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This video video discusses the complexities and challenges associated with applying classical statistical models, particularly in the field of economics. It highlights the importance of applied mathematics for economists conducting empirical work, focusing on concepts such as sampling distributions and the assumptions inherent in statistical models.

The author critiques the classical progression model, noting that it is often taught despite its limitations, particularly in the context of economic data, which does not always adhere to the model's assumptions. Issues such as autocorrelation and multicollinearity in time series data are addressed, explaining how these factors violate the assumptions of independence used in classical statistics.

This video video emphasizes the importance of being honest with students about the applicability of statistical models in economics, advocating for a deeper understanding of data randomness and the necessity of employing more realistic assumptions. Additionally, it touches upon the challenges of heteroscedasticity in data and suggests ways to mitigate such issues in analysis.

Overall, the text expresses a need for a more comprehensive and realistic teaching approach in economics and statistics, acknowledging that although simplified models might be useful for teaching purposes, they may not accurately represent the complexities of real-world data analysis.

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Well, it's, it's applied mathematics,

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but it's, things that are of interest to economists

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or let's say particularly economists who want

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to do empirical work and, and work with data

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and test hypotheses

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and all that kind of thing, which it is assumed that,

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the university trained economists should know.

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So most people have to go through a, a course of this type,

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more or less technical.

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Um, my book is on the more technical side, deliberately,

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it's aimed as at, for,

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more mathematically inclined students if you, if you will.

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It's, it's a, it's an important idea,

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this sampling distribution.

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And I think that it has to do in many cases, really

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with the thought experiment, you have to, the,

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the investigator has to say to himself,

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I've got a sample here, which I collected by.

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It might be even be a time series,

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or it might be a sample survey,

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or might be a bit of panel data.

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Something like that says,

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how do I interpret this from a statistical point of view?

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What does it, what does it mean?

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So you have to imagine

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that it's a drawing from a distribution, a distribution

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of many such samples, which you can independently draw from.

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And often, this is of course, how the world works.

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That, that if you, if you, if you look at a sample of,

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of households, typically somebody will have gone

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to the trouble of, of making a random drawing of, of,

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of postal codes

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or something of that kind and picking them out.

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Uh, so, but sometimes this is,

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this is, this is more theoretical.

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I mean, I, or, or it's basically it's a fiction.

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Uh, if you are thinking of a time series of,

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stock price, well, that just happened once in the world.

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It's not gonna happen again in the same way.

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Um, so except in a, in a parallel universe,

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if you like your quantum theory, that that way, then,

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you know, it could be like that.

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So that's what a sample distribution is.

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Of course, we could still describe it by making some,

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some assumptions more

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or less sensible, in order

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to describe this distribution.

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And then we can begin to interpret our drawing

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and how we understand the randomness in, in the dataset

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that we actually possess.

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I do take this quite seriously in my book,

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because of course, the,

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the classical progression model is a nice bit of nonsense.

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Um, it sets some assumptions, that,

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really don't hold, at least for economic data.

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They hold in, in, in, in, in certain situations.

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Uh, experimental science, you can imagine setting some,

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some parameters of an experiment then observing

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the outcome of the experiment.

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But economics is by

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and large, non-experimental in character.

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So, so that really doesn't work.

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Um, nonetheless, it's taught to generations

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of undergraduates,

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and I've never heard of any single one that's ever said,

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please, sir, this emperor has no clothes.

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This makes no sense, you know, which clearly shows

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that people don't understand the other topics that they're,

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that they're taught now.

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I mean, the reason, of course, that we do this,

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by the way, the, you know, the peculiarity of the,

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of the assumptions is this idea of the fixed progressor

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that when you draw this, this sample, every time you do it,

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you get a new dependent variable.

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But the X is on the other side. They're always the same.

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That's the assumption that you have to make.

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And of course, it makes no sense, so that if you,

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if you draw a sample of some economic behavior,

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everything is new, you know, the, the income

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of the consumer, the, the expenditures of the consumer,

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you know, because the consumer is drawn randomly

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and all his characteristics or her characteristics.

