IVX-AR: serially correlated errors

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

  1. 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).
  2. 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.
  3. Keep the grid point with the smallest residual variance; report that fit’s IVX Wald statistics.
  4. 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.