3. For a sample of 676 individuals, we estimated models to...

3. For a sample of 676 individuals, we estimated models to test for a tradeoff between minutes per week spent sleeping \textit{(sleep)} and minutes per week spent working \textit{(totwrk)}. \textit{log} denotes natural logarithm. v) (s) If the sample correlation between \textit{log(totwrk)} and \textit{yngkid} is not zero, would you drop one of these two independent variables from regression (2)? Whv or why not? vi) (ts) Consider regression (2). Suppose that A spends 5000 minutes on work per week and \textit{yngkid} = 1. B spends 4000 minutes on work per week and \textit{yngkid} = 0. What is the predicted difference in their \textit{log(sleep)}? vii) (ts) Consider the following estimated model with a quadratic term: $\hat{\text{sleep}} = 3608 - 21.490\text{age} + 0.301\text{age}^2$. At what age is sleep minimized? viii) (As) Which Gauss-Markov assumption has been violated in the following regression: $\log(\text{sleep}) = \beta_0 + \beta_1 \log(\text{totwrk}) + \beta_2 \log(\text{totwrk}^2) + u$? Explain your reasoning. What happens if you use Stata to run this regression?

Transcript

00:01 So we're given an incomplete table of the result of a regression analysis and we were given the equation that sleep is equal to beta naught, the constant, plus beta one age, plus beta two, my young kid.

00:30 And we have of course our error term, plus the error.

00:34 And we want to complete the missing elements of the table and then figure out what's going on here.

00:40 So let's do the sum of squares first.

00:42 So we have the sum of squares of the residual, i think this is listed as a, and then we've got b listed here for the adjusted square, and then c is right here.

00:54 So let's do this first.

00:56 So the sum of squares is such that the total sum of squares is equal to the sum of squares of the model plus the sum of squares of the residual.

01:07 So if we find the residual, what we do, we just take the difference of the total minus the model.

01:13 So we do sst minus ssn and that's going to give us ssr.

01:18 And the value we get, so we take this number minus that number, we get one, three, eight, zero, six, four, seven, four, one point nine.

01:35 Good.

01:39 And then for b, the adjusted r squared, this is a formula.

01:48 We'll call it r bar squared, the adjusted r squared.

01:52 That's usually the same.

01:54 It's one minus one minus r squared, which we have right here, multiplied by n minus one over n minus p minus one.

02:06 So p is the number of parameters.

02:08 So there are two parameters in our model, right? age and young kid.

02:13 So let's write this out.

02:14 So we get one minus one minus this number squared, 0 .0084 squared, multiplied by, so n minus one, n is this, the total number of observations.

02:28 So 706 minus one all over 706 minus two minus one, because p is our number of parameters.

02:36 And when we do that, we end up with is 0 .0056.

02:47 And the last thing, c.

02:51 So we have the coefficient standard, we have our test statistic.

02:55 So the way the coefficient standard and test statistics are related is that t is equal to the coefficient, we'll say beta y for young kid over the standard error of y.

03:07 So t, we're given 0 .44, the coefficient 23 .07262 all over this standard error term.

03:19 So what we do, multiply by the standard error, divide by this t value, so we get the standard error y is equal to 23 .07262 all over 0 .40.

03:32 And what we get is 52 .472 or 0 .4377.

03:46 Great.

03:49 Now, we're going to get this confidence interval.

03:54 Here's listed as d, this whole thing is d, this confidence interval is d.

03:58 So our confidence interval is calculated in the following way.

04:03 So this 95 % confidence interval, and this is for our age coefficient, beta age.

04:20 So we take our estimated coefficient for age, plus minus t, 0 .05 over 2, with our degrees of freedom, multiplied by the standard error of beta hat.

04:38 So, our age, we'll say, this degrees of freedom is right here, 703...

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