An important issue in canonical correlation analysis not rigorously addressed in the literature is the magnitude of the canonical correlation required to claim existence of one or more common factor. Previous research has tended to arbitrarily select a benchmark value [2,9]. Since our approach requires making strong statements about the existence of common factors, we need to be more precise. To do so, we perform a simulation to determine the appropriate benchmark canonical correlation level for the stock and REIT data.
From the estimated four and eight principal components for stocks and REITs, respectively, we follow the canonical correlation methodology to calculate the first four canonical variates for both stocks and REITs. We begin with one common factor. The following steps are then repeated for two, three, and four common factors. For one common factor, each stock return is regressed on the first stock canonical variate and each REIT return is regressed on the first REIT canonical variate to determine their respective alphas, betas, and errors:
Stock:RSit =αiS+βiSVtS+itS
REIT:RRj t =αjR+βjRVtR+Rj t. (3) where βiS is the loading of stock i on the first stock canonical variate VtS and βjR is the factor loading on the first REIT canonical variate VtR. The errors are
236 D. W. Blackburn and N. K. Chidambaran
represented byitS for stocks andj tR for REITs. We do not require any particular level of canonical correlation between canonical variates in this step.
We now create simulated returns with the feature of having a common factor. We do this by replacing the canonical variatesVtS andVtR in Eq. (3) with a proxy for the common factorFtC:
Stock: ˆRSit =αiS+βiSFtC+itS
REIT: ˆRRj t =αjR+βjRFtC+j tR.. (4) The alphas, betas, and errors remain the same as in Eq. (3) only the factor is changed.
Constructing the simulated returns in this way preserves many of the statistical features of the true stock returns such as heteroskedasticity, serially correlated errors, and cross-correlated errors. However, the simulated returns now share a common factor. The proxy used for the common factor is derived from the sum of the stock and REIT canonical variates:
FtC= VtS+VtR
"
V ar(VtS)+V ar(VtR)+2Cov(VtS, VtR)
(5)
where the denominator normalizes the factorFtCto have unit variance.10This proxy for the common factor maintains many of the desirable statistical features observed in the canonical variates.
Using the simulated returns RˆitS and Rˆj tR, principal components are estimated and canonical correlation analysis is used to determine the simulated canonical correlations when common factors exist and the number of common factors is known. Since we have embedded common factors into the simulated returns, we should expect canonical correlations much closer to one; however, estimation error in calculating principal components caused by finite samples, heteroskedasticity, and autocorrelation may cause the first canonical correlation to be large, but less than one. We perform this simulation for each quarter of our sample period using the same set of returns—same time-series length and cross-section—as used in the study. The benchmark levels are unique to this particular study.
Results are plotted in Fig.1. Figure 1a is the plot of the first four canonical correlations when there is only one common factor. The bold, solid line is the first canonical correlation, and as expected, is close to one with a time-series average, as shown in Table2-Panel A, of 0.954. The standard error, 0.045, is listed below the mean. All other canonical correlations are much smaller with time-series averages of 0.521 for the second canonical correlation (light, solid line), 0.367 for the third canonical correlation (dashed line), and 0.215 for the fourth canonical correlation
10We also used the stock canonical variates and REIT canonical variates as common factor proxies and the results did not change.
Measuring Market Integration: US Stock and REIT Markets 237
a b
d c
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1985-1 1986-1 1987-1 1988-1 1989-1 1990-1 1991-1 1992-1 1993-1 1994-1 1995-1 1996-1 1997-1 1998-1 1999-1 2000-1 2001-1 2002-1 2003-1 2004-1 2005-1 2006-1 2007-1 2008-1 2009-1 2010-1 2011-1 2012-1 2013-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1985-1 1986-1 1987-1 1988-1 1989-1 1990-1 1991-1 1992-1 1993-1 1994-1 1995-1 1996-1 1997-1 1998-1 1999-1 2000-1 2001-1 2002-1 2003-1 2004-1 2005-1 2006-1 2007-1 2008-1 2009-1 2010-1 2011-1 2012-1 2013-1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1985-1 1986-1 1987-1 1988-1 1989-1 1990-1 1991-1 1992-1 1993-1 1994-1 1995-1 1996-1 1997-1 1998-1 1999-1 2000-1 2001-1 2002-1 2003-1 2004-1 2005-1 2006-1 2007-1 2008-1 2009-1 2010-1 2011-1 2012-1 2013-1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
1985-1 1986-1 1987-1 1988-1 1989-1 1990-1 1991-1 1992-1 1993-1 1994-1 1995-1 1996-1 1997-1 1998-1 1999-1 2000-1 2001-1 2002-1 2003-1 2004-1 2005-1 2006-1 2007-1 2008-1 2009-1 2010-1 2011-1 2012-1 2013-1
Fig. 1 Simulated canonical correlations. The first four canonical correlations between REIT and stock principal components are estimated each quarter from 1985 to 2013 using simulated returns.
The simulated returns are modeled to include one, two, three, and four factors common to both markets. The bold solid line is the first canonical correlation, the thin solid line is the second canonical correlation, the dashed line is the third, and the dotted line is the fourth canonical correlation
(dotted line). This is exactly what is expected. With only one common factor, it is expected that we find one canonical correlation close to one with the remaining much lower.
In Fig.1b, we plot the time-series of canonical correlation when we simulate returns with two common factors. We notice two different affects. First, the second canonical correlation, with a time-series average of 0.896, is much larger than in the one common factor case when its average was 0.521. Second, the average first canonical correlation is slightly larger and has noticeably less variance. Meanwhile, the time-series average for the third and fourth canonical correlations is nearly unchanged. The presence of two common factors increases the signal-to-noise ratio in the system thus making it easier to identify common factors.
Figure 1c for three common factors and Fig.1d for four common factors tell similar stories. Common factors are associated with high canonical correlations.
238 D. W. Blackburn and N. K. Chidambaran
Table 2 Mean canonical correlations
Panel A: average canonical correlations from simulated returns
CC 1 CC 2 CC 3 CC 4
One common factor 0.954 0.521 0.367 0.215
0.045 0.135 0.114 0.094
Two common factors 0.967 0.896 0.368 0.216
0.032 0.037 0.114 0.091
Three common factors 0.972 0.928 0.875 0.216
0.027 0.026 0.034 0.091
Four common factors 0.977 0.941 0.909 0.864
0.022 0.024 0.025 0.035
Panel B: average canonical correlations from actual returns
Eight REIT PCs vs Four stock PCs 0.752 0.514 0.371 0.216
Eight REIT PCs vs FFC four-factors 0.731 0.452 0.316 0.199 We use canonical correlation analysis [2] to measure the presence of common factors between the US stock and REIT markets. In Panel A, we simulate return data to have one to four common factors to determine a canonical correlation level expected when common factors exist. The standard error is listed below each mean canonical correlation. In Panel B, we calculate the time- series mean canonical correlation from actual stock and REIT returns each quarter from 1985 to 2013. For REITs, we use the first eight principal components to represent REIT factors. Two different proxies for stock factors are used—the first four principal components (PC) and the four Fama-French-Carhart factors (FFC)
From Table2-Panel A, we see the dramatic jump in average canonical correlation from 0.216 to 0.864 when moving from three to four common factors. These results clearly demonstrate that for our particular system, common factors are associated with high canonical correlations. The benchmarks we use in this paper to claim statistical evidence for the presence of one or more common factors is the 95% lower confidence level below the time series canonical correlation mean. The benchmark threshold levels for the first, second, third, and fourth canonical correlations, used in this paper are 0.87, 0.82, 0.81, and 0.79, respectively.