Question

9 (i) Let $eta_0$ and $eta_1$ be the intercept and slope from the regression of $y_i$ on $x_i$, using $n$ observations. Let $c_1$ and $c_2$, with $c_2 eq 0$, be constants. Let $ ilde{eta}_0$ and $ ilde{eta}_1$ be the intercept and slope from the regression of $c_1 y_i$ on $c_2 x_i$. Show that $ ilde{eta}_1 = (c_1/c_2)eta_1$ and $ ilde{eta}_0 = c_1 eta_0$, thereby verifying the claims on units of measurement in Section 2.4. [Hint: To obtain $ ilde{eta}_1$, plug the scaled versions of $x$ and $y$ into (2.19). Then, use (2.17) for $ ilde{eta}_0$, being sure to plug in the scaled $x$ and $y$ and the correct slope.] (ii) Now, let $ ilde{eta}_0$ and $ ilde{eta}_1$ be from the regression of ($c_1 + y_i$) on ($c_2 + x_i$) (with no restriction on $c_1$ or $c_2$). Show that $ ilde{eta}_1 = eta_1$ and $ ilde{eta}_0 = eta_0 + c_1 - c_2 eta_1$. (iii) Now, let $ ilde{eta}_0$ and $ ilde{eta}_1$ be the OLS estimates from the regression $log(y_i)$ on $x_i$, where we must as- sume $y_i > 0$ for all $i$. For $c_1 > 0$, let $hat{eta}_0$ and $hat{eta}_1$ be the intercept and slope from the regression of $log(c_1 y_i)$ on $x_i$. Show that $hat{eta}_1 = eta_1$ and $hat{eta}_0 = log(c_1) + eta_0$. (iv) Now, assuming that $x_i > 0$ for all $i$, let $ ilde{eta}_0$ and $ ilde{eta}_1$ be the intercept and slope from the regression of $y_i$ on $log(c_2 x_i)$. How do $ ilde{eta}_0$ and $ ilde{eta}_1$ compare with the intercept and slope from the regression of $y_i$ on $log(x_i)$?

          9 (i) Let $eta_0$ and $eta_1$ be the intercept and slope from the regression of $y_i$ on $x_i$, using $n$ observations. Let
$c_1$ and $c_2$, with $c_2 
eq 0$, be constants. Let $	ilde{eta}_0$ and $	ilde{eta}_1$ be the intercept and slope from the regression
of $c_1 y_i$ on $c_2 x_i$. Show that $	ilde{eta}_1 = (c_1/c_2)eta_1$ and $	ilde{eta}_0 = c_1 eta_0$, thereby verifying the claims on units of
measurement in Section 2.4. [Hint: To obtain $	ilde{eta}_1$, plug the scaled versions of $x$ and $y$ into (2.19).
Then, use (2.17) for $	ilde{eta}_0$, being sure to plug in the scaled $x$ and $y$ and the correct slope.]
(ii) Now, let $	ilde{eta}_0$ and $	ilde{eta}_1$ be from the regression of ($c_1 + y_i$) on ($c_2 + x_i$) (with no restriction on $c_1$ or $c_2$).
Show that $	ilde{eta}_1 = eta_1$ and $	ilde{eta}_0 = eta_0 + c_1 - c_2 eta_1$.
(iii) Now, let $	ilde{eta}_0$ and $	ilde{eta}_1$ be the OLS estimates from the regression $log(y_i)$ on $x_i$, where we must as-
sume $y_i > 0$ for all $i$. For $c_1 > 0$, let $hat{eta}_0$ and $hat{eta}_1$ be the intercept and slope from the regression of
$log(c_1 y_i)$ on $x_i$. Show that $hat{eta}_1 = eta_1$ and $hat{eta}_0 = log(c_1) + eta_0$.
(iv) Now, assuming that $x_i > 0$ for all $i$, let $	ilde{eta}_0$ and $	ilde{eta}_1$ be the intercept and slope from the regression
of $y_i$ on $log(c_2 x_i)$. How do $	ilde{eta}_0$ and $	ilde{eta}_1$ compare with the intercept and slope from the regression
of $y_i$ on $log(x_i)$?

9 (i) Let eta0 and eta1 be the intercept and slope from the regression of yi on xi, using n observations. Let
c1 and c2, with c2
eq 0, be constants. Let ildeeta0 and ildeeta1 be the intercept and slope from the regression
of c1 yi on c2 xi. Show that ildeeta1 = (c1/c2)eta1 and ildeeta0 = c1 eta0, thereby verifying the claims on units of
measurement in Section 2.4. [Hint: To obtain ildeeta1, plug the scaled versions of x and y into (2.19).
Then, use (2.17) for ildeeta0, being sure to plug in the scaled x and y and the correct slope.]
(ii) Now, let ildeeta0 and ildeeta1 be from the regression of (c1 + yi) on (c2 + xi) (with no restriction on c1 or c2).
Show that ildeeta1 = eta1 and ildeeta0 = eta0 + c1 - c2 eta1.
(iii) Now, let ildeeta0 and ildeeta1 be the OLS estimates from the regression log(yi) on xi, where we must as-
sume yi > 0 for all i. For c1 > 0, let hateta0 and hateta1 be the intercept and slope from the regression of
log(c1 yi) on xi. Show that hateta1 = eta1 and hateta0 = log(c1) + eta0.
(iv) Now, assuming that xi > 0 for all i, let ildeeta0 and ildeeta1 be the intercept and slope from the regression
of yi on log(c2 xi). How do ildeeta0 and ildeeta1 compare with the intercept and slope from the regression
of yi on log(xi)?

Added by Aaron S.

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