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The exam May 29 is now corrected
You find it at the student's expedition
I gave 6 credits for problems 1–6 and 9 credits for problem 7.
Marks: A: 41–45cr. B: 36–40cr. C: 31–35cr. D: 26–30cr. E: 22–25cr. (There were none eligible for FX.)
The re-exam with short answers
The assignment projects
are now examined, and the resuls are recorded in LADOK. Those who wrote reports on paper can find them at the student's expedition.
The exam is corrected.
You find it at the student's expedition.
The different problems were given different weights (credits):
1+2: 10cr. 3: 6cr . 4: 6 cr. 5: 9 cr. 6: 4cr. 7: 5cr. 8: 5cr.
Marks: A: 39–45cr. B: 34–38cr. C: 28–33cr. D: 23–27cr. E: 18–22cr.
Thursday 28/2–13
I finished the presentation of ANOVA with my criticism of the heroic assumptions, and pointed out Welch’s t-test as an alternative. I also mentioned that method as an alternative to Wilcoxon’s rank sum test.
Tuesday 26/2–13
I finished the demosntration of exercise 36. A student noted that when we added “interaction”, the sum of SS did'nt add up to the total SS. I thought for a while that I had messed up the numbers, but I hadn’t! The reason is that the “interaction” covariate has a positive (1/8) covariance with the temperature and the detergent covariates, not zero. The sample covariance between a temperature and a detergent covariate is however zero.
The property that the sum of the individual SS equals the total SS (the SS of the dependent) is true if any pair of covariates from different sources has sample covariance zero, but not (necessarily) otherwise. An ANOVA model with no interactions and an equal number of replications will satisfy the condition that any pair of covariates from different sources has sample covariance zero, so in this case the of SS of the various sources will add up to the total SS. The jargon is that we have an “orthogonal design”.
Finally, I talked about “random models”.
BIC is defined on p.41 in the booklet. Maybe I made a typo on the board.
Thursday 14/2–13
I demonstrated some exercises: 13, 23, 28, 29b, 32 and 40.
Tuesday Jonas will demonstrate Logit regression, and more!
Deadline for assignment projects is Monday Mars 11
Sorry for erlier mistake!
Tuesday 12/2–13
I finished the presentation of White’s “robust errors” and 2SLS. Now you are prepared for the first assignment project. See below for details. The calculations should be done in EXCEL.
Earlier today Jonas made some demonstrations on EXCEL and and also discussed some exercises. In exercise 34 he identified two problems: 1) There is a selection bias in the way the unemployed individuals are sampled; those with large error terms will have a greater probability to be sampled than those with a small (i.e., large negative) error term, and 2) There is a self-selection bias in that participation in a labbour market programme is volunteer.
Thursday 7/2–13
I continued to talk about the important and difficult subject “endogeneity” in different forms and shapes. Next I introduced “heterogeneity” and described its consequences. I described White’s method to calculate a consistent heteroskedasticity robust covariance matrix, and started to describe my method to calculate his robust standard error for a coefficient, but time didn’t permit to finish it completely. I will do that next time.
Some of you have asked me about the assignment projects. You are not yet prepared for them – for the first project you need 2SLS which I will describe shortly. There is no hurry; the deadline is Monday Mars 11 for both projects.
The assignment projects are individual, and you shall write like a short report. Pretend that you as a consultant has got the commission to investigate if returns to schooling is the same for whites and blacks, and that you now report your findings to your client. (But make it reasonably short!)
Tuesday 5/2–13
The first two hours, Jondas demonstrated exercises 11, 14, 21 and 30.
The last two hours, I demonstrated exercises 33 and 28. Next I started to talk about “When Everything is Not so Perfect”. I described the problem with multicollinearity and defined endogeneity and gave a few examples. Next Thusrday I will continue with this (very important) issue.
Brief answers to some exercises:
7. 1.99
8. 4.52⋅10-4
10. 0.1095
11. 0.9677
28c. Yes, it reduces the standard error of the estimate of β1.
Tuesday 29/1–13
I finished the derivations of the variouos formulae leading up to the t-distributions if the ratios (estimated_coeffficient–true_value)/Standard_error_of_coefficient. Then I gave examples on the use of interaction covariates and e.g. a covariate squared. I talked about prediction and the “useful trick” to calculate a prediction and a prediction interval.
