Joint Tests of CAPM Alphas with Serially Correlated Returns
Summary
The document describes a factor-sorted portfolio study that tests individual CAPM intercepts using time-series regressions. It sorts stocks into ten portfolios by a characteristic, estimates each portfolio's excess return against the market excess return, and adjusts individual coefficient inference for heteroskedasticity and autocorrelation with a Newey–West covariance estimator. The reported pattern includes negative estimated alphas at the low end and positive alphas in several high-ranked portfolios.
The research question is whether all portfolio alphas are jointly zero, given that the classical Gibbons–Ross–Shanken test relies on restrictive error assumptions. The author seeks a less restrictive joint test and asks about its assumptions and implementation, but the document contains no answer to that question. Its code demonstrates inference for one portfolio at a time; it does not provide a joint test or establish that Newey–West corrections alone solve the joint-testing problem.
Key ideas
- The CAPM regression estimates an intercept for each characteristic-sorted portfolio.
- Newey–West covariance estimates can adjust individual coefficient tests for serial correlation and heteroskedasticity.
- Individual alpha significance does not by itself test whether all portfolio alphas are jointly zero.
- The document raises concerns about the GRS test assumptions but supplies no alternative joint-test procedure.
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Full text
# CAPM: Testing for alphas jointly equal to zero
# CAPM: Testing for alphas jointly equal to zero
For my project, I need to assess if a certain factor X leads to a CAPM-Anomaly. First, I sorted the monthly stock return (sample size: 500+ observations) according to the X factor in 10 decile portfolios ("DP"). Afterwards, I performed for each decile portfolio a linear regression to evaluate if each alpha is significantly different from zero. The linear Regression for the i-th decile portfolio is: $$R_{i,t}-R^f_t = \alpha_i + \beta_i(R^m_t-R^f_t)+ \epsilon_{i,t}$$
Since I'm dealing with a time serie, I will adjust for heteroskedasticity and autocorrelation with the Newey-West (1987) correlation corrector with 6 lags.
Results: The decile portfolios DP1, DP2 have significant negative alphas and DP8, DP9, DP10 a significant positive alpha. However, I would primarily be interested if/not all alphas are jointly equal to zero. I already saw posts that discussed the GRS-Test proposed by Gibbons, Ross, and Shanken (1989) but this test has the major drawback that it assumes normally distributed uncorrelated homoskedastic error terms. Somebody mentioned a test with the Chi-Squared-Distribution but didn't elaborate what the assumptions were let alone how to perform it in R.
I would be so grateful if somebody would know a less restrictive test for evaluating if all alphas are jointly equal to zero. Specifically, I would be interested in the underlying assumptions of the test (preferably with a source) as well as the concrete code in R (if it doesn't causes too much inconvenience) :D
Here is the code which I have so far (for the first decile portfolio DP1)
```
library(sandwich)
library(lmtest)
#Performing Linear Regression
L1 <- lm(DP1 ~ MktRf, data = DATA)
#Building NeweyWest Covariance Corrector
NW_VCOV_L1 <- NeweyWest(L1, lag = 6, prewhite = F, adjust = T)
#Performing Covariance Test
coeftest(L1, vcov = NW_VCOV_L1)
```
which yields following Output
```
t test of coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.00327902 0.00068035 -4.8196 1.799e-06 ***
MktRf 0.92165566 0.02334994 39.4714 < 2.2e-16 ***
---
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
```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.