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So that's the problem with it, is that it's not really,

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it's, it is not real.

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Why do we teach it to undergraduates?

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Because basically it involves statistical concepts, which,

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which are conveyable without too much advanced theory.

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And as I expl try to explain in my, in my little book,

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this is reasonable

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because if you just extend

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to those more realistic assumptions,

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letting the data be fully random, for example,

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but tell some stories about it, then

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the properties which you prove vigorously under the,

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classical progression model are I'll sort of roughly, right?

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You know, and then as a fallback, there's acetic theory,

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which says that if you've got a big enough sample,

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the world looks pretty much like

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the classical progression model.

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Not exactly, but,

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but the same sort of conclusions can be drawn.

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So, so there are various get outs.

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So basically, this is why it's a sad paradox.

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You, you teach these things to students

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because you believe that this, it can be explained in the,

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in the framework of a, of a 10 week course

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to non mathematical people,

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which most economists of course are.

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Um, so, you know, we get by, but,

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but I think it should, it's always good to be honest

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to students and tell them what they're teaching them.

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And that's what I, that's what I've always tried

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to do, for better or worse.

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Well, the ga markoff theorem,

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because is the thing that everybody, has

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to remember about, and they think,

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but these two pals, Gauss

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and Markov, he said, no, actually one died in the same year.

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The other was born. So they weren't really well connected,

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but they did work on the same kind

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of things, wonderful people.

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They both were, it's a,

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it's a result which is proved rigorously under the framework

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of the classical progression model.

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And it says that lease squares is a good thing to do.

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Um, provided you can show that these squares is unbiased,

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which it is under the classical aggression model, then Gauss

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and markoff tell you jointly that your

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estimator will have the minimum

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Possible variance in the,

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and we've gotta be careful here what we mean by that.

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Because in a fixed sample, which is

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what we're talking about, but obviously a repeated sample,

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technically, so we're discussing the distribution

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of our estimator under, repeated sampling,

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then it's, it's going to be efficient.

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That's, that's the technical term.

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We, we say it's, it's, it's the, so,

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so we say it's best in the class of linear,

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I didn't mention, that's

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of course talking about Lee Squares best in the class

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of linear, unbiased estimators.

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Blue is what we teach the students,

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and that's what the gas Markle Theum says.

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So the gas markle theorem, once again, not entirely true,

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but as true as we might wish, is

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what one can say auto correlation is the property

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that regression disturbances are related.

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Our correlated, you know, they move together.

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They're not an independent drawing, which is what you want.

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That's, that's what the, the gas progression model says

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that your disturbances are independently drawn.

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And, and so if that's not true, then of course the,

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for example, the gas markoff, the, no longer,

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no longer applies.

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Um, and, also your,

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if you are doing inference yourt statistics will be,

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will be biased and things like that.

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Um, the trouble with this is

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that when do you encounter order correlation?

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Usually in time series,

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because then there's an ordering to the data,

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which makes it sensible to, to believe

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that there may be a relationship

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between successive observations, you know,

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because they come in, in,

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adjacent periods of time.

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So you tend to get auto correlation in the context

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of time series.

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But of course, time series violate the assumptions

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of the classical progression model with

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or without auto correlation.

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So, so if you're thinking,

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although it's taught to students, not mine, I might say,

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but, pretty often as something to look out

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for in the world of the classical progression model,

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the truth is that, that it doesn't really apply.

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Um, it's, it's a more advanced topic than,

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than you really can think.

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Um, but then of course, what we have

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to face is the students are very often given practical

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exercises on time series.

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This is, this is, this is a fairly, of course,

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time service data are not difficult to, to find.

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You just pull 'em out of the blue book and, and, and, and,

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and, and, and so on.

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So very easy to give students practical exercises using time

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series, even though the, the theory

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that they've been taught doesn't really work.

