Chapter 1 Linear Econometrics for Finance
Figure 1: Predicting returns over alternative horizons.
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Chapter 1 Linear Econometrics for Finance
Multiple linear regression with pre-determined regressors
We consider the model
where X is a fixed (non stochastic) n × k matrix of regressors (n observations and k
regressors) and ε is an n-vector of error terms. Assumptions:
1. E(ε) = 0. On average, the errors are zero.
2. V ar(ε) = σ2In, where In is the identity matrix of dimension n. (a) The errors are homoskedastic (same variances on the diagonal of V ar(ε)) and (b) uncorrelated (the off-diagonal elements, i.e., the covariances, are equal to zero).
More explicitly, write
y1 x11 x21 …xk1β ε1 1
y2 x12 x22 …xk2β2 ε2 y3 = x13 x23 … xk3 + ε3
… … … … … … …
y xx…x βk ε n1n2nkn n
y1 y2
y3 =
are linear combinations of the k regressors contained in the fixed matrix X. On top of 2
… yn
β1×11 +β2×21 +…+βkxk1 β1×12 +β2×22 +…+βkxk2 β1×13 +β2×23 +…+βkxk3 …
ε1 ε2
+ ε3 ,
β1x1n + β2x2n + … + βkxkn
where, again, E(ε) = 0 and V ar(ε) = σ2In. All we are saying is that the Y observations
… εn
Chapter 1 Linear Econometrics for Finance
the linear combination, there is an error vector. The error vector has mean zero and a variance-covariance matrix given by
σ2 0 .. .. 0
0 σ2 0 .. .. Var(ε)=E(εε′)=σ2In = .. 0 σ2 0 .. .
.. .. … … …
0 .. .. 0 σ2
Note: the first column of the X matrix could just be a column of ones. This is the case when there is an intercept in the regression (β1 would therefore be the intercept).
Given the assumptions, it is easy to show that E(Y ) = Xβ and V ar(Y ) = σ2In. In fact,
E(Y ) = E(Xβ + ε) = E(Xβ) + E(ε) = Xβ, V ar(Y ) = V ar(Xβ + ε) = V ar(ε) = σ2In.
1 The ordinary least squares (OLS) method
We need to estimate parameters.
β = argmin(yi−x′iβ)2
= arg min(Y − Xβ)′(Y − Xβ)
= (X′X)−1X′Y
Linear Econometrics for Finance
−1n x1iyi
x21i x1ix2i … …
x1ix2i …
x2i … … …
… x2iyi i=1 .
i=1 nn
… … … … …x2kixkiyi
The least squares method chooses the vector β by minimizing the squared differences
around the multivariate line x′β. Proof.
C(β)=(Y −Xβ)′(Y −Xβ)
∂C(β) = 0⇒−2X′(Y−Xβ)=0 ∂β
derivative is positive. Hence, we confirm that we have a minimum.
1.1 Definitions: fitted values and estimated residuals
Fitted values (Y). These are the values on the estimated line.
By setting the first derivative equal to zero, we obtain the OLS β estimator. The second
∂2C(β) ∂β∂β′
y βx +βx +…+βx
1 111221 kk1
y βx +βx +…+βx
2 112222 kk2
Y=y=βx+βx+…+βx
3 113223 kk3
…
β1x1n +β2x2n +…+βkxkn 4
Linear Econometrics for Finance
Residuals (ε). These are the differences between the true Y values and the fitted values
33 ε=Y−Y= y − y
… …
From the first-order conditions of the minimization problem we can write
X ′ ( Y − X β ) = 0 ⇒ X ′ ε = 0
or, equivalently (less compactly, but more intelligibly),
1.2 Properties
y y 22
n x ε
This is like saying that the residuals are orthogonal to the X observations! Since the
fitted values are linear combinations of the X observations (see above), it also says that the residuals are orthogonal to the fitted values. In other words,
… n
Linear Econometrics for Finance
Y = X β = X ( X ′ X ) − 1 X ′ Y = P Y
ε = Y − Y = Y − X ( X ′ X ) − 1 X ′ Y = ( I − X ( X ′ X ) − 1 X ′ ) Y = M Y
Y ε = 0 .
Important: Note that, if there is an intercept in the regression, then the first element in
X′ ε becomes ε = 0. In other words, the sample mean of the residuals is zero, if there i
is an intercept in the regression!
