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University of California, Los Angeles - ECON 103 BEGINNING ECON103 EXAM PART 1: TRUE/FALSE/EXPLAIN 1)When we drop a variable from a model, the total sum of squares (TSS) increases

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University of California, Los Angeles - ECON 103

BEGINNING ECON103 EXAM

PART 1: TRUE/FALSE/EXPLAIN

1)When we drop a variable from a model, the total sum of squares (TSS) increases.

2. If X and Y are independent, the conditional probability density function f_(X|Y ₎(X|Y ) is equal to the marginal probability density function f_X(X)
3. The Adjusted R-squared (R¯²) is always greater than or equal to the R-squared (R²).

4. 

    is an unbiased estimator of µ

 

5. In the expression Pr(Y = 1) = ( ₀ + ₁X) from a probit model, ₁ cannot be negative, since probabilities have to lie between 0 and 1.
6. If the true model is Y = ₀+ ₁X₁+ ₂X₂+" but you omit X₂ and estimate Y = ₀+ ₁X₁+", your estimate of ₁ will always be biased.
   1. The first one is that the omitted variable (in this case X₂) causes Y . Since the question says that, according to model considered, X₂ is one of the variables that explains Y , this condition is likely to hold in this case.
   2. The second condition is that the omitted variable is correlated with the variable whose coe cient may be biased. As long as X₂ is not correlated with X₁, the estimate of ₁ will still be unbiased even if we omit X₂.

7. In the following model, Wage = ↵₀+↵₁Educ+↵₂Female+↵₃Black+↵₄Female?Educ+u, to check whether the returns to education are the same for males and females you would have to test a joint hypothesis with an F test.
8. Consider the model Consumption = ₀ + ₁Wage + ". The sample regression function estimated with OLS gives you the average (or expected) value of Consumption for each value of Wage.
9. Suppose you run a test of the hypothesis H₀ : ₁ = 0 against the two-sided alternative H₁ : ₁ =6 0. Your t-statistic takes the value -2.001. You therefore reject the null at the 10 % significance level.
10. A random variable X can only take on the following values: 0 with probability b; 10 with probability 4b; 20 with probability 4b; and 100 with probability b. Therefore b must be equal to 0.1

PART 2

You estimate the following linear probability model by OLS:

J_i =   ₀ + ₁GPA_i + ₂Female_i + ₃GPA_i ? Female_i + ₄Black_i + ₅GPA_i ? Black_i + "_i,

where J_i is a binary variable equal to 1 if student i obtained a job o↵er within 2 months of

graduating from UCLA and equal to 0 if he didn’t.

1. Explain how you would test the null hypothesis that the probability of finding a job within 2 months of graduation is the same for men and women.
2. Explain how you would test the null hypothesis that the increase in the probability of finding a job within 2 months of graduation associated to a 10-point increase in the GPA is the same for men and women.

3. Explain whether the following statement is true and why: If V ar(") is not constant, OLS will give you biased estimates of the model coe cients. (5 points.)
4. Somebody points out to you that your estimate of ₁ is biased, as there is an omitted variable that hasn’t been included in the regression. Provide an example in which this statement would be true, specifying which two conditions must hold.
5. You re-estimate the model adding as a regressor the variable that was causing the bias. Based on your answer to the previous point, would you expect your new estimate of ₁ to be higher or lower than before?

PART 3

A researcher is studying the relationship between the time spent watching TV and people’s attitudes towards immigration.

She has interviewed a random sample of individuals, asking them how many hours of TV they watch every week and whether they think more immigrants should be allowed into the country.

The following table is a summary of her data. Cell entries (except in the final row) are column percentages:

		Hours spent watching TV ( H )
		Less than 5	5 or more	All
More immigrants should be admitted ( A )	Yes No	0.42 0.58	0.19 0.81	0.34 0.66
Sample size		200	100	300

Suppose the researcher has asked you for help running her econometric analysis and has provided you with her data. Follow the steps below to analyze them:

1. Write down the linear regression model you would use to estimate the e↵ect of hours of TV watched on the willingness to allow more immigrants into the country (call this e↵ect ₁).

We’d expect ˆ₁ to be negative, as increasing the numbers of hours of TV watched seems to decrease the tolerance for immigrants.

2. You read a study showing that poorer people spend more hours watching TV. Would this cause you to worry about the presence of bias in your estimate of ₁? In what direction would the bias go? (Explain all assumptions you make about the signs of any correlation).
3. You doubt people can remember accurately the exact time they have spent watching TV.

Would this raise concerns about your estimate of       ₁? Why? .

4. You worry that the e↵ect of hours watching TV on the willingness to admit more immigrants may not be linear. How would you change the regression in point 1. to allow for a non-linear e↵ect? How would you test whether the relationship is linear?
5. You realize that your dependent variable can be expressed as the following dummy variable:

:

 

6. For the probit regression above you obtained ˆ₀ = 0.50 and ˆ₁ = 0.01. What is the probability that a person who spends 2 hours a week watching TV will want more immigrants to be admitted? What is the same probability for a person who spends 10 hours a week watching TV? Which of the 2 probabilities is higher? (You don’t need to provide the exact values, just set up the computation)

PART 4

You are trying to determine the e↵ect of social security on individual saving decisions in order to advise the President. You have data for a representative sample of the population that includes information on the level of social security benefits, individual income, and annual savings.

