Use the data in HTV to answer this question. See also Computer Exercise C 10 in Chapter 3 . (i)

step 1 So this is another problem with statistical inference. And this time we have an interesting regr

Use the data in HTV to answer this question. See also...

Key Concepts

Ordinary Least Squares (OLS) Regression

OLS is a statistical method used to estimate the parameters of a linear regression model by minimizing the sum of squared differences between the observed and predicted values. It provides a framework for understanding the relationship between one dependent variable and one or more independent variables, forming the basis for inference in many regression analyses.

Hypothesis Testing in Regression Models

This concept involves testing whether the estimated coefficients are statistically different from zero (or some other specified value), thereby gauging the significance of individual predictors. It includes t-tests for single parameters and F-tests for multiple parameters, helping determine if the proposed relationships in the model hold in the population.

Testing for Nonlinear Relationships

Testing for nonlinear relationships, such as comparing a linear to a quadratic relationship, involves including transformations of explanatory variables (e.g., squared terms) in the regression model. The aim is to ascertain whether a non-linear functional form provides a better fit to the data than a strictly linear model.

Parameter Restrictions and Equality Testing

This involves testing hypotheses that restrict parameters to be equal or set to specific values, commonly using techniques like the Wald test. It allows researchers to statistically evaluate constraints such as whether two predictors have identical effects on the dependent variable.

Joint Significance Testing

Joint significance testing assesses whether a group of variables, when considered together, contribute significantly to explaining the dependent variable. Typically, an F-test is used to determine if the added explanatory variables improve the model, indicating their collective explanatory power.

Multicollinearity and Variable Transformation

Multicollinearity refers to a situation where two or more explanatory variables in a regression model are highly correlated, which can distort the estimated coefficients. Using transformations like averaging correlated variables can mitigate multicollinearity, leading to more stable and interpretable estimates.

Causal Interpretation and Confounding Factors

This concept addresses the interpretation of estimated relationships as causal effects, highlighting the importance of considering potential confounding variables, omitted variable bias, and the direction of causality. It reminds analysts to be cautious when inferring causation from regression results, especially when dealing with observational data.

Transcript

00:01 So this is another problem with statistical inference.

00:06 And this time we have an interesting regression equation that we're going to look at that has a bunch of linear terms that we're used to working with.

00:16 But we also have the addition of a quadratic term or a squared term.

00:21 So you'll read about this in the problem, but just a heads up, we're adding a slight new element to this problem.

00:29 So part one of this problem asked you to estimate the following regression model.

00:38 And as i typically do, i'll just write it out in the function form.

00:44 And here the dependent variable is the education of the individual in the sample.

00:53 That would be the highest level of education completed by that person.

00:59 And their education is a function of both of functions.

01:03 Their parents, highest completed education, so their mother's education and their father's education, and as well as their own ability, and this is a measure that's hard to explain and not really in the data, but just think of it as innate ability.

01:25 And also, here's the quadratic term i was mentioning at the beginning, is ability squared.

01:35 And why would we have? and why would we have? this quadratic term in here.

01:42 And you can just think of this as, well, someone's ability might influence or increase their eventual educational attainment over their life.

01:55 But you could also think that potentially there's decreasing returns to the ability you have as a person from birth.

02:07 So at really high levels of ability, your educational attainment won't increase as much as it was if you had lower levels of ability compared to someone who had a little bit more than you.

02:21 So just that decreasing returns to ability type concept is captured in this quadratic term here.

02:29 So that's just some background, not relevant for actually solving the problem.

02:36 Par one asked you to report the results of this regression here in the usual form.

02:42 And so i'll just write out the intercept first.

02:48 I didn't include the intercept here, but the intercept i'll just write here.

02:52 So here's the coefficient for the intercept and then it's standard error.

03:01 And then i'll go along and start with the mother's education variable here.

03:08 So 0 .19.

03:09 It's a positive.

03:11 Positive coefficient.

03:12 Sense so if you have a mother with more education you're likely to also have a higher educational attainment if you're in this sample how about the father's education so also you should get a positive coefficient here which also makes sense both of these coefficient standard errors tell us that they're both these educational variables are statistically significant for sure.

03:44 How about the ability? so we'd hope this would be positive, and it is in a .401 there, with a, again, a small standard error, so definitely statistically significant here.

