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**Unformatted text preview: **STAT8007
Statistical Methods in Economics and
Finance
Chapter 4 Regression with Stochastic
Predictors Department of Statistics and Actuarial Science
The University of Hong Kong Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 1 / 64 An asset pricing model
4.1 Regression with Stochastic Predictors Recall assumption (A2) or (B2) in Gauss-Markov Theorem: Predictor(s) is (are) xed or non-stochastic.
However, regressions with stochastic predictors are frequently encountered in economics and nance.
For example, in a Capital Asset Pricing Model Rt − RF t = α + β(RM t − RF t ) + εt ,
the predictor RM t − RF t is observed and vary together with the regressor, and is hence
Dr. Z.Q. Zhang ([email protected]) (4.1) stochastic. STAT8007 2019-20 1st Semester 2 / 64 Implication of stochastic predictors
We use a simple LIM as an illustrative example to study the
implication of the stochastic predictor Xi : Y i = β0 + β1 X i + εi .
The OLS estimator of β1 is Pn
Pn
xi (Yi − Y )
xi yi
i=1
b
= i=1Pn
β1 = Pn
2
2
j=1 xj
j=1 xj
Pn
P
xi Yi Y ni=1 xi
i=1
= Pn
− Pn
2
2
j=1 xj
j=1 xj
Pn
xi Yi
= Pi=1
.
n
2
j=1 xj
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 3 / 64 Unbiasedness of βb1
Yi = β0 + β1 Xi + εi ,
Pn
Pn
xi (β0 + β1 Xi + εi )
xi Yi
i=1
b
= i=1 Pn
β1 = Pn
2
2
j=1 xj
j=1 xj
Pn
Pn
Pn
xi
xi εi
xi (xi + X)
i=1
i=1
Pn
= β0 Pn
+ β1
+ Pi=1
n
2
2
2
xj
j=1 xj
j=1 xj
Pn j=1
xi εi
.
= 0 + β1 (1 + 0) + Pi=1
n
2
j=1 xj Substituting So, βb1 is biased if
E Dr. Z.Q. Zhang ([email protected]) !
Pn
x
ε
i
i
Pi=1
6= 0.
n
2
j=1 xj
STAT8007 2019-20 1st Semester 4 / 64 Consistency of βb1
Notice that under some weak conditions, Pn
plimn→∞
where Q j=1 n x2j = Q, is a non-zero constant, and plim stands for converges in probability.
If E(xi εi ) = 0 for all i, then Pn
plimn→∞
Consequently, βb1 i=1 xi εi n = E(xi εi ) = 0. is consistent: Pn
xi εi /n)
0
b1 ) = β1 +
Pi=1
plim(β
= β1 + = β1 .
n
2
plim(
Q
j=1 xj /n)
plim( Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 5 / 64 Troublesome variable
However, βb1 is inconsistent if E(xi εi ) 6= 0. How reasonable an assumption is E(Xi εi ) = 0 (i.e. Xi and εi are uncorrelated)? Unfortunately, this assumption is often
terrible in practice.
When an predictor is correlated with the error term, i.e., E(Xi εi ) 6= 0, we call the predictor a Dr. Z.Q. Zhang ([email protected]) STAT8007 troublesome variable. 2019-20 1st Semester 6 / 64 Omitted variables
4.2 Sources of Troublesome Variables 1. Omitted variables: wrongly omitted predictor(s) from a true model which is (are) contemporaneously correlated with
other predictor(s).
As an illustrative example, suppose that stochastic predictors, X1 and X2 , Y depends on two and the true model is Yi = β0 + β1 X1i + β2 X2i + εi .
We further assume that both
with ε X1 and X2 (4.2) are uncorrelated (not troublesome). Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 7 / 64 Omitted variables
However, suppose that one predictor, actually regress Y on X1 X2 , is omitted. We only, that is, Yi = α0 + α1 X1i + ηi .
