\begin{algorithmic}[1]
\State Construct the IV $z_{t-1}$ by equation (\ref{mulivz}).
\State Construct the IV estimators $\hat{\beta}_{ivx}$, $\hat{\beta}_a$ and $\hat{\beta}_b$ using the full sample and two subsamples by equations (\ref{defdeftwo2}), (\ref{muldef2new}) and (\ref{muldef3new}).
\State Eliminate the DE: Construct the weighted IV estimator $\hat{\beta}_l \equiv (W_1-W_2-W_3)^{-1}(W_1 \hat{\beta}_{ivx}-W_2\hat{\beta}_a -W_3 \hat{\beta}_b)$, where $W_1=\sum\nolimits_{t=1}^{T} \bar{z}_{t-1} x_{t-1}^\top $, $W_2=S_a \sum\nolimits_{t=1}^{T_0} \left( z_{t-1}- \frac{1}{T_0}\sum\nolimits_{t=1}^{T_0} z_{t-1} \right) x_{t-1}^\top$ and
$W_3=S_b \sum\nolimits_{t=T_0+1}^{T} \left( z_{t-1}- \frac{1}{T-T_0}\sum\nolimits_{t=T_0+1}^{T} z_{t-1} \right) x_{t-1}^\top$.
\State Control size for both SD and WD predictors:
\begin{itemize}
 \item Reduce the DiE: Construct the estimator $\hat{\beta}_m = \hat{\beta}_{l}+ {B_m} {W_z} T^{-(1 -\delta) / 2} \frac{K+1}{2}{\hat{\rho}_{u v^*}}/ \sqrt{-2 c_z}$, where
${B_m}\equiv \left( \sum_{t=1}^T \tilde{z}_{t-1} x_{t-1}^\top \right)^{-1}\left( \sum_{t=1}^T \tilde{z}_{t-1} \tilde{z}_{t-1}^\top \hat{u}_t^2 \right)^{1/2}\left(\operatorname{I_K}+ {W_z}\hat{\varpi}_l\hat{\varpi}_l^\top{W_z}^\top \right)^{1/2}$.
 \item Reduce the VEE: Construct the estimator for the asymptotic covariance matrix of as $\operatorname{\widehat{Avar}}(\hat{\beta}_m ) \equiv H_l \left(\operatorname{I_K}+{W_z}\hat{\varpi}_l\hat{\varpi}_l^\top {W_z}^\top \right) H_l^\top$, where
$
H_l = \left( \sum_{t=1}^T \tilde{z}_{t-1} x_{t-1}^\top \right)^{-1}\left(\frac{T}{T-2K-1} \sum_{t=1}^T \tilde{z}_{t-1} \tilde{z}_{t-1}^\top \hat{u}_t^2 \right)^{1/2}
$.
\item \textcolor{black}{Reduce the size distortion induced by the variance of the estimator of $u_t$: Use Lagrange-multiplier principle to obtain $\hat{u}_t= y_t - \hat{\mu}_s - x_{t-1}^\top \hat{\beta}_s$ by the constrained OLS estimation
$(\hat{\mu}_s,\hat{\beta}_s)^\top  = \arg\, \min_{\mu,\beta} \left( y_t - \mu - x_{t-1}^\top \beta\right)^2,
\ s.t. \quad R\beta =r_J$.}
\end{itemize}
\State Construct the test statistic ${Q_m} \equiv \left(R \hat{\beta}_m -r_J \right)^\top \left[R\operatorname{\widehat{Avar}}(\hat{\beta}_m )R^\top\right]^{-1} \left(R \hat{\beta}_m -r_J \right)$ with SD and WD predictors for $H_0:R\beta =r_J$.
\If{Focus on the one-sided marginal test $H_0:\beta_i=0$ vs $H_a:\beta_i>0$ and $H_0:\beta_i=0$ vs $H_a:\beta_i<0$ }
 \State Construct the t-test statistic ${Q_m^t} \equiv \frac{R \hat{\beta}_m -r_J}{ \left[ R\operatorname{\widehat{Avar}}(\hat{\beta}_m )R^\top\right]^{1/2}}$.
\EndIf
\end{algorithmic}

\begin{algorithmic}
[1]
\State Construct the IV $z_{t-1}$ by equation (\ref{mulivz}).
\State Construct the IV estimators $\hat{\beta}_{ivx}$, $\hat{\beta}_a$ and $\hat{\beta}_b$ using the full sample and two subsamples by equations (\ref{defdeftwo2}), (\ref{muldef2new}) and (\ref{muldef3new}).
\State Eliminate the DE: Construct the weighted IV estimator $\hat{\beta}_l \equiv (W_1-W_2-W_3)^{-1}(W_1 \hat{\beta}_{ivx}-W_2\hat{\beta}_a -W_3 \hat{\beta}_b)$, where $W_1=\sum\nolimits_{t=1}^{T} \bar{z}_{t-1} x_{t-1}^\top $, $W_2=S_a \sum\nolimits_{t=1}^{T_0} \left( z_{t-1}- \frac{1}{T_0}\sum\nolimits_{t=1}^{T_0} z_{t-1} \right) x_{t-1}^\top$ and
$W_3=S_b \sum\nolimits_{t=T_0+1}^{T} \left( z_{t-1}- \frac{1}{T-T_0}\sum\nolimits_{t=T_0+1}^{T} z_{t-1} \right) x_{t-1}^\top$.
