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Residual-augmented IVX

Source: vignettes/ivx-ra.Rmd
ivx-ra.Rmd

ivx_ra() implements the residual-augmented IVX estimator of Demetrescu & Rodrigues (2022), a bias-reduced version of IVX in the spirit of Amihud & Hurvich (2004).

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:

\(c\) paper \(t_{ivx}\) ivx paper \(\tilde t_{ivx}^{\mu_0}\) ivx_ra
0 0.116 0.129 0.054 0.064
10 0.088 0.089 0.055 0.064
20 0.074 0.072 0.055 0.064
30 0.066 0.057 0.053 0.055
40 0.064 0.057 0.050 0.059
50 0.061 0.051 0.050 0.056

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.