1. You believe that the total savings of an individual (S) are determined as a function of social security benefits (SSB), their income (I), and by the state’s location in the country ( there are 4 possible locations: Northeast (N), South (S), Midwest (M) and West (W)). Write down a regression model for the determination of total savings according to your beliefs.
2. Describe how you would test the following hypotheses. For each one of the hypotheses below, write down the corresponding statistic and any other regression you may need to run beyond the one used in

Part 5  

 

The following dataset contains information on students in a class.  The data is as follows:

 

 

  Obs:   680

 

  attend                   classes attended out of 32

  termgpa                  GPA for term

  priGPA                   cumulative GPA prior to term

  frosh                    =1 if freshman   soph                     =1 if sophomore   junsen                   =1 if junior or senior

 

Note that all students are either a freshman, sophomore, junior, or senior, but no student can be more than one.

  sum

 

    Variable |       Obs        Mean    Std. Dev.       Min        Max -------------+--------------------------------------------------------       attend |       680    26.14706    5.455037          2         32      termgpa |       680       2.601     .736586          0          4       priGPA |       680    2.586775    .5447141       .857       3.93        frosh |       680    .2323529    .4226438          0          1         soph |       680    .5764706    .4944814          0          1 

 

reg termgpa frosh soph

 

      Source |       SS       df       MS              Number of obs =     680

-------------+------------------------------           F(  2,   677) =    7.29

       Model |  7.76373509     2  3.88186754           Prob > F      =  0.0007

    Residual |  360.633802   677  .532693946           R-squared     =  0.0211

-------------+------------------------------           Adj R-squared =  0.0182

       Total |  368.397537   679  .542558964           Root MSE      =  .72986

 

------------------------------------------------------------------------------      termgpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------        frosh |  -.1359634   .0864241    -1.57   0.116    -.3056549    .0337281         soph |   .1224413   .0738685     1.66   0.098    -.0225977    .2674802        _cons |   2.562008   .0640129    40.02   0.000      2.43632    2.687695

------------------------------------------------------------------------------

 

 

1. What is the interpretation of the constant in the above regression?

 

 

 

 

 

 

 

 

 

2. What is the interpretation of the coefficient on soph?

 

 

 

 

 

 

 

 

 

 

3. What is the interpretation of the coefficient on frosh?

 

 

 

. reg termgpa attend

 

      Source |       SS       df       MS              Number of obs =     680

-------------+------------------------------           F(  1,   678) =  309.40

       Model |  115.436802     1  115.436802           Prob > F      =  0.0000

    Residual |  252.960735   678  .373098429           R-squared     =  0.3133

-------------+------------------------------           Adj R-squared =  0.3123

       Total |  368.397537   679  .542558964           Root MSE      =  .61082

 

------------------------------------------------------------------------------      termgpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------       attend |   .0755857   .0042971    17.59   0.000     .0671484     .084023        _cons |   .6246566   .1147732     5.44   0.000     .3993031    .8500102

------------------------------------------------------------------------------ 

. reg termgpa attend priGPA

 

      Source |       SS       df       MS              Number of obs =     680

-------------+------------------------------           F(  2,   677) =  371.21

       Model |  192.687884     2  96.3439422           Prob > F      =  0.0000

    Residual |  175.709652   677  .259541584           R-squared     =  0.5230

-------------+------------------------------           Adj R-squared =  0.5216

       Total |  368.397537   679  .542558964           Root MSE      =  .50945

 

------------------------------------------------------------------------------      termgpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------       attend |   .0463716   .0039639    11.70   0.000     .0385886    .0541547       priGPA |   .6848606   .0396966    17.25   0.000     .6069173    .7628038        _cons |  -.3830615   .1121398    -3.42   0.001    -.6032451   -.1628779

------------------------------------------------------------------------------

 

 

4. In the first regression on this page, what is the interpretation of the coefficient on attend?

 

 

 

 

 

 

 

 

5. Describe why you think the coefficient on attend decreased when we included priGPA in the regression.

 

Omitted variable bias.

 

 

Hence the coefficient on attend in the first regression was biased due to the omission of the cumulative GPA (it was biased upwards, i.e. was too high). 

 

Once we separately controlled for the effect of the cumulative GPA, the (now unbiased) coefficient on attend became lower.

. reg termgpa attend priGPA frosh soph

 

      Source |       SS       df       MS              Number of obs =     680

-------------+------------------------------           F(  4,   675) =  186.13

       Model |  193.220488     4   48.305122           Prob > F      =  0.0000

    Residual |  175.177049   675  .259521554           R-squared     =  0.5245

-------------+------------------------------           Adj R-squared =  0.5217

       Total |  368.397537   679  .542558964           Root MSE      =  .50943

 

------------------------------------------------------------------------------      termgpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------       attend |   .0456313   .0039993    11.41   0.000     .0377788    .0534837       priGPA |   .7018152   .0421139    16.66   0.000     .6191251    .7845053        frosh |   .0869786   .0620993     1.40   0.162    -.0349525    .2089097         soph |   .0321174   .0516699     0.62   0.534    -.0693357    .1335705        _cons |   -.446286   .1212573    -3.68   0.000    -.6843729   -.2081992

------------------------------------------------------------------------------ 

 

6. What is the interpretation of the coefficient on priGPA?

 

 

7. would you reject the null hypothesis that the coefficient on attend is equal  to zero?