Note that the impact of X1 on Y in Model (4.3), be dierent from its actual value,
Denote the OLSE of
combination of α1 by α
b1 = (4.3) β1 Pn α1 , would in Model (4.2). i=1 w i Yi , a linear Yi s. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 8 / 64 Omitted variables
The unbiasedness requires that α1 ≡ E(b
α1 ) = E n
X !
w i Yi = n
X i=1 = n
X E(wi Yi ) i=1 E(α0 wi + α1 wi X1i + wi ηi ) i=1 = α0 n
X E (wi ) + α1 i=1 n
X E (wi X1i ) + n
X i=1 E(wi ηi ), (4.4) i=1 which indicates that n
X E (wi ) = 0, i=1
Dr. Z.Q. Zhang ([email protected]) n
X E (wi X1i ) = 1, i=1 and n
X E(wi ηi ) = 0.(4.5) i=1
STAT8007 2019-20 1st Semester 9 / 64 Omitted variables
Compared with the true coecient of E(b
α1 ) = E n
X !
w i Yi = i=1 = n
X n
X X1i , E(wi Yi ) i=1 E[wi (β0 + β1 X1i + β2 X2i + εi )] i=1 = β0 n
X E (wi ) + β1 i=1 = β1 +β2 n
X E (wi X1i ) + β2 i=1
n
X E (wi X2i ) n
X E (wi X2i ) i=1
by Eq. (4.5). i=1
The last term in the above equation,
called the β2 Pn i=1 E(wi X2i ), is omitted variables bias (OVB). Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 10 / 64 Measurement error
2. Measurement error: mis-measurement of predictors. Measurement error might arise because:
The data are awed. For example, survey respondents
mis-remember how long they've worked at their current jobs).
The data/variables are mis-used. The economist does not
properly understand the nature of X and uses an imperfect proxy. For example, using total taxable income as a measure
of total income. Notice that mis-measuring Y does NOT lead to measurement error bias, though it does increase the variance of the error
term, and thus increases standard error.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 11 / 64 Measurement error
Suppose we are interested in a simple linear regression model Y i = β0 + β1 X i + εi ,
where X is uncorrelated with Suppose that the predictor M = X + ν, X ε (not troublesome). is mis-measured as in which the measurement error uncorrelated with X or ε. ν is The actual model becomes Yi = α0 + α1 Mi + εi .
α1 is then
P
P
mi y i
mi Yi
α
b1 = P 2 = P 2 .
mi
mi The OLS estimator of Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 12 / 64 Measurement error
Yi = β0 + β1 Xi + εi and mi = xi + νi in it, we
P
P
P
mi (β0 + β1 Xi + εi )
mi xi
mi εi
P 2
α
bi =
= β1 P 2 + P 2
m
mi
mi
P
P
P 2 iP
x i νi
xi εi
νi ε i
x
= β1 P i2 + P 2 + P 2 + P 2 .
mi
mi
mi
mi Substitute have Dividing all numerators and denominators by the sample size n, and letting n → ∞, we obtain plimn→∞ α
b1
where σx2 and σν2 Dr. Z.Q. Zhang ([email protected]) = β1 σx2
σx2
=
β
.
1 2
2
σm
σx + σν2 are variances of
STAT8007 X and ν, respectively. 2019-20 1st Semester 13 / 64 Measurement error
The asymptotic bias is then Asym.Bias(b
α1 ) = plimn→∞ (b
α1 − β1 ) = −β1 σν2
.
σx2 + σν2 The relative bias is
Bias(b
α1 ) β1
This is a bias ≈− σν2
< 0,
σx2 + σν2 towards 0 always. It is an (4.6) attenuation bias. A small amount of random measurement noise νi will not bias the estimate very much. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 14 / 64 Measurement error
Example 4.1: Friedman hypothesized that consumption Yi permanent is the average level of consumption for a household with a given permanent income Xi . To accommodate this, he would like to estimate Yi = βXi + εi . (4.7) However, the permanent income is unobservable in practice
and is usually replaced by current income Mi in the regression model (4.7).
Consequently, the OLS estimator βb suers a measurement 2 error bias. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 15 / 64 Measurement error
Example 4.2: Suppose you are advising policy makers on the eect of a one-time tax rebate on consumption.
Using a cross-section of 50,000 households, you regressed
reported current consumption against reported current
income.