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But you know, that's all right off all,

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they're only undergraduates.

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So, so, so we expect a little,

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inconsistencies in what we tell them, but,

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but, I don't, you know, but, but,

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but the truth is out there.

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So, I don't think it's a big issue really.

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But anyway, that's what I have

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to say rather extensively about all the correlation is

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that it's,

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it's a problem which you are going to encounter.

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However, there's one other thing to say is that dealing

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with it, traditionally there have been various fixes

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for auto correlation, which are best avoided

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because they can make things worse rather than better.

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Uh, but that's another story to go into, which makes

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for an interesting exercise for this,

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for, you know, for students.

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Um, but I have to say that

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because my book is, is Elementary,

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it says nothing about disorder correlation at all.

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It doesn't, doesn't deal with time series

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because it, it tries to keep everything on a realistic,

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even keel.

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Now, you also mentioned multiline.

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That's the other multi-syllable word that we have to teach.

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Um, it just says the data are no good.

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So, you know, I don't think you should take it too seriously

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or maybe that your model is no good

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because you've, you've piled more explanatory variables into

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it than it can really cope with,

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because they're basically telling you, you know,

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it's the same data, it's the same information process in

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different ways, and you get, that's what you,

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how you get co linearity.

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The, the x's in your equation are,

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are too closely related to one another.

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And this means that a matrix, you want to invert,

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I won't go into those technicalities too much,

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but, are, may, may fail to be invertible or,

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or your standard errors become, unacceptably large

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and horrible things like that happen.

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Um, but it basically means either you are using the data in

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the wrong way, which means your model is no good,

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or it means that the data are no good

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and you've gotta, you know, extend your sample in some way.

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Uh, but, but it's not, but, but,

237
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but there is no simple fix for it.

238
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Hetero elasticity, which is the third one you mentioned,

239
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is a little more interesting

240
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because you, you can realistically get

241
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that in the classical progression model

242
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because it's something that could apply,

243
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in a cross section, which basically is,

244
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is when all the data have the same time,

245
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but they're, they're for different, different agents

246
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or different objects, over that time.

247
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So supposing you're looking at a cross section of firms,

248
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well, you may have some very big firms

249
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and some little corner shops in the same sample, and, and,

250
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and you are really mixing, two different types of types

251
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of agent who aren't really comparable.

252
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And so, if some of them have, you know,

253
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their profits measured in, in billions

254
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and others have their profits measured in hundreds

255
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or thousands, this poses a problem

256
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because the, the,

257
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the residuals for these different observations may have very

258
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different variances, is what it it comes down to.

259
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So, so it's a part

260
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of the classical progression model assumptions is

261
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that all the variances of your observations are the same

262
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that they, there are, they're a homogeneous collection.

263
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So there are ways to fix this.

264
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For example, you would, you would not look at profits,

265
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you would look at profits as a percentage of revenue,

266
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and then you get something which will be easily comparable

267
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between the big agent and the small agent.

268
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So, you know, that, that's just, so, so

269
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that's something which it is useful to discuss,

270
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with, with, with students.

271
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And you can do sometimes do something about it.

272
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But, but,

273
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you may need more sophisticated methods, unfortunately.

274
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And then, and, and,

275
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and then you are involved in, non-linearities,

276
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you'll need asso theory to interpret what you've done and,

277
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and so on and so on.

278
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So, not so easy.

