Use the data in TWOYEAR for this exercise. (i) The variable...

Use the data in TWOYEAR for this exercise. (i) The variable stotal is a standardized test variable, which can act as a proxy variable for unobserved ability. Find the sample mean and standard deviation of stotal. (ii) Run simple regressions of jc and univ on stotal. Are both college education variables statistically related to stotal? Explain. (iii) Add stotal to equation (4.17) and test the hypothesis that the returns to two- and four-year colleges are the same against the alternative that the return to four-year colleges is greater. How do your findings compare with those from Section 4-4? (iv) Add stotall to the equation estimated in part (iii). Does a quadratic in the test score variable seem necessary? (v) Add the interaction terms stotal jc and stotal univ to the equation from part (iii). Are these terms jointly significant? (vi) What would be your final model that controls for ability through the use of stotal? Justify your answer.

Use the data in TWOYEAR for this exercise.
(i) The variable stotal is a standardized test variable, which can act as a proxy variable for
unobserved ability. Find the sample mean and standard deviation of stotal.
(ii) Run simple regressions of jc and univ on stotal. Are both college education variables
statistically related to stotal? Explain.
(iii) Add stotal to equation (4.17) and test the hypothesis that the returns to two- and four-year
colleges are the same against the alternative that the return to four-year colleges is greater. How
do your findings compare with those from Section 4-4?
(iv) Add stotall to the equation estimated in part (iii). Does a quadratic in the test score variable
seem necessary?
(v) Add the interaction terms stotal jc and stotal univ to the equation from part (iii). Are these terms
jointly significant?
(vi) What would be your final model that controls for ability through the use of stotal? Justify your
answer.

Key Concepts

Model Specification

Model specification involves selecting the appropriate variables and functional forms for a regression model to ensure that it accurately reflects the data-generating process. A well-specified model controls for confounding factors, tests for non-linearities and interactions, and yields reliable estimates for inference.

Interaction Terms

Interaction terms in a regression model are used to assess whether the effect of one independent variable on the dependent variable depends on the level of another variable. They help in uncovering more complex interdependencies between variables that a model with only additive effects might not reveal.

Quadratic Specification

Quadratic specification adds a squared term to the regression model to capture potential non-linear relationships between the predictor and the outcome variable. This is important for detecting and modeling curvature in the data, which a simple linear term might miss.

Proxy Variables

Proxy variables are used in regression analysis to stand in for unobserved or difficult-to-measure factors. They allow researchers to control for latent attributes, such as ability or motivation, that may otherwise confound the estimated effect of other variables in a model.

Simple Linear Regression

Simple linear regression is used to model the linear relationship between a single predictor and a dependent variable. It helps in assessing the strength and direction of the association between these two variables, and in making predictions based on the estimated relationship.

Descriptive Statistics

This concept involves summarizing data using measures such as the sample mean and standard deviation to capture the central tendency and dispersion of a variable. Understanding these statistics is crucial for describing data distributions and identifying potential outliers before further analysis.

Hypothesis Testing in Regression

Hypothesis testing in regression involves assessing whether the estimated coefficients are statistically different from zero or from each other. This process typically uses t-tests or F-tests to determine if the relationships observed in the data are likely to be real or are due to random chance.

Transcript

00:01 All right, hello everybody.

00:03 Today we're going to be doing some more modeling.

00:06 So first things first, as always, you want to make sure you have the woldridge package installed.

00:12 This package contains all our data.

00:15 And then you want to just select the woldridge library.

00:18 Make sure we're using that.

00:19 We're going to be using one equation or one, we're going to be referencing one previous equation.

00:26 And that's going to be log wage.

00:28 Actually, sorry, let me read it.

00:29 It's going to be l wage, as it's referred to in the data.

00:31 Set is equal to jc plus university plus experience.

00:35 It's good to have that written down so we can refer back to it.

00:38 And all right, let's get started.

00:40 So first we're talking about the s total variable, which refers to standardized test total.

00:46 And it's a proxy variable for unobserved ability.

00:49 So our first thing we want to do is we want to find our sample mean and standard deviation.

00:54 Where we're going to do that is very simply, oh sorry, i forgot the data set.

00:58 So our dataset is 2 year, that'll select that.

01:02 We want to view 2 year.

01:05 We can view that now over here.

01:07 You can see all these different things.

01:09 If i scroll, you will see the various different categories.

01:13 So, to find the mean in standard deviation, very easy function, we're just going to take the mean of the s total column in 2 year, and then we're going to take the standard deviation of 3.

01:28 The s total column in two year.

01:31 And perfect, right there, very quickly, we have our mean and our standard deviation.

01:35 So now we want to run some basic regressions.

01:38 We want to run a regression of j .c on s total and of university on s total.

01:44 And we want to find out, are they statistically related? so that's pretty simple.

01:49 We're going to say our j .c regression is going to be a linear model.

01:54 And it's very simply, our formula is just going to be that of j .c.

01:58 On x total, y on x, obviously referencing the two -year data set.

02:03 And then our university regression is going to be the exact same thing, but instead of j -c, we're going to write university.

02:10 And then, again, dataset is two year.

02:13 I take the summary of these two, j -c -reg, and summary of univ -reg.

02:22 You will see, for our j -c -reg, this s -total is not statistically significant.

02:28 Right it's 0 .31.

02:31 You will also see for university however it is statistically significant you can see that from the three asterisk here showing it's significant at the 0 .1 % confidence level so it's obviously significant at 5 or also from the fact that our probability is less than 2 times 10 than negative 16th so okay it is statistically significant in determining university but not jc all right, cool.

03:00 So now we want to take our equation from before up here, this l -wage to j -c -university and experience, and we want to add s -total into that.

03:14 So our primary model is going to be lm, l -wage, j -c -plus university, plus experience, and now we're going to add in s total.

03:27 Our data is still two -year.

03:30 When we take the summary of our primary model, we will get this.

03:35 Again, notice, everything is significant at the 5th and also at the 0 .1 % confidence levels.

03:42 But there seems to be no real difference in the conclusion drawn because, you know, both j .c.

03:50 And university are still statistically significant with or without s total in the model.

03:55 And if you want me to really click do that, i can actually just do summary of lm, l -o -h, j -c -plus university, plus experience, data equals two -year.

04:08 So you can see that even without s total, all of these are still statistically significant, right? okay, cool.

04:18 So now we want to test the hypothesis that the returns are the same against the alternative that the return to a four -year college is greater.

04:31 So there are a few ways of doing this.

04:35 The one i like is actually a little method where our formula will change to make our lives easier.

04:44 So instead of saying jc plus university plus experience plus s total, we're going to have jc plus, sorry, let me check the column name.

04:55 Again.

04:56 Just don't want to mess it.

04:58 So you'll see we have somewhere in here we should have a, yeah, we have a variable called total college, which basically adds together our jc and our university, right, the number of years in each of these.

05:16 That variable combines those two variables...

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