Ch1 Complete Test Bank + Practice Test -6 Real Stats Using

PRACTICE Midterm Exam

These practice questions cover Chapters 1 through 6 in Real Stats

NOTES AND GUIDELINES:

2. Show your work for partial credit. Explanations should be brief, and calculations may be approximate.

Good luck!

1. The results below are based on data on gas price and consumption for a single country over 128 months. We are interested in how the price of gas, per capita disposable income, and miles per gallon affect per capita expenditure on gas. The regression equation is

Gas_t = ₀ + ₁price_t + ₂income_t + ₃miles_t + _t

where gas_t is the per capita real expenditure on gasoline at time t; price_t is the real price of gasoline at time t; income_t is the per capita real disposable income at time t; and miles_t is average miles per gallon of cars at time t.

reg gas price income miles

Source | SS df MS Number of obs = 128

-------------+------------------------------ F( 3, 124) = 1476.85

Model | 1.78454603 3 .594848676 Prob > F = 0.0000

Residual | .049944845 124 .000402781 R-squared = 0.9728

-------------+------------------------------ Adj R-squared = 0.9721

Total | 1.83449087 127 .01444481 Root MSE = .02007

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gas | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

price | -.1385602 .0109847 -12.61 0.000 -.1603019 -.1168185

income | .9985463 .0154034 64.83 0.000 .9680586 1.029034

miles | -.5181275 .0173898 -29.79 0.000 -.5525468 -.4837082

_cons | -1.514543 .1171849 -12.92 0.000 -1.746485 -1.282601

1. Interpret the coefficient on price, including statistical significance.
2. Do you think the errors will be correlated? If so, what is the consequence? If not, why not?
3. Notice that there is no variable for church attendance. Discuss the likely consequence of omitting this variable for the coefficient on price.
4. Suppose we doubled the sample size (by getting a sample size of twice as many months). What are the consequences for (i) coefficient estimates and (ii) precision of coefficient estimates?
5. Suppose that we estimated the following equation:

Participation= _ + _ Education + _Male + _Male*Education

where participation is an index of political participation and then plotted the fitted values for men and women in the plot below.

Male

Female

Education

1. Is _greater than, equal to or less than zero? [3 points]
2. Is _greater than, equal to or less than zero? [3 points]
3. Is _greater than, equal to or less than zero? [3 points]
4. The following results are based on life expectancy data for 33 countries. Life expectancy is measured in years. Expenditures are overall health expenditures. We also have data on GDP per capita (measured in thousands of US 2010 dollars) and variables indicating region (EastEur = 1 for countries from Eastern Europe, NorthAm =1 for countries from North America,etc).
   1. The following results are from a difference of means test. (i) What is the life expectancy for people in Eastern Europe? (ii) What is the life expectancy for people in the rest of the world? (iii) Is the difference statistically significant (explain briefly)?

reg lifeexpectancy EastEur

Source | SS df MS Number of obs = 33

-------------+------------------------------ F( 1, 31) = 13.49

Model | 61.4479564 1 61.4479564 Prob > F = 0.0009

Residual | 141.197516 31 4.55475857 R-squared = 0.3032

-------------+------------------------------ Adj R-squared = 0.2808

Total | 202.645472 32 6.332671 Root MSE = 2.1342

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lifeexpect~y | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

EastEur | -3.337913 .9087699 -3.67 0.001 -5.191361 -1.484464

_cons | 80.48077 .4185487 192.29 0.000 79.62713 81.3344

• 1. We get the following results when we add a control variable for GDP per capita. Sketch life expectancy as a function of GDP per capita for Eastern European and non-Eastern European countries. Be sure to include every coefficient in your sketch. The sketch does not have to be to scale, but you need to have correct signs of slopes/intercepts and you need to indicate what each coefficient indicates on the two lines.

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lifeexpect~y | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

EastEur | -2.283396 .8911585 -2.56 0.016 -4.103384 -.4634071

GDPPC | .0504931 .0172679 2.92 0.007 .0152275 .0857588

_cons | 78.40963 .8015952 97.82 0.000 76.77256 80.04671

• 1. We next added an interaction between Eastern Europe and GDP per capita (a variable named EastEurGPDPC). What is the estimated effect of GDP per capita on life expectancy in Eastern Europe?

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lifeexpect~y | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

EastEur | -4.772011 1.740877 -2.74 0.010 -8.332504 -1.211518

GDPPC | .0434839 .017324 2.51 0.018 .0080524 .0789154

EastEurGDPPC | .1163334 .0705772 1.65 0.110 -.0280132 .2606799

_cons | 78.69714 .7988712 98.51 0.000 77.06326 80.33101

• 1. Suppose the errors in the above model are heteroscedastic. What is the consequence? Be specific about what elements of the output are affected and how.
  2. Suppose that someone notes that obesity rates are not included in the model. What two conditions need to be true for this to be a problem? Discuss whether they will be true in this model.

