variant; its OLS-demeaned \(\mu_1\) variant is much more biased and “not recommended for testing”), and the residuals must be demeaned before augmentation — without that the term \(\gamma \bar{\hat\varepsilon}\) survives and the bias reduction fails. Both are built in and neither is optional.
• The remaining bias is \(-\gamma\) times the bias of the AR coefficients, so it grows with \(|\gamma|\) and with the persistence of the predictor; it is much smaller than for plain IVX but not zero.
• No bootstrap: ivx_boot() does not accept ivx_ra objects. For horizon > 1 the Kostakis et al. (2015) intercept correction is not applied (it is not derived for the transformed regression), and fitted values and residuals are those of the transformed regression.
• The AR order is selected by AIC in levels, which the authors argue copes with both the stationary and the integrated case; ar fixes it instead.

References

• Demetrescu, M., & Rodrigues, P. M. M. (2022). Residual-augmented IVX predictive regression. Journal of Econometrics, 227(2), 429–460.
• Demetrescu, M., Rodrigues, P. M. M., & Taylor, A. M. R. (2023). Transformed regression-based long-horizon predictability tests. Journal of Econometrics, 237(2), 105316.
• Amihud, Y., & Hurvich, C. M. (2004). Predictive regressions: A reduced-bias estimation method. Journal of Financial and Quantitative Analysis, 39(4), 813–841.