Interpreting Alpha Terms in a Dual-Beta Regression
Article Quant Q&A · Author: Jason Guevara
Summary
The document asks how to interpret the intercept and dummy-variable coefficient in a dual-beta regression of Apple returns against the market’s positive and negative returns. The stated model assigns separate slopes to up and down market conditions and uses a dummy that is one in up or unchanged periods. In the reported regression, the dummy coefficient is the up-state alpha, while the intercept represents the down-state alpha under this parameterization.
The answer suggests a way to check the interpretation: include a separate down-state indicator, defined as one minus the up-state dummy, and fit the model without a general intercept. This makes both state-specific alpha terms explicit. The example uses historical daily data over a limited period, but provides no diagnostics, significance estimates, or discussion of data alignment and risk-free-rate construction. The interpretation depends on the stated coding of the dummy and regression specification.
Key ideas
- In the given specification, the coefficient on the up-state dummy is the up-market alpha.
- The intercept serves as the down-market alpha when the up-state dummy is zero.
- Adding a down-state indicator makes both conditional alpha terms explicit.
- A model with both state indicators should omit the general intercept to avoid redundant terms.
Tags
Full text
# Interpret alpha's on Dual-Beta Model regression Results
# Interpret alpha's on Dual-Beta Model regression Results
I am trying to calculate the Dual-Beta for Apple (AAPL) by running a regression against the Spyder's ETF (SPY) & using the 10-yr Risk Free rate. The formula for the dual beta is:
($r_{AAPL}-r_f) = \alpha^+D + \beta^+( r^+_m - r_f )+\alpha^-(1-D) + \beta^-( r^-_m - r_f ) $
and can be found here: http://en.wikipedia.org/wiki/Dual-beta
I downloaded AAPL & SPY data from Yahoo Finance and the 10-Year Rate from FRED and put them in Excel. The dates are from 2013-10-31 to 2014-10-31.
```
[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
```
For the regression `AAPL-Rf` is the dependent variable and `R+` is the `Up-Beta market risk premium`, `R-` is the `Down-Beta risk premium` , and `Dummy+` is `1` when SPY was positive or unchanged, and `0` otherwise (or negative).
The regression results came back as:
`AAPL-Rf =- 0.0003 + 0.4059 * [R+] + 0.6111 * [R-] + 0.0036 * [Dummy+]`
I am a bit confused on the `alpha's`... The model describes that positive alpha is multiplied by the Dummy Variable which in this case its: `0.0036 * [Dummy+]` but what about the downside alpha? would that be: `- 0.0003` ?
Any input would be helpful....
## Answer by James (score 1, accepted)
https://quant.stackexchange.com/a/15280
Yes, the downside alpha is -0.0003. You can confirm that result if you include (1-D) explicitly as one more covariate, but then you'll have to request a model with no intercept.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.