1.3 Properties: continued
where M and P are symmetric and idempotent matrices. They are symmetric, since M = M′ and P = P′ (try showing it …). They are idempotent since M = MM and P = P P (try showing it …).
Geometrically, P is the matrix which projects Y on the space spanned by the columns of X (recall, Y is a linear combination of the regressors). M is a matrix which projects Y on the space orthogonal to the space spanned by the columns of X (recall ε and Y are orthogonal).
1.4 Partitioned matrices
X = [ X1 | X2 ], n×k1 n×k2
where k = k1 + k2. We are simply separating the full matrix X into two sub-matrices. Then,
Chapter 1 Linear Econometrics for Finance
β = ( X ′ X ) − 1 X ′ Y
X′X X′X−1X′Y
=11121 X 2′ X 1 X 2′ X 2 X 2′ Y
There are two cases.
(1) Assume X1′ X2 = 0 (the regressors in the first block and in the second block are
orthogonal). Then,
X 1′ X 1 0 − 1 X 1′ Y β=0X2′X2 X2′Y
( X 1′ X 1 ) − 1 0 X 1′ Y = 0 ( X 2′ X 2 ) − 1 X 2′ Y
−1 (X′X) X′Y β
=111=1. (X′X)−1X′Y β
Thus, we can run two separate regressions (one on X1 and one on X2) to obtain the least squares estimates.
(2) Assume X1′ X2 ̸= 0. Then, we can show that −1
β (X′MX) X′MY 1=12112,
β (X′MX)−1X′MY 221221
where M2 = In − X2(X2′ X2)−1X2′ and M1 = In − X1(X1′ X1)−1X1′ .
Intuition: Focus on β1. This is like running several regressions. First, regress the first
column of X1 on X2. Obtain the residuals. Now, regress the second column of X1 on X2.
Obtain the residuals. Keep going until you reach the last column of X1. Collect the k1
columns of residuals in a new matrix X1 = M2X1. Regress Y on the matrix of residuals
to obtain β1.
In English, first you want to purge X1 of the effect of X2 and compute the component of X1 which is orthogonal of X2 (the residual matrix), then you want to regress Y on the residual matrix.
Important: multiple regression does this automatically. You never really go through these steps. If you know X2 and only care about β1, just put X2 in your regression!
[X1|X2] 1 = X1β1 + X2β2
′ −1 ′
β1 = X1X1 X1Y
= (X1′ M2X1)−1 X1′ M2Y.
= (M2 + P2)X1β1 + X2β2
= M2X1β1 + P2X1β1 + X2β2
= M2X1β1 + X2(β2 + (X2′ X2)−1X2′ X1β1)
= M2X1β1+X2c β
= [M2X1|X2] c1 .
Notice that β1 has not changed. However, the regressors (and the second set of param- eters) have changed. The two blocks are now orthogonal! Hence, I can find β1 by just running a regression of Y on M2X1 = X1 :
′ −1 ′
β1 = X1X1 X1Y
= (X1′ M2′ M2X1)−1 X1′ M2′ M2Y
= (X1′ M2M2X1)−1 X1′ M2M2Y 8
Linear Econometrics for Finance
Chapter 1 Linear Econometrics for Finance = (X1′ M2X1)−1 X1′ M2Y,
by the symmetry and idempotency of M2.
2 The statistical properties of OLS
= ( X ′ X ) − 1 X ′ Y ⇒
= (X′X)−1X′ (Xβ + ε)
= (X′X)−1(X′X)β + (X′X)−1X′ε
= β + (X′X)−1X′ε.
E(β) = E(β + (X′X)−1X′ε) = β + (X′X)−1X′E(ε)
The OLS estimator is unbiased. Interpret: whatever the true parameter β is, if the
model is true (i.e., if Y = Xβ+ε), then the OLS estimator will deliver the right parameter
(on average). This said, there is some sampling variation around the expectation. Hence,
we need to talk about the variance of β.
(1) The expected value of β.
Chapter 1 Linear Econometrics for Finance
(2) The variance of β.
We can also write
V ar(β) = E[(β − E(β))(β − E(β))′]
= E[(β − β)(β − β)′]
= E (X′X)−1X′ε (X′X)−1X′ε′
= E (X′X)−1X′εε′X(X′X)−1
= (X′X)−1X′E(εε′)X(X′X)−1
= σ2(X′X)−1X′InX(X′X)−1
= σ2(X′X)−1X′X(X′X)−1
= σ2(X′X)−1.
