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CPS2134 Yutaka Kuroki et al.
larger exposures on SMR factors tends to decrease visitors around the year
end.

Figure 2. Time series plots for the retail demand fundamental factors.

Next, we investigate the relationship between the fundamental factors and
the Principal Components (PC). Table 1 reports the correlation coefficients
among these factors and seven principal components. According to this table,
st
1 PC is the weekly seasonal patterns that explains 93% of the overall
variations. The MKT explains other seasonal patterns which could not be
captured by PC1. The SMB correlated with PCs3 and 4 indicates those
components are the source of size variations. Similarly, the SMB is correlated
with PCs3, 5 and 6, and the corresponding principal components indicates the
variations arise in unusual variations in restaurants visitors.

Table 1. Correlation matrix for the observed series, factors, and PCs.
, MKT SMB SMR PC1 PC2 PC3 PC4 PC5 PC6 PC7
, 1 0.45 -0.07 0.01 0.93 0.28 0.13 0.05 0.13 -0.02 0.01
MKT 0.45 1 -0.13 0.19 0.27 0.36 0.12 0.05 0.17 -0.18 0
SMB -0.07 -0.13 1 -0.23 0.01 0.21 -0.43 -0.12 -0.22 0.06 -0.17
SMR 0.01 0.19 -0.23 1 0.06 0.14 0.41 0 -0.37 -0.49 0.04

2. Cross-Sectional Regression and its Performance of Prediction
In this Section, we will present the followings:

➢ Models and Assumptions on the multifactor models.
➢ Model selection together with tests for model assumptions are
investigated.
➢ Interpretations for estimated model parameters are presented.

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