View Gujarati 06.01

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Question:
06.01 Explain carefully the meaning of each of the following terms:
a. Population regression function (PRF).
b. Sample regression function (SRF).
c. Stochastic PRF.
d. Linear regression model
e. Stochastic error term (ui).
f. Residual term (ei).
g. Conditional expectation.
h. Unconditional expectation.
i. Regression coefficients or parameters.
j. Estimators of regression coefficients.

[my adds]
k. What does “Stochastic” in “Stochastic PRF” mean, why does it matter?
l. Is linear regression model redundant, does not regression model imply “linear?”
m. What is the difference between the error and the residual?
n. Given the linear SRF: Y (est) = intercept + slope (X), where is the conditional expectation and the unconditional expectation?
i. In a two-variable regression, how many “estimators of regression coefficient” are there?

Answers:

06.01 a
It states how the population mean value of the dependent variable is related to one or more explanatory variables.

06.01 b
It is the sample counterpart of the PRF.

06.01 c
It tells how the individual Y are related to the explanatory variables and the stochastic error term u, in the population as a whole.

06.01 d
A model that is linear in the parameters, the Bs.

06.01 e
It is a proxy for all omitted or neglected variables that affect the dependent variable Y. The individual influence of each of these variables is random and small so that on average their influence on Y is zero.

06.01 f
It is the sample counterpart of the stochastic error term

06.01 g
The expected value of Y conditional upon a given value of X. It is obtained from the conditional (probability) distribution of Y, given X.

06.01 h
The expected value of an r.v. regardless of the values taken by other random variables. It is obtained from the unconditional, or marginal, probability distributions of the relevant random variables.

06.01 i
The B coefficients in a linear regression model are called regression coefficients or regression parameters.

06.01j
The bs, which tell how to compute the Bs, are called the estimators. Numerical values taken by the bs are known as estimates.

06.01k
With the stochastic “noise” term (u), the PRF is deterministic: Y = B1 + B2*X
Econometrics may not be our favorite field, but at least they have the good sense to know they cannot model everything, unless some unnamed folks we know. So addition of the noise term is a sort of wise humility that acknowledges the model does not capture everything (I think of it as the “humility term”. In a way, econometrics is a deep dive into the “u” term):

PRF (stochastic) = Y B1 + B2*X + random error/noise (u)

Gujarati: Stochastic Process: A random or stochastic process is a collection of random variables ordered in time

Perhaps you are wondering about the root of stochastic? Gujarati has us covered:
“The following discussion is based on the term “stochastic” comes from the Greek word “stokhos,” which means a target or bull’s-eye. If you have ever thrown darts on a dart board with the aim of hitting the bull’s-eye, how often did you hit the bull’s-eye? Out of a hundred darts you may be lucky to hit the bull’s-eye only a few times; at other times the darts will be spread randomly around the bull’s-eye.”

06.01l
No, linear regression is not redundant. We are studying OLS regression.
First, OLS is not the only way to develop linear estimators;
Second, there are non-linear regression methods (although often, we can transform non-linear into linear and avoid…)
Please recall that by “linear regression,” we mean linear in the parameters not the variables. (could imagine this as a test question; e.g., Is Y = B1 +B2*X^2 a linear regression model? Yes! Parameters B1 and B2 are linear even as variable (X^2) is not! counterintuitive…)

06.01m
The error (u) attaches to the PRF; the residual (e) attaches to the SRF.

06.01n
The predicted Y is a conditional mean!
Y = B1 + B2*X really means….
E(Y | X) = B1 + B2*X; i.e., the expected Y conditional on a given X

The unconditional Y is the average Y; if you are given no information about X, what is your best estimate of Y? The average Y
Similarly, unconditional X is average X.
Don’t forget the OLS line alwasy passes through the point (average X, average Y)

06.01o
Two: slope estimator and intercept estimator
(the X and Y are variables)

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