2.4 Units of Measurement and Functional Form
-Two important econometric issues are:
1) Changing measurement
-When does scaling variables have an effect on
OLS estimates?
-When does scaling variables have no effect on
OLS estimates?
2) Functional Forms
-How do natural logs affect linear regressions?
-How do functional forms impact elasticity?
2.4 Units of Measurement and Functional Form
• Consider the following model:
squiˆrrels i  500  0.5treesi
(ie)
-where the number of squirrels and trees are
in single units
-if trees=2000 then the predicted number of
squirrels becomes 1500
-if there are no trees, there are 500 squirrels
-how does this change if squirrels are
measured in hundreds (ie: divided by 100)?
2.4 Units of Measurement and Functional Form
-If squirrels are measured in hundreds, then
zero trees would produce 500 squirrels, so
B0hat=5 (divided by 100)
-If there are 2,000 trees, then B1hat*2000
must equal 10 (15-5, or 1,000 squirrels)
-Therefore B1hat=0.005 (divided by 100)
squiˆrrels i  5  0.005treesi
(ie)
2.4 Acorns are for the squirrels
Therefore
When the dependent variable is
multiplied (divided) by a constant…
Multiply (divide) the OLS intercept
and slope by the same constant.
IE: Change the y – change OLS the
same way.
2.4 Units of Measurement and Functional Form
• Consider the following model:
treehuˆggersi  500  0.5treesi
(ie)
-where the number of treehuggers and trees
are in single units
-if trees=800 then the predicted number of
treehuggers becomes 900
-if there are no trees, there are 500
treehuggers
-how does this change if trees are measured
in hundreds (ie: divided by 100)?
2.4 Units of Measurement and Functional Form
-If trees are measured in hundreds, then
zero trees would produce 500 treehuggers,
so B0hat=500 (nothing has changed)
-If there are 800 trees, then trees=8 and
8(B1hat) must equal 400 (900-500)
-Therefore B1hat=50 (multiplied by 100)
squiˆrrels i  500  50treesi
(ie)
2.4 Acorns are for the squirrels
Therefore
When the independent variable is
multiplied (divided) by a constant…
Divide (multiply) the OLS slope (not
intercept) by the same constant.
IE: Change the x – change OLS
slope the opposite way.
2.4 Units of Measurement and Functional Form
How does R2 (goodness of fit) change when a
variable is scaled?
-It doesn’t
-R2 calculates how much of the variation
in y is explained by x
-this doesn’t depend on scaling
-a similar “best fit line” is drawn through
data points, regardless of scaling
2.4 Functional Form
Thus far, we have focused on LINEAR
relationships
-Linear relationships don’t capture all of
the possible interaction between variables
-Linear relationships assume that the first
x has the same impact on y as the last x
-changing impacts can be captured
through the use of NATURAL
LOGARITHMS
2.4 Log-Lin Model
When a variable has an increasing (percentage)
impact on y, the log-lin model is appropriate:
log( y)   0  1 x  u
(2.42)
-note that log(y) indicates the natural log of y
-if we assume that u doesn’t change,
%y  (1001 )x
(2.43)
-note as x increases, y increases, therefore this
equation expresses INCREASING return
2.4 Log-Lin Model
Assume that absence does make the
heart grow fonder:
log( fondness)  3  0.5absence  u
(i.e)
-assuming (for simplicity) that u=0, 2 days
absence causes fondness of e4 (54.6) while 10
day’s absence causes fondness of e8 (2,981)
-therefore, given another day of absence:
%fondness  (1001 )x  50%
(2.43)
2.4 Lin-Log Model
Therefore:
-the 3rd day of absence increases
fondness by 27.3
-the 11th day of absence increases
fondness by 1,490.5
-we have INCREASING RETURNS
