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Chapter 2 Linear Econometrics for Finance
Multiple linear regression without pre-determined regressors (an asymptotic approach)
yt =xtβ+εt =β1x1t +…+βkxkt +εt.
• We postulate a linear relation between the variable y and the k variables in x (x is
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a vector).
• As discussed earlier, this is the most widely used (and abused) econometric model in finance.
1. The estimation of linear functions is easier than the estimation of nonlinear functions (you do not want to go through the complications posed by nonlinear specifications – which nonlinear specification to use? – unless necessary).
2. Any nonlinear function can be approximated (by Taylor expansion).
3. Most finance and economics models are linear.
• Note: We changed the subscripts from n to t. This is to signify that we are dealing with time series data. Each observation corresponds to a time period. Earlier, each observation corresponded to an “individual”. Time series data cannot be “pre- determined” (as in Chapter 1). We therefore have to assume that the regressor matrix is stochastic, rather than pre-determined.
Assumptions:
1. E(εt|xt) = 0.
Conditional on the regressors (recall, they are stochastic now), the expectation of
the error term is zero.
2. (a) E(ε2t |xt, xt−1, …εt−1, εt−2, …) = σε2 = constant. Also, (b) E(εt|xt, xt−1, …εt−1, εt−2, …) =
Assumption 2(a) is a homoskedasticity condition on the error terms. Assumption
2(b) implies that the εs are uncorrelated of the xs and past εs. 1
Chapter 2 Linear Econometrics for Finance
Note that Assumption 1 and Assumption 2 are the analogue of Assumption 1 and 2 in Chapter 1. Assumption 1 is, as we discuss next, an assumption of “correct specification of the model”. Assumption 2 will be later relaxed (see Chapter 3).
Interpretation. Write the tautology:
y = E(y|x) + {y − E(y|x)}
= E(y|x) + ε,
where ε = y − E(y|x). Then, by construction, E(ε|x) = 0. Now, assume E(y|x) = x′β. In other words, we believe that the conditional mean of y given x, i.e., E(y|x), can be represented as a linear function of x, namely x′β. So, E(ε|x) = 0 is a condition of correct (linear) specification.
1 Estimation
• Given observations (yt, xt), the ordinary least squares (OLS) estimate of β, denoted
by β, is the value which minimizes the sum of the squared residuals around the line, i.e.,
β= argmin(yt −x′tβ)2.
• This is not our usual way to express the OLS estimator. If X is the matrix of regressors and Y is the regressand’s vector, we usually write β = (X′X)−1X′Y . The two expressions are equivalent. However, the expression in Eq. (1) clarifies the role played by sample moments of the data. Since empirical finance largely relies
• Solution:
T −1 T
β= xx′ xy . (1)
tttt t=1 t=1
Chapter 2 Linear Econometrics for Finance
on asymptotic (i.e., T → ∞) arguments and, as we will show below, asymptotic arguments apply to sample moments, this expression is highly convenient.
• Note: If k = 2 and the first regressor is a column of ones, i.e., in the typical univariate case, then
as always.
β =y/T−β x/T=Y−βX, 0t1t1
yt −Y(xt −X)
(xt −X)2 t=1
• Note: fitted values and residuals can be defined in the usual way. Needless to say, they have the usual properties (see Chapter 1).
2 The asymptotic properties of β
2.1 Consistency
T −1 T
β= xx′ xy tttt
−1 T T
t=1 −1 T
x x′β+ε ttttt
x x′ β + x x′ tttttttt
x ε t=1 t=1
In other words,
Linear Econometrics for Finance
−1 T xε .
tt tt t=1 t=1
T −1 T tttt
β−β= xx′ xε .
For β to be a “good” estimator of β, the difference β − β ought to be “small”.
T β−β= xtx′t/T
−1 T xtεt/T .
Aside: The Law of Large Numbers, LLN.
