# Labs for Unit B1: Correlation and Regression

## 1 Lab 1

### 1.1 Correlation

Give Stata the following command to load a file, which contains six
pairs of variables, `x1`

and `y1`

, `x2`

and `y2`

etc:

use http://teaching.sociology.ul.ie/so5041/labs/correl

First, graph all six pairs in scatterplots: `scatter x1 y1`

etc. What
sort of association do you see in each case (positive, negative, none,
strong, weak)? Make a guess what the value of the correlation
coefficient might be (write it down).

Then for each graph, get the correlation coefficient: e.g., `corr x1 y1`

. How
do the reported correlation coefficients correspond with those you
guessed?

### 1.2 Linear Regression

The following code will run a regression with county-level crime rate as the outcome, and county-level median income as the explanatory variable (the data are from Florida).

clear use http://teaching.sociology.ul.ie/ws/data/floridacrime scatter crime income reg crime income

Run the code and examine the output.

- Write out the \(Y = a + bX\) equation
- Report R
^{2} - Test the hypothesis that income is associated with crime
- Predict crime for income = 20 and income = 30
- Draw the regression line (on paper)
- Calculate the predicted value and residual (error term) for Columbia county and relate them to the observed value

You can verify your predicted values and the line, by getting Stata to do the work:

predict ypred list income crime ypred line ypred income || scatter crime income

### 1.3 NLSW:

Execute the following commandsto load the *National Longitudinal
Study of Women* data set that comes with Stata.

clear sysuse nlsw88

Generate a new variable containing the natural log of wage:

gen lw = ln(wage)

For technical reasons, the log of variables like wage often work better in regression models. Considering the following list of variables:

`age`

`ttl_exp`

, total lifetime work experience`grade`

, years of education`union`

, whether a member of a union

Let's consider wage (logged) as the "dependent variable", to be
explained by the others (ignoring `union`

for the moment as it only has
two values). Create scatterplots for log-wage (on the Y-axis)
compared with each of the other variables. Consider the correlations too
(e.g., `corr ttl_exp lw`

). Can you see much of a
relationship?

Now do regression analyses:

`reg lw`

varname

replacing *varname* with each of the other variables **one at a time** as the
independent. There are two key things to look at: the
R^{2} figure and the parameter estimate (`Coef.`

for the
independent variable, along with its significance). Which variables
affect wage much? Do any not affect it at all?

Interpret the results: in each case ask the question, "what happens to the predicted value of income, if the value of X were to change by one unit?". For two different values of the independent variable (X) calculate the predicted value of income – see where these fall on the scatterplot, and see where the regression line would lie. Does it seem like a good summary of the relationship?

If R^{2} is big, the independent variable "explains" the
dependent variable "a lot". However, it is possible for R^{2} to
be small and yet for the independent variable to a systematic effect
(i.e. very low p-value for significance): this independent variable may
be only one thing among many that affects the dependent variable.

### 1.4 Union effects

Test the effect of `union`

on logged wage. Note that `union`

in a
binary variable (i.e., yes/no represented as 1/0). Use a t-test in
the first instance, and then fit a regression. Compare the results.

Do the same relating `grade`

to `union`

. Note
that unionised workers tend to earn more and be better educated. Could
it be that the union effect is simply due to them being better educated?
That is, for workers with similar education does union status matter?

Fit the wage/grade regression for unionised and non-unionised workers separately, and think about the results (make scatterplots too). Do:

reg lw grade if union==0 predict p0 reg lw grade if union==1 predict p1

Having saved the predicted values, we can plot the two regressions simultaneously:

line p0 p1 grade || scatter lw grade

The `||`

syntax allows us to combine plots. The following example
exploits this a little more to make a plot that distinguishes by union
even better:

line p0 p1 grade || scatter lw grade if union==0 || scatter lw grade if union==1

### 1.5 Two explanatory variables

You can also fit a model with both union status and grade explaining
wage. Fit a regression with both `grade`

and `union`

as explanatory variables. Interpret the
parameter estimates.

Compare your results to the previous separate regressions, and the t-test.