-Note: a Log-lin model can also be
expressed:
(  0  1 x  u )
ye
2.4 Log-Log Model
Recall from Econ 299 that elasticity is
calculated as:
 ln( y )

 ln( x)
-if constant elasticity is theoretically important to
a model, a log-log functional form ensures that
elasticity is constant as B1:
log( y)   0  1 log( x)  u
2.4 Scaling and Dependent Logs
• Consider a Log-Lin model where the y value is
multiplied by c:
log( yi )   0  1 xi  ui
log( yi )  log( c)   0  1 xi  ui  log( c)
log( cyi )  [  0  log( c)]  1 xi  ui
log( cyi )   0  1 xi  ui
-scaling a dependent variable in log form
changes the intercept but does not affect the
slope
2.4 Units of Measurement and Functional Form
Different Functional Forms are Summarized as Follows:
Model
Function
Interpretation of B1
Lin-Lin
y=f(x)
∆y= B1∆x
Lin-Log
y=f(log(x))
∆y= (B1/100)%∆x
Log-Lin
Log(y)=f(x)
Log-Log
%∆y= 100B1∆x
*also call semi-elasticity
Log(y)=f(log(x)) %∆y= B1%∆x
2.4 Units of Measurement and Functional Form
Notes:
1) Even though non-linear variables are
included in models (ie: log(y) or y2), the
models are still considered “Linear
Regressions” as they are linear in the
parameters B1 and B2
2) Non-linear variables make interpreting B1
and B2 more complicated
3) Some estimated models are NOT linear
regression models
2.5 Expected Values and Variances of
the OLS Estimators
• This section will, using classical Gauss-Markov
Assumptions, find 3 OLS properties:
1) OLS is unbiased
2) Sample Variance of OLS Estimators
3) Estimated Error Variance
This will be done viewing B0hat and B1hat as
estimators of the population model:
y   0  1x  u
Gauss-Markov Assumption SLR.1
(Linear in Parameters)
In the population model, the dependent variable,
y, is related to the independent variable, x, and
the error (or disturbance), u, as
y   0  1 x  u
(2.47)
Where B0 and B1 are the population intercept and
slope parameters, respectively.
Gauss-Markov Assumption SLR.1
(Linear in Parameters)
Notes:
1) In reality, x, y and u are all viewed as
random variables
2) Since OLS needs only be linear in B1 and
B2, SLR.1 is far from restrictive
Given an equation, an assumption must
now be made concerning data
Gauss-Markov Assumption SLR.2
(Random Sampling)
We have a random sample of
size n, {(x,y): i=1,2,…..n},
following the population
model in equation (2.47).
Gauss-Markov Assumption SLR.2
(Random Sample)
We will see in later chapters that random
sampling can fail, especially in time series data
but also in cross-sectional data
-Now that we have a population equation and an
assumption about data, (2.47) can be rewritten as:
yi   0  1 xi  ui , i  1, 2,...., n.
(2.47)
Where ui captures all unobservables for
observation I and differs from uihat
-to estimate B0 and B1 we need a 3rd assumption:
Gauss-Markov Assumption SLR.3
(Sample Variation in the
Explanatory Variable)
Sample outcomes of x,
namely, {xi, u=1,….,n}, are
not all the same value.
Gauss-Markov Assumption SLR.3
(Sample Variation in the
Explanatory Variable)
-This assumption ensures that the denominator
of B1hat is not zero
-This assumption is violated if:
-The variance of x is zero
-The standard deviation of x is zero
-The minimum value of x is equal to the
maximum value
-Although we can now obtain OLS estimates, we
need one more assumption to ensure
unbiasedness
Gauss-Markov Assumption SLR.4
(Zero Conditional Mean)
The error u has an expected value of
zero given any value of the
explanatory variable. In other words,
E (u | x)  0
Gauss-Markov Assumption SLR.4
(Zero Conditional Mean)
Given our assumption about random
sampling, we can further conclude:
E (u i | x i )  0 i 1, 2,...n .