Consider an IID random sample {xt} from a distribution with mean μ and variance σ2 < ∞. Note, we are not making assumptions on the probability distribution of the individual observations xt. We are simply saying that they have the same expected value
(μ) and the same variance (σ2). T
xt Write X = t=1
The sample mean is unbiased for the true expected value μ. TT
. Let us now compute the expected value and the variance of X. We
xt E(xt) Tμ
E(X) = E t=1 = t=1 = = μ. T T T
xt Var(xt) Tσ2 σ2 Var(X)=Vart=1 = t=1 = = .
TT2 T2T
Chapter 2 Linear Econometrics for Finance The sample mean has a variance which goes to zero as the number of observations in-
creases. In other words V ar(X) → 0 as T → ∞.
Note: because (1) the sample mean is, in expectation, the same as the expected value and (2) the variance of the sample mean (which is a measure of variability around the expected value) goes to zero, the sample mean converges to the expected value. What does it mean to converge to the expected value? Next, we talk briefly about modes of covergence.
Convergence in probability: Let g1, g2, g3, ...., gT be a sequence of random variables 123T
xt xt xt xt
(for example, t=1 , t=1 , t=1 , ...,t=1 ). We say that gT converges in probability to a
123T constant c as T → ∞ and write
if, ∀ε > 0,
PlimgT =c,
lim Pr(|gT − c| > ε) = 0. T→∞
Converges in mean-squared: Let g1, g2, g3, …., gT be a sequence of random variables. We say that gT converges in mean-squared to a constant c and write
m.2. gT → c
E(gT−c)2 →0. T→∞
Note: convergence in mean-squared implies convergence in probability. Now, let us return to the sample mean. Notice that
2 2σ2 Var(X)=E(X−E(X)) =E(X−μ) = T →0,
where the first equality is the definition of the variance, the second equality depends on 5
Chapter 2 Linear Econometrics for Finance
the sample mean being an unbiased estimator of μ and the third equality was derived earlier. Given the third equality,
We say …
X → μ = E(x)
X→p μ=E(x).
(1) … that the sample mean converges in probability to μ = E(x) as the number of observations increases. This is also called a “weak law of large numbers”.
(2) … that the sample mean is a “consistent” estimator for μ as the number of observations increases. It converges to μ, in probability, as the sample grows.
Important for our purposes: In our discussion above, we assumed an IID sample with σ2 < ∞. The result is, however, general. Sample means converge to expectations under fairly mild assumptions like stationarity (an IID sample is clearly stationary) and bounded second moments. We will therefore apply the result more generally.
Back to our problem:
xtεt/T →p
E(xtεt), E(xtx′t).
xtx′t/T →p
Linear Econometrics for Finance
β−β → E(xtxt) E(xtεt)
T→∞ ′−1
E(xtxt) E(E(xtεt|xt)) ′−1
E(xtxt) E(xtE(εt|xt)) = 0.
(The second equality derives from the “law of iterated expectations”.) Hence, β is “con- sistent” for β. Alternatively, β converges “weakly” to β.
2.2 Asymptotic normality
• The OLS estimator β is, of course, a random variable.
• We saw that, in the limit (as T → ∞), β converges to β (a number).
• For different samples, we obtain different estimates. β has a probability distribution.
• Note: we have already seen what the distribution of β is when assuming that the errors are normal. Now, we are not making the assumption of normality of the error terms. Our reasoning is asymptotic for T → ∞.
• Why do we care? Suppose we want to test whether β = β0, i.e., where β0 is some true β. In order to test this hypothesis, we need to know how far β can be from β0 and compare this value to the actual distance of β from β0.
• More generally, hypothesis tests rely on the sampling distribution of β.
What is the probability distribution of β?
Chapter 2 Linear Econometrics for Finance
Aside: The Central Limit Theorem, CLT.
Consider an IID random sample {xt} from a distribution with mean μ and variance
σ2 < ∞. Again, we are not making assumptions on the probability distribution of the individual observations xt. We are simply saying that they have the same expected value (μ) and the same variance (σ2). We know that
2. V ar(X) = σ2
By the Central Limit Theorem, we also know that “averages” are “approximately” normally distributed as T → ∞. Note that the key word is “approximate”. The larger the sample size, the better the approximation. Thus, we can write
d σ2 XYN μ,T ,
where Yd means “approximately distributed as.” We note that this result is coherent with the convergence in probability of the sample mean to μ. As the number of observations grows, the variance decreases to zero and the sample mean becomes a more and more accurate estimator of the expected value μ.