Finally I pointed out the differences between structural interpretation of a model versus a predictive interpretation.
Here are some exercises you can do. Jonas will demonstrate some of them next Thursday:
11, 12, 19, 21, 27.
You may also test if returns to education is the same for males and females using the cps85.xls data as I demonstrated today.
Earlier exams has been asked for by some of you. here is a link but note that I do not consider these to be part of the course material! Hence I or Jonas will not answwer questions about these unless we find it appropriate.
Thursday 24/1–13
I described how to calculate confidence intervals for the coefficients in an OLS regression. Then I started with the theory: assumptions underlying the linear model, and the method of least square. I derived the “normal equations” and got about half way of the derivation of the standard errors of the coefficient estimates.
The last two hours Jonas did some explanation of confidence intervals for parameters in general, and did some demonstrations in EXCEL.
Tuesday 22/1–13
I made an error during the last 20 minutes. The (student) t-distribution is “defined” as the ratio
where X ∈N(0,1) and Y∈χ2(f) and X and Y are independent. It follows that
T2 is F(1;f)-distributed.
Sorry for the mistake!
The first two hours I talket about the LR test when we have a compsite hypothesis, like the example on three Poisson intensities in my booklet. I mentioned that the usual χ2 tests can be viewed as approximatios to the LR-test. We analysed the ”Sex differences“ article somewhat more extensively than in the booklet, and in that context I explained Bonferroni's method to test several hypotheses.
We skipped for the moment the various Welch t-tests, and started with multivariate linear regression. We took a simple example where we wanted to assess if there is (or rather was) a gender discrimination as to wages. I udes data from the Current Population Survey from 1985 to demonstrate a simple model, and described the various outputs from EXCEL (LINEST).
Finally I talked about the relations between the four distributions Normal, χ2, Student's t and F-distribution. (This is where I made the mistake; I forgot the 1/f in the numerator under the qsuare root sign – see above.)
If you want a more extensive technical text
on econometrics, you can look att this text by Bruce Hansen.. It focuses on the technical details of regression analysis, very little on modeling issues.
Look out for the notion “accept the null hypothesis.” – read carefully what he says on “When a test accepts a null hypothesis…” on p.165!
Short answers to some of the problems Jonas solved Thursday 17/1–13
44. The p-value for equal incidents is 0.0118, so we reject that hypothesis. The point estimates are 7.20% for A and 7.28% for B, so we conclude thaat B has the higher incidence.
45. At a risk level of 10%: the probability that B tastes better than A is higher than the reverse.
At a risk level of 5%: no conclusion.
47. With the “exact” method outlined in exercise 46: p=0.2520.
48. At an expermiental error rate of 0.0003 we conclude that the first data in exercise 7 come from a distribution with lagrer mean value than the second. We can not conclude anything more at an experimental error rate of 0.05.
1. What you should know is that
(AB)t = BtAt, and
(AB)-1 = B-1A-1 and that
(At)-1 = (A-1)t.
Ans.: Ct.
2. A-1C.
The LR-interval for a binomial proportion
Our friend Rickard Norlander has procuced this diagram. Compare with figures 5 and 11 in the paper "Interval estimation…"
Tuesday 15/1–13
I talked a lot on hypotheses testing, especially testing for a parameter value and confidence intervals, see the section “Testing parameter values” in my boolket “Topics on Applied Mathematical Statistics”. I gave two examples on the Likelihood Ratio Test; first on the parameter of an exponential distribution, later for the proportion in a binomial distribution where we compared with the “exact” test and the “classical” procedure for confidence intervals.
This Thursday Jonas will talk about the exercises 42–48, and also show how linear regression is done i EXCEL, and he will also show how we can solve problems 7 and 8 by linear regression on “dummy variables”. Time will not permit all this, so we will see how far he gets.
Here is the paper on sex differences
I refer to as [3] in the first chapter in my booklet.
Estimating a binomial proportion
Here is a paper in “Statistical Science” which shows how bad the standard procedure for estimating a confidence interval for the probability in a binomial distribution is. This method is implemented on TI:s pocket calculators, and is also the method in Blom's et. al. book. The “exact” method I describe in my booklet is the “Clopper-Pearson” method in this paper. As you can see, it is the most conservative.
on some matrix algebra that you may find helpful. It is written by my collegue Gunnar Englund.
