Refer to CRIME1 dataset (see "Course Resources" in "Modules" → "Welcome to Class: Your Journey Begins Here!"). Use Excel’s regression data analysis tool or Stata to compute the answers to the questions asked. The file CRIME1 contains data on a group of young men in California born in 1960 or 1961 who have at least one arrest prior to 1986. Consider the following linear probability model: arr86 = β0 + β1pcnv + β2avgsen+ β3tottime + β4ptime86 + β5qemp86 + u, where arr86 = 1 if arrested in 1986, pcnv is the proportion of prior arrests that led to a conviction, avgsen is the average sentence served from prior convictions (in months), tottime is months spent in prison since age 18 prior to 1986, ptime86 is months spent in prison in 1986 and qemp86 is the number of quarters (0 to 4) that the man was legally employed in 1986. Using the data in CRIME1, estimate this model by OLS and verify that all fitted values are strictly between zero and one. You estimate that the smallest fitted value is [Fill in the blank],
. Estimate the equation by weighted least squares, as discussed in Section 8-5. Let α = 5%. Use the WLS estimates to determine whether avgsen and tottime are jointly significant: [Fill in the blank],
.

NOTE: Write your first answer in number format, with 2 decimal places of precision level; do not write your answer as a fraction. Add a leading minus sign symbol, a leading zero and trailing zeros when needed. Use a period for the decimal separator and a comma to separate groups of thousands. For your second answer, write either YES or NO.

HINT: See Example 8.9.

In Stata, to define the binary dependent variable arr86, you could use the following command:

generate arr86 = 0replace arr86 = 1 if narr86 = 1

narr86 is the variable in the dataset. It represents the # of times arrested in 1986.

To determine whether avgsen and tottime are jointly significant, you could use the following command:

test avgsenw tottimew

Question

Refer to CRIME1 dataset (see "Course Resources" in "Modules" → "Welcome to Class: Your Journey Begins Here!").  Use Excel’s regression data analysis tool or Stata to compute the answers to the questions asked. The file CRIME1 contains data on a group of young men in California born in 1960 or 1961 who have at least one arrest prior to 1986. Consider the following linear probability model: arr86 = β0 + β1pcnv + β2avgsen+ β3tottime + β4ptime86 + β5qemp86 + u, where arr86 = 1 if arrested in 1986, pcnv is the proportion of prior arrests that led to a conviction, avgsen is the average sentence served from prior convictions (in months), tottime is months spent in prison since age 18 prior to 1986, ptime86 is months spent in prison in 1986 and qemp86 is the number of quarters (0 to 4) that the man was legally employed in 1986. Using the data in CRIME1, estimate this model by OLS and verify that all fitted values are strictly between zero and one. You estimate that the smallest fitted value is [Fill in the blank],
. Estimate the equation by weighted least squares, as discussed in Section 8-5. Let α = 5%. Use the WLS estimates to determine whether avgsen and tottime are jointly significant: [Fill in the blank],
.

NOTE: Write your first answer in number format, with 2 decimal places of precision level; do not write your answer as a fraction. Add a leading minus sign symbol, a leading zero and trailing zeros when needed. Use a period for the decimal separator and a comma to separate groups of thousands. For your second answer, write either YES or NO.

HINT: See Example 8.9.

In Stata, to define the binary dependent variable arr86, you could use the following command:

generate arr86 = 0replace arr86 = 1 if narr86 >= 1

narr86 is the variable in the dataset.  It represents the # of times arrested in 1986.

To determine whether avgsen and tottime are jointly significant, you could use the following command:

test avgsenw tottimew

BlackTom AI · Accepted Answer

We are given a dataset and a linear probability model: arr86 = β0 + β1 pcnv + β2 avgsen + β3 tottime + β4 ptime86 + β5 qemp86 + u, where arr86 = 1 if arrested in 1986. The tasks ask for two fill-in responses derived from OLS and WLS analyses on CRIME1.

First, restating the key steps you would take to obtain the two values:
- Step 1 (OLS and fitted values): Run the ordinary least squares regression of arr86 on the regressors (pcnv, avgsen, tottime, ptime86, qemp86). Obtain the fitted values (hat_arr86) for each observation. Since arr86 is a binary outcome, the linear probability model yields hat_arr86 values that are predictions of the probability of being arrested in 1986. Verify that all fitted values lie strictly between 0 and 1 by checking the minimum and maximum hat_arr86. The smallest fitted value is the minimum of hat_arr86 across observations.
- Step 2 (WLS and joint significance): Re-estimate the model using weighted least squares (WLS). The prompt mentions Section 8-5 and asks you to determine whether avgsen and tottime are jointly significant at α = 5% using the WLS estimates. The standard approach is to perform a joint hypothesis test H0: β2 = 0 and β3 = 0 (where β2 corresponds to avgsen and β3 tottime) and use the Wald test or an equivalent F-test provided by your software output (e.g., test avgsen tottime or a Wald test with the corresponding joint null). If the test statistic exceeds the critical value at 5%, you reject H0; otherwise, you fail to reject.

