Okay. Good morning. This morning we are honored to have Professor June you to give a lecture, so may I introduce Professor June to you first?
Professor professor team is currently u mdf chair professor of finance Economics, and being of faculty of the Business Administration at the University of Mccall, he received his Phd.
In economics from university of western Ontario in nineteen ninety-eight. His primary research interest of financial economics in a magic theory and empirical asset pricing.
He provides education, a editorial service for general optimal metrics and economic theory. He was awarded fellow of journal, of Economics.
In in twenty eleven. Inaugural fellow of society, of financial economic metrics in twenty twelve certificate for highly cited research of journal of Asian economics in twenty, sixteen et cetera.
Work has appeared in Journal of Analytics, international Economic Review Review of financial Studies, quantitative economics, economic theory is a metric theory general of business and economic statistics added them in the science et cetera okay, now, let's welcome professor June.
Thank you for coming. Oh, thank you, Professor, Full, for the current introduction, and thank you for attending this summer school.
I think this is a very good opportunity for you, and also for me to explore.
You know a new territory. I actually give a two meaning lectures in total six hours, two or three years ago, on a topic very much related to this.
One. So today, the topic is continuous on fraction process. So last time I was giving many lectures on a discreet time.
Fractional processes, and so relative to the lost lecture. This lecture is more, it's more cutting age, so it's it means that many research funding that I'm going to present.
Only。
Become available over the last couple of years so it's a very new research area.
I hope you know this two lecture can help you to understand this. The state of the odd in the continuous fraction of processes.
So as you know that we are going to have two mini lecture today. So why in the morning wine in the afternoon, so we will have a three hours in the morning and three hours in the afternoon and I was told we need.
To take a break half an hour break in the morning, and a half. Now break in the afternoon, so we start the lecture at eight, forty, five.
So we was, and the first half in the morning at ten o'clock, and then we start the second half at ten, forty, ten, thirty, and then we finished the second half at eleven, forty, five, so that's the that's the plan for the morning, and then accordingly for the afternoon.
If you have any questions during my presentation, just raise your hand. I guess somebody will pass a mic to you so that you can ask question.
I think that's the best way to to interact with me to help you to understand what I'm going to talk about.
I'm going to make assumption that you have learned linear algebra and have a enough knowledge in calculus, and the probability theory.
So basically, I treat you as a you know, postgraduate student. Okay, but if there is a knowledge gap somewhere during my presentation, feel free to ask me, then I try my best to answer your question.
Of course, right. There's no way I can, you know, give you.
Four treatment from zero from Grant, because, you know, we only have a six hours in total. So that's why I have to make assumption that you have some basic knowledge.
So the title of these two lectures is a estimation, inference and prediction of a continuous term, fractional processes.
So let me first talk about the motivation. Why, I'm I'm going to doing that.
But before that, and I mean introduce you. Some basic concept tools that we need to based upon.
So this is a continuous time model. It's basically a tom-stu model right?
So that means we have a Tom shares. So let me call Thomas with y T.
What's happening?
Looks like it's Internet problem.
Must be Internet, Robert.
That's okay. Sorry. Sorry for the Internet. Probably.
Do you guys have a slides?
No, you don't have slice.
Oh, it doesn't matter.
Ok，可能我们需要抽联系啊。
Yes, already there. Yeah.
还是？
Okay, thank you. Sorry for the Hassle. So, because we are dealing with a Tom series.
So I'm going to assume that Tom'sius is yt where T. Is running from one to capital T.
Small T is running from one to capital t. So the first concept that I'm going to introduce is a Covariance stationary process.
So if yt is a covalent stationary process, it must satisfy three conditions. The first condition is that the mean is a constant.
It's a finite, and it's constant. We call it new.
I just call it later, right?
And then the second condition is the variance of yt is also constant, for finite, because of that I'm going to assume it's a sigma square.
Which is also denoted as a gamma zero. For the reason which will be clear in in the second, so we have a constant, mean, constant variance, and then the next one is the covariance between yt.
And yt minus k. Where k is any integer number? Okay? It's also a constant with respect to tea and it's a finite, and we call it a gamma K.
So this Gamma K.
Sdr.
Is the covariance with auto-ovariance, and because the lag is a key, so we call it k slack auto-ovariance.
Our case call auto-overance so clearly, if you choose K. To be zero, then we know yt minus k becomes yt.
So that's why the covariance becomes a variance. That's why I call variance, Gamma zero.
Because that means, you know, conditions three. Can include condition two. As a special case, of course, here I restrict the cannot be zero.
Okay, so that means, you know, in you know, to some to summarize the three conditions we require.
The random variable yt has a constant, mean, constant variance also the covariance covariance structure.
Of this process remain the same over time. Okay, so that's the first concept and the gamma case call auto-vance case.
All the Auto-variance, and if we normalize the auto covariance by Gamma zero, we have just a new and notation.
Okay is called autocorrelation, function by cosy in okay must be always bounded between minus one and one.
比如。
Cannot be.
个人的。
So that's why is normalized quantity. Take a taking value between minus one and one.
And if ro k is a positive, it means that yt and the yt minus k tend to be possibly correlated. If roca is an active, then yt, and the yt minus k tend to be negatively coordinated does that make any sense okay, so that's you know the first payoff concept.
And for stationary and a Gothic times with I have no time to explain. I got this city, but just want to give you this result for a station and a Goddess series.
The all those quantities like mean variance, and the auto covariance can be consistently estimated by the sample countboard, for instance, the variance which is a population variance can be consistently estimated.
By the sample variance, or something closely related to sample, to sample variance. So this is basically a sample counterpart and case order.
Auto correlation function can also consistently estimated by the sample counterpart. Okay, so you see that if you divided by numerator, by T.
That's the sample coverence. If you divide by denominator by T.
That's a sample variance. So the ratio, of course, is the auto-relation function at the case.
Order. Another concept, which is very usually highly useful.
啊sdr。
Spectral density, the spectral density of a covariance stationary process.
Is basically the Fourier transform of the auto-overances. So let me use a gamma Y of K.
So this guy to denote the auto-overance of a stochastic process.
Then we use this foria transformation of Gamma Kay. We get a function. We call it F of a number.
So F of number. Now, of course, it's an effort. Number is a for process. Y.
So this guy is cost spectrum density. Well, number Ps. Here, that's the exponential of minus.
I cut K. No, so k is basically the lag Nance i is basically a complex number is the imaginary number is, I is basically square root of minus one.
So we, if you learn calculus, then you know, this is basically the Fourier transformation.
Of auto-coverances. So the spectral density is fully transformation of auto covariances.
Fourier transformation always exist. And actually fully transformation is a one to one mapping, so it means that if you know the auto-coverances, then the spectral density is uniquely determined and vice versa, if the spectral density, is even then the all the other auto covariances, are given so this one one mapping basically suggest that spectral
density contain the same amount of information as auto-coverances.
Okay? Because of the Fourier transformation, the property of a Fourier transformation. In fact, this number here is called Fourier Frequency.
The reason is called fruit. Frequency is because this function, which is the exponential function which can be written as a sign plus cosine.
So we know, sign and cosine functions are periodical functions.
不要佛。
We have a frequency number here, and they usually this we only we only study the property of the spectral density.
Over one period of a frequency, for instance because the assigned the cosine function have a period of two pi.
So usually we restrict num dot be between either minus by to pi or zero to two pi. So if you know the spectral density, over a period of two, pi, then you understand the spectral density over the entire domain for pi okay, and if you, assume the frequency number to be zero, so that means this is zero and that means this is zero, so explanation of zero take a value of one as
you know that means f of zero times two pi equals the summation of a Gamma K.
K. Is moving from minus infinity to plus infinity. So what do we have is basically the summation of varies and the covariances of all order.
In economics. This is extremely important quantity for statistical inference that's called so long run.
Variance basically can be obtained, as from the spectrum density evaluate at zero frequency.
Okay. So this is very easy thing to see, and also we can easily check that F of a number is the symmetric about it, or a region.
