ECONOMETRICS
LECTURE 5A
Trang Le
LECTURE SLIDES TOPICS
 OLS asymptotics
 Data scaling
 More on functional form
 Average Partial Effects
 Bad Controls
 Key references
 Wooldridge Chs 5.1-5.2 (excluding 5.2a), Ch 6.1 and 6.2
2
OLS PROPERTIES - OVERVIEW
 So far properties derived for OLS hold for any sample
size – finite sample properties
 Expected values/unbiasedness under MLR.1 – MLR.4
 Variance formulas under MLR.1 – MLR.5
 Gauss-Markov Theorem under MLR.1 – MLR.5
 Exact sampling distributions/tests under MLR.1 – MLR.6
 Properties of OLS that hold in large samples –
asymptotic properties
 Consistency under MLR.1 – MLR.4
 Asymptotic normality/tests under MLR.1 – MLR.5
 MLR.6 no longer required
3
WHAT ARE ASYMPTOTIC PROPERTIES?
 Remember the population line (image this image has
infinite data points)
4
Population regression line
1 = −1.5
WHAT ARE ASYMPTOTIC PROPERTIES?
 With very few observations an “unlucky“ draw can lead
to very bad estimation
5
WHAT ARE ASYMPTOTIC PROPERTIES?
 As the sample gets bigger even with an unlucky draw we
are never so far away from the real parameters
6
WHAT ARE ASYMPTOTIC PROPERTIES?
 In statistical terms this means that as the sample gets
bigger the variance of the OLS estimates gets smaller
(̂1)=3 > (̂1)=6
 We can see it mathematically:
̂1 = 2∑=1 ( − ̅)2
The noise remains the same but the signal increases
7
CONSISTENCY
 Informal definition:
 If we continue increasing the sample can we get to the point
where (̂1)=∞ = 0 and so ̂1 is no longer a random
variable but is just equal to 1.
 Formal definitionPr � − < → 1 > 0 → ∞
 Alternative shorthand notation: � =
 Estimator converges in probability to the true value
 Interpretation:
 Consistency probability that estimate is arbitrarily close to
true population value can be made arbitrarily high by
increasing sample size 8
CONSISTENCY...
9
CONSISTENCY OF OLS
 Theorem 5.1 (Consistency of OLS)
− ⇒ ̂ = , = 0,1, … ,
 Special case of simple regression model (can show)
̂1 = 1 + 1, 1
 1, & 1 are population quantities
 As 1 = 0 ⇒ 1, = 0 ⇒ ̂1 is a consistent
estimator
10
ASYMPTOTIC NORMALITY
 In practice, normality assumption MLR.6 often
questionable
 If MLR.6 does not hold results of t or F tests may be wrong
 Fortunately OLS estimates are normal in large samples
even without MLR.6
 Usual t or F tests still work approximately if sample size is
large enough
 Theorem 5.2 (Asymptotic normality of OLS)
 Under MLR.1-MLR.5
̂ −
̂
~ 0,1
�2 = 2 11
ASYMPTOTIC NORMALITY...
 Practical consequences
 Construct t-statistics as before
 In large samples t-tests are valid irrespective of whether
MLR.6 holds or not
 Even when MLR.6 holds t-distribution is close to N(0,1)
distribution in large samples
 Similarly proceed as before with confidence intervals & F-tests
 Note: MLR.1 – MLR.5 are still necessary
 Especially homoskedasticity
12
MODELLING INFANT BIRTHWEIGHT – SMALL SAMPLE
13
 Relate birthweight to cigarette smoking & family
income
�= 116.97(1.05)− .63.092 + .093.029
= 1,388,2 = .0298
 Does distribution of look normal?
 Does it matter for inference here?
 What happens if use only first 694 obs?
�= 116.65(1.5)− .58.39 + .09.02
= 69,2 = .039
0
.0
05
.0
1
.0
15
.0
2
D
en
si
ty
0 100 200 300
birth weight, ounces
MODELLING INFANT BIRTHWEIGHT - UNITS
 What happens if birthweight is converted from ounces to
grams? (Hint: 1 ≈ 28 )
�= 116.97(1.05)− .63.092 + .093.029
= 1,388,2 = .0298
= −5.06
 Compared with
�= 3275.3(29.)− 12.982.56 + 2.60.82
= 1,388,2 = .0298
= −5.06
 What if change of units occurred when had specified log = 0 + 1 + 2 + ? 14
MORE ON FUNCTIONAL FORM: LOGS
 Convenient percentage/elasticity interpretation
 Slope coefficients of logged variables are invariant to
rescalings
 Taking logs often useful
 Eliminates/mitigates problems with outliers
 Helps to secure normality & homoskedasticity
 Variables that should not be logged
 Those measured in units such as years or percentage points
 Those that take on zero or negative values
15
MORE ON FUNCTIONAL FORM: QUADRATICS
 Consider wage equation
�= 3.73(.35)+ .298.0 − .00612.0009
= 526,2 = .093
 Marginal effect of experience
∆
∆
= .298 − 2 .0061
 Marginal effect depends on level of experience
 It makes no sense to evaluate each of the estimates
separately - what would be the ceteris paribus assumption?
