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- Assumption A2 about the LRM
- The sample information is generated from the PRE.
- That is, the linear relationship between X and Y is the actual relationship
between the variables in the data set. This is a critical assumption.
If at some point we don't think that we can write down the form of the
relationship between X and Y, then it is difficult to say anything about
the relationship.
- Assumption A3 about the LRM
- The variable X is non-random across
*samples*. - That is, our analysis will
presume that if another sample were taken from the PRE then:
the value
would be the same as before (but
would differ because of a different draw of
);
the value of
would be the same as before
(but
would differ because of a different draw of
);
etc.
We are
**not**assuming that X is constant*within*a sample, but*across*samples. And we are explicitly assuming - When the sample comes from a real experiment, the values of X are under the
control of the researcher. In these cases,
**A3**is not difficult to maintain. For example, recall the example of sampling young Canadians and asking about their sex (obviously exogenous) and the number of cigarettes they smoke (obviously endogenous). Suppose person 1 in the sample is male ( = 0) and person 2 is female ( = 1). If we think about taking a second sample of 600 people, then A3 implies that would again equal 0 and would again equal 1. We are**not**assuming that person 1 is the same person in each sample, but simply the same sex in each sample. Therefore, the endogenous variable, , would differ across samples, but according to A3 the exogenous variable is assumed to be the same.- With most economic data, another sample of data would most likely involve different values of X. For example, if we re-study the issue of minimum wages five years later and used data from the intervening years, we would most likely have new values of X that didn't appear in the earlier sample.
**A3**would then appear to be a difficult to take seriously. But all of our results can be derived from much weaker assumptions. In particular, the actual assumption that we need to make is- Assumption A3*
- See covariance. That is, the important part of
**A3**is not that is the same across samples. If it is the same, then is non-random across samples and a non-random variable (i.e. constant) always have zero covariance with any other random variable, in this case the disturbance term . So**A3**implies**A3***, but the reverse is not true.**A3***is the assumption that really matters for our purposes, but it is much simpler to start with**A3**and think of X as non-random. - With most economic data, another sample of data would most likely involve different values of X. For example, if we re-study the issue of minimum wages five years later and used data from the intervening years, we would most likely have new values of X that didn't appear in the earlier sample.

- Assumption A4
- for
- The disturbance term has mean of 0.
- Assumption A5
- for
- (The disturbance terms in any two observations
have no covariance, and
hence are uncorrelated.)
- Assumption A6
- for
- Each disturbance term has the
*same*variance, an unknown population parameter .)- Assumption A7
- for
- (That is, the disturbance follows the normal distribution for each observation in the sample.)

**A5** and **A6** are more important than **A4**. **A5**
says that the different
observations in the data set do not have statistically related disturbance
terms. Notice, we didn't say "independent," because independence is a stronger assumption than
0 covariance. In fact, independence
between two random variables is very hard to test, but checking or testing
whether the disturbance terms have 0 covariance is not difficult.

**A6** says that each of the disturbance terms has the same variance,
.
It assumes that each observation has equal variance around the PRE. This
means each observation provides the same information (in a sense made clear
later) about where the PRE is located. If, instead, some
observations had lower variances than others,
then the low variance observations tend to be closer to the PRE. When
trying to find the PRE (by estimating
and
)
we would want to put more weight on the low variance observations.
The term *ordinary least squares* really
means *equally weighted least squares*, and Assumption **A6**
about the distribution of
therefore relates to the performance of OLS.

Notice that we are taking the conditional
expectation of
, conditional upon knowing the value of
.
Equation E7 is another way to think of the LRM. It says that we
are assuming a linear relationship between the exogenous
variable
and the expected value
of the random
variable
. This is an ordinary linear relationship;
there are no random variables left in E7 because
is a population
parameter. Taking expectations wipes out the influence of the random
factor
.

It is then perfectly legal and reasonable to take derivatives in E7:

We interpret the slope coefficient
as a derivative, *the rate of change in
as the exogenous variable X
changes*. In our minimum wage example,
would determine the rate
at which expected employment (across provinces) changes as the minimum wage changes.
Notice that the disturbance term u has no effect on this interpretation. In
others, the * level * of
is moved around by the disturbance terms, but
measures how
would change with
.

Document Last revised: 1997/1/5