We can write the same result in a number of different ways:
X −μ Yd N (0, 1)
By simply standardizing ... √TX−μYd N(0,σ2)
The sample mean converges to μ at a speed of convergence of
of convergence to zero of the standard deviation of the sample mean.
Tt=1(xt−μ) Yd N(0,σ2)
T . This is the speed
Chapter 2 Linear Econometrics for Finance
Differences from the mean (suitably standardized) converge to a normal random variable. Notice, in fact, that
Tt=1(xt − μ) Tt=1 E(xt − μ)
E √ = √ =0.
So, it makes sense for the limiting normal random variable to be centered at zero. In fact, we are averaging de-meaned random variables. Also,
T (xt −μ) T Var(xt −μ) Tσ2 t=1 t=1
Var √T = T =T=σ. Hence, we can also write:
T (x −μ) d t=1 t
By standardizing by
distribution is a well-defined number (which does not diverge to infinity or converge to zero). Because the variance is “balanced”, the limiting distribution is well-posed. The representation in Eq. (2) will be used heavily in what follows.
Important for our purposes: In our discussion above, we assumed an IID sample with σ2 < ∞. The result is, however, general. Standardized (by √T) sample means of de-meaned random variables converge to normal random variables under fairly mild assumptions like stationarity (an IID sample is clearly stationary) and bounded second moments. We will therefore apply the result more generally.
Important: The weak law of large numbers (WLLN) is about the convergence of sample means, i.e. objects like Tt=1 xt , to the corresponding expected value. The central limit
T (x −μ) t=1 t
YN 0,Var √ (2) TT
T, we are guaranteeing that the variance of the limiting
theorem (CLT) is about the convergence of standardized (by de-meaned observations to zero-mean normal random variables.
T) sample means of
Back to our problem. Recall:
T β−β= xtx′t/T
Linear Econometrics for Finance
−1 T xtεt/T .
From the CLT (using the representation in Eq. (2)):
T T (xtεt − E(xtεt)) (xtεt − E(xtεt))
T T Yd N0,Vart=1 √
T T xtεt xtεt
t=1 d t=1 √T YN0,Var √T
since, by the law of iterated expectations, E(xtεt) = 0. Now, write
1T Var √ xtεt
T t=1 1 T
= T Var(xtεt) (3) t=1
= T1 E(xtx′tε2t)−E(xtεt)E(x′tεt) (4)
= σε2 E (xt x′t ) (5)
• Eq. (3) derives from Assumption 1 and Assumption 2(a). Notice, in fact, that Cov(xtεt,x′t+jεt+j) = E(xtεtx′t+jεt+j) − E(xtεt)E(x′t+jεt+j) = E(xtεtx′t+jεt+j) be- cause E(xtεt) = E(xtE(εt|xt)) = 0 given Assumption 1. In addition, E(xtεtx′t+jεt+j) = E(xtεtx′t+jE(εt+j|xt+j,xt,εt)) = 0 for j > 0 given Assumption 2(a). As mentioned,
Chapter 2 Linear Econometrics for Finance Assumption 2(a) is an assumption of uncorrelatedness of the error terms.
• Eq. (4) is the matrix analogue of the “alternative variance formula” (V ar(x) = E(x2) − (E(x))2)
• Eq. (5) derives from E(xtεt) = E(xtE(εt|xt)) = 0 (Assumption 1) and E(xtx′tE(ε2t |xt)) = σε2E(xtx′t), which is the result of the homoskedasticity condition in Assumption 2(b).
= N0,σ2Σ−1 εx
√d 2−1 T β−β YN(0,σεΣx ).
Putting, now, everything together, we obtain
T −1T √′√
Tβ−β= xtxt/T T xtεt/T
d ′ −1 2
Y E(xtxt) N 0,σεΣx = Σ−1N0,σ2Σ
We can estimate σ2 and Σ−1 by using sample moments: εx
ε → σ , T−k tT→∞ε
t=1 1xtx′t →p Σx.