Suppose your results found that the marginal propensity
to consume was too small to justify the tax rebate. In
other words, your results suggest that the tax rebate in
question will NOT have a large enough impact on
consumption to justify the policy. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 16 / 64 Measurement error
However, a proponent of the measure argues that you
should be regressing consumption against Friedman's
permanent income instead of current income. Assess this argument: If your regression suers from measurement error, then the true eect of the tax rebate
on consumption is larger than what you found. The
coecient with attenuation bias is too small to justify the
policy. The larger, true coecient might or might not be 2
large enough to support the tax rebate. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 17 / 64 Simultaneous causality
3. Simultaneous causality, also called jointly determined
variables or endogenous variables, means that both X
and Y are simultaneously and jointly determined by some common factor(s).
The classic example of simultaneous causality in economics is supply and demand. Both prices and quantities adjust until supply and demand are in equilibrium. A shock to
demand or supply causes both prices and quantities to move.
Thus, any attempt to estimate the relationship between
prices and quantities (say, to estimate the demand elasticity)
suers from simultaneous causality.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 18 / 64 Simultaneous causality
Example 4.3: Demands and supplies for wheat. The quantity demanded for wheat is a function of the price
consumers pay and the income of the population: D
D
QD
i = β0 + β1 Pi + β2 Ii + εi .
As we expected, β1 < 0 and β2 > 0. The quantity of wheat supplied is a function of the price
suppliers receive and the weather (which aects crop yields): QSi = α0 + α1 PiS + α2 Wi + εSi ,
where α1 > 0 as expected. In equilibrium, S
QD
i = Qi and PiD = PiS . Let's focus on the demand equation.
Dr. Z.Q. Zhang ([email protected]) STAT8007 Is PiD correlated with εD
i ?
2019-20 1st Semester 19 / 64 Simultaneous causality
Suppose that there is a positive shock to demand,
This shock makes QD
i εD
i > 0. greater than usual. S
S
QD
i = Qi . Qi has to be greater
S
supply equation, Pi must increase. In equilibrium, also. To balance the Suppliers must be paid a higher price to supply the greater demanded
quantity.
In equilibrium, PiD = PiS . The consumers must pay a higher price to enjoy the higher quantity of wheat they demand. Thus, D
D D
a positive shock εD
i induces a higher Pi , E(Pi εi ) > 0. The demand shock and the price are positively correlated. 2 OLS will be inconsistent.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 20 / 64 Endogenous and Exogenous variables
When we have a system of equation(s) (as with supply and
demand), all the variables that are jointly determined or
aected by a same factor or factors are called variables. endogenous Price and quantity are endogenous variables, they are jointly
aected by some common shocks. Variables that are determined outside the system of
equations are called exogenous variables. The weather is an exogenous variable.
The population's income is also exogenous. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 21 / 64 Lagged dependent variables
4. Using k > 0) lagged dependent variables (e.g., Yt−k
as predictors is another for some potential source of correlation between a predictor (predictors) and the error term.
NO troublesome variable exists in the absence of serial
correlation in {Yt }. However, lagged dependent variables present a problem in the presence of serial correlation. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 22 / 64 Lagged dependent variables
For example, consider the following regression model with
AR(1) disturbances: Yt = α0 + α1 Yt−1 + εt ,
From the regression equation,
Especially, Yt−1 depends on Yt εt = φεt−1 + νt .
depends on for all t. εt−1 . From the equation for disturbances,
Therefore, both the predictor
jointly aected by the shock Dr. Z.Q. Zhang ([email protected]) εt Yt−1 εt depends on εt−1 . and the response Yt , are εt−1 . They are endogenous. STAT8007 2019-20 1st Semester 23 / 64 Instrumental variable
4.3 Instrumental Variable Estimation When there is (are) a (some) troublesome variable(s) in a
regression model, we can use a (some) instrumental
variable(s) to tease out the trouble part(s) (correlated with disturbance) of the troublesome variable(s).
An instrumental variable, or, an instrument: Z should be a variable that
(i) contemporaneously correlated with the troublesome variable X : Corr(Z, X) 6= 0;
(ii) contemporaneously uncorrelated with the disturbance ε, E(Zε) = 0.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 24 / 64 Instrumental variable
Example 4.4: When the economist is worried about measurement error, a good choice of instrument is simply a dierent measure of the same variable. The new measure may have its own errors, but these errors
are unlikely to be correlated with the mistakes in the rst
measure, or with any other component of ε. For example, Ashenfelter and Rouse were studying the eect
of education on earnings.
