IVX-AR: serially correlated errors • ivx

library(ivx)

ivx_ar() implements the IVX-AR procedure of Yang, Long, Peng & Cai (2020), which extends the Kostakis et al. (2015) test to predictive regressions whose errors are serially correlated — the situation the authors document for housing price index returns.

Model

\[ y_t = \mu + \beta' x_{t-1} + u_t, \qquad u_t = \sum_{j=1}^{q} \phi_j u_{t-j} + \varepsilon_t, \]

with persistent predictors as in vignette("ivx"). Serial correlation in $u_t$ invalidates the IVX Wald statistic’s $\chi^2$ limit (and the Newey–West correction inside it is designed for the predictor’s innovations, not for $u_t$). Applying the AR filter $\phi(L) = 1 - \sum_j \phi_j L^j$ to both sides,

\[ \phi(L)\, y_t = \phi(1)\mu + \beta' \phi(L)\, x_{t-1} + \varepsilon_t, \]

gives a predictive regression with white-noise errors and the same slope $\beta$, to which IVX applies. Since $\phi$ is unknown and its estimate from the residuals is biased under persistence, the paper profiles over it.

Procedure

Fit the plain IVX regression and fit an AR($q$) model to its OLS residuals, choosing $q$ by an information criterion (ar = "auto", ar_ic, ar_max) or fixing it (ar = q).
For every point $\phi$ of a grid around the AR estimates (ar_grid, default $\pm 0.3$ in steps of 0.02 per coefficient), quasi-difference $y_t$ and $x_{t-1}$ with $\phi(L)$ and refit IVX.
Keep the grid point with the smallest residual variance; report that fit’s IVX Wald statistics.
Additionally test $H_0: \phi_1 = \dots = \phi_q = 0$ with the Wald statistic Wald_AR (see ac_test_wald()); ar = 0 reduces to ivx().

m <- ivx_ar(hpi ~ log(res) + cpi, data = ylpc)
m
#> 
#> Call:
#> ivx_ar(formula = hpi ~ log(res) + cpi, data = ylpc, horizon = 1)
#> 
#> Lag Selection:
#> Auto (bic) with AR terms q = 4
#> 
#> Coefficients:
#>   log(res)         cpi  
#>  0.0018317  -0.0002078
summary(m)
#> 
#> Call:
#> ivx_ar(formula = hpi ~ log(res) + cpi, data = ylpc, horizon = 1)
#> 
#> Auto () with AR terms q = 4
#> 
#> Coefficients:
#>            Estimate Std. Error t value Wald Ind Pr(> chi)  
#> log(res)  0.0018317  0.0012533   1.461    2.136    0.1439  
#> cpi      -0.0002078  0.0001075  -1.932    3.732    0.0534 .
#> ---
#> Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#> 
#> Joint Wald statistic:  4.316 on 2 DF, p-value 0.1156
#> Multiple R-squared:  0.03437,    Adjusted R-squared:  0.02281
#> Wald AR statistic: 133.6 on 4 DF, p-value < 2.2e-16

The estimated AR coefficients of the selected grid point are in m$coefficients_ar, and the automatic order selection can be replaced:

m$coefficients_ar
#>        ar1        ar2        ar3        ar4 
#>  0.4062933 -0.1139760  0.3459206  0.2142420
coef(ivx_ar(hpi ~ log(res) + cpi, data = ylpc, ar = 1))
#>      log(res)           cpi 
#>  0.0010780144 -0.0001155392

The ylpc dataset is the authors’ quarterly US housing data.

Caveats

The grid search is over $q$ coefficients with the default 31 points each, so cost grows as $31^q$; keep ar_max small or fix ar.
ar = "forecast" delegates order selection to forecast::auto.arima() and needs that package.
The tuning and standard-error options of ivx() (beta, cz, bandwidth, robust) are passed through to every grid refit.
ivx_boot() does not accept ivx_ar objects: the bootstrap schemes are defined for the untransformed regression.
When the residuals show no serial correlation (ac_test() on an ivx fit is a quick check) ivx() is the better choice; the profiling adds noise.

References

Yang, B., Long, W., Peng, L., & Cai, Z. (2020). Testing the predictability of US housing price index returns based on an IVX-AR model. Journal of the American Statistical Association, 115(532), 1598–1619.
Kostakis, A., Magdalinos, T., & Stamatogiannis, M. P. (2015). Robust econometric inference for stock return predictability. Review of Financial Studies, 28(5), 1506–1553.