Replicating Rapach et al. (2016)

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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.

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