Table 1 Estimates of the parameters with mean squared errors (in parentheses) with the normality assumption for errors
Case | Parameter | TRUE | BSM (two-step) | BSM (one-step) | USM (one-step) | CC |
|---|---|---|---|---|---|---|
1 | \(\alpha _{11}\) | 4.5 | 4.54 (0.20) | 4.54 (0.20) | 1.95 (2.55) | 4.54 (0.19) |
\(\alpha _{12}\) | − 0.6 | − 0.60 (0.03) | − 0.60 (0.03) | − 0.30 (0.31) | − 0.60 (0.03) | |
\(\alpha _{21}\) | 1 | 1.05 (0.25) | 1.05 (0.25) | – | 0.69 (0.34) | |
\(\alpha _{22}\) | 0 | − 0.02 (0.07) | − 0.02 (0.07) |  | 0.10 (0.10) | |
\(\beta _{1}\) | − 1 | − 0.80 (0.22) | − 0.86 (0.23) | − 0.85 (0.19) | − 0.80 (0.23) | |
\(\beta _{2}\) | 1.5 | 1.44 (0.07) | 1.48 (0.05) | 1.49 (0.06) | 1.43 (0.07) | |
\(\rho _{01}\) | − 0.5 | − 0.01 (0.39) | − 0.31 (0.28) | − 0.35 (0.37) | – | |
\(\rho _{02}\) | 0 | 0.00 (0.00) | − 0.22 (0.38) | – | – | |
\(\rho _{12}\) | 0 | 0.05 (0.34) | 0.05 (0.37) | – | – | |
\(\sigma _{00}\) | 1 | 0.96 (0.06) | 1.05 (0.11) | 1.10 (0.11) | 0.97 (0.06) | |
2 | \(\alpha _{11}\) | 2 | 2.07 (0.17) | 2.07 (0.17) | 2.96 (0.98) | 2.07 (0.17) |
\(\alpha _{12}\) | − 0.2 | − 0.21 (0.02) | − 0.21 (0.02) | − 0.45 (0.25) | − 0.21 (0.02) | |
\(\alpha _{21}\) | 5 | 4.96 (0.72) | 5.02 (0.64) | – | 3.92 (1.11) | |
\(\alpha _{22}\) | − 0.7 | − 0.69 (0.08) | − 0.69 (0.09) | – | − 0.43 (0.27) | |
\(\beta _1\) | − 1 | − 0.96 (0.19) | − 1.01 (0.11) | − 1.04 (0.11) | − 0.83 (0.19) | |
\(\beta _2\) | 1.5 | 1.48 (0.12) | 1.49 (0.03) | 1.52 (0.03) | 1.41 (0.09) | |
\(\rho _{01}\) | − 0.5 | − 0.32 (0.49) | − 0.26 (0.42) | − 0.60 (0.21) | – | |
\(\rho _{02}\) | − 0.5 | − 0.10 (0.56) | − 0.47 (0.30) | – | – | |
\(\rho _{12}\) | 0.5 | 0.63 (0.29) | 0.47 (0.55) | – | – | |
\(\sigma _{00}\) | 1 | 1.67 (2.73) | 1.01 (0.10) | 1.03 (0.09) | 0.91 (0.11) | |
3 | \(\alpha {11}\) | 4.5 | 4.68 (0.45) | 4.68 (0.45) | 3.13 (2.61) | 4.68 (0.45) |
\(\alpha {12}\) | − 0.5 | − 0.52 (0.05) | − 0.52 (0.05) | − 0.32 (0.33) | − 0.52 (0.05) | |
\(\alpha {21}\) | − 3 | − 3.06 (0.26) | − 3.04 (0.27) | – | − 3.06 (0.26) | |
\(\alpha {22}\) | 1 | 1.03 (0.09) | 1.02 (0.09) | – | 1.03 (0.09) | |
\(\beta {0}\) | − 1 | − 0.65 (0.60) | − 1.00 (0.25) | − 0.38 (0.69) | − 0.20 (0.81) | |
\(\beta _{1}\) | 1.5 | 1.43 (0.11) | 1.50 (0.06) | 1.37 (0.13) | 1.36 (0.14) | |
\(\rho _{01}\) | − 0.5 | 0.12 (0.81) | − 0.42 (0.36) | 0.63 (1.13) | – | |
\(\rho _{02}\) | 0.5 | 0.12 (0.56) | 0.63 (0.10) | – | – | |
\(\rho _{12}\) | − 0.5 | − 0.50 (0.10) | − 0.51 (0.21) | – | – | |
\(\sigma _{00}\) | 1 | 1.01 (0.17) | 1.05 (0.08) | 0.99 (0.14) | 0.91 (0.10) |