回歸
解釋雙 Beta 模型回歸結果的 alpha
我試圖通過對 Spyder 的 ETF (SPY) 進行回歸併使用 10 年無風險利率來計算 Apple (AAPL) 的 Dual-Beta。雙重貝塔的公式是：
( $ r_{AAPL}-r_f) = \alpha^+D + \beta^+( r^+_m - r_f )+\alpha^-(1-D) + \beta^-( r^-_m - r_f ) $
可以在這裡找到：http ://en.wikipedia.org/wiki/Dual-beta
我從 Yahoo Finance 下載了 AAPL & SPY 數據，從 FRED 下載了 10 年利率，並將它們放入 Excel。日期為 2013 年 10 月 31 日至 2014 年 10 月 31 日。
[AAPL-Rf] [R+] [R-] [Dummy+] -0.01% -0.01% 0.00% 1 -0.52% 0.23% 0.00% 1 1.28% 0.35% 0.00% 1 -0.25% 0.00% -0.32% 0 -0.29% 0.50% 0.00% 1 -1.65% 0.00% -1.27% 0 1.56% 1.34% 0.00% 1 -0.30% 0.01% 0.00% 1 0.17% 0.00% -0.21% 0 0.12% 0.80% 0.00% 1 1.43% 0.49% 0.00% 1 -0.62% 0.43% 0.00% 1 -1.22% 0.00% -0.36% 0 0.17% 0.00% -0.23% 0 -0.89% 0.00% -0.32% 0 1.17% 0.80% 0.00% 1 -0.27% 0.49% 0.00% 1 0.76% 0.00% -0.11% 0 1.81% 0.02% 0.00% 1 2.33% 0.24% 0.00% 1 1.83% 0.00% -0.07% 0 -0.88% 0.00% -0.27% 0 2.69% 0.00% -0.44% 0 -0.25% 0.00% -0.02% 0 0.51% 0.00% -0.44% 0 -1.41% 1.11% 0.00% 1 1.13% 0.25% 0.00% 1 -0.16% 0.00% -0.37% 0 -0.75% 0.00% -1.13% 0 -0.15% 0.00% -0.34% 0 -1.11% 0.00% -0.02% 0 0.54% 0.62% 0.00% 1 -0.46% 0.00% -0.33% 0 -0.78% 1.70% 0.00% 1 -1.15% 0.00% -0.12% 0 0.83% 0.57% 0.00% 1 3.76% 0.53% 0.00% 1 -0.43% 0.21% 0.00% 1 -0.67% 0.50% 0.00% 1 -0.69% 0.00% -0.01% 0 -1.00% 0.00% -0.02% 0 1.15% 0.47% 0.00% 1 -1.41% 0.00% -0.97% 0 -2.23% 0.00% -0.02% 0 0.53% 0.00% -0.30% 0 -0.72% 0.61% 0.00% 1 0.62% 0.01% 0.00% 1 -1.30% 0.05% 0.00% 1 -0.67% 0.27% 0.00% 1 0.51% 0.00% -1.34% 0 1.97% 1.08% 0.00% 1 1.98% 0.53% 0.00% 1 -0.57% 0.00% -0.14% 0 -2.49% 0.00% -0.43% 0 1.53% 0.29% 0.00% 1 0.44% 0.06% 0.00% 1 0.83% 0.00% -0.83% 0 -1.84% 0.00% -2.14% 0 0.81% 0.00% -0.50% 0 -8.35% 0.59% 0.00% 1 -1.14% 0.00% -0.97% 0 -0.21% 1.06% 0.00% 1 0.15% 0.00% -0.60% 0 0.18% 0.00% -2.26% 0 1.43% 0.70% 0.00% 1 0.75% 0.00% -0.13% 0 0.57% 1.31% 0.00% 1 1.38% 1.23% 0.00% 1 1.77% 0.17% 0.00% 1 1.29% 1.09% 0.00% 1 -0.01% 0.04% 0.00% 1 1.56% 0.50% 0.00% 1 -0.09% 0.55% 0.00% 1 0.36% 0.11% 0.00% 1 -1.60% 0.00% -0.67% 0 -1.17% 0.59% 0.00% 1 -1.12% 0.00% -0.12% 