Principles of Econometrics (721066S), Solutions to Exercises 2
Santtu Karhinen, TA
Exercise 1.
a. Null hypothesis is the hypothesis to be tested. Then, data is used to compare the null
hypothesis to a second hypothesis called alternative hypothesis. The alternative hypothesis
specifies what is true if the null is not.
b. Considering the estimations we are doing, we can set hypotheses for our slope coefficient 𝛽!
for example as
𝐻! : 𝛽! = 𝛽!,! 𝑣𝑠. 𝐻! : 𝛽! ≠ 𝛽!,! (two-­‐sided)
𝐻! : 𝛽! = 𝛽!,! 𝑣𝑠. 𝐻! : 𝛽! < 𝛽!,! one-­‐sided
For example, in the student-teacher/test score problem, people can presume that smaller
classes provide a better learning environment. Then, it that example, 𝛽! would be negative,
so smaller classes lead to higher scores. Therefore, it could make sense to test for
𝐻! : 𝛽! = 𝛽!,! 𝑣𝑠. 𝐻! : 𝛽! < 𝛽!,! one-­‐sided
But in practice, one-sided alternative hypotheses should be used only when there is clear
reason for doing so. Therefore, one-sided tests are not often used by econometricians.
c. It is a set of values that cannot be rejected using a two-sided hypothesis test with a 5%
significance level. Also, it has an interval that has a 95% probability of containing the true
value of 𝛽! , meaning that in 95% of possible samples that might be drawn, the confidence
interval will contain the true value of 𝛽! . Then, because this interval contains the true value
in 95% of all samples, it is said to have a confidence level of 95%.
d. A regressor (variable) can be binary, which means that it can only have two values, 0 or 1.
For example, gender (female = 1, male = 0), small class size (depending on definition, small
= 1, large = 0) and so on.
e. Error term is homoskedastic if the variance of the conditional distribution of error term 𝑢!
given 𝑋! , 𝑣𝑎𝑟 𝑢! 𝑋! = 𝑥 , is constant for all i and does not depend on x. If the variance isn’t
constant across x, the error term is heteroskedastic.
Exercise 2.
a. 𝐴𝐻𝐸 = 1.08 + 0.60×𝐴𝑔𝑒. So on average, earnings increase by 0.60 dollars per hour when workers age increases by 1 year. b. A prediction for 28-­‐year-­‐old is 1.082275 + 0.604986×28 = 18.02. For 31-­‐year-­‐old the prediction is 1.082275 + 0.604986×31 = 19.84. c. 𝑅 ! is 0.029, so age explains only a small fraction (2.9%) of the variability in earnings across individuals. d. Correlation between age and average hourly earnings is 0.170364. In 2 variable regression the 𝑅 ! is equal to the square of the correlation coefficient 0.17! = 0.029 e. The standard error for the intercept is 0.40 and for the slope coefficient 1.17. The t-­‐statistics for the slope coefficient is then !.!"
!.!"
= 15.02, which has a p-­‐value of 0. Therefore, the null hypothesis of 𝛽! = 0 can be rejected at the 10%, 5% and 1% levels. f. 0.60 ± 1.96×0.04 = 0.53 to 0.68. g. In case of only bachelor degree (bachelor = 1), the estimated regression is 𝐴𝐻𝐸 = −4.44 + 0.92×𝐴𝑔𝑒 1.77 0.06 The pointed t-­‐statistics is calculated as !.!"
!.!"
= 15.13, which has a p-­‐value of 0.000, so the null hypothesis can be rejected at 10%, 5% and 1% levels. h. In case of only bachelor degree (bachelor = 1), the estimated regression is 𝐴𝐻𝐸 = 6.52 + 0.30×𝐴𝑔𝑒 1.25 0.04 The pointed t-­‐statistics is calculated as i.
!.!"
!.!"
= 7.02, which has a p-­‐value of 0.000, so the null hypothesis can be rejected at 10%, 5% and 1% levels. The difference in the estimated slope coefficients is 𝛽!,!"##$%$ − 𝛽!,!"#!!"!!!" = 0.92 − 0.30 =
0.62. The standard error for the estimated difference is 𝑆𝐸 𝛽!,!"##$%$ − 𝛽!,!"#!!"!!!" =
(0.06! + 0.04! )!/! = 0.07. The 95% confidence interval for the difference is then 0.62 ±
1.96×0.07 = 0.48 to 0.76. Exercise 3. a. Plot a scatterplot of growth and trade share. There seems to be a weak positive relationship. 10
8
6
growth
4
2
0
-2
-4
0.0
0.5
1.0
1.5
2.0
tradeshare
2.5
b. Malta is the outlying observations with a trade share almost 2. c. With all observations included, the estimated regression is 𝐺𝑟𝑜𝑤𝑡ℎ = 0.64 + 2.31×𝑇𝑟𝑎𝑑𝑒𝑆ℎ𝑎𝑟𝑒 Here the intercept is 0.640265 and the slope 2.306434. The coefficient on tradeshr is 2.31, so an increase in trade share by one unit is associated with an increase in the average annual growth rate of real GDP from 1960 to 1995 of 2.3 percentage points. For example, consider Argentina, to which the average tradeshr is 0.15, that is the share of trade in economy. Suppose now if tradeshr variable increases by one, from 0.15 to 1.15, what does that mean? It means that openness in economy increase 15 % to 115%! So, in this case it is much more meaningful to consider the effect via 0.01 increase in tradeshr variable, that is 1 percentage point increase in openness yields to 0.023 percentage point increase in growth. d. Then, the predicted growth for a country with 𝑇𝑟𝑎𝑑𝑒𝑆ℎ𝑎𝑟𝑒 = 0.3 is given by 0.64 + 2.31×0.3 = 1.333 And the prediction for a country with 𝑇𝑟𝑎𝑑𝑒𝑆ℎ𝑎𝑟𝑒 = 0.7 is given by 0.64 + 2.31×0.7 = 2.257 e. 95% confidence intervals You can also calculate the 95% confidence interval by hand using the formula: 𝛽! ± 1.96𝑆𝐸 𝛽! = 2.31 ± 1.96×0.663 f.
Malta excluded. Now the estimated regression is 𝐺𝑟𝑜𝑤𝑡ℎ = 0.96 + 1.68×𝑇𝑟𝑎𝑑𝑒𝑆ℎ𝑎𝑟𝑒 The similar predictions as in part d) yield 𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑𝐺𝑟𝑜𝑤𝑡ℎ 𝑇𝑟𝑎𝑑𝑒𝑠ℎ𝑎𝑟𝑒 = 0.3 = 0.96 + 1.68×0.3 = 1.464 𝑃𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑𝐺𝑟𝑜𝑤𝑡ℎ 𝑇𝑟𝑎𝑑𝑒𝑠ℎ𝑎𝑟𝑒 = 0.7 = 0.96 + 1.68×0.7 = 2.136 Dropping the outlier results in a moderate decline in the slope, from 2.31 to 1.68, and a pretty substantial increase in the standard error, from 0.66 to 0.87. These two changes combined make the t-­‐statistic fall from 3.48 to 1.94, so that the slope coefficient is no longer significant at the 5% level (although it is at the 10% level). Also, the 𝑅 ! falls from 0.12 to 0.04 when the outlier is dropped. Overall, the single outlier observation has a big effect on the results. g. The nature of its trade does not involve production. Many goods coming into Malta as imports are immediately transported to other countries as exports (transshipment). The large value of trade share therefore doesn’t accurately represent the value of economic activity related to the trade in Malta, and because it is highly influential in the estimation, Malta should be excluded from the analysis.