所以。
For that reason we only need to know the behavior of F of when namda is between zero and the pi, because once you know that then you know the behavior of the spectrum density when lambda is between minus pi and zero.
So far, so good.
Okay. So, of course, the definition of the long run variance assume that the auto-coverances is some of them.
Okay, and often this condition is violated, if the long run variance is not summable, it means that the summation of the auto covariances goes to infinity, so in that case it means that the spectral density evaluate at zero frequency diverse is infinity okay, so for stationary times yes, if the auto covariances are absolutely summable that means longer.
Variance exist we call it. This Thomas was, has a short memory.
However, if auto-coverances are not summable, then we say this time service has a long memory.
Clearly for short memory, process, spectral density at frequency. Zero is a finite. That's all.
As I already mentioned, for long memory process. That's infinity.
So I'm now going to give you a few examples, so that you can appreciate. The the concept that I just mentioned.
So the the process I'm going to give to you are all discrete-time models.
Later we are going to introduce, continue some models. So the first example is the discreet auto-regressive process with a leg one we call it a R.
One or a means auto, regression or auto-regressive. So the mathematical definition of a one model is yt equals R for yt, minus one.
就是。
I'm going to assume offer. Take a value less than one in absolute value and absolutely is iid random, variable, with mean zero variance.
Sigma, epsilon square. So that's the error term. So it can be easily shown that.
Yt is a stationary process. Covariance stationary process.
In fact, we can show that the expected value of yt is zero, which is constant finite. The variance of yt is a sigma, a personal square divided by one minus Alpha Alpha square which is also constant Finite and the Auto Covariance at the J's order Gamma J equals Rv to J times the variance of Yt which is this guy.
哎， therefore auto， function， function。
At the J's order is.
阿尔法特。
When J. Is positive, of course Gamma Jay, equal to Gamma minus J. Because it's a covalent stationary process.
They are for rode J. So because of this three expressions, variance, mean variance, and covariance, then we basically show that the ar one model is covariance stationary and the correlation of course we already mentioned is a gamma is a offer.
To J. So that means the autocorrelation function. Is a geometric series is our offer to J.
Offer is a number less than one. So offer to J. Is that your metric series we know to your magic series, decay towards zero very quickly, as J.
Increases, and we also know that geometric series is a sumable. Yes, I'm absolutely summable.
叫佛。
A one model with take a value less than one.
Is as a short memory.
Make sense to you. Okay, we can also prove that the spectral density of a one model.
Take this expression.
Okay, so you can replace number by zero. If you do that, then let me let me see if I can.
If I can write anything.
Maybe I.
So basically two.
Two排。
Because I don't know how to do so. Two pi times f of zero. So you move two part to the left side.
So what do you have is basically sigma square?
Sigma.
这个完minus阿富。
Because the nominal is zero. So this E. Zero is one e. To zero is one.
So you have a one minus R of whole square. So that's basically the implication of this expression.
So we would expect this whole thing, on the on the left hand, on the right hand side it's the long run-wide variance of ar one model.
You can easily verify that, because you know the variance, you know the auto covariance.
So you add everything together. Remember, J. Has to be running from minus infinity to plus infinity.
So when J. Is zero is variance, and then you need to exhaust all possible values of J.
If you do that, you will get this number. I'm not going to verify for you, you know you can do it yourself, so that's the you know.
Spectral density of a one model, and we can in practice offer is unknown.
So what are you observe? Is yt since office? I know you what you can do. You can estimate all of our from the data.
A very well known and easy estimator is the ordinary leisure square estimator. So suppose the data is, you know, is Y.
Tt is running from zero to capital t then the least. The square estimate of a offer we quite alpha hat.
Take this expression, okay, and it's very easy to show that this estimator consistently estimate all of so that means overhead converge in probability to our phone.
Also the asymptotic distribution of our half is given by this equation square root of t, which is sample size times of hat, minus alpha.
So this is a centered oh, as estimator, multiplied by square root of some size, converge in distribution to normal, meaning is zero variance is one minus alpha, squared, so that's basically as some sort of theory is actually, quite easy to get this as some data.
Theory. What do you need to do is basically you, you minus off ahead by Alva, then you eventually this yt becomes absolute key.
Okay, and and then you basically apply law of large number to the denominator. Apply a central limit theory for station and a Gothic sequence to the numerator.
So you get normality. For the quantity of stairs. You you get constant in the limit for the quantity downstairs, and then you'll use a Slasky Theorem to combine the meeting result of upstairs and the downstairs, to get a some target on that so i'm not going to prove that this is usually done in thomas with
course.
Another example is an a one model.
我们。
One means, like Nance one. So the ma one model is given by this mathematical equation.
Yt equals absolute t minus beta, absolute t minus one where absentee again is iid with mean zero various sigma sigma epsilon square.
For this process, regardless of the of the value of Beta, we can obtain the mean of yt, which is zero variance of yt, which is given by Sigma Square.
Times one plus beta squared and then gamma, one, which is a first all the other covariance is minus beta times, Sigma, absent square.
So that's basically the mean, finite constant. There is finite, constant, and the auto covariances.
Finite, constant. Of course, the first order is not potentially non-zero, but higher order all the covariances are all zero.
As a result of that, the first auto order acf take an expression of minus beta, divided by one plus beta square.
Rho j equals zero for any J. Bigger than one.
So M. One model is also covariance stationary for any beta sorry there was a typo, not for any theta for any beta, and the auto-relation function caught off of the first Lap.
If, in addition, we assume Beta, take a value between minus one and one, then Ma. Process, and may.
One process is also invertible. So in that case we can have a auto-regressive, auto-regressive representation.
Okay. The spectral density of Ma. One model is given by this mathematical expression. So that's the spectral density of a one model.
Again. If you assume that number take a value of zero, then you can get the.
啊。
Which is a sigma square at times, one minus the beta o square. Hey?
Of course we can make a more complicated a one process more make a more complicated ma process, and then we can combine Ma.
And and component together. So in that case we can have our pq process.
So for our pq process. Basically, we have a piece of the point inomial which we call R.
For l Alpha L. Is a piece of the polynomial. It is A, and then we can have a beta l times absentee.
So Alpha Al comes to yt equals beta l times the absolute t. So beta l is a queues order polynomial function of lag, nance, lag operator.
So if queue is a zero P. Is one, then you have an ar one model. If a P.
Take a value of zero queue. Take a value of one, then you have an a one model, but in general he may not be zero.
Q. May or be zero, but P. And the Q. Are both integers.
So when that happened, we have a Pq model.
In order to impose stationarity and the invertibility condition, we assume the roots of Alpha L.
Equal zero, and the beta l equal to zero lies outside of the unit circle. So that's for ensuring the you know.
Importability and the stationarity that's similar to the requirement we impose here our for less than one.
For a one model, and the Beta less than one for M. One model. So it's basically extension of these two conditions to a higher order case, and the stationary I'm a Pq.
Model, although it's more complicated, but we can show that the auto correlation function geometrically decreased to zero.
Eventually okay, in particular, we can prove the the geometric decay rate for the auto-relation function, since the auto-relation function for geoet geometrically decayed towards zero.
Eventually, so the first, few, the first few other terms the finite order term, the behavior for the first few auto-coverances.
Are irrelevant. When we talk about some ability, because you only have a finite number of terms.
So the sumability is determined by high order behavior of auto correlation function, which we know high order, high order, behavior of auto-variance is that your magic series?
Therefore, for stationary mpq model.
第二。
Auto, covariance are absolutely summable. It means that I'm a Pq. Model stationary.
On my pq model. How's that short memory?
Okay. So that's the implication of of this property that auto-relation function.
Geometrically decrease to zero. And the spectrum then steer for Almah.
Pq model is given by this mathematical equation.
Okay. It's a bit more complicated, but it's still very easy, you know, function to deal with.
还是？
Wide applications, a wide range of applications in economics, in finance, and the in many other area of you know, displays also in many areas.
Scientific areas like us, you know, engineering.
However.
Is easy to show that for meaning economic time, space, financial time, space, also, other times, with in natural sciences.
