The efficient market hypothesis
Stats 413: Applied Regression Analysis
Winter 2017
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The US Treasury bills market
• US Treasury bills are essentially IOU’s by the US government that
promises the pay the holder of the bill a specified amount (the face
value) on a future date (the maturity date). Investors make a profit by
buying the bill at a discount. For example, if you purchased a $1000
T-bill that matures at the end of month t at the start of the month
for $990, you can redeem the T-bill at the end of the month for $1000,
making a $10 profit.
• The US government sells T-bills to raise money to fund its operations.
They are attractive to investors because they are a very low-risk investment. Unless the US government defaults on its debts, investors
will be able to redeem their T-bills at maturity. The flip side is the
rate of return on T-bills is lower than that of many other higher-risk
investments.
• Notation:
– vt is the price of a T-bill at the beginning of month t. The price
is at most the face value of the bill.
– Rt is the nominal rate of return on the bill over month t. By
nominal, we mean unadjusted for inflation. For example, if you
purchased a $1000 T-bill that matures at the end of month t at
the start of the month for $990, your rate of return over month t
is
1000 − 990
Rt =
≈ 0.01.
990
– Pt is the Consumer Price Index (CPI) at the beginning of month
t. The change in the CPI is a measure of inflation.
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– πt+1 is the inflation rate over month t:
πt+1 =
Pt+1 − Pt
.
Pt
– π
bt+1 is the forecast of the inflation rate over month t.
– t+1 is the forecast error: πt+1 − π
bt+1 .
– rt+1 is the real rate of return over month t:
rt+1 =
vt
1
Pt+1 − Pt
vt
Pt
=
1 + Rt
− 1 ≈ Rt − πt+1 .
1 + πt+1
The nominal rate of return Rt is not adjusted for inflation that
occurs over the maturity term. The real rate of return rt+1 adjusts
for inflation.
– b
rt+1 is the forecast of the real rate of return:
b
rt+1 =
1 + Rt
− 1 ≈ Rt − π
bt+1 .
1+π
bt+1
The subscript on non-hat variables indicates the first month in which
the variable is known. For example, the inflation rate πt+1 is only
known at the beginning of month t + 1 because it depends on Pt+1 .
On the other hand, the rate of return on the bill Rt is known at the
beginning of month t because it only depends on the price vt and face
value of the bill.
• To keep things simple, we only consider T-bills whose maturity term
coincides with the sampling frequency of the data (monthly).
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Fama’s efficient market hypothesis
• Fama’s efficient market hypothesis posits:
– rational expectations: the forecast of the inflation rate assimilates all available information:
π
bt+1 = E πt+1 | It , It := {Rt , Rt−1 , . . . , πt , πt−1 , . . . },
where the information set It includes all the variables known at
the beginning of month t.
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– constant real interest rate forecasts: the forecasts of the real
rate of return is constant:
b
rt+1 = r0 for all t.
• The first condition is a formal statement of the hypothesis that the
market “fully assimilates” the information at time t: the conditional
expectation πt+1 is the minimum MSE predictor that depends only on
the information in It .
• Under Fama’s hypothesis, the forecast error sequence {t } is a MDS.
Indeed, we have
E t+1 | It = E πt+1 − π
bt+1 | It
= E πt+1 | It − E π
bt+1 | It
= E πt+1 | It − E πt+1 | It (rational expectations)
= 0.
(2.1)
At this point, we wish to appeal to the tower property to conclude
E t+1 | t , t−1 , . . . = E E t+1 | It | t , t−1 , . . . = 0.
To justify the first step, we need to show that It contains at least as
much information as {t , t−1 , . . . }. The formal way of establishing
this is to show s , s ≤ t is a function of the variables in It . Indeed, we
have
s = πs − π
bs = πs − E πs | Is , s ≤ t
which is a function of the variables in It . We conclude {t } is a MDS.
• The intuitive interpretation of (2.1) is the forecast error t+1 does not
depend on It . After all, if the forecast assimilates all the information
in It , then forecast error should not depend on It .
• Since MDS’s are serially uncorrelated, one possible test of Fama’s
hypothesis is to test whether the forecast errors are in fact serially
uncorrelated. Testing for serial correlation is a topic that is taught in
courses on time series analysis, and we skip it here.
• The real interest rate has a constant expected value and is serially
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uncorrelated. We have
rt+1 = Rt − πt+1
= Rt − π
bt+1 + π
bt+1 − πt+1
= r̂t+1 − t+1
(definition of r̂t+1 and π̂t+1 )
= r − t+1 .
(constant real interest rates)
(2.2)
Since {t } is a MDS and a MDS is centered and serially uncorrelated,
we have
E rt = r + E t = r,
cov(rs , rt ) = E (rs − r)(rt − r) = E s t = 0, s 6= t.
• To derive a testable implication from Fama’s hypothesis, we combine
the definition of πt+1 with (2.2) and (2.1) to obtain
E πt+1 | It = E Rt − rt+1 | It
(definition of rt+1 )
= E Rt − r + t+1 | It
(2.2)
(2.3)
= Rt − r + E t+1 | It
= Rt − r.
