Econometrics: L2
Multiple Regression Model
Sung Y. Park
Chung-Ang Univ.
Motivation
Consider the following example:
wage = β0 + β1educ + β2exper + u,
where exper is years of labor market experience.
wage is determined by the two independent variables
β1: the effect of educ on wage holding all other factors affecting
wage (our interest)
just as with simple regression, we make assumptions about how u is
related with independent variables
Note that we have to assume educ and exper are uncorrelated in
the simple regression (why )
Motivation
A model with two independent variables:
y = β0 + β1×1 + β2×2 + u,
β0: the intercept
β1: measures the change in y with respect to x1, holding
other factors fixed
β2: measures the change in y with respect to x1, holding
other factors fixed
Note: In Economics, this is used a lot: ceteris paribus, keeping all
else constant.
Motivation
Quadratic functional relationships:
cons = β0 + β1inc + β2inc
2 + u,
note that consumption depends on only income x1 = inc
and x2 = inc
2
here, is β1 the ceteris paribus effect of inc on cons No!
(why )
the change in consumption with respect to the change in
income (the marginal propensity to consume)
cons
inc
～ β1 + 2β2inc.
Motivation
Key Assumption:
E (u|x1, x2) = 0
for any values of x1 and x2, the average unobservable is equal
to zero
note that the common value 0 is not crucial as long as β0 is
included in the model
E (u|educ, exper) = 0 other factors affecting wage are not
related on average to educ and exper
E (u|inc, inc2) = 0 E (u|inc) = 0 (inc2 is redundant)
Motivation
More generally…
y = β0 + β1×1 + β2×2 + β3×3 + · · ·+ βkxk + u,
there are k independent variables in the model but k + 1
parameters
the parameters other than the intercept (β0): slope
parameters
u: error term of disturbance
Motivation
An example:
log(salary) = β0 + β1 log(sales) + β2ceoten + β3ceoten
2 + u,
β1 : ceteris paribus elasticity of salary with respect to sales
if β3 = 0, 100β2: the ceteris paribus percentage increase in
salary when ceotan increases by one year
if β3 6= 0, the effect of ceoten on salary is more complicated
(later)
note that it is nonlinear relationship between salary and sales
but linear in βj
OLS estimates
OLS estimates: Choose, simultaneously, β 0, β 1 and β 2 that make
n∑
i=1
(yi β 0 β 1xi1 β 2xi2)
2
as small as possible.
first index i refers to the observation number
second index represents different independent variable
xij : the i-th observation on the j-th independent variable.
OLS estimates
General case:
min
β0,β1,··· ,βk
n∑
i=1
(yi β0 β1xi1 β2xi2 · · · βkxik)
2
leads to k + 1 linear equations in k + 1 unknown β 0, β 1, · · · , β k :
First order conditions:
n∑
i=1
(yi β 0 β 1xi1 β 2xi2 · · · β kxik) = 0
n∑
i=1
xi1(yi β 0 β 1xi1 β 2xi2 · · · β kxik) = 0
…
n∑
i=1
xik(yi β 0 β 1xi1 β 2xi2 · · · β kxik) = 0
OLS estimates
Interpretation:
y = β 0 + β 1×1 + β 2×2
β 1 and β 2 have partial effect (ceteris paribus) interpretations
From the above equation we can get the predicted change in y
given x1 and x2
y = β 1 x1 + β 2 2
when x2 is fixed ( x2 = 0)
y = β 1 x1.
when x1 is fixed ( x1 = 0)
y = β 2 x2.
OLS estimates : “Partialling out”
Interesting Formulas:
y = β 0 + β 1×1 + β 2×2