Their data (saved in twins.csv) came from a survey of twins. They were concerned that individuals might mis-report their
own years of schooling, leading to measurement error biases.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 25 / 64 Instrumental variable
However, Ashenfelter and Rouse had two separate measures
for each individual's years of schooling.
The survey asked each individual to list both his/her own
years of schooling, and also the years of schooling for his/her
twin.
The twin's report of an individual's schooling served as an
instrumental variable for the individual's self-report. 2 Instrumental variables are not always as apparent as in
Example 4.4. Working out reasonable instruments needs deep
understanding of natural of the problem.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 26 / 64 Instrumental Variable Estimation
Consider a simple LIM: Y = β0 + β1 X + ε,
where X is a troublesome variable: The OLS estimator of the slope β1 (4.8) Corr(X, ε) 6= 0.
is x0 y
βb1OLS = 0 ,
xx
where x = (x1 , · · · , xn )0 and (4.9) y = (y1 , · · · , yn )0 are vectors of deviated observations. βb1OLS is biased and inconsistent. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 27 / 64 Instrumental Variable Estimation
Suppose that Z is an instrumental variable for Corr(Z, X) 6= 0,
We may use and X, that is, E(Z, ε) = 0. Z to tease out the troublesome part of X through conditional expectation. That is, to transform X into e = E(X|Z).
X
After the conditional expectation, or, the
transformation, e
X correlated with Z. Since Z can be regarded as part of is uncorrelated with uncorrelated with
Dr. Z.Q. Zhang ([email protected]) ltering ε too. e
X e
ε, X is the STAT8007 X that is fully is (expected to be) clean part of X .
2019-20 1st Semester 28 / 64 Instrumental Variable Estimation
Transformation of Model (4.8) (in deviations forms): 1 ≤ i ≤ n. E(y|z) = β1 E(x|z) + E(ε|Z), Conditional expectations can be estimated by (4.10) projections. That is, \ = z 0 y,
E(y|z)
where \ = z 0 x,
E(x|z) \ = z 0 ε,
E(ε|z) z = (z1 , · · · , zn )0 . Note that z0ε ≈ 0 estimator for because E(εZ) = 0. We may get a new β:
\
E(y|z)
z0y
βb1IV =
= 0 .
zx
\
E(x|z) Dr. Z.Q. Zhang ([email protected]) STAT8007 (4.11) 2019-20 1st Semester 29 / 64 Instrumental Variable Estimation
Estimator βb1IV is called the estimator.
Compare the IV estimator
in Eq. (4.9): all
Substituting x0 (not x) y = β1 x + ε instrumental variable (IV)
βb1IV with the OLS estimator are replaced with βb1OLS z0. into Eq. (4.11), we have z0ε
βb1IV = β1 + 0 .
zx
It can be immediately seen that the IV estimator βb1IV is asymptotically unbiased and consistent. Remark: the instrument itself is not (and cannot be) used as a predictor of the regression.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 30 / 64 Instrumental Variable Estimation
The previous IV estimation approach can be generalized to a
multiple LIM model with p predictors: Y (n×1) = X (n×(p+1)) β ((p+1)×1) + ε(n×1) .
Suppose that the last two predictors,
troublesome variables, while
the constant column of
Let
an Z p−1 X) X p−1 and (4.12) X p, are X j , 0 ≤ j ≤ p − 2 (X 0 = 1 is are non-troublesome. be an instrument for X p−1 , and Zp for X p. Dene instrumental matrix
Z (n×(p+1)) = (1, X 1 , · · · , X p−2 , Z p−1 , Z p ). Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester (4.13)
31 / 64 Instrumental Variable Estimator
The IV estimator is then b IV = (Z 0 X)−1 Z 0 Y .
β
Again, replace
Again, b IV
β X0 in (4.14) b OLS = (X 0 X)−1 X 0 Y
β with Z 0. is asymptotically unbiased and consistent. Finally, both the t-tests and the F -tests can be applied for hypothesis testing. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 32 / 64 Model identication
We will not always be so happy to have one instrument for
one troublesome variable, for which we say that the model is exactly identied.