(2.1)
Since Rt ∈ It , we have
E πt+1 | Rt = E E πt+1 | It | Rt = Rt − r,
which suggests a one-for-one relationship between inflation and the
nominal interest rate.
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Is the US T-bills market efficient?
• In summary, Fama’s hypothesis suggests the regression coefficient of
Rt is one. Before we test this restriction, we check the conditions of
the linear model are implied by Fama’s hypothesis.
T
• Linearity: Let yt = πt+1 and xt = 1 Rt . By (2.3), we have
E πt+1 | It = xTt β ∗ + t ,
T
β ∗ = −r 1 .
• Predetermined predictors: The predetermined condition is
1
t+1
E
t+1 = E
= 0.
Rt
Rt t+1
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Thus we need to show E t = 0 and E Rt t = 0. The former holds
because {t } is a MDS. The latter holds because
E Rt t = E E Rt t | It
= E Rt E t | It
(Rt ∈ It )
=0
(2.1)
• Ergodicity and stationarity: The stationarity condition is problematic because the nominal interest rate Rt , as shown in Figure 1, has
an upward trend. For now, we ignore this empirical evidence to the
contrary and proceed. It is hard to check the ergodicity condition in
practice, and we just have to assume it.
• The second moment of the predictors is
" #
T
1
1
1 Rt
E
=E
.
Rt Rt
Rt R2t
Its determinant is
2
E R2t − E Rt = var Rt ,
which is non-zero.
• We know {t , t−1 , . . . } ⊆ It . Thus
{xt t , xt−1 t−1 , . . . } ⊂ {xt , xt−1 , . . . , t , t−1 , . . . } ⊆ It .
By the tower property, we have
E xt+1 t+1 | xt t , xt−1 t−1 , . . .
= E E xt+1 t+1 | It | xt t , xt−1 t−1 , . . .
(It has more information than {xt t , xt−1 t−1 , . . . })
= E xt+1 E t+1 | It | xt t , xt−1 t−1 , . . .
(Rt ∈ It )
= 0,
which implies {xt t } is a MDS.
• Having checked that the conditions of the linear model are plausible, we
proceed to test H0 : β2∗ = 1. We use the dataset in [?] which includes
the one-month T-bill rate and CPI from February 1950 to December
1990. Figure 1 shows the data.
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Figure 1: Inflation πt and interest rates Rt (Figure 2.3 in [Hay00])
• To replicate Fama’s results, we restrict to the data from January 1953
to July 1971 (n = 223). We use the heteroskedasticity-robust estimator
of the asymptotic variance
−1 P
P
−1
1
1
T
2 x xT
T
d βbn = 1 P
ˆ
Avar
x
x
x
x
i
i
i
i
i
i
i
i∈[n]
i∈[n]
i∈[n]
n
n
n
= n(XT X)−1 XT diag(ˆ
2 )X(XT X)−1 ,
which leads to heteroskedasticity-robust standard errors. By default,
R does not return heteroskedasticity-robust standard errors, and we
must ask for them specially.
lm(formula = inf ~ TB1, data = fama)
Residuals:
Min
1Q Median
3Q
Max
-7.1928 -1.9386 -0.2488 1.8967 8.4651
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.8678
0.4332 -2.003
0.0464 *
TB1
1.0147
0.1127
9.003
<2e-16 ***
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
1
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Figure 2: Real interest rates rt (Figure 2.4 in [Hay00])
Residual standard error: 2.843 on 221 degrees of freedom
Multiple R-squared: 0.2364, Adjusted R-squared: 0.2329
F-statistic: 81.79 on 1 and 221 DF, p-value: < 2.2e-16
• We see that 1 is well within two standard errors of βb2 , which implies
the two-sided z-test of H0 easily accepts.
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Subsequent developments
• Studies after the publication of Fama’s paper showed that the one-forone relationship between inflation and interest rates is limited to the
period between the end of World War II and 1979. Further, Figure 2
shows that the second part of Fama’s hypothesis, that the real interest
rate is constant, is implausible post-1979.
• If we restrict to the data post-October 1979, the interest rate coefficient
in (2.3) is significantly smaller than one.
lm(formula = inf ~ TB1, data = afterFama)
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Residuals:
Min
1Q
Median
-11.4406 -2.8608 -0.6087
3Q
Max
2.6677 16.8705
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 1.57500
0.69340
2.271
0.024 *
TB1
0.66478
0.09997
6.650 2.1e-10 ***
--Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1
1
Residual standard error: 4.347 on 231 degrees of freedom
Multiple R-squared: 0.1391, Adjusted R-squared: 0.1354
F-statistic: 44.6 on 1 and 231 DF, p-value: 1.782e-10
• This raises the question of why the one-for-one relationship between
inflation and interest rates vanishes in the post-1979 data. One possible explanation by Mishkin is the inflation rate affects interest rate
gradually. Thus transient fluctuations of inflation have little effect on
interest rates, but sustained trends in inflation affect interest rates.
• We see in Figure 1 that there is a sustained upward trend from 1950 to
1979, which is also the period when there is strong correlation between
inflation and interest rates.
References
[Hay00] Fumio Hayashi. Econometrics. Princeton University Press, 2000.
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