Mathematics
UBC ECON 322 ?Applied Econometrics Exercises

ECON 322

The University of British Columbia

ECON

Question Description

Iβm working on a Mathematics exercise and need support.

applied econometrics.

• Basic estimator properties (chapter 3.1)
• OLS assumptions (chapter 4)
• Impact of variance of regressor and error terms on the SE of the regression slope (chapter 4)
• Linear regression: interpretation of the results (Stata output provided). Intercept, continuous and dummy regressors, linear prediction, residual calculation, R2, statistical inference.

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Studentβs Name Midterm Spring 2020 Do cell phones distract drivers and cause accidents? Worried that this is happening, many states have passed legislation to reduce distracted driving. By 2012, some states had banned handling cell phone use while driving illegal and/or had banned texting while driving. Explore the relationship between cell phones and traffic fatalities using statistics drawn from a data sample on 50 US states in 2012. Codebook for cell phone data Variable name Description State State name state_numeric State name (numeric representation of state) numberofdeaths Number of traffic deaths per year cell_subscription Number of cell phone subscriptions (in thousands) population Population within a state (in numbers of individuals) total_miles_driven Total miles driven within a state for that year (in millions of miles) cell_ban = 1 if it is illegal to operate a handled cell phone while driving = 0 otherwise = 1 if it is illegal to text while driving = 0 otherwise text_ban Summary Statistics . sum numberofdeaths cell_subscription population total_miles_driven cell_ban text_ban Variable Obs Mean numberofde~s cell_subsc~n population total_mile~n cell_ban 50 50 50 50 50 670.92 6000.78 6265634 59304.86 .2 text_ban 50 .68 Std. Dev. Min Max 675.9649 6729.329 7000304 60495.32 .404061 59 518 576412 4792.082 0 3398 35616 3.80e+07 326271.7 1 .4712121 0 1 Although we donβt know how many people are using their phones while driving, we can find the number of cell phone subscriptions in a state (in thousands). 1 Studentβs Name Discuss the estimates of the simple regression run under the assumption of homoscedasticity of the error terms: . reg numberofdeaths cell_subscription Source SS df MS Model Residual 18433286.8 3956212.88 1 48 18433286.8 82421.1017 Total 22389499.7 49 456928.565 numberofdeaths Coef. cell_subscription _cons .0911447 123.9805 Std. Err. .0060947 54.64417 t 14.95 2.27 Number of obs F(1, 48) Prob > F R-squared Adj R-squared Root MSE = = = = = = 50 223.65 0.0000 0.8233 0.8196 287.09 P>|t| [95% Conf. Interval] 0.000 0.028 .0788906 14.11103 .1033989 233.85 1.1) Are the estimates of the slope and intercept statistically significant? If yes, At what level of significance ? 1.2) Interpret the estimates of the intercept and slope (see the codebook above for the measurement units of the dependent and explanatory variable). Does the sign of the estimated marginal effect of cell subscription look reasonable? 2 Studentβs Name 2.1) Write down the model estimated in (1). 2.2) Name the method of estimation. 2.3) Show the value function minimized under this method. 3 Studentβs Name 3) Report and interpret the explanatory power of the regression in (1). 4) Report the explained, residual, and total sums of squares from (1). Show how the Rsquared of 0.82 is calculated. 4 Studentβs Name 5) State the null hypothesis underlying the Breush- Pagan test below: Breusch-Pagan / Cook-Weisberg test for heteroskedasticity Ho: Constant variance Variables: fitted values of numberofdeaths chi2(1) Prob > chi2 = = 71.26 0.0000 Are the estimates of the standard errors reported in (1) valid? Explain. 5 Studentβs Name 6) What happens to the coefficient on cell phone subscriptions once the population control is added in the regression? Do you suspect endogeneity of the explanatory variable in (1)? Why or why not? . reg numberofdeaths cell_subscription population Source SS df MS Model Residual 19169497.5 3220002.19 2 47 9584748.74 68510.685 Total 22389499.7 49 456928.565 numberofdeaths Coef. cell_subscription population _cons -.2108858 .0002909 113.9346 Std. Err. .0923033 .0000887 49.91417 t -2.28 3.28 2.28 Number of obs F(2, 47) Prob > F R-squared Adj R-squared Root MSE P>|t| 0.027 0.002 0.027 = = = = = = 50 139.90 0.0000 0.8562 0.8501 261.75 [95% Conf. Interval] -.3965761 .0001124 13.52023 -.0251955 .0004694 214.3489 6 Studentβs Name 7) State (mathematically) the exogeneity assumption required for validity of the OLS estimates. Does it hold in (1)? 7 Studentβs Name 8) What happens to the coefficients on cell phone subscriptions and population size once the total miles driven is controlled for? . reg numberofdeaths cell_subscription population total_miles_driven Source SS df MS Model Residual 20645516.8 1743982.86 3 46 6881838.94 37912.6708 Total 22389499.7 49 456928.565 numberofdeaths Coef. cell_subscription population total_miles_driven _cons .0024648 -.0000742 .0188321 4.34595 Std. Err. Number of obs F(3, 46) Prob > F R-squared Adj R-squared Root MSE t .0767069 .0000882 .0030182 41.07542 P>|t| 0.03 -0.84 6.24 0.11 0.975 0.404 0.000 0.916 = = = = = = 50 181.52 0.0000 0.9221 0.9170 194.71 [95% Conf. Interval] -.1519381 -.0002518 .0127568 -78.33458 .1568678 .0001033 .0249074 87.02648 Why? Consider a supplementary regression below and discuss what kind of relationship was the estimate on