-this is read “for all 1=1, 2,….n”
-given SLR.2 and SLR.4, we can derive the
properties of OLS estimators as conditional on
xi’s values
-given these 2 assumptions, nothing is lost in
derivation by assuming xi is nonrandom
2.4 OLS is Unbiased
In order to prove OLS’s unbiasedness, B1hat must
first be algebraically manipulated:
ˆ1 
 (x  x)y
 (x  x)
i
i
2
(2.49)
i
-By a familiar mathematical property.
-Substituting out yi and restating the
denominator gives us:
(note that SSTx is not the same as SST)
ˆ1
(x


i
 x)(  0  1x i  u i )
SSTx
(2.50)
2.4 OLS is Unbiased
Using summation properties, the numerator
becomes:
 (x i  x)0   (x i  x)1x i   (x i  x)ui (2.51)
  0  (x i  x)  1  (x i  x)x i   (x i  x)ui
-Which is simplified using the properties:
 ( x  x)  0
i
and
 ( x  x) x   ( x  x)
i
i
i
2
 SSTx
2.4 OLS is Unbiased
Returning to our B1hat estimate, we now have:
ˆ1  1
(x


i
 x)u i
SSTx
1
 1 
SSTx
d u
i
i
(2.52)
-Which indicates that the estimate of B1 equals B1
plus a term that is a linear combination of
errors
-Conditional on values of x, B1hat’s randomness is
due solely to the errors
-Note:
i
i
d x x
Theorem 2.1
(Unbiasedness of OLS)
Using assumptions SLR.1 through SLR.4,
E(ˆo )  0 and E(ˆ1 )  1
(2.53)
for any values of B0 and B1. In other
words, B0hat is unbiased for B0 and
B1hat is unbiased for B1.
Theorem 2.1 Proof
Since expected values are conditional on
samples of x, and SSTx and di are
functions only of xi, they are
nonrandom in conditioning. Therefore:
E ( ˆ1 )  1  E[(1/SST)  d i ui ]
 1  (1/SST)  E(d iui )
 1  (1/SST)  d i E (ui )
 1  (1/SST)  d i 0 1
Theorem 2.1 Proof
-this is proved from the fact that each ui
(conditional on sample x’s) is zero from
SLR.2 and SLR.4
-”since unbiasedness holds for any
outcome on {x1, x2,…,xn}, unbiasedness
also holds without conditioning on {x1,
x2,…,xn}
-unbiasness of B1hat is now
straightforward:
Theorem 2.1 Proof
ˆ0  y  ˆ1 x   0  1 x  u  ˆ1 x
  0  ( 1  ˆ1 ) x  u
E ( ˆ0 )   0  E[( 1  ˆ1 ) x]  E[u ]
   E[(   ˆ ) x]
0
1
1
 0
Since we already proved that
E ( ˆ1 )  1 therefor e E( ˆ1 - 1 )  0
Theorem 2.1 Notes
-Remember that unbiasedness is a
feature of the sample distributions of
B1hat and B2hat
-if we have a poor sample, our OLS
estimates would be far from the true
values
-if any of our 4 initial Gauss-Markov
assumptions are not true, OLS’s
unbiasedness fails
Assumption Failure
-If SLR.1 fails (y and x are not linearly
related), very advanced estimation
methods are needed
-Failure of SLR.2 (random sampling) is
discussed in Chapters 9 and 17
-common in time series and possible
in cross-sectional data
-If SLR.3 fails (x’s are all the same), we
cannot obtain OLS estimates
-If SLR.4 fails, OLS estimators are biased,
which can be corrected
SLR.4 Failure
-If x is correlated with u, we have
spurious correlation
-the relationship between x and y is
influenced by other factors connected
with x
-note that some vague connection is
always possible but not statistically
significant
SLR.4 Failure Example
-Saskatchewan instituted a hypothetical drunk
driving awareness (DDA) campaign as an
alternative to jail time for DUI
-It was found that the relationship between DUI’s
and enrolment in the program is as follows:
ˆ
DUI i  23  0.78DDAi
-Even though the program looks to have failed, it
is due to a spurious correlation:
-the existence of drunk drivers both increases the
number of DUI’s and the enrolment in the
program