04:00 And finally, the square of ability is positive there.

04:08 So this is actually interesting.

04:09 So i hypothesized just a minute or two ago that there might be decreasing returns to a ability, meaning that at really high levels of ability, you don't really, you're not predicted to have that much higher educational attainment.

04:29 However, looks like i might have been a little wrong because this is a positive coefficient and it is statistically significant.

04:37 So that would actually represent increasing returns to ability.

04:42 So at high, higher levels of ability, you actually are predicted to have even higher.

04:49 And higher increases in your educational attainment across individuals.

04:55 That's interesting, very interesting result there.

04:59 So that's the first part of part one.

05:02 The other part they want you to do in this first little part is test the null hypothesis, the following null hypothesis, that education is linearly related to ability.

05:19 So the null hypothesis you can think of as, that the coefficient for ability squared equals zero, right? so the non -hypothesis is that education is just linearly related to ability.

05:40 So that's another way of saying that or representing that is that the coefficient on that the quadratic term is zero, which gives the alternative is that the relationship is quadratic.

05:56 So just change just a little bit to say that the alternative hypothesis is that this coefficient on the quadratic term does not equal zero.

06:10 So to test this hypothesis, you have to do one of our favorite tests.

06:15 We use this a lot is the f test, right? so you have to run your unrestricted regression, which includes the quadratic term.

06:25 So it includes the squared ability term and also run the restricted regression without the quadratic term and just plug in the f statistic formula, those sums of squared residuals.

06:39 So i'll just tell you what your f statistic should be.

06:44 Actually, i'll write out the actual numbers that i got here.

06:47 And we'll look at the f statistic together.

06:50 So your calculation should look somewhat like this.

06:57 This is the sum of squared residuals here for the restricted regression.

07:02 So without the quadratic term, and then subtract off of that, the sum of squared residuals.

07:12 Let's see if i can write this okay.

07:13 There we go.

07:15 Sum of squared residuals for the unrestricted regression, so that's including the quadratic term.

07:22 We only have one restriction here.

07:24 So just put up one here.

07:27 And then in the denominator again, as we always do, we repeat the sum of squared residuals for the unrestricted regression and divided by our sample size minus our independent variables plus 1.

07:52 So that would be.

08:00 And so once you do all this calculation, you should get the lf statistic of, let me just do this really quick.

08:17 All right.

08:17 It does kind of look like this will be a rather large f statistic at this point, so i'm expecting this to be relatively large.

08:27 And sure enough, so let's just say this is about equal to 37.

08:32 Should equal about 37.

08:35 So i'll circle that.

08:38 So if the f statistic is that large, that's obviously going to reject the null hypothesis.

08:46 So we can reject the null and accept the alternative hypothesis, which is that the quadratic term should be in the regression.

08:59 And another way of saying that is that education is quadratically related to ability, right? which we could have guessed up here just from this coefficient, the fact that it's statistically significant.

09:14 But that's a good check.

09:17 All right.

09:19 So part two, you have to do a similar, kind of a similar exercise, but this time we're testing whether, oh, let's what color, let's read, whether beta 1 equals beta 2, which is saying, does a year of mother's education have the same impact on an individual's education.

09:49 Does that the same impact as a year of father's education.

09:52 So looking at whether these coefficients are equal, the coefficients on these two variables.

10:01 So what you should do for part two is, first of all, run the regression.

10:11 So just run the regression from part one, the way that it's laid out.

10:16 Make sure you include the quadratic ability term, right? because we found out from part one that we rejected that null hypothesis.

10:27 So just run the regression as it's given.

10:32 And after that, i'm not sure what software package you use, but there should be some sort of option for this in the packages you use.

10:43 I use stata.

10:45 Something you can do in the syntax is just test whether two coefficients are equal.

10:51 And the syntax, and the syntax is as following.

10:54 So for our purposes, you would go test mother or mud, m -o -t -h education.

11:07 So test, m -e -e -e -e -e -e -e -e -w -e -e -e -w -e -e -e -e -e -e -e -e -gation, because those are the variable names in the dataset.

11:23 So that should just be in the command line in stata.

11:26 And if you're in another software package, there should be another way to do that sort of test and try and look that up if you don't know it.

11:35 But run that test, and what should pop out of that test, and this is what state it gives out...

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