Notice that, because we are employing asymptotic arguments, using ε or ε T−k t T t
is irrelevant in the limit (as T → ∞). This is because k is fixed.
Important observation for empirical work. The estimated variance of the OLS estimator becomes:
Linear Econometrics for Finance
21 −1 2 −1 σε T xtxt T
t=1 2 ′ =σε xtxt
which has a very familiar look, right? If Assumptions 2(a) and 2(b) are satisfied, this theory justifies using the standard errors reported by most softwares (like e-views) without adjustments.
Note: Consistency and asymptotic normality would still hold if:
• The variance of the εts depends on x – i.e., Assumption 2(a) is violated – more on
this in Chapter 3.
• The εts are serially correlated – i.e., Assumption 2(b) is violated – more on this in Chapter 3.
• Also, the xts can be serially correlated.
• If Assumption 2(a) and 2(b) were violated, only the form of the asymptotic variance
of the estimator would change. See Chapter 3.
Important condition, however: E(εt|xt) = 0 – correct (linear) specification of the model. We will continue to assume it throughout.
3 Asymptotic inferential theory: testing 3.1 A single restriction
Test the hypothesis H0 : c′β = γ against HA : c′β ̸= γ. Recall our discussion in Chapter 1.
• Example: H0 : β1 = 2β2. Then,
c′ = [1,−2,0,0,.,.],
2 ′ −1 =σε(XX) ,
γ = 0. • Example: H0 : β3 = 0. Then,
Linear Econometrics for Finance
c′ = [0,0,1,0,.,.], γ = 0.
Construction of the test: Because β is asymptotically normally distributed (as T → ∞), we can write
d Σ−1 c′βYN c′β,σ2c′ x c
d Σ−1 c′β−c′βYN 0,σ2c′ x c
c′β−c′β d
Σ−1 YN(0,1)
σ c′ x c T
and, under the null hypothesis H0 : c′β = γ, c′β−γ d
Again, note that
−1 Y N(0,1). σ c′Σx cH0
2 −1 σε T xtxt
σεΣx t=1 2
2 ′ −1 =σε(XX) ,
T = T =σε
i.e., σε2 and Σx can be estimated using sample analogues, thereby leading to the usual standard errors.
2 −1 Important: Asymptotically, replacing σε Σx
Linear Econometrics for Finance
with σε Σx does not matter, i.e., the asymp-
totic distribution is still approximately normal. One never invokes the t distribution (like in Chapter 1) when using asymptotic arguments.
Implementation: For a 5% level test, reject the null hypothesis if
c′β−γ −1 >2.
σ c′Σx c T
TestthehypothesisH0 : R β = r againstHA :Rβ̸=r. Asbefore,qisthe (q×k)(k×1) (q×1)
number of restrictions.
• Example: H0 :β2 =β3 =…=βk =0.Then,
3.2 Multiple restrictions
010 0010
0 0 … 0 =[0k−1,1|Ik−1],
001 r = 0,
where 0k−1,1 is a k − 1 column vector of zeros and Ik−1 is the identity matrix of dimension k − 1.
Construction of the test: Because β is asymptotically normally distributed (as T → ∞), we can write
√ d 2−1′ TR β−β YN(0,R σεΣx R).
Chapter 2 Hence,
Linear Econometrics for Finance
√2−1′−1/2 d
T R σεΣx R R β−β →N(0,1).
TRβ−β′Rσ2Σ−1R′−1Rβ−βYd χ2. εxq
Under the null H0 : Rβ = r, then TRβ−r′Rσ2Σ−1R′−1Rβ−rYd χ2
′2−1′−1 d 2 T R β − r R σ ε Σ x R R β − r Y χ q ,
′ 2 −1 −1 R β − r R σ ε Σ x R ′ R β − r
Important: Again, estimating σ2Σ−1 does not affect the asymptotic distribution! One
never invokes the F distribution (like in Chapter 1) when using asymptotic arguments. Implementation: for a 5% level test, reject when
R σ ε Σ x T
Important observation for empirical work
R ′ R β − r
R σε (X X)
−1 ′−1 R Rβ − r
where χ2q,0.05 is the value such that P(χ2q < χ20.05) = 0.95.