阿盟。
Are not good models, so I'm going to give you one example. So that's our motivation of today's lectures.
In practice auto-relation function often does not geometrically decreased to zero.
The spectral density of.
发？
Thomas Fis often go to infinity, as the frequency goes to zero.
Ok. This two, finding.
Acts each other. If other covariance goes to zero very slowly slower than to your magic series. It may imply that the auto-coverances are not summable if the auto coherences are not summable it means that the spectral density at the zero frequency is not finite.
Okay. So. But we know the implication of our stationary armor model is geometric series for other covariance so that means, in that case you can not use the stationary model to describe such auto-varance.
In practice, so to show the relevance of this two statement, let me get a data for you.
So this is a very important data series is Sp. Five hundred, which is our most important index.
Financial for Equity, Financial Index, S. P. Five hundred. So I take a daily Sp.
Five hundred. I get to realize Volatility Rv. So this realized volatility basically captured the fluctuation of Sp.
Five hundred, or the risk of a sv. Five hundred, and then I take a lot.
And the data series is between the beginning of two thousand, and the the end of nineteen twenty, nineteen.
So in total we have a more than five thousand observations, which I plot in discount, so you can see that, and the P.
Five hundred daily realized variance. Flocks, rate.
I take a look so it's not the the original realized Volatility series is a log, realized volatility series.
But you can still see the fluctuation.
哎。
So the question I'm going to ask is, Can we use armor model to describe the movement?
的。
Ivy, so I'll set that question.
I plot.
The sample a cf. Remember, I say that a caf can be estimated by the sample counterpart, so I estimate each covariance by the sample counterpart, and now I plot the sample code the auto-relation function as a lack nance J.
So I allow Jay to.
Take a to take a value between zero and the five hundred. So the shortest lagness is zero, which is variance.
In fact, it's a acf. So at the zero lack. The value must be one, because it's gamma zero divided by Gamma zero right?
And then up to five hundred. So five hundred basically is a gamma five hundred divided by Gamma zero.
Okay, so if we use our model to describe such a data, such, we should expect a eventual geometric series, because that's a property of our stationary.
I'm a Pq. Model.
This doesn't look like a stage geometric series at all. Why so? Let me give you a two plot?
These are superimposed to two functions. The first function is a point nine nine to J.
That's the geometric series. The second function is another geometric series, zero point nine five to J.
So I have a two geometric series. You can see that the first geometric series decayed too quickly.
Relative to the red curve. Radical is the is the acf implied by the data.
And the other curve is a point nine nine five to J. Although it describes this part of the Acaf.
Reasonably okay, but at the beginning is too high.
So basically these two. When I compare this three graph three curve, it tells us.
We cannot use geometric series.
To approximate the Rat. So this just simply suggests that we cannot use.
Our model for to capture an auto-relation function like what data tells us.
Okay. We can do a similar exercise for the spectrum density. We can ask them at spectral density, and then, if we do that, although I didn't do that, do that here, but if you do that then you can see the spectral density will go, to gets bigger and bigger as a lambda getting closer and closer, to zero, so that means spectral density approach to
infinity, as a frequency approach zero. Again, that suggests that I'm a model in is in adequate.
For this time space.
Not only for the you know. S. P. Five hundred, realize volatility.
We see something like that for many other economic financial time, space.
We see similar properties.
Okay? As early as in the nineteen sixties, Clive Granger, who get Nobel prize in economics, he publish a paper argue that for many macroeconomic times with the spectral density increases in definitely as a frequency decreases.
As we thought of that, he argued that meaning Economic Time Series can all be well modeled.
By hama model.
So that's a motivation.
发？
Of of that today's talk. So far, so good. Yes, yes, all of the.
Can you? Can you wait until you get Mike, so that other can hear so I don't know whether is it a possible job?
Just phenomenon is due to the structure change, because I noticed that so long conference sample.
Okay. Very good question. So let me repeat your question so that I I understand. So, the question being imposed is that it's possible to use a structure change model to get all the covariance functions like that of course it's it's possible that's not stationary once you have a structure.
Break you have a non-stationarity, and once you have a non-stationarity, you can get almost any kind of shape or from other, but if you have a complicated structure, breaks in your model, it means that a model.
Are less useful. Okay, think about it. If you have a constant structure, break, how can you forecast the future?
Possibly I should use the sample size as the sum after the break. But you still, if you have so many structure, break, you don't know when, when another break who had taken place in the new future, how big is the break when the break would happen as a result of that if you fail to take into account, of the that possibility your forecast will be seriously wrong so that's why you
know, still very important to think about this alternative way to structure, break for modern in time, space, behavior.
Okay, very good question. Another question. So give a round the standpoint.
I don't know whether it's possible to test whether the Asf is the decaying at geometric rate, or I think it's like to test a shape of a function.
But as a function so I don't know whether that test is possible. That's exactly what we're going to do.
We are not directly test the auto-relation function as a geometric rate, but we are going to formulate a model which, you know Nast.
Jr. The model whose acf is geometric series as a special case. We're gonna estimate rather than parameter, and make statistical inference about that parameter.
Therefore we automatically design a test to suit your purpose. Oh, thanks. Any other question.
Yes.
Okay, thank you, Professor, for such a great summary on discrete Time Series analysis, and actually, I lend all these stuff by myself, by reading Hamilton Textbook.
So there is a question always confused, confused me. That what is the practical meaning or the economic meaning?
When we say that the Lambda or Omega goes to zero. So what is the practical meaning of that?
Could you please use it, I should have I should have discussed that number is called frequency.
Right. So even number take a value close to zero in means that extremely low frequency, no frequency means, you know.
You sample. The sampling interval is a long. So when Namda take a big value means high frequency.
所以有sample的data。
Often. So that's the economic meaning.
Okay。这样。
Of course you would expect when I'm gonna take a different value. You have a different amount of information.
Hi.
In the past after grandeur, establish the fact that many economics cannot be modeled by our stationary armor model.
In particular, he argued that many economic talks has a long memory.
There are two alternative approaches to model a time space of financial terms with that has a long memory.
The first approach is called discrete time, fractional models, which is what I talk about in the last summer school.
Two to three years ago.
There's a not approach to the first post is a dominant approach, so most of the people are using the first approach.
Okay. The second approach is call continuous. Continuous start, Tom, fraction, approach this approach, although, was invented in the sixties by a very famous mathematician, Mendebot, but and never receive much attention.
I'm tier. A few years ago in finance. So people basically revive that literature.
A few years ago.
And more and more people are attracted into that digital, this digital. Okay, steer, discreetam approach is dominant up to now, but I see more and more research into the continuous approach.
So that's why I like to give a a lecture on continuous approach.
This is an area that I'm not actually trying to make a contribution to.
So we call it continues some fraction of models. So I'm going to introduce a few continuous time.
Fractional models. Let me, before I introduce you at first continuous time. Fraction of models.
Let me first give you a continuous model, which is a standard, continuous sound model. You probably know this model is called primary motion B.
Aim.
Promotion was invented by Brah Brown is actually a biologist from from Uk.
He studied the movement of a particles in liquid within, inside the liquid, and if he found the the movement of a liquid is a purely random, is unpredictable.
And so, a few years later, there are two scientists, one, the first one is known as the Upper Einstein.
He use pon a motion to study the physical motion.
In. He published a paper in ninety O. Five.
That use upon a motion in physics. In the meantime, almost independently, I I'm sure it's independent.
There's a French financial economies. His name is he used Brandon motion to model the stock price, which is not surprising why he's doing that.
If you think from now from today's angle, because stock price, we all know that many people believe the stock market is sufficient market if it's efficient, then it's unpredictable that's exactly match match the property of our brand and motion so bachelor actually use Brandon motion to model the movement of the stock price so you can see the brown
emotion goes all the way back to more than one hundred years ago. So let me formally define brown and motion.
Promotion is a Gaussian process with a zero mean, with a auto covariance function given by this equation.
So you have a Wt. Ws, which is a Boundary motion. Observe at two time point T. And S.