In case that we have fewer instrument variables than
troublesome variables, the model is said to be identied. under As an example, for Model (4.12), if we have only one
instrument, Z p, but don't have Z p−1 , then the model is under identied. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 33 / 64 Model identication
In this case, we can only replace Xp by Z p, but remain un-replaced in the denition of instrumental matrix
(4.13), to Z, X p−1
Eq. partially correct the inconsistency. The resulted IV estimator b IV
β would be still inconsistent. If we have more instrument variables than troublesome
variables, and each troublesome variable has at least one
instrument, we say that the model is over identied. Use Model (4.12) again as the example. Suppose that we
have one more instrument Dr. Z.Q. Zhang ([email protected]) Z p+1 STAT8007 (for either X p−1 of X p ). 2019-20 1st Semester 34 / 64 Model identication
To suciently make use of instrumental information, we may
add Z p+1 to the instrumental matrix to have Z (n×(p+2)) = (1, X 1 , · · · , X p−2 , Z p−1 , Z p , Z p+1 ).
The consistent IV estimator becomes a (4.15) generalized method of moments (GMM) estimator: b GM M = (X 0 P Z X)−1 X 0 P Z Y ,
β
where PZ is a projection matrix (P 0Z = P Z and (4.16) P 2Z = P Z ): P Z(n×n) = Z(Z 0 Z)−1 Z 0 .
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester (4.17)
35 / 64 Two-Stage Least Squares (2SLS)
When the regression equation is exactly identied or over identied, we can apply IV estimation. In practice, however, econometricians use a slightly simpler use newly constructed instruments to
replace the predictors in OLS. procedure. They This strategy requires a two-stage process, and is called Two-Stage Least Squares (2SLS or TSLS).
To introduce two steps of 2SLS, we consider the multiple
LIM model (4.12) as an example, and suppose that the model
is over identied with an instrumental matrix dened by Eq.
(4.15).
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 36 / 64 Two-Stage Least Squares (2SLS)
Stage 1: Construct new instruments.
Regress each column of X on X j = Zδ j + η j , Z:
j = 0, 1, · · · , p. The OLS estimators are bj = (Z 0 Z)−1 Z 0 X j ,
δ j = 0, 1, · · · , p. The predicted values (new instruments) are bj = Z(Z 0 Z)−1 Z 0 X j ,
cj = Z δ
X
Write all cj 's
X j = 0, 1, · · · , p. into a matrix: c = Z(Z 0 Z)−1 Z 0 X = P Z X.
X
Remark: Actually,
Dr. Z.Q. Zhang ([email protected]) cj = X j
X
STAT8007 for j = 0, 1, · · · , p − 2.
2019-20 1st Semester 37 / 64 Two-Stage Least Squares (2SLS)
Stage 2: Estimate β.
Regress Y on the newly constructed instruments c:
X c + ξ.
Y = Xβ
The 2SLS estimator of β is then the OLS estimator of β in this stage: b 2SLS = (X
c0 X)
c −1 X
c0 Y
β
= [(P Z X)0 P Z X]−1 (P Z X)0 Y
= (X 0 P 0Z P Z X)−1 X 0 P 0Z Y
b
= (X 0 P Z X)−1 X 0 P Z Y = β GM M . Remark: Notice that in Stage 2 of 2SLS, the errors (4.18) ξ 6= ε. Hence, the e.s.e.'s are neither the OLS ones nor the IV ones, t-tests fail to work here. Fortunately, F -test (without calculating the e.s.e.'s) still apply.
Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 38 / 64 Hausman's specication test
Both IV and 2SLS estimations are somewhat complicated
procedures, requiring us to nd enough valid instruments to
identify the regression equation.
Furthermore, unless the instrument
variable X Z and the troublesome are highly correlated, using instrumental variables will increase the variance of the estimator.
We would prefer using OLS when we can. OLS is consistent
(and asymptotically ecient) when there is no troublesome
variable, even that predictor(s) is (are) random. Dr. Z.Q. Zhang ([email protected]) STAT8007 2019-20 1st Semester 39 / 64 Hausman's specication test
Hausman proposed a test, referred to as Hausman's specication test, for the null hypothesis
H0 : No troublesome vari...

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