Understanding Sampling Distributions in Economics

00:00:00 - Well, it's, it's applied mathematics, but it's, things that are of interest to economists or let's say particularly economists who want to do empirical work and, and work with data and test hypotheses and all that kind of thing, which it is assumed that, the university trained economists should know. So most people have to go through a, a course of this type, more or less technical. my book is on the more technical side, deliberately, it's aimed as at, for,
00:00:36 - It's, it's a, it's an important idea, this sampling distribution. And I think that it has to do in many cases, really with the thought experiment, you have to, the, the investigator has to say to himself, I've got a sample here, which I collected by. It might be even be a time series, or it might be a sample survey, or might be a bit of panel data. Something like that says, how do I interpret this from a statistical point of view?
00:01:06 - So you have to imagine that it's a drawing from a distribution, a distribution of many such samples, which you can independently draw from. And often, this is of course, how the world works. That, that if you, if you, if you look at a sample of, of households, typically somebody will have gone to the trouble of, of making a random drawing of, of, of postal codes or something of that kind and picking them out. so, but sometimes this is, this is, this is more theoretical.
00:01:40 - if you are thinking of a time series of, stock price, well, that just happened once in the world. It's not gonna happen again in the same way. so except in a, in a parallel universe, if you like your quantum theory, that that way, then, you know, it could be like that. So that's what a sample distribution is. Of course, we could still describe it by making some, some assumptions more or less sensible, in order
00:02:08 - And then we can begin to interpret our drawing and how we understand the randomness in, in the dataset that we actually possess. I do take this quite seriously in my book, because of course, the, the classical progression model is a nice bit of nonsense. it sets some assumptions, that, really don't hold, at least for economic data. They hold in, in, in, in, in certain situations. experimental science, you can imagine setting some,
00:02:43 - the outcome of the experiment. But economics is by and large, non-experimental in character. So, so that really doesn't work. nonetheless, it's taught to generations of undergraduates, and I've never heard of any single one that's ever said, please, sir, this emperor has no clothes. This makes no sense, you know, which clearly shows that people don't understand the other topics that they're, that they're taught now.

Challenges of Teaching Classical Statistical Assumptions

00:03:13 - by the way, the, you know, the peculiarity of the, of the assumptions is this idea of the fixed progressor that when you draw this, this sample, every time you do it, you get a new dependent variable. But the X is on the other side. They're always the same. That's the assumption that you have to make. And of course, it makes no sense, so that if you, if you draw a sample of some economic behavior, everything is new, you know, the, the income of the consumer, the, the expenditures of the consumer,
00:03:47 - and all his characteristics or her characteristics. So that's the problem with it, is that it's not really, it's, it is not real. Why do we teach it to undergraduates? Because basically it involves statistical concepts, which, which are conveyable without too much advanced theory. And as I expl try to explain in my, in my little book, this is reasonable because if you just extend to those more realistic assumptions,
00:04:20 - but tell some stories about it, then the properties which you prove vigorously under the, classical progression model are I'll sort of roughly, right? You know, and then as a fallback, there's acetic theory, which says that if you've got a big enough sample, the world looks pretty much like the classical progression model. Not exactly, but, but the same sort of conclusions can be drawn. So, so there are various get outs.
00:04:50 - You, you teach these things to students because you believe that this, it can be explained in the, in the framework of a, of a 10 week course to non mathematical people, which most economists of course are. so, you know, we get by, but, but I think it should, it's always good to be honest to students and tell them what they're teaching them. And that's what I, that's what I've always tried to do, for better or worse. Well, the ga markoff theorem, because is the thing that everybody, has
00:05:21 - but these two pals, Gauss and Markov, he said, no, actually one died in the same year. The other was born. So they weren't really well connected, but they did work on the same kind of things, wonderful people. They both were, it's a, it's a result which is proved rigorously under the framework of the classical progression model. And it says that lease squares is a good thing to do. provided you can show that these squares is unbiased, which it is under the classical aggression model, then Gauss
00:05:55 - estimator will have the minimum Possible variance in the, and we've gotta be careful here what we mean by that. Because in a fixed sample, which is what we're talking about, but obviously a repeated sample, technically, so we're discussing the distribution of our estimator under, repeated sampling, then it's, it's going to be efficient. That's, that's the technical term. We, we say it's, it's, it's the, so,