 

 

8. Would you reject the null hypothesis that the coefficient on priGPA is different from 1? 

 

 

 

 

 

 

9. Would you reject the null hypothesis that the coefficients on attend, priGPA, frosh and soph are all jointly equal to zero?

 

 

.  gen fr_attend=frosh*attend

 

. gen soph_attend=soph*attend

 

. reg termgpa attend priGPA frosh soph fr_attend soph_attend

 

      Source |       SS       df       MS              Number of obs =     680

-------------+------------------------------           F(  6,   673) =  124.43

       Model |  193.744059     6  32.2906766           Prob > F      =  0.0000

    Residual |  174.653477   673  .259514825           R-squared     =  0.5259

-------------+------------------------------           Adj R-squared =  0.5217

       Total |  368.397537   679  .542558964           Root MSE      =  .50943

 

------------------------------------------------------------------------------      termgpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------       attend |   .0552681   .0081861     6.75   0.000     .0391948    .0713414       priGPA |   .7026836    .042133    16.68   0.000     .6199557    .7854115        frosh |   .4749743   .2936508     1.62   0.106    -.1016077    1.051556         soph |   .3148917   .2459288     1.28   0.201    -.1679882    .7977717    fr_attend |  -.0150505   .0111231    -1.35   0.176    -.0368907    .0067897  soph_attend |  -.0109867   .0093063    -1.18   0.238    -.0292595    .0072861        _cons |  -.6957827   .2217804    -3.14   0.002    -1.131247    -.260318

 

 

THE FOLLOWING QUESTIONS REFER TO THE REGRESSION ON THIS PAGE

 

10. Set up the F-statistic for the null hypothesis that the interaction terms are all equal to zero.  What is the critical value you would compare it to (use a 5% level of significance, and check the critical values provided on the first

page of the exam)?  

 

 

 

 

 

 

 

11. What is the interpretation of the interaction term fr_attend?  

 

 

12. In the regression above, what is the standard error on the coefficient on attend?

 

 

 

13. What is the Adjusted R-squared?

 

 

 

 

 

 

14. What is the Total Sum of Squares?

 

 

 

 

15. What is the coefficient on frosh?

 

 

 

 

Part 6

 

Consider again the following regression (for the same variables as defined in the previous exercise):

 

. reg termgpa attend

 

      Source |       SS       df       MS              Number of obs =     680

-------------+------------------------------           F(  1,   678) =  309.40

       Model |  115.436802     1  115.436802           Prob > F      =  0.0000

    Residual |  252.960735   678  .373098429           R-squared     =  0.3133

-------------+------------------------------           Adj R-squared =  0.3123

       Total |  368.397537   679  .542558964           Root MSE      =  .61082

 

------------------------------------------------------------------------------      termgpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------       attend |   .0755857   .0042971    17.59   0.000     .0671484     .084023        _cons |   .6246566   .1147732     5.44   0.000     .3993031    .8500102

------------------------------------------------------------------------------

 

 

You are concerned that the coefficient of attend may be biased due to an omitted variable. You decide to run an Instrumental Variables (IV) regression. Your instrument for “attend” is the variable “distance”, which measures how far the students live from College.

 

The following is part of the output from your first-stage regression:

 

. reg attend distance

 

------------------------------------------------------------------------------       attend |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------

    distance |   -.089651   .0029648            _cons |   1.566222   .4534568 

------------------------------------------------------------------------------ 

1. Is “distance” a relevant instrument? Why?

 

 

 

 

2. Which other condition should “distance” fulfill for it to be a valid instrument?  

 

 

 

      Source |       SS       df       MS              Number of obs =     680

-------------+------------------------------           F(  1,   677) =  

       Model |  122.687884     1   96.3439422          Prob > F      =  0.1027

    Residual |  245.709652   677  .259541584           R-squared     =  0.1230

-------------+------------------------------           Adj R-squared =  0.1216

       Total |  368.397537   679  .542558964           Root MSE      =  .10945

 

------------------------------------------------------------------------------      termgpa |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval] -------------+----------------------------------------------------------------   attend_hat |   .0463716   .0339639            _cons |  -.3830615   .1121398    

------------------------------------------------------------------------------

 

 

 

3. What is the 95% confidence interval around the coefficient on attend_hat?

 

 

 

 

4. Is the coefficient on attend_hat statistically significant?

 

 

 

 

 

 

 

 

 

5. Do these results provide evidence that the coefficient on attend was biased (before you instrumented the variable)?

 

 

 

 

 

 

6. Would you worry about the standard errors provided in the last regression? Why?

 

 

 

 

 

 

 

 

 

 

 

 

7. How could you obtain a valid standard error for the coefficient on attend_hat?