V ar(β) = σ2 1 X′X −1
Interpret. The variance of β depends directly on the variance of the error terms σ2 and
inversely on the “variability” of the X observations, i.e., X′X . It also depends inversely n
on the number of observations. Notice that, when the number of observations increases without bound (i.e., when n → ∞), the distribution of the β estimator becomes more and more concentrated around the expected value β. We will return to this idea in Chapter 2.
2.1 The Gauss-Markov Theorem
The OLS estimator β is BLUE (best linear unbiased). For any estimator β which is linear (in the observations Y ) and unbiased, it turns out that
V ar(β) ≥ V ar(β).
Proof. Consider the generic estimator β = AY , where A is a k × n matrix. Note that β
is linear in Y. Let us compute its expected value. 10
Chapter 1 Linear Econometrics for Finance
E(β) = E(AY )
= E(AXβ+Aε)
= AXβ+AE(ε) = AXβ.
Thus, AX = Ik for β to be unbiased. So, β = AY (with the restriction AX = Ik) is a generally specified unbiased and linear estimator of β. Note:
Now, we need to show that
V ar(β) ≥ V ar(β).
V ar(β) = E((β − E(β))(β − E(β))′) = E((β − β)(β − β)′)
= E(Aεε′A′)
AA′ − (X′X)−1
= AA′ − AX(X′X)−1X′A′
= A(In − X(X′X)−1X′)A′
Chapter 1 Linear Econometrics for Finance
by the fact that AX = Ik and M is symmetric and idempotent. Is AMM′A′ positive semi-definite? Write
z′AM(AM)′z ′
for any real vector z ∈ Rk. Because z′AMM′A′z ≥ 0 for any conformable real z, AMM′A′ is positive semi-definite and AA′ − (X′X)−1 ≥ 0.
2.2 Estimation of σ2
We (almost) use the empirical variance of the estimated residuals:
ε ε σ = n − k = n − k .
2 i=1 2222
Statistical property: σ is unbiased for σ (i.e., E(σ ) = σ ). Proof. Recall
ε = M (X β + ε) = M ε.
21′1′′ E(σ) = n−kE(εε)=n−kE(εMMε)
= 1 E(ε′Mε) = 1 E(trε′Mε) n−k n−k
Linear Econometrics for Finance
1 E(trMεε′) = 1 n−k n−k
tr (ME(εε′)) n−ktr(MIn)=n−ktr(M)
σ2 ′−1′σ2 ′−1′
n−ktr(In−X(XX) X)=n−k(trIn−tr(X(XX) X)) σ2 ′ −1 ′ σ2
n−k(n−tr (XX) XX )=n−k(n−tr(Ik)) σ2
n−k(n−k)=σ2.
The result relies on the symmetry and idempotency of M. It also relies on the properties
of the trace (for a review, refer to Chapter 0).
3 Exact inferential theory: testing
ε→d N(0,σ2In).
The symbol →d signifies “distributed as”. Because the error terms are normally distributed and Y is a linear combination of normal random variables (with Xβ deterministic), we have
Y →d N(Xβ,σ2In).
Note: we are imposing strong restrictions on the error terms. Not only are we saying that they are mean zero, homeskedastic (same variance) and uncorrelated, we are also saying that they are normally distributed. This will lead to an exact inferential theory. We will see later what we mean by “exact”. In Chapter 2, normality will be relaxed. In Chapter 3, we will relax normality, homoskedasticity and uncorrelatedness.
3.1 Classical testing problems
Linear Econometrics for Finance
(1) Single linear restriction:
or, equivalently,
H0 : c′β = γ
H0 :cjβj =γ.
j=1 Example: Standard t-test on the j th parameter.
H0 :βj =0. 0
.. c =
1 (this is the jth spot) …
H0 : R β = r q×kk×1 q×1
and γ = 0.
(2) Multiple linear restrictions:
with q ≤ k. Here q is the number of restrictions, k is always the number of parameters. Example: Standard F-test on the slope parameters (excluding the intercept).
H0 :β2 =β3 =…=βk =0.
Chapter 1 Linear Econometrics for Finance
3.2 Implementation
010… 0 0 1 0 …
k−1×k R = … … 0 … 0
0 …01 0
0 r =….
k−1×1
Y →d N(Xβ,σ2In). β = (X′X)−1X′Y.