Challenges of Teaching Regression Analysis

00:06:24 - I didn't mention, that's of course talking about Lee Squares best in the class of linear, unbiased estimators. Blue is what we teach the students, and that's what the gas Markle Theum says. So the gas markle theorem, once again, not entirely true, but as true as we might wish, is what one can say auto correlation is the property that regression disturbances are related. Our correlated, you know, they move together.
00:06:55 - That's, that's what the, the gas progression model says that your disturbances are independently drawn. And, and so if that's not true, then of course the, for example, the gas markoff, the, no longer, no longer applies. and, also your, if you are doing inference yourt statistics will be, will be biased and things like that. the trouble with this is
00:07:26 - Usually in time series, because then there's an ordering to the data, which makes it sensible to, to believe that there may be a relationship between successive observations, you know, because they come in, in, adjacent periods of time. So you tend to get auto correlation in the context of time series. But of course, time series violate the assumptions of the classical progression model with or without auto correlation.
00:07:55 - although it's taught to students, not mine, I might say, but, pretty often as something to look out for in the world of the classical progression model, the truth is that, that it doesn't really apply. it's, it's a more advanced topic than, than you really can think. but then of course, what we have to face is the students are very often given practical exercises on time series. This is, this is, this is a fairly, of course,
00:08:27 - You just pull 'em out of the blue book and, and, and, and, and, and, and so on. So very easy to give students practical exercises using time series, even though the, the theory that they've been taught doesn't really work. But you know, that's all right off all, they're only undergraduates. So, so, so we expect a little, inconsistencies in what we tell them, but, but, I don't, you know, but, but,
00:08:56 - So, I don't think it's a big issue really. But anyway, that's what I have to say rather extensively about all the correlation is that it's, it's a problem which you are going to encounter. However, there's one other thing to say is that dealing with it, traditionally there have been various fixes for auto correlation, which are best avoided because they can make things worse rather than better. but that's another story to go into, which makes for an interesting exercise for this,

Challenges in Economic Data Analysis and Teaching

00:09:29 - but I have to say that because my book is, is Elementary, it says nothing about disorder correlation at all. It doesn't, doesn't deal with time series because it, it tries to keep everything on a realistic, even keel. Now, you also mentioned multiline. That's the other multi-syllable word that we have to teach. it just says the data are no good. So, you know, I don't think you should take it too seriously or maybe that your model is no good
00:10:01 - it than it can really cope with, because they're basically telling you, you know, it's the same data, it's the same information process in different ways, and you get, that's what you, how you get co linearity. The, the x's in your equation are, are too closely related to one another. And this means that a matrix, you want to invert, I won't go into those technicalities too much, but, are, may, may fail to be invertible or,
00:10:33 - and horrible things like that happen. but it basically means either you are using the data in the wrong way, which means your model is no good, or it means that the data are no good and you've gotta, you know, extend your sample in some way. but, but it's not, but, but, but there is no simple fix for it. Hetero elasticity, which is the third one you mentioned, is a little more interesting because you, you can realistically get
00:11:01 - because it's something that could apply, in a cross section, which basically is, is when all the data have the same time, but they're, they're for different, different agents or different objects, over that time. So supposing you're looking at a cross section of firms, well, you may have some very big firms and some little corner shops in the same sample, and, and, and you are really mixing, two different types of types of agent who aren't really comparable.
00:11:35 - their profits measured in, in billions and others have their profits measured in hundreds or thousands, this poses a problem because the, the, the residuals for these different observations may have very different variances, is what it it comes down to. So, so it's a part of the classical progression model assumptions is that all the variances of your observations are the same that they, there are, they're a homogeneous collection.
00:12:06 - For example, you would, you would not look at profits, you would look at profits as a percentage of revenue, and then you get something which will be easily comparable between the big agent and the small agent. So, you know, that, that's just, so, so that's something which it is useful to discuss, with, with, with students. And you can do sometimes do something about it. But, but, you may need more sophisticated methods, unfortunately. And then, and, and, and then you are involved in, non-linearities,

Complexities in Statistical Interpretation and Application

00:12:41 - and so on and so on. So, not so easy.

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