> χq,0.05,
The statistic for multiple restrictions that we derived is generally called “Wald statis- tic.” Call it W.
Chapter 2 Linear Econometrics for Finance
Note: if Assumption 2(a) and 2(b) are satisfied, the classical F statistic is effectively equal to
where q is the number of restrictions. Let us go back to the example in Chapter 1 to
convince ourselves.
Regression, liquidity and asymmetric information: continued
Earlier, we tested the assumption that the coefficients associated with log turnover and the number of analysts are jointly equal to zero.
We used a classical F test with 2 restrictions. Using asymptotic arguments, we could have also used a Wald (or Chi-square) test. With two restrictions, the value of the Chi-square statistic is double that of the F statistic. Needless to say, the critical values to which we should be comparing this value are the critical values of the Chi-square random variable rather than those of the F random variable.
F and : Untitled
Test Statistic
F-statistic Chi-square
4.979229 9.958458
Probability 0.0088
Std. Err. 0.002913
Null Hypothesis: C(4)=0, C(6)=0 Null Hypothesis Summary:
Normalized Restriction (= 0)
Value 5.04E − 05
−0.098714 Restrictions are linear in coefficients.
Chapter 2 Linear Econometrics for Finance
Which statistic to use and which critical values to use? If you are not comfortable with normality of the residuals, you should be using the asymptotic theory we just presented. For most applications of practical relevance in finance, the number of observations is large enough for asymptotic arguments to be very reliable.
Another observation for empirical work
You could of course re-express the test in terms of restricted and unrestricted residuals. You could also re-express the test in terms of restricted and unrestricted R2. So,
2−1 σ ε Σ x
R β − r R
R β − r
T ε ε − ε ε
R2 − R∗2 1−R2
∗ ∗22 where ε are the residuals from imposing the restriction Rβ = r and R is the R from
imposing the restriction Rβ = r. Re-cap :
• We have specified the conditional mean of y as a linear function of x
• We have estimated the parameters by OLS
• We have shown the asymptotic properties of the OLS parameter estimates
• Consistency: As T increases, β converges to β in probability
• Asymptotic normality: As T increases, the distribution of β converges to that of a normal random variable with an estimable asymptotic variance.
• We can test whether βj = 0 (or any other value). 17
Chapter 2 Linear Econometrics for Finance
More generally, we can test, asymptotically, linear (single or multiple) restrictions on the parameters.
The Capital Asset Pricing Model (CAPM) and Re- gression
The CAPM model:
E(Ri,t −Rf)=βiE(RM,t −Rf),
βi = Cov(Ri,t −Rf,RM,t −Rf). Var(RM,t −Rf)
Ri,t is the return on asset i at time t
Rf is the return on a risk-less asset
RM,t is the return on the market portfolio at time t
Interpretation.
• The expected return that a generic risky asset i provides in excess of the risk-free return depends on a risk premium.
• The risk premium comprises two components: a quantity of risk βi and a price of risk E(RM,t − Rf ).
• There is only one source of risk: market fluctuations.
• Exposure to market fluctuations are compensated.
• The compensation is proportional to the market risk exposure βi = Cov(Ri,t,RM,t). V ar(RM,t)
Chapter 2 Linear Econometrics for Finance
• βi can be interpreted as the sensitivity of the return i to market risk. If βi = 0, then asset i is not exposed to market risk. If βi > 0, then asset i is exposed to market risk and E(Ri,t) > Rf , since E(RtM − Rf ) > 0.
• The quantity E(RM,t − Rf ), i.e., the market risk premium, has historically been very high. No reasonable economic model can justify such a large premium for holding stocks versus the risk-less asset. This is typically called “the equity premium puzzle”. See Mehra and Prescott (1985) and Mehra and Prescott (2003).
In order to estimate βi, we run the regression
Ri,t −Rf =αi +βi(RM,t −Rf)+εi,t,
where αi, the regression intercept, can be interpreted as the pricing
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