ECON20110/30370 Econometrics Semester 1 2019/20 Midterm Examination

For questions 1-16 consider the following regression model that relates the traffic accident fatality rates to a range of variables. The data are for 47 US States and observations are for 1988.

fat = β0 + β1  unrate + β2  ydr + β3  mpp + β4  jail + u                      (1)

where  fat  is the traffic accident fatality  rate  (per  100,000  inhabitants),  unrate  is state unemployment rate (measured in percentage points, i.e.  2 = 2%), ydr is the percentage of young (aged 15-24) drivers (i.e.   10 = 10%) amongst the driving licence holders, mpp measures the average miles travelled per inhabitant per year in the state, jail is a dummy variable =1 if the there is a mandatory jail sentence for a drunk driving conviction in the state (and 0 otherwise).

On the next page you can find some summary statistics and the regression output for the model in equation (1).

The summary statistics for these variables are shown in the following output

>  summary(fat_data[,c("mrall","unrate","yngdrv","mpp","jaild")])

fat

Min .      :12 .31 1st  Qu . :16 .27 Median  :20 .15 Mean      :20 .73 3rd  Qu . :24 .77 Max .      :32 .36

unrate

Min .      :  2 .40 1st  Qu . :  3 .95 Median  :  5 .20 Mean      :  5 .46 3rd  Qu . :  6 .55 Max .      :10 .90

ydr

Min .      :  7 .314 1st  Qu . :15 .592 Median  :16 .237 Mean      :16 .228 3rd  Qu . :17 .117 Max .      :22 .072

mpp

Min .      :  5790 1st  Qu . :  8043 Median  :  8538 Mean      :  8618 3rd  Qu . :  9297 Max .      :11812

jaild

Min .      :0 .0000 1st  Qu . :0 .0000 Median  :0 .0000 Mean      :0 .2979 3rd  Qu . :1 .0000 Max .      :1 .0000

The model is estimated on 47 observations using OLS reported in the following R output,

Call:

lm(formula  = mrall  ~  unrate  +  yngdrv  + mpp  +  jaild,  data  =  fat_data)

Residuals:

Min            1Q   Median           3Q         Max

-6 .3176  -2 .4686  -0 .5525    2 .0059    7 .0864

Coefficients:

Estimate  Std .  Error  t  value  Pr(>|t |)

(Intercept)  -1 .254e+01    5 .447e+00    -2 .302    0 .02637

unrate             8 .986e-01    2 .991e-01      3 .004    XXXX

ydr                   2 .965e-01    2 .454e-01      1 .208    0 .23375

mpp                   2 .675e-03    4 .834e-04      5 .533  1 .86e-06

jaild                1 .687e+00    1 .201e+00      1 .404    0 .16770

---

Signif .  codes:    0  ***  0 .001  **  0 .01  *  0 .05  .  0 .1      1

Residual  standard  error:  3 .597  on  42  degrees  of  freedom

Multiple  R-squared:    XXXX,Adjusted  R-squared:    0 .5328

F-statistic:  14 .12  on  4  and  42  DF,   p-value:  2 .212e-07

The correct interpretation: a number such as 4.9e _ 2 is written in scientific notation, it indicates 4.9 × 10 −2  = 0.049. We saw this when we looked at R output.                              Scientific notation is a standard tool taught as part of the School Maths curriculum; for a

further explanation of scientific notation see:  https://www.national5maths.co.uk/scientific- notation/,                                                                                                                              https://www.bbc.co.uk/bitesize/guides/zxsv97h/revision/1,                                                  https://www.mathsisfun.com/numbers/scientific-notation.html.                                            The way computers express numbers in scientific notation is with the ”e” or ”E” which you also saw in the exam, 4.9 × 10 −2  = 4.9e _ 2 = 4.9E _ 2 = 0.049.

If you incorrectly interpreted e as exp:  so thought 4.9e _ 2 meant 4.9 × exp(_2) you would (incorrectly) get 4.9 × 0.135 = 0.66.

If you incorrectly ignored everything after the e then you would have interpreted 4.9e _ 2 as 4.9 you would (incorrectly) get 4.9.

1. Which of the following statements about the estimated coefficients is correct?

A.  For an increase in 1 percentage point of unemployment rate (say from 4% to 5%) the fatality rate increases by 9.09 (deaths per 100,000).

B. As the percentage of young drivers drops by 1 percentage point (say from 13% to 12%) the fatality rate drops by 2.9 (deaths per 100,000).

C.  For an increase in 1 percentage point of unemployment rate (say from 4% to 5%) the fatality rate increases by 0.909 (deaths per 100,000).

D.  For an extra 1,000 miles travelled per person per year in a state, the fatality rate increases by 0.0027 (deaths per 100,000).

