$($i$)$

Use OLS to estimate a linear probability model to determine the likelihood that a young male was arrested during $1986$: arr$86=$$0+$$1$pcnv$+$$2$avgsen$+$$3$tottime$+$$4$ptime$86+$$5$qemp$86+$uarr$86=$$0+$$1$pcnv$+$$2$avgsen$+$$3$tottime$+$$4$ptime$86+$$5$qemp$86+$u$,$ where narr$86$narr$86$is the binary variable. If a man was arrested during $1986,$ arr$86=1$arr$86=1,$ otherwise arr$86=0$arr$86=0.$

The estimated equation is as follows: arr$86=$arr$86^=0\mathrm{.}4434+(-0\mathrm{.}1597)$pcnvpcnv$+0\mathrm{.}0061$avgsenavgsen$+(-0\mathrm{.}0016)$tottimetottime$+(-0\mathrm{.}0226)$ptime$86$ptime$86+(-0\mathrm{.}0431)$qemp$86$qemp$86.$

Obtain the fitted values from the regression.

The smallest$$fitted value is approximately $0\mathrm{.}428$and the largest$$fitted value is approximately $1.$

$($ii$)$

Estimate the following equation by weighted least squares using weights $1$hi$1$hi$^,$ where hi$=$arr$86$i$(1$arr$86$i$)$hi$^=$arr$86$i$^1$arr$86$i$^.($Hint: Run the regression without an intercept and note that $0$$0^$is a coefficient of $1$hi$1$hi$^.)$

The estimated equation is as follows: arr$86*=$arr$86*^=$int$*$int$*+$pcnv$*$pcnv$*+$avgsen$*$avgsen$*+$tottime$*$tottime$*+$ptime$86*$ptime$86*+$qemp$86*$qemp$86*.$