Doesn't need to be integer anymore, because it's continuous on model. So t and s can be anything.
Then the covariance between Wt. And the Ws. Is given by the minimum value between t and s so because it's a Gaussian process, you know, Gaussian process means the marginal distribution is normal right?
So if the distribution is normal, then we know, we only need to know the mean and variance to understand the whole thing.
Right, so we only need the first two moments to determine.
All the properties of a Gaussian process. That's why this is, you know, clear definition.
So we have a Gaussian process. We know the meaning is zero. We know the auto covariance function.
Okay. You probably have a better understanding of a discreet, random walk.
But emotion is basically the info limit of the random walk process. So the random work process can be written as a Y.
Th equals y t minus one times h plus apps along t H. Where absolute ph is id normal with mean zero variance h.
So if you assume H. Go to zero, then yt basically converge to Barnot motion.
So that's basically, this is a discreet time representation of a front of motion, and the brown emotion is the infield limit of a random walk.
No, that's the variance must be zero. There must be a variance must be the same as a sampling interval.
H, so as h go to zero. Of course, this guy, the the variance of this distribution goes to zero, and means that Absalom th will converge to zero.
If this converge to zero as H. Equals zero, it means that why th will take a value. The same as a yt minus one H.
So that means the sample pause of a brown, and motion is continuous in probability, almost surely okay, so part of motion is a continuous term process, and it's a non-stationary, because it's random walk we know random walk is non-stationary so you may ask and this looks like a yeah, one model.
Yeah, it is a one model, but the Ar one. Coefficient is one right? So the here, if you put an off from here, I say, you know as we did, you know, one model, the offer take advantage of one.
Previously we assume of a lesson one so that's a stationary, a one model.
Now we have a non-stationary, a one model which is random one. And then, if you look at in increment of a brown emotion so like you know, ytm minus yt minus one.
H.
但是。
The increment is not only in stationary, but also independent, because is iid. So it's independent increment.
The the increment is a random variable, which is distributed as normal with the mean zero variance.
H. That's another property, and already mentioned the sample path is continuous, and also for notation of convenience.
We denote the change in Brandon motion over an infinite decimal interval.
阿拉伯。
So dwt basically denotes the change of bundle motion over and.
So a typical sample pause for part of motion is given here. The bronze motion is restricted, is assumed to start at zero.
When t equal to zero. So the initial point of bundle motion is always at the origin, and then this is a typical sample pause.
It wound around, so the sample pause wound around.
You cannot predict the direction of the movement of a primary motion, because it's iid normal with mean zero.
So it's not predictable.
That's why we have an efficient market.
How can we simulate a sample? Pause? The simulation from Brandon? Motion is basically basically come from this equation?
So I don't know if you use all. If you use our let me explain to you.
How do I typically simulate from the from promotion? Let me go all the way to to the end of.
To the end of my slides. Well, I give all code to simulate from Brandon motion.
Remember the increment of a brown emotion is the Iid normal with mean zero variance. H. So let me use.
This is our code. Okay, let me use the order to denote the sampling interval, like what I call small H.
And is a sample size. How many observation you simulate you when you use a computer, you can not simulate continuous on observation computer doesn't deal with is continuous sample integration.
Computer can only deal with discreet time, but you can imagine if Delta is so small it's all almost like a continuous observation.
And then what you need to do is you first need to simulate. Iid normal. So this, all known is a simulation of Iid normal.
How many id normal you simulate? Yeah, the mean is zero variance is a square root of age, which we call the auto, and then the initial point is zero.
So you are pants, you know. You put a zero in the beginning of this factor which I call X. And then why is basically cumulative sum of X.
That's basically the brown motion, because promotion is the increment of a bonnet.
Motion is id normal. So if you have a id normal, what do you need to do is accumulate.
Some of id normal. You get a bunch of motion. Okay? And then you can, you know, plot this.
Why, which is a Brandon motion to get your sample pause. That's exactly what I'm what I'm going to do.
Okay, so let me go to all my, all code, and then let me tell you this is my code that I just explained to you in detail.
I copy and pace into all, and anybody use our yeah. I think most of you use all right.
If you use other software, you know it's also fine. This is only a two line of code, and then I'm going to.
Choose a delta to be point zero zero one, and to be one thousand. So I'm going to simulate one thousand observations.
From us, Brandon, and then I plot it.
You see. So this is a typical sample pause of a brown and motion.
So it's very easy to simulate from Parna motion.
I would strongly suggest you whenever you see a model you try to simulate data from the model.
You get some feeling about the dynamics of the model. So that's the first continuous number, and of course Brandon motion doesn't generate long memory because it's non-stationary bye.
Okay, and it's very easy to achieve stationarity from Barna motion. You take first difference.
Once you take a first difference, you get iid sequence. You completely get rid of the you know.
Correlation structure in bonnet motion. As a result of the first difference you get iid sequence.
Id means independent and identically distributed.
Okay, now let me introduce the first fractional, continuous time fractional process.
Which I already mentioned to you, which was to mend the Broad and Venice in a ninety, sixty, eight paper.
Linda Bot is a very well known mathematician from Yale. He invented multiple things.
That's become useful in finance. This is one of them. Fractional bonnotion we call Fb.
Aim. F is a means fractional, B. A. Means brown emotion.
So it's also Gaussian process with mean zero. So so far is the same, however, the auto covariance.
Has this structure?
So it's not. It's not only in a function of T. And S. But we also have another prime that we call H.
This age has a name, is called Host, the parameter H. You, O. S. T.
Hurst. Mathematical notation for fraction of our emotions is called B.
Soup h of T, that's how we usually call fractional bonding motion. B. Super H.
Of T, indexed by the host parameter. You may ask why we we called H. The host Parameter.
It's because Hurst is a hydrologist. He studied the river level of a Nile it's the longest river.
On earth. So he studied the river level of Niles over time, and then he plot.
He plotted the auto correlation function and found the auto-relation function of a Nile Revo.
River, decay. Very slowly. He doesn't know how to model it. He just basically report that you know observation.
So in honor of that discovery people call H. The host Parameter. That's the reason.
So you see that Menda broad define a Gaussian process with mean, with auto-coverist fraction.
Therefore, this is a uniquely defined Stochastic process. We know the mean. We know the virus.
We know it's a Gaussian process. Therefore, you know everything.
Okay, and H. Is required to take a value strictly between zero and one T. And S.
Are ending. Two things doesn't need to be integer that. Basically, they represent Tom just like a brown emotion.
If the host Parameter, is zero point five, then what do you have?
If T. If H. Is zero point five, then I'm going to work out, I'm going to find out the yeah for the auto-ovariance that's going to be zero point five Tom's.
的是。
Am I right?
So is t nice plus s mine is absolute value I don't know how to write absolute value.
In.
Maybe here yeah okay t minus s.
So the right hand side of the other covariance becomes this, so you can see that whenever t bigger than s what do we have?
If t is bigger than s we get t. If t is smaller than s what we get we get s sorry we get s.
If our T. Is smaller than s we get t. So that basically tells this is a minimum between t and the S.
Am I right? What we already saw this auto-ovariance function, that's for fun of motion.
Therefore fraction of our emotion becomes brown and motion. One H. Is half.
Agree.
And there are many other interesting properties for fraction upon a motion, but before I talk about those properties.
If H. Is not half then we don't have. You know, this nice, fractional.
This is nice minimum tns, as us and our auto reception. Right? You would expect fraction of bundle motion will be more complicated, much more complicated than brown and motion.
One age is not half which I'm going to study in a minute.
There are, I already mentioned, fraction of Barnot motion, as a you know, many interesting properties, like Brandon, motion, fraction of the motion is always long stationary for any age.
We know if H. Is point five point five you have a Brandon motion which is non stationary. Okay, also, like a promotion.
We have a self similarity. That means b h evaluate that at where A is a constant T is a Tom.
Can in distribution is equivalent to.
A的，他们。
So you can move a out, so that means be H. You've added at is similar to the original fraction of bundle motion.
You just scale it by a to the power H.
