The Gallup Poll frequently surveys the electorate to quantify the public's opinion of the
president.Since 1945, Gallup settled on the following wording of its presidential poll:
"Do you approve or disapprove of the way (name)is handling his job as president?"
Gallup has not changed its presidential question since then, and respondents can answer
"approve," "disapprove," or "no opinion."
You want to see how this approval rating is related to the Michigan index of consumer
sentiment (ICS).The monthly survey, conducted with a minimum sample of 500, asks
people if they feel "better/worse off" with regard to current and future conditions.
(a)To estimate dynamic causal effects, you collect quarterly data from 1962:I - 1998:II for
the United States.You allow a binary variable for each presidency to capture the intrinsic
popularity of the President.Furthermore, you eliminate observations that include a
change in party for the presidency by using a binary variable, which takes on the value of
one during the first quarter of the year after the election.Finally, a friendly political
scientist provides you with (i)an "events" variable, (ii)a "Vietnam" binary variable, and
(iii)a "honeymoon" variable, which measures the effect of a higher popularity of a
president immediately following the election.(The coefficients of these variables will not
be reported here.)
Assuming that consumer sentiment is exogenous, you estimate the following two
specifications (numbers in parenthesis are heteroskedasticity- and autocorrelation-
consistent standard errors)

\begin{array}{l}\begin{array} { l } \widehat { \text { Approval } _ { t } } = 26.08 + 0.178 \times \text { ICS } _ { t } + 0.232 \times \text { ICS } _ { t - 1 } ; R ^ { 2 } = 0.667 , \text { SER } = 7.00 \\\quad\quad\quad\quad\quad\quad\text { (8.83) } ( 0.120 )\quad\quad\quad\quad\quad(0.135) \\\end{array}\\\begin{aligned}\widehat { \text { Approval } _ { t } = } = & 26.08 + 0.178 \times \Delta I C S _ { t } + 0.411 \times I C S _ { t - 1 } ; R ^ { 2 } = 0.667 , S E R = 7.00 \\& \begin{array} { l l } \text { (8.17) } ( 0.120 ) & ( 0.089 )\end{array}\end{aligned}\end{array}

What is the difference between the two specifications? What is the advantage of
estimating the second equation, if any?

Infeasible GLS A XtX _ { t }Xt​ D Assumes the Parameters of the Error Autocorrelation Process to PCs

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Infeasible GLS A $X _ { t }$
D Assumes the Parameters of the Error Autocorrelation Process to PCs

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