16
MORE ON FUNCTIONAL FORM: QUADRATICS...
17
 Specification allows either convex
(U shaped) or concave (inverted U)
relationship
 Estimates indicate concavity
 Wage maximum (turning point) wrt
experience
∗ = ̂12̂2 = 0.2982(.0061)
≈ 2.
 Does this mean return to
experience becomes negative
after 24.4 years?
 Reasonable approximation or
indication of mis-specification?
MORE ON FUNCTIONAL FORM: INTERACTIONS
 Consider following housing price model
= 0 + 1 + 2 + 3 ∗ + 4 +
⇒
∆
∆
= 2 + 3
 Similar to quadratics, interaction terms complicate
interpretation of parameters
 Effect of number of bedrooms depends on level of square
footage
 2 represents effect when = 0
 1 represents effect when = 0
 3 extra effect of on price for each added
18
MORE ON FUNCTIONAL FORM: INTERACTIONS
Similarly you can say that: ∆
∆
= 1 + 3
 Example: House with 2 bedrooms
What is the effect of increasing the size of the house by 1?
∆
∆
= 1 + 23
 Example: House with 3 bedrooms
What is the effect of increasing the size of the house by 1?
∆
∆
= 1 + 33 19
AVERAGE PARTIAL EFFECTS
 In many models size of the effect depends on values of
one or more explanatory variables
 Average partial effect (APE) provides a summary measure
 APE is the effect calculate for the average regressor
 A reporting issue so no right or wrong approach
 Could also report partial effect at a particular value
 Most importantly be clear in what you’re reporting
20
AVERAGE PARTIAL EFFECTS...
 Recall interaction example
= 0 + 11 + 22 + 31 ∗ 2 +
 Is effect of 2 on when 1 = 0 (i.e. 2) of interest?
 Reparametrization may be useful in order to directly
estimate interesting partial effects
= 0 + 11 + 22 + 3(1−1) ∗ (2 − 2) +
⇒
∆
∆2
= 2 + 3(1−1)
 Now 2is effect of 2 on when 1 = 1 & we choose 1
 1 could be sample mean of 1 ⇒ 2 is APE for this example 21
AVERAGE PARTIAL EFFECTS...
 Advantages of reparametrization
 Easy interpretation of parameters
 Choice of 1 at discretion of researcher – could be any
interesting value
 Before & after reparametrization are not different
models
 Simply isolating different parameter combinations to estimate
 From previous example 2 = 2 + 31
 Easy to estimate using original model but calculation of se 2 less immediate
22
OVERCONTROLLING AND BAD CONTROLS
 One way of dealing with omitted variable bias is to
control for many observables.
 But is possible to control for too may variables
 Example: Let our objective be to find what is the effect of
beer taxes on traffic fatalities
 Two important details:
1. This question is about causality (not goodness of fit)
2. We hypothesize that the causal links are:
→ →
23
OVERCONTROLLING AND BAD CONTROLS
 Data on different cities in Australia (with different tax
rates on beer)
 Two econometric models:
= 0 + 1 +
= 0 + 1 + 2 +
Let say that our hypothesis is true then (and ASMPT 1-4):1. 1<0 (the effect of tax)
2. But 1 = 0 (and 2>0 ). Because the policy has and
effect through beer consumption 24
OVERCONTROLLING AND BAD CONTROLS
 We call a bad control
 Given that our objective is to find the effect of the policy
we shouldn’t control for (don’t control for
mechanisms)
 We can control for if instead our objective is to
predict
25
PREDICTION
 Generating predictions from multiple regression models is
straightforward
 In general case where
= 0 + 11 + ⋯+ +
 And want to predict for 1 = 1, … , = then
0 = 1 = 1, … , = = 0 + 11 + ⋯+
�0 = ̂0 + ̂11 + ⋯+ ̂
 What is less clear is ( �0)
 But our reparametrization approach helps again
 0 is a linear function of parameters – see Week #4 26
PREDICTION…
 Our reparameterization approach helps again. Define
θ0 = 0 + 11 + ⋯+
 Then 0 = θ0 − 11 −⋯− , and we could rewrite
the original model as
= θ0 + 1(1 − 1) + ⋯+ ( − ) +
 Regression of on �1 = 1 − 1, … , � = − would
provide estimates of θ0 and , = 1, … ,
 Most importantly would provide ( �0) 27
FINAL THOUGHTS
 Have dealt with asymptotic theory in a somewhat
superficial manner
 Details are somewhat complicated
 BUT important implications in practice that are clear
 This is all the material that will be covered in the midterm