So that's another property called self-similarity. What about sample? Pause the supper pause of a fraction upon a motion is holder continues of all the H.
Minus Epsilon where Absalom is any positive number, is a small positive number can be as small as anything bigger than zero.
So it's an arbitrary small number. So because there's there's some. The continuity.
The whole continuity of a promotion is determined by age, so you would expect.
The smaller the H is the roof of the sample path.
When H. Is a bigger than point five. Fraction of bonnotion is smoother than standard brand of motion.
When H. Is a less than point five.
Fraction of our emotion is a rougher than standard amount of motion, simply because the continuity, the order of continuity.
Is determined by H.
我们。
We know the Barna motion is continuous, but the order of that continuity is up to half.
That's up, you know. The smoothness for standard bundle motion up to half, but for fraction of bundle, motion.
The smoothness is up to age.
Obviously smaller. The H is.
的，的。
So when I say, Ralph is relative to Barna motion, when I say smooth, also relative to body, motion.
So let me explain this three graphs, and then we can take a break. So I simulate data from fractional bonding motion.
With a three different h values. H go to point one point five point nine. I will explain to you.
How did I simulate the three data after the break? So we have a three sample pause.
The sample pause in the middle is for standard random motion. So I already, you know, give you the our code to simulate from standard, broader motion, and this is really a typical part. Sample pause for standard brand emotion, or continuous random wall.
So it has some degree of continuity, but when H. Is point one, you see, the sample path is a much rougher.
Then the one in the middle. When H. Is a point nine, the sample pass is much smaller than the one in the middle, so you can visually see that Ruffin is the smoothness relative to barnot motion.
Okay? All three series are non-stationary.
Is very easy to see non-stationarity from the lost Graph.
It's harder to see the non-stationarity from the first graph, but all three series on non-stationary.
I'm going to take a break thirty minutes break after the break. I will explain to you.
How do I simulate the three graph? Okay, let's take break. Thank you.
好，欢迎大家回回到课堂里。
Or that's continue. Okay, so this is basically the simulated sample path of a fraction of on a motion with different H.
Values. I'm going to explain. How did I simulate data from the fraction of bonding motion?
But before I do that since fractional Barna motion is non stationary, I would like to find a way to achieve stationarity.
So one way to achieve stationarity is to take a first difference. So let me take a first difference of fraction upon a motion at two consecutive Tom, discreet Tom.
Say t and t minus one. You don't have to do at t and t minus one. You can do t.
And s two number to time, point and then, if you scale it by sigma, then I call it y, then yt is known as a fractional Gaussian noise.
Ftm, which is another model. So fraction of Gaussian noise is the increment of fraction, of bound emotion.
Tom's a scale parameter sigma. Okay. So in all of us, fraction of on a motion is accumulated sum of a fractional Gaussian noise, divided by six months, right so in this in discreet time, if you if you regard yt as a error, turn.
但啊。
The reason we can not call it a random mark model is because every term is not necessarily an iid error.
If y is a iid, then this guy is basically a random one, because this guy equals that guy plus and eritor, which is iid.
But it in general is not a that's why we say it in discreet arm, fractional fraction upon a motion is a unit route, because the ar one coefficient is one plus an hour term which is not necessarily Iid so this noise term is not why noise, is a fractional gaussian noise so that's a relationship
between fractional by the motion and the fractional Gaussian noise. So you get fractional Gaussian noise by taking first difference in fraction of bundle motion, or you get fraction of on a motion by take a cumulative sum of a fractional c and noise one can easily show that fractional Gaussian noise is always stationary for the
same h parameter, which is between zero and one and the other covariances of fractional c and noise is given by this equation.
Two. Oh, that's easy to see that if h is half then the covariance becomes zero whenever K.
Is a past is non zero.
In the case nonzero. The covariance becomes zero. You can see that right which is not surprising, because if H.
Is a half. The increment of fractional bonnotion is the same as the increment of function, which is iid.
Therefore, of course there's no covariance. That's why you know the covariance is zero.
But if H. Is not half, then this guy is no longer zero for ending. K.
Doesn't matter how big K. Is as long as H. Is different from half.
但。
Auto-coverances is never zero. What does that mean? That means the fraction of Gaussian noise will always involve serial dependence.
Will never be on why noise? We are always involve serial dependence. You can easily show that if H.
Is bigger than half. This guy is always positive. If a H is less than half this guy, always native, think about it.
If it's an active, that means that yt, and the ytt plus k for anyk where tend to be negatively correlated.
So if yt is a positive yt plus k tend to be positive as well, because corridor correlation is positive.
The stronger, the smaller the Hs. The larger, the probability that yt plus k tend to have the same sign as white tend to have the smaller the ages, the more the higher probability that yt and yt plus k tend to have an active different.
Sign can you have a different sign? Because it's more active. The bigger the h is h is, say point.
Nine. Then this the Covariance tend to take a big number, and hence the covariant, and hence it and a ytt plus k tend to take up same sign that's why one eighties bigger than zero point five it tend to generate a smooth sound sample pause because the increment tend to be positive right similarly when h is less than zero point
five, just two variable. Have a native auto correlation. And hence we have an entire persistence.
This this period we have a positive sign next period. Tend to have a active sign. So it's and that the sign changes.
That's why it has our entire persistence. A rough sample pause. We can even do a approximation when K.
Is large. K. Is a license right because the memory property only did is only determined by the long lag.
When k is a small that doesn't affect the sumability of auto covariance, some ability of our auto-reference is only determined by the property of the auto covariance when K is large so That's why I study the property of a large K when K is large you do you know expansion you do expansion twice then you will
get this approximation, so covariance will be approximated when K. Is large by Sigma Square Times, H.
Times two h. Minus one times k to two h. Minus two. Okay.
So first, let's see.
What happened if H. Is half? This is this approximation doesn't work for H.
E to. But let's see what H. Is what happened. If H.
Is hard, two times half is one, one minus two is active, hard. Okay? So that's the benchmark.
This poximation doesn't work. When H. Is half, because when H. Is half the covariance is zero period.
But I just the reason I want to tell you this benchmark is because I want to study when H.
Is a bigger than how when H. Is bigger than half, then this is a k, two h. Minus two.
Remember when H. Is half we are dealing with a one over K. You see, did you see that? Okay?
Let me write it down.
If H. Is zero point five. Then this guy is a one over K. Acf.
Covariance is is approximately.
Approximately one over K. Right.
And we know one over K. Is not summable.
Do you know that one overca is not summable one over K. If you are, if you add one over K.
Say, K. From one to infinity. You. You are going to get a log N. Log, infinity.
Which is not finite number, which is infinity. Right? But now, if H. Is bigger than half what are you going to do?
You are going to sum k to two h. Minus two, which decays slowly. Then more slowly than one overkay.
Right. If one other case is not sumable, then, of course, K. To two h. Minus two is not sumable. When H.
Is bigger than half.
So this suggests that if H. Take a value between point five and a five and a one, the auto covariances of, if fractional Gaussian noise are not summable.
So this means fraction of Gaussian noise has a long memory. If H. Is bigger than zero point five.
Can you appreciate that? That's why fractional Gaussian noise can be important model.
You can generate normal memory. If H. Less than point five, we can show that the long run errands equal to zero.
我的。
We know variance is always possible. Right? So if you if K. Equal to zero, you get a variance.
If k running from minus infinity to one minus one plus k running from one to plus infinity, everything add up to various.
Minus minus varies. That's why you get zero right. So this quantity, this property, this one.
Oh.
This property.
Basically suggest that variance E calls.
All the covariances, some together.
And baron is a positive number, so the summation of all auto-coverance must be connected.
In fact, every covariance is connected at any order, because we show it here.
So this is, has a very strong implications. Yeah, of course, it's too strong. So that's why the fraction of Gaussian noise may not be.
Idea model in practice.
Okay?
What about variance? Variance of yt, you know? If K. Is zero, you can easily show that, you know you get Sigma Square.
So Sigma Square is the variance of a fraction of thousand noise, and then the when H.
