In equation (4.42) of Chapter 4, using the data set BWGHT, compute the L M statistic for testing

In equation (4.42) of Chapter 4, using the data set BWGHT,...

Key Concepts

Lagrange Multiplier (LM) Test

The LM test is used in hypothesis testing to assess whether certain coefficients in a regression model can be restricted to zero. It operates under the idea of estimating a restricted model and using its residuals to form a test statistic that, under the null hypothesis, approximately follows a chi-squared distribution, thereby allowing researchers to determine if there is evidence against the imposed restrictions.

Restricted vs Unrestricted Models

In regression analysis, the restricted model is one where certain coefficients are set to values (typically zero) as imposed by the null hypothesis, while the unrestricted model freely estimates all coefficients. Comparing these models helps in testing whether the restrictions are valid, often using test statistics like the LM test, the F-test, or the Wald test.

Joint Significance Testing

Joint significance testing involves assessing whether a group of variables collectively has an impact on the dependent variable. Instead of testing each variable individually, hypotheses regarding multiple coefficients are evaluated simultaneously, providing a more comprehensive view of the effect of the combined predictors.

Data Consistency in Model Estimation

Ensuring that the data used in regression models are consistent across the restricted and unrestricted models is crucial. This typically means using only those observations that have complete data for all relevant variables across models, thus avoiding biases or inefficiency that might occur from differing sample sizes when comparing model performance.

Transcript

00:01 Hi everyone.

00:02 So this problem for chapter 5 is just a more practice in doing an f test.

00:12 So looking at whether more than one explanatory variables in your model are jointly significant.

00:20 And i've preloaded our equation.

00:22 I don't usually do this, but i preloaded the equation.

00:25 We'll be estimating onto the screen here.

00:28 And this problem is just very closely tied to equation 4 .42.

00:35 So this goes back to a problem in chapter 4.

00:41 And so this is what it is just looking at body weight here.

00:46 And this is, sorry, birth weight of a child of an infant.

00:52 And looking at that birth weight as a function of how much the mother smoked during pregnancy, that's this variable, the birth order of the child, so where the child fits into the family in general, so how many other kids are surrounding it.

01:13 And then the family income here is the third variable, and then the two variables that are more of interest and we'll be doing our test on is the mother's education, father's education.

01:28 So all five of these explanatory variables are thought to have a potential effect on the birth weight of a child.

01:42 The hypothesis, oh, sorry, first i'll say that the problem asks you to, with this equation, using the data set, bwght, so the data set you'll be using to do this problem is that's its name.

02:07 It's the birth weight dataset.

02:12 The problem asks you to compute the statistic for testing whether mother's education here and father's education, whether they're jointly significant.

02:22 And so when you hear joint significance, you should immediately think, all right, we will probably have to do an f test to find a solution to that.

02:37 That problem and estimate estimate that joint significance.

02:43 So the first thing you want to just make sure you stay up front is that you're going to have, we're going to have two restrictions in the model, the two restrictions.

02:58 And the restrictions again are going to be the mothers and the father's education.

03:06 Okay.

03:12 And your null hypothesis also you can state, and your null hypothesis could basically just be stated as beta 4 equals 0, comma, so and beta 5 equals 0, right? so once you have that all set up, that's where, oh, and you can also, something else you can, can also do is specify and just state the degrees of freedom.

03:59 And you're going to want to look at the degrees of freedom for the unrestricted model, which is this model that i've already written up here.

04:06 But the degrees of freedom are going to have n, so your sample size, n minus six.

04:18 So let's put that in there.

04:22 And has that six number come about i'll just write equals n minus k minus one where k is the number of explanatory variables in the unershicketed model so n minus 5 minus 1 okay we've stated all those things up front that'll all be useful as we go through the problem and computer f statistic to look at whether those two education variables are jointly significant the problem also want you to really think hard about making sure that when you estimate the restricted model for the f statistics so when you do let's just say when you think about the number of observations which i'll just say up front here you can figure this out pretty easily hopefully the the number of observations in this sample is 1 ,191 so you have that.

05:36 And just remember that that sample size is in the unrestricted model.

05:41 So we haven't taken the education variables out of the model yet.

05:46 There's all still included.

05:51 And the problem is just asking you to make sure that you use that same number of observations when you estimate the restricted model.

06:02 That's the way it has to work.

06:04 So i guess i can write this by saying, you just also have to make sure you use that also in the restricted model, right? and that's because some observations for which, some observations for the mothers and fathers education here, some of them might be missing data for one or both of these.

06:38 So that's why you would have potentially a greater number of observations in the restricted model.

06:47 So just make sure you use those two in both of those unrestricted and restricted models use the same number of observations.

06:58 All right, so once we have all this set up, what we need to do is just write our f so i'll just write it out explicitly for us and we want to use the r squared form of the s statistic here, which i'm not sure i've done my videos before.

07:19 I typically use the sum of squared residuals form but here we will use the r squared form so the r squared form of the f statistic is as follows so you have the r squared from your unrestricted model is that first term here minus the r squared from the restricted model so that's taking out those two education variables divided by q and that's a number of restrictions, and so that will equal two in our case, divided by 1 minus, again, the r squared from the unrestricted model, the full model, then that is divided by the degrees of freedom in the unrestricted models.

08:12 So again, we've already set ourselves up.

08:15 We've found out some information such as this degrees of freedom.

08:19 Let me just write out really quickly what these will be, so q is going to be two.

08:26 This degrees of freedom we found out up here n minus 6 right so 1 ,191 minus 6 so you should have the degrees of freedom will be 1 -185 here and now we just have to run those regressions right so the first regression you'll run let's write this out so we are clear on what we have to do here so there are let's just say regression sub -e you are.

09:04 So the unrestricted regression is the full model, right? it's the first one you want to run.

09:15 And then the second, let's put a little one here.

09:19 It's the first regression you want to run.

09:22 Second regression you want to run is the regression, the restricted regression.

09:28 So the regression dropping the two as education variables...

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