0 0.42% 0.55% 0.00% 1 -1.05% 0.00% -0.05% 0 -0.92% 0.00% 0.00% 1 1.97% 0.51% 0.00% 1 -0.28% 0.25% 0.00% 1 0.27% 0.00% -0.71% 0 0.66% 1.40% 0.00% 1 0.19% 0.08% 0.00% 1 -0.30% 0.22% 0.00% 1 -0.07% 0.04% 0.00% 1 0.09% 0.00% -0.06% 0 0.96% 0.00% -0.50% 0 0.10% 0.01% 0.00% 1 -1.13% 0.00% -1.13% 0 -1.13% 0.00% -0.29% 0 0.38% 0.90% 0.00% 1 0.88% 0.71% 0.00% 1 -0.03% 0.00% -0.54% 0 -0.49% 0.58% 0.00% 1 0.78% 0.00% -0.40% 0 1.17% 0.00% -0.42% 0 1.06% 0.47% 0.00% 1 -0.97% 0.00% -0.73% 0 -0.43% 0.00% -0.21% 0 -0.13% 0.48% 0.00% 1 -0.03% 0.81% 0.00% 1 0.91% 0.66% 0.00% 1 0.16% 0.32% 0.00% 1 -0.71% 0.00% -0.14% 0 -1.30% 0.00% -1.19% 0 -1.59% 0.00% -1.11% 0 -0.02% 0.40% 0.00% 1 1.31% 1.07% 0.00% 1 -1.31% 0.00% -2.11% 0 -0.75% 0.00% -0.91% 0 0.39% 0.78% 0.00% 1 -0.71% 0.68% 0.00% 1 0.20% 1.04% 0.00% 1 1.13% 0.13% 0.00% 1 1.17% 0.34% 0.00% 1 0.09% 0.45% 0.00% 1 -1.32% 0.00% -0.24% 0 7.87% 0.19% 0.00% 1 0.73% 0.00% -0.82% 0 3.79% 0.31% 0.00% 1 -0.31% 0.46% 0.00% 1 -0.38% 0.29% 0.00% 1 0.22% 0.00% 0.00% 1 0.18% 0.00% -0.15% 0 1.39% 0.19% 0.00% 1 -1.10% 0.00% -0.88% 0 -0.36% 0.58% 0.00% 1 -0.19% 0.00% -0.11% 0 -0.43% 0.14% 0.00% 1 1.23% 0.97% 0.00% 1 0.15% 0.08% 0.00% 1 0.02% 0.00% -0.48% 0 -0.86% 0.00% -0.88% 0 1.46% 0.34% 0.00% 1 1.16% 0.36% 0.00% 1 0.02% 0.00% -0.64% 0 0.26% 0.83% 0.00% 1 0.14% 0.24% 0.00% 1 1.12% 0.39% 0.00% 1 1.84% 0.61% 0.00% 1 -0.27% 0.00% -0.08% 0 1.80% 0.51% 0.00% 1 -0.38% 0.16% 0.00% 1 -0.70% 0.11% 0.00% 1 1.40% 0.00% -0.06% 0 1.12% 0.19% 0.00% 1 0.38% 0.65% 0.00% 1 -0.28% 0.47% 0.00% 1 1.58% 0.10% 0.00% 1 0.57% 0.00% 0.00% 1 -0.41% 0.00% -0.35% 0 -1.70% 0.00% -0.72% 0 -1.10% 0.30% 0.00% 1 0.99% 0.07% 0.00% 1 -0.14% 0.27% 0.00% 1 0.10% 0.73% 0.00% 1 -0.35% 0.10% 0.00% 1 -1.05% 0.20% 0.00% 1 -0.10% 0.00% -0.04% 0 -0.62% 0.00% -0.61% 0 0.08% 0.45% 0.00% 1 0.59% 0.00% -0.08% 0 1.17% 0.19% 0.00% 1 1.03% 0.00% -0.06% 0 0.63% 0.66% 0.00% 1 -0.05% 0.09% 0.00% 1 0.57% 0.49% 0.00% 1 2.05% 0.00% -0.36% 0 -0.66% 0.00% -0.65% 0 0.04% 0.44% 0.00% 1 -0.38% 0.00% -0.40% 0 