Is less than zero point five. The summation of all all auto-coverances equals minus Sigma squared.
Okay, how can we simulate sample? Pause from fractional Gaussian noise?
I already ask this question, how can we simulate from fraction of bonding motion? The idea is, we first simulate from fashion and noise, and then once we have a fractional Gaussian noise fraction of our emotion is accumulated sum of a fractional Gaussian noise right so the in terms of simulation it would be conceptually easier to simulate
from this guy first, and then we can simulate the fraction upon the motion as a cumulative song.
But how can we simulate from fractional Gaussian noise? Clearly fraction of Gaussian noise is gal.
Same process meaning is zero. Auto-reference is given here.
Right. So if I know a stochastic process is a normally distributed, the mean is zero.
I know the various Covariance, all the element in variance covariance. So what I do.
Okay, so let me, you know. Tell you a very simple fact.
So, if x is a stochastic process is a multivariate, you know a random, variable.
It's it follows a normal distribution with a vector zero as a mean with, say, a as a variance covariance.
Matrix.
Right then. I know.
Then, I know, suppose, suppose A. And meet and Chaloski decomposition?
就是。
然后。
I get a standard normal. I get a multi-arian, normal variance to be identity matrix.
So, in order to simulate from fractional Gaussian noise, you first need to simulate from multivariate independent, normal, which is very easy to do, and then you need to multiply by chelsea decomposition. You get basically the fractional gaussian noise.
Okay, that's exactly. What are we going to do?
So you see that I have this our code. I first see simulate from Iv.
Normal. I treat that as a multi-varian normal. How many normal end of them? Oh, before that, this is all function.
So I still have a delta have an end which is the same as before. But now I have a H.
So user need to need to input the value of H. And then the first code of the first line of code is to simulate multi vary and normal distribution with a mean zero.
With, you know. Various stands.
And then I choose the sigma to be an identity. Matrix, and then I'm going to give up. So sigma is is a M. By M.
Matrix.
Dc就是。
And then I'm going to give each element in this variance covariance, matrix and definition.
So this definition is exactly the same as the covariance matrix of the fraction of Gaussian noise.
It's a half Tom's Kappa, plus one to two age, plus k minus one to two H.
Minus two times, you know the difference of the index to the two.
H. So that's basically the definition at the property of the auto covariance function of a fractional Gaussian noise.
So that I do a do loop. I running from one to N. J. Running from one to end.
Then Sigma now is the for a variance covariance, matrix of a fractional Gaussian noise.
Then I need to do decomposition. C. H. O.
L. Of sigma means Shallowski decomposition, I multiply that a pre-multipilized telescope, decomposition by X. X.
Is a multi-arian, normal, where the covariance matrix variance measures of X is a diagonal matrix.
But I I multiply this by Chaloski decomposition. Then I change it to as serially dependent, you know.
Process, I. And then the initial condition is zero, because fraction of bundle motion need to start at zero with a location zero.
So I call that Z. Then y is accumulated some of Z. So the last, you know, couple of lines are the same as the as how I assimilate from brown and motion, but an emotion is a cumulative sum of Iid sequence.
But Z is no longer id sequence because of this telescope composition.
Bye and that's how I how am I going to simulate data from fraction upon a motion in particular?
These are fractional Gaussian noise. Why is a fractional bonnet?
Motion. So let me put that in in the all code.
Simulate fraction upon the motion.
So I just copy and paste everything into my all editor.
Okay, simulate?
Functional button motion. So I have a three parameter again. I'd like like before I change it out to be point zero zero one, and to be one thousand, and then H.
To be zero point one.
So you see, that's the sample. Pause from fraction of on a motion. Of course you can plot, you know, fractional thousand noise.
That means you plot Z, but I plot fraction upon a motion. What about when H.
Is zero point nine.
So you see, very smooth.
Okay, again. The reason the intuition why we have a smooth sample pause is because the increment are possibly created.
One H. Is point nine when the the previous one is so raw is because increment tend to be negatively correlated.
And persistent. That's why, you know, cumulative. Some tend to have a rough sample pause.
Okay?
So. So that's the yes, you know the fractional calls and noise. Let me continue.
What about spectral density? The spectrum density of a fraction of Gaussian noise is given by this equation.
Three. I won't be able to, you know, derive this, because this requires some time to do it.
If you are interested in how this is derived, you can look at the textbook, Baron nineteen ninety-four.
Proposition two point one. They report the spectral density for fracture, no noise.
He give a spectral density, the expression of fracture density. When Lambda is between zero and two pi again, if you want to study the spectral density, one python between minus pi the number is between minus five that's also fine okay, so you can see that spectrum density, of a fraction.
Of Gaussian noise is quite bit more complicated. Then the arm model. Why? Because this is involve a infinite sum.
This calculation of infinite sum require some, you know, approximation. You have to approximate it one way or the other.
We are going to come back to this later how to calculate spectral density of fraction, of goals and noise.
Computer does not I do this infinite summation, because this invite summation doesn't have an analytical solution.
So you need to, you know, either truncate or you. Do a you know some alternative approximation as before, you can.
Do you know a low frequency approximation? So if number go to zero, you can approximate this spectral density as a a power function of Namda?
So it's a number to one minus two H. Again, take a h equal to half as a benchmark.
If H. Is half, then this is basically namda to the power of zero.
Bye. So of course that's a finite number. So if H. Is less than half, what are you going to have?
You are going to have two Thompson number, which is smaller than half even now I'm gonna go to zero. What's going to happen?
So the power, one minus two H. Will be positive in that case. So Namda, with a positive power as a number go to zero, will converge to zero.
So spectral density converge to finite number. In fact, we know we will converge to zero. Why?
Because long romance is zero. This should converge to long run. Marriage.
Okay. That's the first thing. What about H. Is a positive.
What about H is bigger than half? If H. Is bigger than half, then one minus two h. Two h.
Will be, that's it so, not to an active number as an active power as number go to zero.
We are diverged to infinity. Can you see that if H is a positive, if H.
That's why we say that fraction of Gaussian noise has a long memory property, because at zero frequency. The spectrum is the doesn't exist.
And when normalizing near zero, the spectral density can actually be approximated by by this guy.
So that's approximation. Order the above approximation of Namda is good even for relative large frequencies.
Okay, since the spectral density of Fda involve infinite sum. So if you want to calculate spectral density, kes are needed.
Okay. So this is what I already said at a frequency near zero F. Of number behave like a number two, one, one minus H.
So when H. Is a is a bigger than point five if a number diverges as a number converted to zero, hence Fgn has a long memory.
When H. Is bigger than point five. So that's one way to achieve stationarity from fraction of bound and motion.
There is another way to achieve, to to achieve stationarity from fraction upon a motion. This is our much more complicated than by taking first difference but in terms of a modeling this is a much more realistic so I'm going to explain but before I explain that let me introduce you the Ou process Owen Stan Ollenberg process ou process.
Is a stochastic process, continuous number based on standard promotion. So wt is a standard boundary motion.
所以可以可以。
So dyit, which is increment of equals kappa times mu minus ytt times dt.
Plus sigma dwt.
Where we assume initial condition. Why is a W. Zero? So we start from Y.
Zero. And then after that the process evolve according to this continuous time, model, since this is a differential equation, we can solve differential equation.
Is a stochastic differential equation, but it's we still can treat it as a differential equation.
We can solve it. The solution of this differential equation has analytical solution, because it's a very simple function.
Differential equation. It has a Nia drift term, and the diffusion term, which is a stochastic term, is a constant multiply by the derivative of our primary motion.
So this is a one of the simplistic, most simplistic stochastic differential equation, so you can, if you know how to solve ordinary differential equation analytically, then you is, then it's easy for you to find a solution.
To process. So that's the solution. So yt equals e to minus kappa t.
Times y zero times this term plus sigma times this stochastic term, so you can see this is a increment of a brown and motion, this is a weight, you know, increment of bonnotion.
Is iid. So it's a summation of a way to the average of a iid sequence.
So, of course this is a random component, and that this is basically the deterministic component.
