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QAF week 2 Linear Regression Models and The Capital Asset Pricing Model,…
QAF week 2 Linear Regression Models and The Capital Asset Pricing Model
CAPM model
deterministic economic model
statistical model: simple linear regression model
parameter β
: measure of exposure of the returns on asset i to movements in the market, relative to risk-free asset.
intercept α
: abnormal return to i in addition to the asset's exposure to the excess return on the market.
8 assumptions
model is estimated from a
random sample
T>2
statistically independent
but
identically distributed
.
Each random error has
zero conditional expected value
conditional variance
of the random error is
constant
(does not depend on t)
random errors in any two different time periods are
uncorrelated.
ε(t) is serially uncorrelated.
independent
variable is not
constant
. not perfect multicollinearity.
Under first 5, the OLS estimators are best linear unbiased estimator.
random errors are
uncorrelated
with independent variables.
random errors are
normally distributed
random errors are
stationary
and
weakly dependent.
total sum of squares(SST) = SSR + SSE
Regression sum of square(SSR): systematic risk
Error sum of square(SSE): idiosyncratic risk
R² : proportion of total risk that is systematic
1-R²: proportion of the total risk that is indiosyncratic
descriptive statistic output
sample
mean
> market: outperform the market
sample
std.dev
> market: more volatile than the market
-
skewness
< 0 (negative): return skewed to the left
kurtosis
> 0: fat tails, leptokurtic
small
normtest.p
shows reject null hypothesis of normal distribution. Shapiro-Wilk test for normality.
Comment on estimated CAPM output
sample coefficient of determination(R²)
is about 0.83, suggesting that this simple linear regression accounts for about 83% of the variation in ER.cnsrm in the sample at hand.
F-test:
overall significance of the regression model. H0:β = 0, HA: β≠0. or H0: ρ² = 0, HA: ρ²≠0. Where ρ² is the population coefficient of determination. p-value for this test is practically zero, reject and conclude that
1)this regression model is significant. 2)R² is significantly positive
significant beta risk -
ER.Mkt parameter:
one percentage point increase of the excess return to the market is expected to be accompanied by an about 0.932 pp rise of the excess return to the consumer portfolio.
intercept α:
H0: α = 0, HA: α≠0. p-value for α t-test is 0.106. main null hypothesis even at 10% significance level and conclude that consumer portfolio has an insignificant α-risk.
do not interperate
insignificant coefficients, but for α: when the return to the market is 0, the excess return to the consumer portfolio is about 0.116 pp.
Test on CAPM residuals
Pattern:
residuals have cycle and variance appears to vary over time.
White(W) test for heteroskedasticity
. H0: homoskedasticity, HA:heteroskedasticity. P-value is practically zero, reject H0.
Breusch-Godfrey(BG) LM test for autocorrelation
in error terms. H0: no autocorrelation up to order 6. HA:some 1-6 order autocorrelation.p-value<0.01, reject H0.
Jarque-Bera(JB) test for normality
of the error term.H0: normal distribution. HA: non-normal distribution. p-value practically zero, H0 can be rejected.
Lagrange Multiplier (LM) test for ARCH errors
: whether squared error terms are autocorrelated. H0: no ARCH effect. HA: ARCH effect.
Ramsey's Regression Specification Error Test(RESET): detect general functional form misspecification.
H0: correct functional form, HA: incorrect functional form. p-value > 0.1, so H0 cannot be rejected.
HAC standard error
Two t-test to check whether
Cnsmr portfolio conservative.
1.H0: β=0, HA:β>0. 2.H0:β=0, HA:β<1. p-value are for
two-tail
, check sign of t-value matches alternative hypothesis. compare half of the reported p-value to the significant level. Both reject H0, conclude β are significantly positive.
general F-test for linear equality restrictions: H0:β = 1, HA: β≠1
.two tail test, since β=0.932<1 and p-value/2 is practically zero, reject H0 conclude Cnsmr portfolio is significantly smaller than one.
Multi-factor CAPM
Small minus Big(SMB)
High minus low(HML)
Robust minus weak(RMW)
Conservative minus Aggressive(CMA)
F-test: H0: β2 = β3 = β4 = β5 = 0 H1: at least one of β2,β3,β4,β5 is different from zero. p-value practically zero, strong significant.
