Palgrave Handbook of Econometrics: Applied Econometrics

John DiNardo 133

Consider a case where the true state of the world can be characterized simply by
the following:

y=β 0 +β 1 T+β 2 X+ , (3.10)

where, for simplicity, theβare unknown parameters, theXare things that “cause”
yand are observed, are things that “cause”ybut are not observed, andT=
1 (received treatment).
For the non-Bayesian, one of the benefits of randomization is that theXvariables
available are usually very inadequate. Also is some convolution of omitted vari-
ables and functional form misspecification: it is not generally plausible to make
a statement like “ follows the normal distribution,” although statements like
that are often found in the literature. Hence, though one could write down a
“likelihood,” it isn’t necessary for the non-Bayesian.
A caricature might make this clear: it isnotthe case that “on the first day, God
createdyand made it a linear deterministic function ofTandX; on the second day,
in order to make work for econometricians, God appended a normally distributed
error term with mean 0.”
Indeed, in a randomized controlled trial (RCT), when the experimenter can inter-
vene and assignTrandomly, the “model” the experimenter estimates is often
much less complicated:

y = β 0 +β 1 T+. (3.11)

For purposes of estimation onecouldwrite down a normal likelihood for this
model:

y=

1

σ

√

2 π

exp

(

−

yi−β 0 −β 1 Ti

2 σ^2

)

. (3.12)

With this likelihood, one could then specify prior beliefs about the fixed parameters
β 0 andβ 1 , stipulate the form of heterokedasticity (that is, that the variance of
was a constant for all observations, or model the heteroskedasticity), and so on.
After seeing the data a Bayesian could update his/her beliefs about the values of
these two parameters. Note that, in this formulation, there appears to be nothing
special about the likelihood to distinguish it from any other comparison of means

• nothing tells us, for example, thatTwas assigned randomly.
  Nonetheless, writing down the likelihood seems a bit bizarre for the non-
  Bayesian. For example, if the treatment,T, was a nicotine patch andywas some
  outcome like “quit smoking successfully,” no one thinks that only the patch mat-
  ters and nothing else that can be observed matters – clearly the price of cigarettes,
  social norms, and so on, play a role. Indeed, available covariates are usually not
  used except to “test” the validity of the design. Specifically, in repeated samples:

E[y 1 ]−E[y 0 ]=β 1 (3.13)

E[X 1 ]−E[X 0 ]= 0 (3.14)

E[ 

1 ]−E[ 

0 ]=0, (3.15)