So that will basically change the the location. But this is basically determined. The various. Okay from that solution.
One can easily show that whenever you assume y zero to be normally distributed with mean mu variance equal to Sigma, squared, divided by two kappa with a copper positive yt is a stationary, and I got it you can actually see that since yt is related to y zero this way.
Y. T minus one must be also related to y zero in a similar way. So yt and the yt minus one must be related to each other.
In fact, yt and yt minus one is related manually. So that means, you know, we have a auto-regressive representation in discrete time, for all you process.
Although there are been more mass involved. But I don't want to show you all this mask, otherwise I don't have a time to to introduce other models.
So the discreet-m, exact, discrete-time representation at two consecutive time point between I minus one delta, and the I delta is given by this equation ar. One.
Model. So why i delta is an near function of a yi minus one delta. Well, the ar one coefficient.
We used to quite offer or Beta. This coefficient is now a function of a Carpa and the delta delta is a sampling interval.
Kappa is the parameter here. This is called mean reversing parameter.
啊。
Parameter in the drift function. So you see that the other parameter doesn't show in the errors in the a one coefficient only in copper Show, up and the exponential function.
Of copper, and then this is the intercept. This is the error which I already say, that is basically weighted.
Average of Iid. Sequence, therefore, the error turn is expected to be iid sequence because of a independent increment property of brown and motion.
So that's why it's a one model, because the noise is a wide noise. The relationship between the two consecutive observation is Nina.
So this is a Gaussian ar, one model why is a Gaussian? Because this is normally distributed, and then it's weighted. The average.
Of a normal distribution, therefore absolute they are. The noise is no more distribution. So that's why it's a Gaussian Ai, one model with an ar one coefficient alpha equal to the exponential.
然后。
Okay. So if Delta is very close to zero, all of our take a value close to zero, close to one, so that means we have a slow mean reversion.
If take a value far away from zero. Then our power will take a value far away from one.
That means we have a strong. Okay.
If you sample, very frequent, that means delta go to zero. Delta is a sampling interval.
Right. If you sample very frequently, that means Delta is taking a small number. Our price is going to be closer to one.
Coding copper constant by shrinking delta towards zero. You were shrink of a value towards one. You push the value of to one.
So that's basically the discreet representation. And then, because this is a one model, we can easily get the auto covariance function which is given by this function sigma square divided by two K. By E.
To the minus carba times. The absolute value of K. This is a case order, other Covariance.
This is a geometric series. Actually, it's continuous to model. So we, the continuous time correspondence of a geometric series is a exponential decay.
When copies, capacity?
So that function dk towards zero very quickly.
So this model basically correspond to stationary ar one model. So you will expect all those properties in Ar.
One in the in a one model.
Can be found.
In the o u process. Okay? So that's the process which doesn't allow fractional property because it's such, you know, exponential decay.
So the auto covariance is a sum of them. What about you know?
You replace the standard bundle motion wt. By fractional motion, holding everything else constant.
Then what what are you? What are you going to get is a fractional process.
So this is a fraction of you, which is a very new model to the literature. Okay, if a kappa is a positive, and then, if you assume initial condition, is a normal distribution with a with a variance given by this complicated guy then we can show that yt is a stationary.
Okay. So this so Baron is determined by Sigma Carpa age, and then a gamma function gamma function is a special function which is defined by integer.
Okay, it's a special function, so we can more or less treat as a gamma function as a analytical function, because it's very easy to calculate.
So that's the case when copies are.
第三。
Because increment of a fractional boundary motion is no longer iid. So if you have a one representation which is given here, the exact discreet-on representation is given here, where, although we it looks like we have an ar one representation, however, the error, term which is a weighted, average of our fraction, of gaussian noise, is not Iid so this is this
is more like fractional Gaussian noise rather than wide noise, so the noise is not.
Iid therefore this is no longer an ar one model, strictly speaking, isn't a, you know, is a model where noise term is not.
Iid. The noise involves zero.
Dependence.
If you shrink down towards zero, one can show that absolute idlot behave like a fraction of Gaussian noise, so that means, if H.
Is a bigger than point five. We have a positive serial correlation in error. If if H.
Is less than zero point five. Then we have native serial correlation in error. Of course, if it is half, then arithm is a wide noise, therefore the discreet representation is an ar one model which is not surprising, because if h is a half then Fou becomes O you.
Right, and you would expect if H. Is a bigger than half. We have a long memory here, because this guy behave like a fraction of Gaussian noise, and we know fraction of God.
Gaussian noise has a long memory. When H. Is hard, which one age is bigger than hard, therefore the fraction ou process can also have a long memory.
Property. One H. Is positive. One H. Is bigger than how one age is less than.
好。
Then fou doesn't have a long memory. Property.
Okay. So that's basically is a steel fractional model. But it doesn't produce a lot, doesn't have a long memory property.
Given Kappa is policy given a constant sampling interval.
Okay?
Of course, when kappa is zero, then this term is gone, the drift function is gone.
Then yt becomes sigma times b t h, then yt is basically a scaled round and fractional boundary motion, and hence is non-stationary.
So in discrete time. In this case yt is a unit route with fractional Gaussian noise.
It's not a random walk model. It's a unique room model, with a fraction of Gaussian noise.
Like a fractional thousand noise. We can show that Fou is a holder continuous of all the H.
Up to h can f u is our H. Less than zero point five has a rougher sample pause than that of process.
Holding other, you know. Variables constant are the parameters constant. Oh, is H. Bigger than zero point five has a smooth sample, cost and than that, O you Process.
And in the literature people studied statistical properties of Ev you. And here very recently, for instance, for any positive copper value.
嘎尼亚的。
Give an expression for the auto covariance of fo, which is given here. So you see that A for you has a much, much more complicated auto-coverances than O you, then fractional, Gaussian noise.
This involve an integral one, dimensional integral. Unfortunately, this.
英帝国。
Is why are there is one dimensional integral doesn't have analytical solution, so you can not write it as a analytical function or special function.
So, in order to calculate the auto covariance. One way to do it is to use numerical integration.
Which will have a serious implication, implication in a in many applications which I'm going to, you know.
Come back later. So that's a one very important property for fo. In terms of auto covariance.
So auto-coverance is given by equation. Seven. We are going to come back to this auto-varance.
And then in another paper by Charito and and all show that for any positive Kappa, the Auto covariance of Fo, behave like a auto-revance of a fraction of Gaussian noise when the leg nancy is large so that's why when h is a bigger than point five we have a long memory in fou in fact the auto
Covariance of Ev. You are not absolutely summable. This paper actually is not very easy to read.
Is involve a lot of approximation, you know serious approximation.
And when, from the discrete time representation, when Kappa is fixed, the other go to zero, we know the ar one coefficient will converge to one.
So the exact, discreet time model is a local to uniform model local to unit room, model in discreet time, basically require a one coefficient to converge to one right.
And however, we have a fraction or noise in in the discrete, in the exact, discreet model.
So it's a local to uniform model, with fraction on noise. If a Kappa is not zero, if Kappa is zero, of course it's an exact unit room model but again, with a fractional noise fractional arrow.
So far, so good.
Okay. So as a another property is a spectral density, for of evou this is a continuous sum model for continuous model.
The spectrum. Density is no longer your article function. The spectral density of fou is given by this function, which is actually, you know, a simple function for number between minus infinity and infinity.
That continuous Tom F. You spectral density is given by Sigma, squared, divided by two pi times C.
Of H, where C. Of H. Is given here.
Tom's number to one minus two H. Times Copper Square, plus nominal square to the power of minus one. So that's basically the spectral density of Fo.
And it's easy to see that from this spectral density.
一发啊。
H. Is take a value less than point five when number go to zero. Then f of number will go to zero.
But if ages a pause is a bigger than point five, when I'm not going to zero, if of number.
Now we're diverged to infinity. That's exactly what I said.
对啊。
F you with H. Value bigger than zero point five, has a long memory property.
Okay. So I'm going to make assumption. So because in the end we want to estimate the parameters in F.