\State Control size for both SD and WD predictors:
\begin{itemize}
\item Reduce the DiE: Construct the estimator $\hat{\beta}_m = \hat{\beta}_{l}+ {B_m} {W_z} T^{-(1 -\delta) / 2} \frac{K+1}{2}{\hat{\rho}_{u v^*}}/ \sqrt{-2 c_z}$, where
${B_m}\equiv \left( \sum_{t=1}^T \tilde{z}_{t-1} x_{t-1}^\top \right)^{-1}\left( \sum_{t=1}^T \tilde{z}_{t-1} \tilde{z}_{t-1}^\top \hat{u}_t^2 \right)^{1/2}\left(\operatorname{I_K}+ {W_z}\hat{\varpi}_l\hat{\varpi}_l^\top{W_z}^\top \right)^{1/2}$.
\item Reduce the VEE: Construct the estimator for the asymptotic covariance matrix of as $\operatorname{\widehat{Avar}}(\hat{\beta}_m ) \equiv H_l \left(\operatorname{I_K}+{W_z}\hat{\varpi}_l\hat{\varpi}_l^\top {W_z}^\top \right) H_l^\top$, where
$
H_l = \left( \sum_{t=1}^T \tilde{z}_{t-1} x_{t-1}^\top \right)^{-1}\left(\frac{T}{T-2K-1} \sum_{t=1}^T \tilde{z}_{t-1} \tilde{z}_{t-1}^\top \hat{u}_t^2 \right)^{1/2}
$.
\item \textcolor{black}{Reduce the size distortion induced by the variance of the estimator of $u_t$: Use Lagrange-multiplier principle to obtain $\hat{u}_t= y_t - \hat{\mu}_s - x_{t-1}^\top \hat{\beta}_s$ by the constrained OLS estimation
$(\hat{\mu}_s,\hat{\beta}_s)^\top = \arg\, \min_{\mu,\beta} \left( y_t - \mu - x_{t-1}^\top \beta\right)^2,
\ s.t. \quad R\beta =r_J$.}
\end{itemize}
\State Construct the test statistic ${Q_m} \equiv \left(R \hat{\beta}_m -r_J \right)^\top \left[R\operatorname{\widehat{Avar}}(\hat{\beta}_m )R^\top\right]^{-1} \left(R \hat{\beta}_m -r_J \right)$ with SD and WD predictors for $H_0:R\beta =r_J$.
\If{Focus on the one-sided marginal test $H_0:\beta_i=0$ vs $H_a:\beta_i>0$ and $H_0:\beta_i=0$ vs $H_a:\beta_i<0$ }
\State Construct the t-test statistic ${Q_m^t} \equiv \frac{R \hat{\beta}_m -r_J}{ \left[ R\operatorname{\widehat{Avar}}(\hat{\beta}_m )R^\top\right]^{1/2}}$.
\EndIf
\end{algorithmic}
{
    "title": "robust inference for multiple predictive regressions with an application   on bond risk premia",
    "id": "2401.01064",
    "abstract": "we propose a robust hypothesis testing procedure for the predictability of multiple predictors that could be highly persistent. our method improves the popular extended instrumental variable (ivx) testing (phillips and lee, 2013; kostakis et al., 2015) in that, besides addressing the two bias effects found in hosseinkouchack and demetrescu (2021), we find and deal with the variance-enlargement effect. we show that two types of higher-order terms induce these distortion effects in the test statistic, leading to significant over-rejection for one-sided tests and tests in multiple predictive regressions. our improved ivx-based test includes three steps to tackle all the issues above regarding finite sample bias and variance terms. thus, the test statistics perform well in size control, while its power performance is comparable with the original ivx. monte carlo simulations and an empirical study on the predictability of bond risk premia are provided to demonstrate the effectiveness of the newly proposed approach.",
    "categories": "stat.me econ.em",
    "doi": "",
    "created": "2024-01-02",
    "updated": "",
    "authors": [
        "xiaosai liao",
        "xinjue li",
        "qingliang fan"
    ],
    "affiliation": [],
    "url": "https://arxiv.org/abs/2401.01064"
}
{
    "ToxicityFilter": {
        "text_to_toxicity_attributes": {
            "agsm.bst": {
                "toxicity_score": 0.02308189,
                "severe_toxicity_score": 0.0014400482,
                "identity_attack_score": 0.0073625734,
                "insult_score": 0.011160898,
                "profanity_score": 0.016889455,
                "threat_score": 0.007754337,
                "sexually_explicit_score": 0.014154521,
                "flirtation_score": 0.35954246
            },
            "manuscriptarxiv.tex": {
                "toxicity_score": 0.012126249,
                "severe_toxicity_score": 0.0011730194,
                "identity_attack_score": 0.0038292783,
                "insult_score": 0.0076838774,
                "profanity_score": 0.015488927,
                "threat_score": 0.007598988,
                "sexually_explicit_score": 0.01250316,
                "flirtation_score": 0.21894184
            },
            "manuscriptarxiv.bbl": {
                "toxicity_score": 0.012314741,
                "severe_toxicity_score": 0.0012016296,
                "identity_attack_score": 0.0041067624,
                "insult_score": 0.007797878,
                "profanity_score": 0.015181494,
                "threat_score": 0.007586042,
                "sexually_explicit_score": 0.011264639,
                "flirtation_score": 0.20508216
            }
        },
        "reason": null
    }
}
{
    "num_done": {
        "figure": 0,
        "algorithm": 2
    }
}
{
    "NonTrivialRenderingFilter": {
        "white_pixels_ratio": 91.93296188564327,
        "hash": "070f0f8387830f0f",
        "most_frequent_color_ratio": 91.93296188564327
    }
}