Since, β is a linear combination of normal random variables (the Y s), it is also a normal random variable. Hence,
β →d N ( β , σ 2 ( X ′ X ) − 1 ) 3.2.1 Single linear restriction
Construction of the test:
H0 : c′β = γ.
c ′ β →d N ( c ′ β , σ 2 c ′ ( X ′ X ) − 1 c ) 15
Chapter 1 Linear Econometrics for Finance or
c′β−c′β →d N(0,σ2c′(X′X)−1c) c′β−c′β d
σc′(X′X)−1c → N(0, 1) and, under the null hypothesis H0 : c′β = γ,
c′β−γ d σc′(X′X)−1c →H N(0,1).
This would be our test statistic, if we knew σ. If we knew σ, we could test the null hypoth- esis c′β = γ by checking if the ratio √ c′β−γ falls in the tails of the normal distribution
σ c′(X′X)−1c
(i.e., if it is larger than 2, or smaller than -2, for a 5% level test). Unfortunately, we do
′ not know σ. We estimate σ using σ = ε ε .
We will show that, when we replace σ with σ, the ratio is not standard normal
anymore. It is t-distributed with n − k degrees of freedom. The following result will lead to the finding.
First aside:
εε→d χ2n−k, σ2
where χ2n−k is a chi-squared random variable with n − k degrees of freedom. Proof:
Recall, ε= Mε. Hence,
εε= εM Mε = εMε = εQΛQε σ2 σ2 σ2 σ2
by the Jordan decomposition of the idempotent matrix M (please refer to Chapter 0). Q is the matrix containing the eigenvectors of M. Note that QQ′ = In. Λ is the matrix containing the eigenvalues of M on the diagonal and zeros everywhere else. By
Chapter 1 Linear Econometrics for Finance idempotency, the eigenvalues of M are either 1 or zero. Since the trace of M is n − k, it
turns out that the number of ones is n − k. Now, notice that Q′ε →d N(0, 1 Q(σ2In)Q′) = N(0, In).
n−k ε′QΛQ′ε=Z′ΛZ=zi2→d χ2n−k
variable with n − k degrees of freedom
Second aside: Consider a standard normal random variable. Consider a chi-square random variable with n − k degrees of freedom. Assume the two random variables are independent. Hence,
χ2 = tn−k,
a t distribution with n − k degrees of freedoms. Let us now go back to
σ σ2 Thus, call Q′ε = Z →d N(0,In). This implies,
since the sum of n − k independent normal random variables is a chi-squared random
c′β−γ d σc′(X′X)−1c → N(0, 1)
√ c ′ β − γ
σ c′(X′X)−1c d N(0,1)
−1 = ′ →χ2 =tn−k.
c′β−γ ′ ′
σ c(XX) c εε n−k
between a normal random variable and the square root of an independent chi-squared random variable divided by the number of degrees of freedom (n − k, in this case). As in
Interpret. When we replace σ with its estimator σ, we effectively compute the ratio
Chapter 1 Linear Econometrics for Finance
the second aside, this ratio is distributed as a t-student distribution with n − k degrees of freedom. Thus,
falls in the
tails of the t distribution with n − k degrees of freedom (i.e., if it is larger than t0.025,n−k
– i.e., slightly larger than 2 – or smaller than −t0.025,n−k – i.e., slighly smaller than -2 – for a 5% level test).
Example: Classical t-test (H0 : βj = 0). The relevant statistic is
′ ′ −1 → tn−k.
σ c(XX) c
We now test the null hypothesis c′β = γ by checking if the ratio
c ′ ( X ′ X ) − 1 c
βj −0 t=′ −1
3.2.2 Multiple linear restrictions
Construction of the test:
This implies that
Rβ →d N(Rβ, σ2R(X′X)−1R′) Rβ−Rβ →d N(0,σ2R(X′X)−1R′)
Zq =σ−1R(X′X)−1R′−1/2(Rβ−Rβ)→d N(0,Iq). 18
where (X′X)−1 is the jth spot on the diagonal of the matrix (X′X)−1.
H0 : R β = r . q×kk×1 q×1
Chapter 1 Linear Econometrics for Finance
Zq′Zq =σ−2(Rβ−Rβ)′R(X′X)−1R′−1(Rβ−Rβ)→d χ2q and, under the null hypothesis,
Zq′Zq =σ−2(Rβ−r)′R(X′X)−1R′−1(Rβ−r)→d χ2q. H0
This last result is obvious. The internal product of q independent standard normal random variables is just a chi-squared random variable with q degrees of freedoms.