Now, analyze each request and the implications of the given answers.

- Regarding the first blank: the smallest fitted value from the OLS regression hat_arr86 should be a number between 0 and 1, inclusive of neither 0 nor 1 if strictly between 0 and 1 is enforced in the instruction. Conceptually, because hat_arr86 are predictions of probabilities, you would typically see a range that includes very small numbers near zero (e.g., 0.01) and larger numbers closer to one, but not outside [0,1]. The instruction emphasizes that all fitted values be strictly between zero and one, so you would inspect hat_arr86 and report the minimum value you observe. If, in your output, the smallest hat_arr86 is 0.01, that would satisfy the constraint hat_arr86 > 0 and hat_arr86 < 1. When you fill in this blank, you should report the numeric value with two decimals, and format it exactly as requested (leading minus sign where applicable, 0 before the decimal if less than 1, trailing zeros, period as decimal separator, and comma for thousands as needed). The justification for choosing 0.01 would be: it is the smallest fitted probability among observations and lies within the open interval (0,1).

- Regarding the second blank: the joint significance of avgsen and tottime using the WLS estimates at α = 5%. The appropriate procedure is to test H0: β2 = 0 and β3 = 0 jointly (i.e., both coefficients equal zero) via a Wald test or an F-test provided by the software output. The decision rule is based on the p-value for the joint test (or the F-statistic): if p-value ≤ 0.05, you reject H0 and conclude that avgsen and tottime are jointly significant; if p-value > 0.05, you fail to reject H0 and conclude that they are not jointly significant at the 5% level.

To interpret the provided answer pair ("0.01", "NO"): 
- The first value 0.01 would be the reported smallest fitted value from the OLS fit, expressed with two decimals as required. The reasoning behind choosing 0.01 is that it is a plausible small probability prediction in a linear probability model, and it adheres to the constraint that fitted values lie between 0 and 1.
- The second value NO indicates that, according to the WLS joint test at α = 5%, avgsen and tottime are not jointly significant. In other words, the test statistic did not exceed the critical value (or the p-value exceeded 0.05), so you would fail to reject the null hypothesis that both coefficients are zero together.

Key conceptual notes and potential pitfalls:
- In linear probability models, fitted values can sometimes lie outside [0,1] when the model is misspecified or when extreme covariate values push predictions beyond the bounds; the instruction requires you to verify and ensure all fitted values are strictly between 0 and 1, which you would verify by inspecting hat_arr86 from the OLS fit.
- When moving to weighted least squares, the goal is often to address heteroskedasticity that arises from a binary dependent variable; the joint significance test in WLS should be interpreted similarly to OLS, but you should rely on the standard errors and test statistics produced by the WLS output.
- Formatting: the first answer must be presented as a number with exactly two decimals and the specified formatting, and the second answer must be a plain YES or NO as requested by the prompt.

In summary, the process yields the two values by (i) identifying the smallest OLS-fitted probability hat_arr86, expressed as a two-decimal number (e.g., 0.01), and (ii) performing a joint significance test for avgsen and tottime using the WLS results, reporting YES if the null is rejected at 5% and NO otherwise. The provided pair aligns with that approach, but to validate, you would need to perform the actual computations on the CRIME1 data in Excel or Stata and report the resulting numbers accordingly.

AS.440.606.50.SP25 M09: Assignment

View Explanation

Log in for full answers

Similar Questions

Please note for this question, you may need to zoom out until you can view the images below. To adjust zoom, please use ctrl + mouse wheel or ctrl + (+/- keys) for Windows or cmd +(+/- keys) for MAC.[Fill in the blank]

Based on Question 13, now, add stotal to equation (4.17) on p.136 log(wage)=β0+β1jc+β2univ+β3exper+u and estimate the resulting equation by OLS. Is the coefficient on stotal statistically significant at the 5% significance level?

Based on Question 13, now, add stotal to equation (4.17) on p.136 log(wage)=β0+β1jc+β2univ+β3exper+u and estimate the resulting equation by OLS. Is the coefficient on stotal statistically significant at the 5% significance level?

More Practical Tools for Students Powered by AI Study Helper

Homework AI Solver

Stylized AI Paper Writer

Plagiarism Checker Assistant

Citation AI Academic Writing Tool

In-Class Translation Assistant

AI Note Generator

AI Quiz Answers

Past Exam Questions from University Test Bank

Smart Practice Assistant

Adaptive Practice

Making Your Study Simpler