Return direction forecasting: a conditional autoregressive shape model with beta density

Xie, Haibin; Sun, Yuying; Fan, Pengying

doi:10.1186/s40854-023-00489-z

Financial Innovation

Table 3 Estimates of B-CARS(1,1) model

From: Return direction forecasting: a conditional autoregressive shape model with beta density

	B-CARS(1,1)	bm	tbl	−ntis	−infl	ltr	−svar	tms	−dfy	dp	dy	ep	de
\(\omega\)	\(\underset{(0.041)}{0.108}\)	\(\underset{(0.042)}{0.106}\)	\(\underset{(0.042)}{0.108}\)	\(\underset{(0.031)}{0.060}\)	\(\underset{(0.045)}{0.108}\)	\(\underset{(0.046)}{0.039}\)	\(\underset{(0.044)}{0.071}\)	\(\underset{(0.045)}{0.104}\)	\(\underset{(0.041)}{0.106}\)	\(\underset{(0.041)}{0.108}\)	\(\underset{(0.041)}{0.108}\)	\(\underset{(0.040)}{0.103}\)	\(\underset{(0.043)}{0.108}\)
\(\gamma _1\)	\(\underset{(0.077)}{0.766}\)	\(\underset{(0.078)}{0.767}\)	\(\underset{(0.086)}{0.766}\)	\(\underset{(0.076)}{0.801}\)	\(\underset{(0.080)}{0.766}\)	\(\underset{(0.075)}{0.797}\)	\(\underset{(0.087)}{0.752}\)	\(\underset{(0.080)}{0.769}\)	\(\underset{(0.085)}{0.756}\)	\(\underset{(0.079)}{0.766}\)	\(\underset{(0.079)}{0.766}\)	\(\underset{(0.077)}{0.766}\)	\(\underset{(0.088)}{0.766}\)
\(\tau _1\)	\(\underset{(0.018)}{0.048}\)	\(\underset{(0.018)}{0.048}\)	\(\underset{(0.018)}{0.048}\)	\(\underset{(0.018)}{0.040}\)	\(\underset{(0.018)}{0.048}\)	\(\underset{(0.018)}{0.041}\)	\(\underset{(0.018)}{0.042}\)	\(\underset{(0.018)}{0.047}\)	\(\underset{(0.018)}{0.046}\)	\(\underset{(0.018)}{0.048}\)	\(\underset{(0.018)}{0.048}\)	\(\underset{(0.018)}{0.048}\)	\(\underset{(0.019)}{0.048}\)
\(\beta\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.534}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.534}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)	\(\underset{(0.020)}{0.532}\)
X		\(\underset{(0.015)}{0.002}\)	\(\underset{(0.012)}{0.000}\)	\(\underset{(0.020)}{{0.046}}\)	\(\underset{(0.044)}{0.000}\)	\(\underset{(0.054)}{{\textbf {0.124}}}\)	\(\underset{(0.041)}{0.049}\)	\(\underset{(0.015)}{0.002}\)	\(\underset{(0.018)}{0.009}\)	\(\underset{(0.012)}{0.000}\)	\(\underset{(0.012)}{0.000}\)	\(\underset{(0.015)}{0.006}\)	\(\underset{(0.019)}{0.000}\)
\(R^2\)(%)	0.66	0.66	0.66	1.05	0.66	1.18	0.70	0.67	0.70	0.66	0.66	0.72	0.66

Bold represents a better performance
[1] MLE method cannot be applied to the B-CARS model when \(ur_t\)=0 or 1. We deal with this problem using the following principle: \(ur_{t}=\left\{ \begin{array}{rcl} \underset{t}{max}\{ur_{t}/[1]\} & &\text {if}\ ur_{t} = 1\\ \underset{t}{min}\{ur_t/[0]\} & &\text {if} \ ur_{t}=0\\ \end{array} \right.\) where \(\{ur_{t}/[v]\}\) means that the elements in \(ur_t\) series have been removed if \(ur_t\)=v
[2] Numbers in the parentheses are standard errors
[3] As we do not know whether \(r(x_t)\) is positively or negatively correlated with \(k_{t}\), both \(r(x_t)\) and 1-\(r(x_t)\) are used as exogeneous variables when performing MLE and the one with a higher \(R^2\) is selected to be the final one. Symbol -x means that x is regularized by 1-\(r(x_t)\)
[4] When performing MLE, the initial five \(k_t\)s are set to be the unconditional mean of \(ur_t\)

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