E.  None of the above.

2. What is the residual sum of squares (RSS) for the above regression?

A.  12.93841

B. 608.1052

C. 543.4132

D. 20.73

E.  None of the above.

3.  In addition to the above information you know that the sum of square total (SST) is 1273.856. The value for Multiple  R-squared has been omitted from the above regres- sion output. What is it?

A. 3.633

B. 0.336

C. 0.573

D. 0.533

E.  None of the above.

4. Which of the following statements is correct?

A.  Reject H0  : β2  = 0.18 versus HA  : β1   0.18 at the 5% significance level.

B.  Reject H0  : β2  = 0 versus HA  : β0   0 at the 10% significance level.

C.  Fail to reject H0  : β1  s _0.20 versus HA  : β1  > _0.20 at the 10% significance level.

D.  Fail to reject H0  : β3  = 0 versus HA  : β3   0 at the 5% significance level.

E.  None of the above.

5. You want to test the hypothesis that H0  : β2  = 0.5 versus HA  : β2  < 0.5. Which of the following decision rules is correct when testing at a significance level of 1%.

A.  Reject H0  if the t-statistic is larger than 2.423.

B.  Reject H0  if the absolute value of the t-statistic is larger than 1.96.

C.  Reject H0  if the t-statistic is larger than 1.96.

D.  Reject H0  if the t-statistic is smaller than -2.423.

E.  None of the above.

6. You want to test the hypothesis that H0  : β2  = 0.5 versus HA  : β2  < 0.5. Calculate the t-statistic.

A.  1.208

B. -0.829

C. 0.604

D. 0.829

E.  None of the above.

10. The coefficient on unrete  has been found to be statistically significant.  Which of the following statements is correct?

A.  Changes in the unemployment rate, unrete, cause changes in the fatality rate, }et, if the errors are normally distribution with constant error variance.

B. The unemployment rate, unrete, may be correlated with relevant omitted vari- ables, such as the  proportion of the  population that is  urban.   This  means that the effect of the unemployment rate, unrete, on fatalities, }et, can not be interpreted as causal.

C.  In order to interpret the estimated effect of the unemployment rate, unrete, on fatalities, }et, as causal we require E(uIunrate)  0.

D. There are no circumstances under which we would be able to interpret the estimated effect of the unemployment rate, unrete, on fatalities, }et, as causal.

E.  None of the above.

11. Assume that you want to conduct a joint hypothesis test that none of the explanatory variables but for the unemployment rate (unrate) are statistically significant.  Which of the following is the restricted model with which to conduct the required F-test?

A.  fat = β0 + β1unrate + u

B.  fat = β0 + β2ydr + β3  mpp + β4jail + u

C.  fat = β1unrate

D.  fat = β0 + β1unrate + β4jail + u

E.  None of the above.

12. Assume that you want to conduct a joint hypothesis test that none of the explanatory variables but for the unemployment rate (unrate) are statistically significant.  In addition to the above information you also learn that the R2  of the restricted model is 0.171. What is the F-test?

A.  13.180

B.  14.778

C. 0.943

D. 39.617

E.  None of the above.

13.  Consider the case where the estimated model is

ln(fat) = γ0 + γ1  unrate + γ2  ydr + γ3  mpp + γ4  jail + u                (2)

and we obtain 1   = 0.0467.  Which of the following statements about the estimated coefficient is correct?

A.  For an increase by 1% in the unemployment rate, the fatality rate increases by 0.0467 (deaths per 100,000).

B.  For an increase in 1 percentage point of unemployment rate (say from 4% to 5%) the fatality rate increases by 4.67%.

C.  For an increase in 1 percentage point of unemployment rate (say from 4% to 5%) the fatality rate increases by 0.0467 (deaths per 100,000).

D.  For an increase in 1 percentage point of unemployment rate (say from 4% to 5%) the fatality rate increases by 0.0467%.

E.  None of the above.

14. Which of the following statements is correct?

A. The p-value of a hypothesis test is the probability of the null hypothesis being correct.

B.  In a regression you have discovered an economically important effect if the null hypothesis of no effect can be rejected.

C. A null hypothesis can be rejected if the p-value is larger than the set significance level.

D. When performing a hypothesis test the significance level should always be set to 5%.

E.  none of the above.

15. Which of the following assumptions are not required for the OLS estimator of β2  to be unbiased.

A. The proportion of young drivers is different across states.

B. The variance of the error term is the same for each state.

C. The expected value of the error term, conditional on the proportion of young drivers, is zero.

D. The population model is defined by equation (1).

E. All of the above are required.

16. You receive extra information from another study: A state’s population density has been found to be positively related to fatality rates. Unemployment rates and population density are negatively correlated. Given this information, which of the following statements about the OLS estimator for β1  is correct?

A.  E[βˆ1] = 0

B.  E[βˆ1] = β1

C.  E[βˆ1] < β1

D.  E[βˆ1] > β1

E.  None of the above.