Replicating Rapach et al. (2016)

This article replicates Table A2 of the online appendix to Rapach, Ringgenberg, and Zhou (2016), “Short interest and aggregate stock returns” (Journal of Financial Economics, 121, 46-65), using two functions from forecastdom:

\[r_{t:t+h} = \alpha + \beta\,\mathrm{SII}_t + \varepsilon_{t:t+h}, \qquad r_{t:t+h} = \frac{1}{h}\sum_{j=1}^{h} r_{t+j},\]

at horizons \(h \in \{1, 3, 6, 12\}\). IVX-Wald tests \(H_0: \beta = 0\). The \(\widehat{qLL}\) statistic tests \(H_0: \beta_t = \beta\) for all \(t\).

The data

Monthly U.S. log excess return on the S&P 500 and the standardised linearly-detrended log of the equal-weighted short interest index (EWSI), 1973-01 to 2014-12 (504 observations).

ggplot(rrz2016, aes(date, SII)) +
  geom_hline(yintercept = 0, linetype = "dashed", colour = "grey60") +
  geom_line(colour = "#47A5C5", linewidth = 0.6) +
  labs(x = NULL, y = "SII",
       title = "Short interest index, 1973-2014") +
  theme_minimal()

Replicating Table A2

The original MATLAB program (Compute_IVX_Wald.m in the JFE data archive) calls the IVX-Wald with beta = 0.99, M_n = 0 and the negated SII. RRZ hypothesise that SII negatively predicts returns, and the sign does not affect the Wald statistic. The \(\widehat{qLL}\) test is called with Newey-West truncation L = h. We pass the same arguments to ivx_wald() and qll_hat().

horizons <- c(1, 3, 6, 12)

results <- do.call(rbind, lapply(horizons, function(h) {

  ivx <- ivx_wald(rrz2016$r, matrix(-rrz2016$SII, ncol = 1),
                  K = h, M_n = 0L, beta = 0.99)

  T_ <- nrow(rrz2016)
  P  <- T_ - h
  y_h <- sapply(seq_len(P), function(t) mean(rrz2016$r[(t + 1):(t + h)]))
  X_h <- matrix(rrz2016$SII[1:P], ncol = 1)
  Z_h <- matrix(1, P, 1)
  qll <- qll_hat(y_h, X_h, Z = Z_h, L = h)

  data.frame(h = h, IVX_Wald = ivx$statistic, qLL = qll$statistic)
  
}))

knitr::kable(results, digits = 3, row.names = FALSE,
             col.names = c("$h$", "IVX-Wald", "$\\widehat{qLL}$"))

\(h\)	IVX-Wald	\(\widehat{qLL}\)
1	3.377	-3.721
3	4.513	-4.858
6	4.603	-4.909
12	3.669	-5.016

Critical values (from RRZ 2016, online appendix):

IVX-Wald: 10% = 2.71, 5% = 3.84, 1% = 6.64.
\(\widehat{qLL}\): 10% = \(-7.14\), 5% = \(-8.36\), 1% = \(-11.05\) (reject for small values).

For comparison, the paper reports IVX-Wald = 3.38*, 4.51**, 4.60**, 3.67* and qLL = \(-3.72\), \(-4.86\), \(-4.91\), \(-5.02\). Our values match to two decimal places at every horizon. Conclusion: SII predicts the equity premium at all horizons (significant IVX-Wald) and constant \(\beta\) is not rejected.

References

Elliott, G. and Müller, U. K. (2006). Efficient tests for general persistent time variation in regression coefficients. Review of Economic Studies, 73(4), 907-940.
Kostakis, A., Magdalinos, T. and Stamatogiannis, M. P. (2015). Robust econometric inference for stock return predictability. Review of Financial Studies, 28(5), 1506-1553.
Rapach, D. E., Ringgenberg, M. C. and Zhou, G. (2016). Short interest and aggregate stock returns. Journal of Financial Economics, 121(1), 46-65.