POLS6481-Assignment 1 Solved

In 2020, some political leaders urged re-opening the economy in spite of the Coronavirus. For example, on March 26, President Trump stated the following, “You have suicides over things like this when you have terrible economies. You have death. Probably — and I mean definitely — would be in far greater numbers than the numbers that we’re talking about with regard to the virus.”

Year 
Unemployment rate 
(per million population) 
1968 
3.6
107
1969 
3.5
110
1970 
4.9
115
1971 
5.9
117
1972 
5.6
120
1973 
4.9
120
1974 
5.6
121
1975 
8.5
127
1976 
7.7
125
1977 
7.0
133
1978 
6.0
125
To support this claim, one might cite the article “An Economic Theory of Suicide,” published in the Journal of Political Economy in 1974. Hamermesh and Soss write: “When unemployment rises, individuals’ expectations of future incomes are revised downward. … People will believe future prospects to have diminished and will commit suicide.” To test this claim, examine the following table:  Suicide rate 

Load the data (csv), and use R to find the sample means, sample variances, and sample standard deviations of suicide rate (y), and find the same quantities for unemployment rate (x).
𝑦𝑦 =       𝑥𝑥̅ =    var(y) = sy2 =     var(x) = sx2 =      sy =        sx =

If you wish, you can download the optional file “homework 1 problem 1.pdf” and fill it out.

Use R to find the covariance between the unemployment and suicide rates, and then calculate the correlation (Pearson’s r) between the unemployment and suicide rates.
cov(x,y) =                                            rxy =

Using the quantities you computed for a. and b., obtain the intercept and the slope estimates in for a linear model in which the suicide rate is the dependent variable and the unemployment rate is the independent variable. That is, obtain the intercept and slope estimates in the sample regression function: 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑟𝑟𝑟𝑟𝑟𝑟𝑠𝑠 = 𝜷𝜷𝟎𝟎 + 𝜷𝜷𝟏𝟏𝑠𝑠𝑢𝑢𝑠𝑠𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑢𝑦𝑦𝑢𝑢𝑠𝑠𝑢𝑢𝑟𝑟 𝑟𝑟𝑟𝑟𝑟𝑟𝑠𝑠
𝜷𝜷𝟏𝟏 = cov(x,y) ÷ var(x) =

𝜷𝜷𝟎𝟎 =  𝑦𝑦 – 𝜷𝜷𝟏𝟏𝑥𝑥̅ =

Use R to verify your calculations for the slope and intercept: run the linear regression model, and display the results.
Verify the property that the regression line passes through the point whose Cartesian coordinates are the average unemployment rate and the average suicide rate. You can do this by computing the predicted suicide rate when the unemployment rate is set at its mean.
𝜷𝜷𝟎𝟎 + 𝜷𝜷𝟏𝟏𝑥𝑥̅ =

Use R to compute the fitted values (𝑦𝑦𝑖𝑖) and residual (𝑠𝑠𝑖𝑖) for each observation, filling in the table below – especially the bottom two rows (sums and averages). Round to one decimal point. Verify two important properties of the residuals: E(𝑠𝑠) = 0, and E(𝑠𝑠x) = 0.
 

yi
xi
𝒚𝒚𝒊𝒊
𝒖𝒖𝒊𝒊
xi⋅𝒖𝒖𝒊𝒊
107
3.6
 
 
 
110
3.5
 
 
 
115
4.9
 
 
 
117
5.9
 
 
 
120
5.6
 
 
 
120
4.9
 
 
 
121
5.6
 
 
 
127
8.5
 
 
 
125
7.7
 
 
 
133
7.0
 
 
 
125
6.0
 
 
 
Sum (Σ·) =
 
 
 
        Average (       ·) =

𝑛𝑛
 
 
 
 

 

Are countries that are more inclusive in terms of political representation also more equitable in terms of the income distribution? Download “inequality and representation.csv” which contains data on 20 OECD countries – advanced, industrialized countries that accept the principles of a freemarket economy and representative democracy. Inequality refers to the ratio of the richest 10%’s wealth to the poorest 10%’s wealth, and Representation refers to the percent of seats in parliament occupied by women.

Country
Inequality
Representation
Australia
12.2
28
Austria
6.7
32
Belgium
8.0
36
Canada
9.2
25
Denmark
7.9
37
Finland
5.4
38
France
8.9
14
Germany
6.7
31
Greece
10.0
13
Ireland
9.2
14
Italy
11.4
16
Japan
4.2
11
Netherlands
9.0
34
NewZealand
12.2
32
Norway
6.0
38
Portugal
14.6
21
Spain
10.0
31
Sweden
6.0
45
Switzerland
8.8
25
UnitedKingdom
13.6
19
Using R, estimate the OLS regression in which Representation is the dependent variable, and
Inequality is the independent variable. Is the slope statistically significant? The slope coefficient, _______, tells us that for every one-unit increase in the inequality ratio (e.g., from 5 to 6) the percent of seats occupied by women decreases by roughly _____.

