Campbell and Thompson Restriction

A short description of the post.

Compare Out-Of-Sample Forecast

\[ R^{2}_{OS} = 1 -\frac{(1/ n_2) \sum^{n_2}_{s=1}(r_{n_1+s} - \hat{r}_{i,n_1 + s})^2}{(1/ n_2) \sum^{n_2}_{s=1}(r_{n_1+s} - \bar{r}_{n_1 + s})^2} \]

Monthly U.S. Equity Premium Out-of-Sample Forecasting Results
Individual Economic Variables, 1957:01–2010:12
Economic Variable	Overall	Expansion	Recession
	$$R^2_{OS}(\%)$$	$$R^2_{OS}(\%)$$	$$R^2_{OS}(\%)$$
Panel A: Unrestricted predictive regression forecasts
log (DP)	−0.05 [0.10]	−1.24 [0.42]	2.41 [0.00]
log (DY)	−0.37 [0.07]	−2.28 [0.40]	3.56 [0.00]
log (EP)	−1.88 [0.28]	−2.21 [0.31]	−1.20 [0.38]
log (DE)	−2.04 [0.97]	−1.26 [0.80]	−3.67 [0.97]
SVAR	0.32 [0.17]	−0.02 [0.50]	1.01 [0.16]
BM	−1.74 [0.31]	−2.56 [0.44]	−0.04 [0.28]
NTIS	−0.91 [0.41]	0.50 [0.03]	−3.82 [0.94]
TBL	−0.01 [0.09]	−0.84 [0.30]	1.71 [0.10]
LTY	−1.17 [0.12]	−2.37 [0.38]	1.32 [0.11]
LTR	−0.08 [0.20]	−0.85 [0.63]	1.52 [0.05]
TMS	0.06 [0.16]	−0.40 [0.34]	1.00 [0.09]
DFY	−0.04 [0.59]	−0.06 [0.64]	−0.01 [0.48]
DFR	−0.01 [0.38]	0.12 [0.25]	−0.28 [0.48]
INFL	−0.09 [0.50]	0.10 [0.22]	−0.48 [0.66]
Panel B: Predictive regression forecasts with Campbell and Thompson (2008) restrictions
log (DP)	0.15 [0.07]	−0.92 [0.38]	2.36 [0.00]
log (DY)	0.17 [0.04]	−1.33 [0.40]	3.26 [0.00]
log (EP)	−0.82 [0.24]	−1.19 [0.30]	−0.06 [0.32]
log (DE)	−1.74 [0.98]	−1.19 [0.78]	−2.88 [0.98]
SVAR	0.15 [0.23]	−0.02 [0.50]	0.49 [0.20]
BM	−1.17 [0.29]	−1.68 [0.40]	−0.13 [0.30]
NTIS	−0.91 [0.41]	0.50 [0.03]	−3.82 [0.94]
TBL	0.21 [0.10]	−0.25 [0.27]	1.16 [0.10]
LTY	−0.01 [0.09]	−0.67 [0.29]	1.36 [0.07]
LTR	0.22 [0.12]	−0.47 [0.52]	1.64 [0.03]
TMS	0.12 [0.15]	−0.42 [0.37]	1.23 [0.06]
DFY	−0.01 [0.50]	−0.03 [0.55]	0.01 [0.45]
DFR	−0.16 [0.49]	0.09 [0.27]	−0.68 [0.66]
INFL	−0.06 [0.46]	0.10 [0.22]	−0.38 [0.63]

\[ \text{CDSFE}_{i,\tau} = \sum^{\tau}_{s=1} (r_{n_1 + s} - \bar{r}_{n_1 + s})^2 - \sum^{\tau}_{s=1} (r_{n_1 + s} - \hat{r}_{i,n_1 + s})^2 \]