At this point, we could use the 95th percentile of the chi-squared distribution with q degrees of freedom (χ20.95,q) to test the null hypothesis. If
σ−2(Rβ − r)′ R(X′X)−1R′−1 (Rβ − r) ≥ χ20.95,q
then we would reject the null hypothesis. The problem, again, is that we do not know σ. Just like earlier, we will show that when we replace σ with σ, the distribution of the test statistic changes. In this case, it changes to that of an F random variable with number of degrees of freedom q, n − k (when the test statistic is also divided by the
number of restrictions q).
Third aside: Consider a chi-squared random variable with q degrees of freedom χ2q. Consider a chi-squared random variable with n − k degrees of freedom χ2n−k . Assume the two random variables are independent. Hence,
χ2q /q →d Fq,n−k , χ2n−k/n−k
an F distribution with q degrees of freedom in the numerator and n−k degrees of freedoms in the denominator.
Now, write
σ−2(Rβ − r)′ (R(X′X)−1R′) ′
(Rβ − r)/q
σ2(n−k) 19
Linear Econometrics for Finance
d (Rβ−r)/q→Fq,n−k.
−2 ′ ′ −1 ′−1 = σ (Rβ−r) R(XX) R
Thus, when we replace σ with σ (and divide by q ), rather than using the 95th percentile of the chi-squared distribution, we use the 95th percentile of the F distribution to test.
Example: Classical F-test with an intercept (H0 : β2 = β3 = … = βk = 0). The relevant statistic is
(Rβ) R(X X) R (Rβ)/(k − 1) → Fk−1,n−k 010…
−2 ′′−1′−1 d
0 0 1 0 …
k−1×k R =… … 0 … 0.
Note: All of these tests are “exact” in the sense that they are valid for any number of observations n. Exact tests can be derived only by imposing strong restrictions (like normality) on the error terms. Without normality, this testing framework would not hold. In Chapter 2, we will abandon normality of the error terms and derive tests which are not “exact” (and are, therefore, not valid for any n) but “asymptotic”, i.e., they are valid only when the number of observations goes off to infinity.
4 Regression analysis, liquidity and asymmetric in- formation
We are interested in the relation between the average bid-ask spreads on stocks and the characteristics of the corresponding companies (Stoll, 2000). Download the file spreads- microstructure.xls. The file contains information for the 100 stocks in the S&P 100 index. Our variable of interest (the Y variable) is the bid-ask spread (constructed as an average
Chapter 1 Linear Econometrics for Finance
over the day) – or tradecost – of the S&P100 stocks. The explanatory, or X, variables are:
1. log volatility – The log of the daily return standard deviation
2. log size – The log of the size of the stock. Size is total outstanding number of shares multiplied by share price. Size is measured in thousands of dollars
3. log trades – This is the log of the average number of trades per day
4. log turn – This is the log of the ratio of the average number of shares traded per day (in dollars) over the number of shares outstanding (in dollars)
5. NumberAnalysts – This is the number of analysts following the stock
The same data is used in Bandi and Russell (2007). Consider the following theories of the determinants of bid-ask spreads.
1. Asymmetric information. Stocks with greater degrees of asymmetry in infor- mation (regarding their fundamental value) tend to have wider bid-ask spreads. The number of analysts following a stock is viewed as an asymmetric information proxy. The larger it is, the lower private information, the smaller the spreads. Log turn-over is, also, seen as an asymmetric information proxy. The larger it is, the larger private information, the larger the spreads. (As Stoll, 1989, points out, without informed trading, stocks would be traded in proportion to their shares outstanding. Trading rates in excess of this proportion should be associated with informed trading.)
2. Liquidity. Stocks that trade more frequently and have larger market capitalization (i.e., more liquid stocks) tend to have lower bid-ask spreads. The larger log trades and log size, the larger liquidity, the smaller the spreads. Log turn-over is, also, sometimes seen as a liquidity proxy. The larger it is, the larger liquidity, the smaller the spreads.
3. Fundamental volatility. Stocks that have a higher volatility of fundamental values tend to have larger bid-ask spreads. Higher uncertainty about the underlying stock’s value i
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