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Please round numbers to 4 decimal places. Write down your answers clearly and show all the important steps you used in your calculations. You do not need to provide your programming code in R or STATA.

1 Problem 1 – Empirical Exercise

Download the data set on Guns (with n=1,173 observations provided by Stock and Watson) on NYU Classes (Folder Data). Import the excel file to your statistical software.

Consider the following variables:

- Yit (vio): violent crime rate (incidents per 100,000 members of the population in state i and year t)
- Xit (avginc): real per capita personal income in state i and year t, in thousands of dollars

Consider only observations in the year 1986 throughout the whole problem set!

1.1 Part 1

Consider the following regression model for the year 1986:

Yit=β0+Uit, t=86 (1) 1. Write down the objective function of the least squares method.

Consider only observations of the year 1986.

- Derive the OLS estimator βˆ0 for β0 in regression model (1) as a function of {Yit}.
- Estimate regression model (1) using the OLS regression command from your statistical software. Write down the estimate βˆ0.
- Interpret the estimate βˆ0.
- What is the value of the sample variance of the fitted values, s2ˆ ?
- What is the value of the coefficient of determination of regression R2 in model (1)? Interpret the value of R2 and explain in 1-2 sentences why R2 takes on this specific value.

1.2 Part 2

Consider now the following regression model:

Yit =β0+β1Xit+εit, t=86 withεi|Xit ∼i.i.dN(0,σε2) (2)

- Estimate regression model (2) using the OLS regression command. Write down the estimates βˆ0 and βˆ1.
- Write down the standard errors of the estimators βˆ0 and βˆ1.
- Writedownthegeneral(conditional)distributionoftheestimatorβˆ1|Xit (with distribution parameters).

Note: You do not need to derive the variance formula. You can use the corresponding variance formula from the lecture slides. - Write down the unbiased estimator σˆε2 for the noise variance σε2. (Be careful to distinguish between εˆ and ε !)

- Write down the sum of squared residuals (SSR) of regression model (2).
- Compute σˆε2.
- Compute the 90% confidence interval for β1 (and write down the for- mula that you are using).

Consider the one-sided test with H0 : β1 ≥ 60.

- Write down the (most critical, conditional) distribution of the OLS estimator, βˆ1|Xit, under the null hypothesis.
- Write down the test statistic and its distribution under the null hypothesis.
- Compute the test statistic.
- Compute the p-value [make your approach to the question clear by writ- ing down the corresponding probabilities]. What do you conclude?

1.3 Part 3

Consider now the following regression model:

Yit=β1Xit+εit, t=86 withεi|Xit∼i.i.dN(0,σε2) (3)

- Estimate regression model (3) using the OLS regression command. Write down the estimate βˆ1.
- Write down the standard errors of the estimators βˆ1.
- Compute Y2.
- Write down the total sum of squares (TSS). What is the difference between Y 2 and TSS? Explain why the total sum of squares are i i,86 larger than in model (1).

Hint: When X has no explanatory power in (3), the baseline model (to which the model fit is compared) becomes Y ̃ |β1 =0 = 0 instead of Y ̄ .

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