--- title: "Residual-augmented IVX" output: rmarkdown::html_vignette vignette: > %\VignetteIndexEntry{Residual-augmented IVX} %\VignetteEngine{knitr::rmarkdown} %\VignetteEncoding{UTF-8} --- ```{r, include = FALSE} knitr::opts_chunk$set(collapse = TRUE, comment = "#>") ``` ```{r setup, message = FALSE} library(ivx) ``` `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. ```{r} m <- ivx_ra(Ret ~ DP + TBL, data = kms) m summary(m) ``` 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. ```{r} rbind(ivx = coef(ivx(Ret ~ DP + TBL, data = kms)), ivx_ra = coef(m)) ``` ## 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`. ```{r} sapply(c(1, 3, 12), function(h) ivx_ra(Ret ~ DP, data = kms, horizon = h)$tstat) ``` Under the null with a near-integrated, strongly endogenous predictor ($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.