cell_subscription in (1) picking up? . reg total_miles cell_subscription Source SS population df MS Model Residual 1.7516e+11 4.1619e+09 2 47 8.7581e+10 88551828.2 Total 1.7932e+11 49 3.6597e+09 total_miles_dri~n Coef. cell_subscription population _cons -11.3291 .0193866 5819.252 Std. Err. 3.318462 .00319 1794.5 t -3.41 6.08 3.24 Number of obs F(2, 47) Prob > F R-squared Adj R-squared Root MSE P>|t| 0.001 0.000 0.002 = = = = = = 50 989.04 0.0000 0.9768 0.9758 9410.2 [95% Conf. Interval] -18.00499 .0129691 2209.184 -4.653219 .025804 9429.321 8 Studentβs Name 9 Studentβs Name 9) Given the summary statistic of the dummy variables describing the state policies in 2012 (see the stata table on the first page), provide the marginal sample distributions of the dummies: What percentage of states banned/allowed to operate a handled cell phone while driving? What percentage of states banned/allowed to text while driving? Explain how you derived the distribution. 10 Studentβs Name 10) The model estimated below includes the policy dummies. . reg numberofdeaths cell_subscription population total_miles cell_ban text_ban, r Linear regression Number of obs F(5, 44) Prob > F R-squared Root MSE numberofdeaths Coef. cell_subscription population total_miles_driven cell_ban text_ban _cons .0273334 -.0000723 .015885 -100.7896 -165.2344 150.098 Robust Std. Err. .0908511 .000106 .0024762 52.14998 55.94026 54.31378 t 0.30 -0.68 6.42 -1.93 -2.95 2.76 P>|t| 0.765 0.499 0.000 0.060 0.005 0.008 = = = = = 50 45.14 0.0000 0.9409 173.45 [95% Conf. Interval] -.1557649 -.0002859 .0108945 -205.891 -277.9746 40.63575 .2104317 .0001414 .0208754 4.311745 -52.4942 259.5602 10.1) What are roughly the t-statistic, 95% confidence interval (you may round it up to integers when approximating), and p-value, corresponding to the estimate associated with the cell_ban variable? 11 Studentβs Name 10.2) How efficient are the policies regulating cell phone usage when driving? Did they have an effect? If yes, specify the effect. 12 Studentβs Name 11) The table below shows the outcomes of different variables in Maryland: . sum population cell_subscription cell_ban text_ban total_miles_driven numberofdeaths if state == "Maryland" Variable Obs Mean population cell_subsc~n cell_ban text_ban total_mile~n 1 1 1 1 1 5884563 6116 1 1 56475.59 numberofde~s 1 505 Std. Dev. Min Max . . . . . 5884563 6116 1 1 56475.59 5884563 6116 1 1 56475.59 . 505 505 The number of deaths predicted by the regression in (12) conditional on Marylandβs characteristics is reported below using stata: Adjusted predictions Model VCE : Robust Expression at _cons Number of obs = 50 : Linear prediction, predict() : cell_subsc~n = 6116 population = 5884563 total_mile~n = 56475.59 cell_ban = 1 text_ban = 1 Margin Delta-method Std. Err. 523.155 50.34662 t 10.39 P>|t| 0.000 [95% Conf. Interval] 421.688 624.6219 11.1) Show how this prediction is calculated. 13 Studentβs Name 11.2) Calculate the corresponding residual (approximately, you may round it up to integers). 11.3) What would be a predicted value of the number of deaths should Maryland allow using the cell phones but keeping texting banned? 14 Studentβs Name 12) Consider an estimated model augmented by regressor cell_allowed defined as . gen cell_allowed = 1 - cell_ban Linear regression Number of obs F(3, 46) Prob > F R-squared Root MSE numberofdeaths Coef. cell_subscription population cell_ban cell_allowed _cons -.1459941 .0002324 -317.6107 0 154.0891 Robust Std. Err. .1116929 .0001148 88.75439 (omitted) 60.79421 t P>|t| = = = = = 50 32.50 0.0000 0.8891 232.3 [95% Conf. Interval] -1.31 2.02 -3.58 0.198 0.049 0.001 -.3708201 1.30e-06 -496.264 .078832 .0004636 -138.9574 2.53 0.015 31.71669 276.4615 What assumption underlying validity of the OLS estimator causes Stata to drop this additional regressor? 13) Define (name and state mathematically) the main properties of a good estimator (two main properties evaluating its closeness to the true value of a parameter of interest). 15 Studentβs Name 14) Under which assumptions the OLS estimator qualifies as a good estimator defined in question 13? 16 ...
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1.1
The estimates of the slope and intercept are both significant as it can be seen from the Pvalue. The coefficient for cell_subscription is significant at 1% significance level, while the
intercept is significant at 5% significance level.
1.2
The estimate of the slope is telling us that an increase in 1 thousand cellphone subscriptions
increases the number of traffic deaths per year by 0.0911447 units.
The coefficient for the intercept estimate instead is telling us that the average number of
traffic deaths per year is 123.9805.
The estimate for the slope does not look much reasonable because it is unlikely that, given
the significance of the estimate, 1000 thousand more cellphone subscriptions lead to a
increase of less than 1 traffic death per year.
2.1
The model estimated in (1) is the following one:
ππ’ππππππππππ‘βπ  = π½0 + π½1 πππππ π’ππ πππππ‘ππππ  + ππ
2.2
The method used is the classic Ordinary Least Squares method.
2.3
The OLS tries to minimize the sum of squared errors
π½ππ π‘ππππ‘ππ = ππππππ π(π½)
Where
π

π

π(π½) = β |π¦π β β πππ π½π |2 = ||π β πΏπ·||2
1=1

π=1

3.
The explanatory power of the regression in (1) is given by the R-squared.
The number shown...

damianosaba (316)
Carnegie Mellon University
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