Reducing large datasets to improve the identification of estimated policy rules

Abstract

Monetary policy rules describe how policy interest rates respond to macroeconomic developments. These rules incorporate forward-looking models that require instruments for consistent estimation. The use of standard instruments leads to weak identification of forward-looking rules. We combine principal component analysis with hard thresholding to construct new instruments based on high-dimensional macro data. Component-based instruments enhance the identification of policy rules relative to standard results. The finding is attributed to specific variables in the data.

Appendix

Are component-based instruments valid for the AR test? Kapetanios et al. (2016), KKM hereafter, answer in the affirmative in an extended AR test where factor-based instruments are estimated via principal components that achieve size control under standard regularity conditions.

The proof builds on Eqs. (6) and (7), where $X$ is dropped for notational convenience. The AR statistic for the null hypothesis $b={b}_{h}$ in the structural equation is:

$$AR\left({b}_{h}|Z\right)=\frac{T-n}{n}\frac{{\left(D-Y{b}_{h}\right)}^{^{\prime}}\left[I-M\left(Z\right)\right]\left(D-Y{b}_{h}\right)}{{\left(D-Y{b}_{h}\right)}^{^{\prime}}\left[M(Z)\right](D-Y{b}_{h})}$$

where $M\left( K \right) = I - N\left( K \right)$ and $N\left( K \right) = K\left( {K^{\prime}K} \right)^{ - 1} K^{\prime}$ for any full-column rank matrix $K$.

KKM modify the AR test to accommodate the common factors of a dataset. In this context, if $F$ is a $T\times r$ matrix of factors, the AR statistic takes the following form.

$$AR\left({b}_{h}|F\right)=\frac{T-r}{r}\frac{{\left(D-Y{b}_{h}\right)}^{^{\prime}}\left[I-M\left(F\right)\right]\left(D-Y{b}_{h}\right)}{{\left(D-Y{b}_{h}\right)}^{^{\prime}}\left[M(F)\right](D-Y{b}_{h})}$$

Suppose that $\widehat{F}$ is a $T\times r$ matrix of leading principal components of this dataset. It can be shown that $AR\left({b}_{h}|F\right)$ and $AR\left({b}_{h}|\widehat{F}\right)$ are asymptotically equivalent so that factors can be replaced by their component-based estimates.

To prove $AR\left({b}_{h}|F\right)-AR\left({b}_{h}|\widehat{F}\right)={o}_{p}\left(1\right)$ requires that:

$$\frac{\left(T-r\right)}{r}\frac{{\left(D-Y{b}_{h}\right)}^{^{\prime}}F{\left({F}^{^{\prime}}F\right)}^{-1}{F}^{^{\prime}}\left(D-Y{b}_{h}\right)}{{\left(D-Y{b}_{h}\right)}^{^{\prime}}\left(I-F{\left({F}^{^{\prime}}F\right)}^{-1}{F}^{^{\prime}}\right)(D-Y{b}_{h})}-\frac{\left(T-r\right)}{r}\frac{{\left(D-Y{b}_{h}\right)}^{^{\prime}}\widehat{F}{\left({\widehat{F}}^{^{\prime}}\widehat{F}\right)}^{-1}{\widehat{F}}^{^{\prime}}\left(D-Y{b}_{h}\right)}{{\left(D-Y{b}_{h}\right)}^{^{\prime}}\left(I-\widehat{F}{\left({\widehat{F}}^{^{\prime}}\widehat{F}\right)}^{-1}{\widehat{F}}^{^{\prime}}\right)\left(D-Y{b}_{h}\right)}={o}_{p}\left(1\right)$$

which satisfies if:

$${v}^{^{\prime}}FH{({H}^{^{\prime}}{F}^{^{\prime}}FH)}^{-1}{H}^{^{\prime}}{F}^{^{\prime}}v-{v}^{^{\prime}}\widehat{F}{\left({\widehat{F}}^{^{\prime}}\widehat{F}\right)}^{-1}{\widehat{F}}^{^{\prime}}v={v}^{^{\prime}}F{\left({F}^{^{\prime}}F\right)}^{-1}{F}^{^{\prime}}v-{v}^{^{\prime}}\widehat{F}{\left({\widehat{F}}^{^{\prime}}\widehat{F}\right)}^{-1}{\widehat{F}}^{^{\prime}}v={o}_{p}\left(1\right)$$

under the null hypothesis and

$${Y}^{^{\prime}}FH{({H}^{^{\prime}}{F}^{^{\prime}}FH)}^{-1}{H}^{^{\prime}}{F}^{^{\prime}}Y-{Y}^{^{\prime}}\widehat{F}{\left({\widehat{F}}^{^{\prime}}\widehat{F}\right)}^{-1}{\widehat{F}}^{^{\prime}}Y={Y}^{^{\prime}}F{\left({F}^{^{\prime}}F\right)}^{-1}{F}^{^{\prime}}Y-{Y}^{^{\prime}}\widehat{F}{\left({\widehat{F}}^{^{\prime}}\widehat{F}\right)}^{-1}{\widehat{F}}^{^{\prime}}Y={o}_{p}\left(1\right)$$

under the alternative hypothesis for any non-singular rotation matrix $H$.

The above expressions hold if the following is satisfied.

$$\frac{{F^{\prime}F}}{T} - \frac{{\hat{F}^{^{\prime}} F}}{T} = o_{p} \left( 1 \right)$$

$$\sqrt T \left( {\frac{{F^{\prime}Y}}{T} - \frac{{\hat{F}^{^{\prime}} Y}}{T}} \right) = o_{p} \left( 1 \right)$$

$$\sqrt T \left( {\frac{{F^{\prime}v}}{T} - \frac{{\hat{F}^{^{\prime}} v}}{T}} \right) = o_{p} \left( 1 \right).$$

KKM show that these three conditions are met if factors $F$, endogenous variables $Y$, and the structural error $v$ have finite fourth moments, a non-singular covariance matrix, and satisfy the central limit theorem. These in turn hold under Assumptions 1–3 from KKM and Lemma A.1 from Bai and Ng (2006).

See Figs. 9, 10, 11, 12, and 13.

Fig. 9

All weights in components 1 and 2. Figure plots weights on 109 macro variables for components 1 and 2. Variable codes are from Stock and Watson (2009)

Fig. 10

All weights in components 4 and 5. Figure plots weights on 109 macro variables for components 4 and 5. Variable codes are from Stock and Watson (2009)

Fig. 11

All weights in components 7 and 8. Figure plots weights on 109 macro variables for components 7 and 8. Variable codes are from Stock and Watson (2009)

Fig. 12

All weights in components 11 and 13. Figure plots weights on 109 macro variables for components 11 and 13. Variable codes are from Stock and Watson (2009)

Fig. 13

All weights in components 14 and 15. Figure plots weights on 109 macro variables for components 14 and 15. Variable codes are from Stock and Watson (2009)