0.18% 0.13% 0.00% 1 1.27% 0.49% 0.00% 1 -1.18% 0.00% -0.20% 0 -0.58% 0.36% 0.00% 1 -1.80% 0.00% -1.14% 0 1.42% 1.01% 0.00% 1 -0.52% 0.00% -0.19% 0 0.81% 0.43% 0.00% 1 2.57% 0.22% 0.00% 1 -0.17% 0.00% 0.00% 1 0.65% 0.00% -0.48% 0 1.36% 0.03% 0.00% 1 -0.66% 0.00% -0.44% 0 -0.24% 0.01% 0.00% 1 -2.63% 0.00% -1.98% 0 0.54% 0.00% -0.31% 0 -0.56% 0.72% 0.00% 1 -0.50% 0.00% -0.98% 0 -0.18% 0.02% 0.00% 1 -0.02% 0.00% -0.55% 0 0.27% 1.15% 0.00% 1 1.30% 0.28% 0.00% 1 -0.03% 0.00% -0.15% 0 1.31% 0.67% 0.00% 1 0.26% 0.47% 0.00% 1 0.48% 0.00% -0.03% 0 1.19% 0.83% 0.00% 1 1.37% 0.52% 0.00% 1 0.03% 0.26% 0.00% 1 0.00% 0.29% 0.00% 1 0.73% 0.00% -0.16% 0 0.21% 0.50% 0.00% 1 -0.65% 0.06% 0.00% 1 1.22% 0.00% -0.05% 0 0.11% 0.00% -0.06% 0 0.24% 0.28% 0.00% 1 0.77% 0.00% -0.06% 0 -4.32% 0.00% -0.06% 0 -0.84% 0.00% -0.15% 0 0.86% 0.44% 0.00% 1 -0.63% 0.00% -0.26% 0 -0.38% 0.00% -0.64% 0 3.02% 0.37% 0.00% 1 0.42% 0.11% 0.00% 1 0.22% 0.00% -0.59% 0 -0.04% 0.00% -0.08% 0 -0.77% 0.75% 0.00% 1 0.70% 0.13% 0.00% 1 0.20% 0.52% 0.00% 1 -0.83% 0.00% -0.10% 0 0.09% 0.00% -0.78% 0 1.54% 0.00% -0.58% 0 -0.88% 0.78% 0.00% 1 -3.89% 0.00% -1.62% 0 2.89% 0.79% 0.00% 1 -0.64% 0.00% -0.19% 0 0.63% 0.00% -0.27% 0 -1.58% 0.00% -1.36% 0 0.72% 0.01% 0.00% 1 -0.29% 1.09% 0.00% 1 -0.01% 0.00% -0.12% 0 -0.88% 0.00% -1.55% 0 2.05% 1.74% 0.00% 1 0.21% 0.00% -1.99% 0 -0.29% 0.00% -1.15% 0 -0.92% 0.00% -1.65% 0 -1.07% 0.15% 0.00% 1 -1.24% 0.00% -0.68% 0 -1.33% 0.00% -0.09% 0 1.45% 1.17% 0.00% 1 2.11% 0.96% 0.00% 1 2.67% 1.97% 0.00% 1 0.50% 0.00% -0.72% 0 1.76% 1.16% 0.00% 1 0.37% 0.76% 0.00% 1 -0.11% 0.00% -0.14% 0 1.53% 1.14% 0.00% 1 0.55% 0.00% -0.16% 0 -0.34% 0.63% 0.00% 1因為回歸
AAPL-Rf是因變數，R+是Up-Beta market risk premium，R-是Down-Beta risk premium，並且當 SPY 為正或不變時，Dummy+否則（或負）。1``0回歸結果如下：
AAPL-Rf =- 0.0003 + 0.4059 * [R+] + 0.6111 * [R-] + 0.0036 * [Dummy+]我有點困惑
alpha's……該模型描述了正alpha乘以虛擬變數，在這種情況下它是：0.0036 * [Dummy+]但是下行alpha呢？那會是：- 0.0003？任何輸入都會有所幫助….
是的，下行 alpha 為 -0.0003。如果您將 (1-D) 明確包含為一個協變數，則可以確認該結果，但是您必須請求一個沒有截距的模型。