To continue interpreting the results, the intercept, _______, tells us that _____ percent of seats would be occupied by women if the richest 10% held none of the wealth, which is obviously illogical. Compute a more logical (if implausible) prediction: in a perfectly equitable country, in which the top 10% and bottom 10% hold equal wealth (i.e., the ratio equals 1), roughly _____ percent of seats would be occupied by women.
The next page provides a worksheet for calculating the variances of x and y, and the covariance; it is not necessary to fill in the first twenty rows in the first five empty columns, but at least use R like a calculator to find seven entries at the bottoms of the columns:
Means:      𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝒖𝒖𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝒚𝒚 = ________                   𝑹𝑹𝑰𝑰𝑹𝑹𝑹𝑹𝑰𝑰𝑹𝑹𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑰𝑹𝑹𝑰𝑰 = ________

Variances:          var(Inequality) = _______                                       var(Representation) = _______

Covariance:        cov(Inequality, Representation) = ________

While you are at it, calculate the sums of squared deviations for x and y too.

Using the quantities that you computed in d., obtain the intercept and the slope estimates in for a linear model in which Representation is the dependent variable and Inequality is the independent variable. That is, compute 𝜷𝜷𝟏𝟏 = cov(x,y) ÷ var(x) =           
𝜷𝜷𝟎𝟎 =  𝑦𝑦 – 𝜷𝜷𝟏𝟏𝑥𝑥̅ =

Look at the regression results provided by R, and find the standard error of the regression, sigma = ______ ; and the standard error of the regression slope, s.e.(𝜷𝜷𝟏𝟏) = ______ . Fill in the orange circles on the next page to show where these values come from. Fill in the last three columns of the table on page 5, using R as needed to make life easier.
Use R to construct a scatterplot with Inequality on the horizontal axis and Representation on the vertical axis. Include a fitted line if you can. Next, examine the scatterplot and identify which case is an outlier: ________
Use R to calculate hat values, studentized residuals, and DfFits; report these values only for the apparent outlier. Hat-value (leverage) = ________ for the outlier
Studentized residual (discrepancy) = ________ for the outlier

DfFits (influence) = ________ for the outlier

Use the Table on page 6 and the blank space below it to calculate these values by hand only for the apparent outlier.

Use R to calculate the values of DfBeta for the outlier, and calculate what the intercept and slope should be if you had (i) omitted the outlier or (ii) included a dummy variable for the outlier. i.e., without the outlier, the slope would have equaled ________ and the intercept would have equaled ________
Adapt the code from the lab for week 2 and/or lecture 4, and either omit the outlier from the analysis or include a dummy variable for the outlier. Show your results! Then use the results to answer these three queries:
What is the new intercept without the outlier? ________

What is the new slope without the outlier? ________

Is the slope statistically significant without the outlier? ________

Extra Credit! Use the alternative model you estimated for i., and fill in the last three columns on page 5 for all countries. Attempt to calculate the value of Cook’s D for the outlier based on these columns. (The lecture slides show the equation.) Use R to confirm your calculation.
 

 

Country Inequality Representation                                                   (Fitted)     (Residual)   (Residual)

(y—y) (x—x)•(y—y) y = 𝛽𝛽 + 𝛽𝛽 x u = y-y u = (y-y)

 

 

 

 

Studentized                   Alternative      Squared

Country  Inequality Representation  Leverage     residuals      DfFits                                                   fitted values  Differ.   Differ.

i
xi
yi
+

𝑢𝑢
SST𝑥𝑥
𝑠𝑠𝚤𝚤

𝑟𝑟𝑖𝑖 =

𝜎𝜎 ∙ 1 − ℎ𝑖𝑖
ℎ ti ×1−𝑖𝑖ℎ𝑖𝑖
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 
𝑦𝑦𝚤𝚤 = 𝛽𝛽0 + 𝛽𝛽1xi + 𝛿𝛿̃
yi − 𝑦𝑦𝚤𝚤
(𝑦𝑦𝚤𝚤 − 𝑦𝑦𝚤𝚤)2
Australia
12.2
28
 
 
 
 
 
 
 
Austria
6.7
32
 
 
 
 
 
 
 
Belgium
8.0
36
 
 
 
 
 
 
 
Canada
9.2
25
 
 
 
 
 
 
 
Denmark
7.9
37
 
 
 
 
 
 
 
Finland
5.4
38
 
 
 
 
 
 
 
France
8.9
14
 
 
 
 
 
 
 
Germany
6.7
31
 
 
 
 
 
 
 
Greece
10.0
13
 
 
 
 
 
 
 
Ireland
9.2
14
 
 
 
 
 
 
 
Italy
11.4
16
 
 
 
 
 
 
 
Japan
4.2
11
 
 
 
 
 
 
 
Netherlands
 9.0
34
 
 
 
 
 
 
 
NewZealand
12.2
32
 
 
 
 
 
 
 
Norway
6.0
38
 
 
 
 
 
 
 
Portugal
14.6
21
 
 
 
 
 
 
 
Spain
10.0
31
 
 
 
 
 
 
 
Sweden
6.0
45
 
 
 
 
 
 
 
Switzerland
 8.8
25
 
 
 
 
 
 
 
UnitedKingd
 13.6
19
 
 
 
 
 
 
 
Σ
 

 
 

 
 
 

 
 

 
 

 
 

 
 

 
 

 
Σ/n