• 1. We next report a model with life expectancy as a function of an Eastern Europe dummy, GDP per capita and (health) expenditures. One might expect that GDP per capita and health expenditures are correlated. If so which, if any, results are called into question. If not, why not?

reg lifeexpectancy EastEur GDPPC expenditures

Source | SS df MS Number of obs = 32

-------------+------------------------------ F( 3, 28) = 9.46

Model | 86.4403967 3 28.8134656 Prob > F = 0.0002

Residual | 85.3184104 28 3.04708609 R-squared = 0.5033

-------------+------------------------------ Adj R-squared = 0.4500

Total | 171.758807 31 5.54060668 Root MSE = 1.7456

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lifeexpect~y | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

EastEur | -2.666673 .8661047 -3.08 0.005 -4.440808 -.8925377

GDPPC | .0362486 .0166744 2.17 0.038 .0020927 .0704046

expenditures | .0596671 .1583984 0.38 0.709 -.2647973 .3841314

_cons | 78.60578 1.643044 47.84 0.000 75.24016 81.9714

• 1. Here we report results from GDP per capita as a function of Europe dummy and (health) expenditures. What is the effect of multicollinearity on the variance of the coefficient on GDP per capita in the previous output? Provide a specific value and explain it.

reg GDPPC EastEur expenditures if lifeexpectancy !=.

Source | SS df MS Number of obs = 32

-------------+------------------------------ F( 2, 29) = 3.97

Model | 3003.31835 2 1501.65918 Prob > F = 0.0298

Residual | 10959.3335 29 377.908051 R-squared = 0.2000

-------------+------------------------------ Adj R-squared = 0.1900

Total | 13962.6518 31 450.408123 Root MSE = 19.44

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GDPPC | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

EastEur | -18.96621 8.979425 -2.11 0.043 -37.33119 -.6012208

expenditures | 1.616321 1.73829 0.93 0.360 -1.93888 5.171523

_cons | 26.26179 17.63602 1.49 0.147 -9.807908 62.3315

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• 1. Here is a model of life expectancy with controls for expenditures and GDP per capita. Dummy variables for all regions except North America are included. Interpret the coefficient on Western Europe.

reg lifeexpectancy expenditures GDPPC EastEur LatinAmerica Asia WestEur MidEast

Source | SS df MS Number of obs = 32

-------------+------------------------------ F( 7, 24) = 11.21

Model | 131.521281 7 18.7887545 Prob > F = 0.0000

Residual | 40.2375257 24 1.67656357 R-squared = 0.7657

-------------+------------------------------ Adj R-squared = 0.6974

Total | 171.758807 31 5.54060668 Root MSE = 1.2948

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lifeexpect~y | Coef. Std. Err. t P>|t| [95% Conf. Interval]

-------------+----------------------------------------------------------------

expenditures | .2892163 .1295467 2.23 0.035 .0218452 .5565875

GDPPC | .0229366 .0136133 1.68 0.105 -.0051598 .051033

EastEur | 2.079226 1.159876 1.79 0.086 -.3146406 4.473093

LatinAmerica | 4.265266 1.695908 2.52 0.019 .7650835 7.765448

Asia | 5.820845 1.195759 4.87 0.000 3.35292 8.28877

WestEur | 4.749333 1.039352 4.57 0.000 2.604216 6.89445

MidEast | 6.514199 1.679054 3.88 0.001 3.048802 9.979597

_cons | 72.30463 1.799617 40.18 0.000 68.5904 76.01885

• 1. Suppose we had instead excluded Middle East (and included North America). What would be the (approximate) value of the coefficient on the Eastern Europe dummy variable? Explain how you got the result.

1. Approximate the power of a test for a test of H₀: ₁= 0 vs H_A: ₁>0 with = 0.05. The standard error of is 1.

• 1. What is the value needs to be greater than in order to reject the null hypothesis? [2 points]
  2. Sketch the distribution of when the true value of  =1.6 and N is large? Describe the shape (e.g. what type of distribution?), the mean and the variance of this distribution. [2 points]
  3. Approximately what is the power of the test when the true value of  =1.6? [3 points]
  4. Approximately what is the power of the test when the true value of  =10? [3 points]

1. Circle the Stata syntax to limit a regression to observations for which College (a dummy variable indicating college graduates) equals 1.

[R users: explain how you limit sample in R]

• 1. reg Y X1 X2 if College==1
  2. CollegeReg Y X1 X2
  3. reg Y X1 X2[College==1]
  4. multicollinear reg Y X1 X2

1. Suppose a null hypothesis is: H₀: β₁ =0. If I fail to reject this null, then we know that the true value of the coefficient is equal to zero.

Explanation

True or false

1. An observation is influential if its value of X is very different from the values of X for other observations.

Explanation

True or false

1. Suppose we are designing an experiment in which you can determine the value of all independent variables for all observations. True or false: We want the independent variables to be not correlated with each other.

True or false

Explanation