$3.$ Computer Exercise C$6$

For this exercise, you will be using a variation of CRIME$1,$ which contains cross$-$sectional data on the number of arrests in $1986,$ proportion of prior convictions, time spent in prison during $1986,$ and number of quarters employed in $1986,$ among other variables describing individuals' characteristics.

The following section gives detailed information about the variables within the data set:

Variable Descriptions

narr$86$narr$86=$ number of times arrested, $1986$nfarr$86$nfarr$86=$ number of felony arrests, $1986$nparr$86$nparr$86=$ number property crime arrests, $1986$pcnvpcnv$=$ proportion of prior convictionsavgsenavgsen$=$average sentence length, monthstottimetottime$=$ time in prison since turning $18$ years old $($months$)$ptime$86$ptime$86=$ months in prison during $1986$qemp$86$qemp$86=$ number of quarters employed, $1986$inc$86$inc$86=$ legal income, $1986,$ $$100$sduratdurat$=$ recent unemp durationblackblack$=1$ if blackhispanhispan$=1$ if Hispanicborn$60$born$60=1$ if born in $1960$pcnvsqpcnvsq$=$ pcnv$2$pcnv$2$ptime$86$sqptime$86$sq$=$ ptime$862$ptime$862$inc$86$sqinc$86$sq$=$ inc$862$inc$862$

Open any of the following data files to reference the data from CRIME$1.$ Use the chosen data file to answer each of the following questions.

$$

Open R File

$$

Open Excel File

$$

Open Stata File

$($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$$fitted value is approximately $0\mathrm{.}0120$and the largest$$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$86*.$

$($iii$)$

Use the WLS estimates to determine whether avgsenavgsen$$and tottimetottime$$are jointly significant at the $5\%$ significance level.

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

The variables avgsenavgsen$$and tottimetottime$$jointly significant at the $5\%$ significance level, as the F$$statistic for their joint significance is about$$with p$-$value equal to$.$