You how many parameters you? We have a three parameters, a four parameters, Kappa, me.
So, in order to estimate these are four parameters. We need to have a collect observations, and then we need to know that.
Yeah. Four parameters are uniquely identified in order to identify the four parameters, we make assumptions about the parameter. So we are going to assume the because the spectral density doesn't involve me Mew is the long run mean as we show yeah, the mu is the mean, so if.
And the spectrum doesn't involve me. So I'm not going to talk about the identification of a mute, and the spectral density only involve the other three prime just sigma square h and the Kappa so i'm going to assume the parameter the three parameter copper h and the sigma square is in the Compact.
Subset of. Of of this you know domain. So under this assumption, we can easily show that the spectrum density is uniquely determined by the Prime Minister.
It means that if I change this three prime of any value of this three parameter a little bit, the function will be different.
The value of the function will be different. So the function, the spectral density function is uniquely determined by the three parameters.
We can prove it by the argument of a contradiction.
Okay. Quite easy proof, although I'm not going to prove here. But before I I forgot to mention that we can also simulate data from the and how to simulate is actually easy, because we have auto covariance right so you know that every element.
In the auto covariance in the various covariance, matrix, like what we do is a fractional Gaussian noise, and then we know the meeting is mute.
Because it's stationary, and is a stationary process. We know the mean. We know the every element in the auto covariance.
Of course every Admin need to compute. You need to compute the integration. Numerical, but you can do it, although it's time consuming, and then you have auto-coverances, and then you do compensation you do exactly the same as what we did for simulating fraction of motion so you can simulate fo in the same way so that's how
I did. I simulate data from. So this is three graph, this one, this one, this one. I simulate data with a three different age, and I fix the other parameters.
Karpa, for instance, is a one point three a one mute to be two point one eight nine nine six sigma to be zero point eight four four h.
To be either point seven point five or point one four five three, and the you may ask me why you choose this screening numbers because I got those numbers especially Karpa, Mu, Sigma, and the H.
E. Go to point one four five three from real data. Okay. So when I simulate data from those parameters second, I got plot Bcd.
So you can see this is this is our process it's not fou is a O.
U. Because h is point five right. So oo process. You can see this is a rougher than Barna motion.
Why is Rapha? Because Kappa is not zero anymore? Kappa is a one point three a one.
So that's you know. That's why it generate some. Now, you know, relative to Brandon, motion, not non smoothness, because we have a mean reversion.
Now by relative to promotion. But that's not the key message. The key message is, compare with a process.
These two fractional ou process has a very different sample. Path. Smoothness. For instance, if H.
Is a point seven.
If a is point seven, you see that, then, O you.
If H. Is a point one four. The sample pause is a rougher.
但。
This is our one. H. Is a point one four five three. This is a rougher than this.
This is a smaller than this.
So the benchmark now is not fractional by a motion. The benchmark is O.
U process so relative to process fraction of you can have a smoother, rougher sample path, depending on the value of H.
I also give you a sample pause of a Sb. Five hundred, the log s. Five hundred. Which is this the data I gave to you initially.
So among these three simulated sample paths, which one mimic.
Better are the best. Among these three grow up, which one is closer, is the closest to S.
P. Five hundred realize the volatility. Obviously this one.
That discovery was only made in the last couple of years.
The fo with h value equal to point one five. Roughly can mimic the sample pause can mimic the the realized volatility data better than this, and the app.
Okay, of course we are going to come back to this point later. So this is a spectral density.
I'm going to make assumption, and then, on the this assumption we know the spectral density is uniquely identified.
That means we can estimate the practice from the sample generated from Fou.
Yes, yes, wait a second until the Mike has passed you. Is this method enough for Capture?
The extreme values in the real data. Yeah, simple answer is No.
Because this is Gaussian process. So Gaussian process doesn't allow extreme observations.
Okay. So this process is men to capture persistence, including the long memory or anti persistence.
Okay, okay, thanks. So if you want to capture the extreme observations you need distribution, which is different from normality.
So the I guess the importance of this model, this type of model fractional type of model.
If he's gonna allow long memory. So talking about the river level of the nine.
What does that mean by long memory? Normal means that yt we remember what happened very long time ago, say a hundred years ago?
So in practical sense it means that although one hundred years ago, we have a flaw, and it never happened.
Never have a flawed after one hundred. The last flow. So you know, over the last one hundred we don't have a flock one hundred years.
We don't have a flawed anymore. It doesn't mean you can even know what happened one hundred years ago, because it has a long memory.
You can still have a.
装。
Probability of having another flock because of the long memory.
That's exactly the implication, the most serious implication of law, memory.
Okay. If you use a short memory data model like Ama model, you basically, you know what happened long time ago, because the geometric series is so slow, so quickly, but I caught into the long memory.
No memory model. You should not. So. That's the implication.
It's not about stream value is about persistence, memory.
What happened in finance, although we have a ninety, nineteen thirty-three, Great Depression that happened.
Can we? You know that we cannot, because if we have a long memory, that crash will happen again.
That's basically the implication. So as a trader, you should be very careful about managing your risk.
You should not forget about that episode.
但是很难进行中国的民主，中国的民主化进程。
Okay?
Of course, in practice we only have a discrete time observation. So that's why we need to get the spectrum density of discreetly Sampled Fo, so other sudden now normalize within one period say, zero, to two, pi or minus five minus five to buy so that's the mathematical expression for the spectral density of discrete
sample for you. Unfortunately, you can see that this function involve a infinite sum like fraction of Gaussian noise.
Fractional thousand noise is always discreet on model, because it's the first increment, is the increment of a fractional bottom motion.
So it's a discreet model by construction. Similarly, here we have an infinite sum, and in order to calculate spectral density, we call it F.
Delta. The reason why I got Delta here is because still there is a sampling interval so automatically we will have a discreet time.
So, in order to calculate Delta, you need to approximate the spectrum density.
Of course, a natural way to approximate spectral density is to truncate the lab.
I'm going to come back to this point later, and then we know the spectrum.
The two spectrum density are related to the auto covariance. In this way basically fully transformation.
So this is a Fourier transformation when f function is not periodic, this is a Fourier transformation.
When function is periodic, and the period has two pi. So both functions are fourier transformation of, you know, fully inversion of the article variance, and we know Fourier theorem tells us the mapping is one to one so if this guy can uniquely identify the three parameters are uniquely identified by the three parameters then this guy is also
uniquely identified by the three parameters. So we have a unique identification in the spectrum density of discrete sample.
F Ou. So that's basically the introduction of the motivation and the introduction of of our models continue some models.
So now I'm going to move to the estimation method and discuss the corresponding as some talk theory.
So there are alternative, estimation, methods for fraction of bundle, motion, or fraction of Gaussian noise, because these two models are basically equivalent.
One is the increment the ry is accumulated some. So if you know how to estimate prime there from one model, you basically know the others.
So I'm going to combine these two models into the same section to talk about the estimation method.
So sometimes I only talk about estimation, of fraction, of bundle motion. Sometimes I talk about estimation.
For fractional Gaussian noise. You know one, you know the other one. Okay, so let me first assume that a Delta, I. Xi.
Delta. So this guy where I is running from one to end, to be the discrete sample from fractional bonnot motion.
Okay? I'm going to talk about what happened. If Xi. Delta is a sample pass for is a discrete sample from F. Gn.
Later, the first estimation method that I'm going to talk about is a coof method.
Change of frequency, method in essence. This method is a mathematical.
You know math a moment, right? So you basically use a one moment to identify one parameter to ask me one parameter.
So this moment condition is obtained from the change of frequency.
So the idea is if you have a fraction of barn, a motion, you take a second.
All the difference at the highest frequencies.
And then you square it. Take a cumulative sum, and these were basically determined a smoothness of a fraction of modern motion, and then you do the same thing at by skipping one observation.
So you still do second all the difference. But you skip one observation when you sample one.
So this is actually, you see that relative to this second, all the difference you change to four, you change one to two.
So that's why you change frequency. And then once you do this square and summation, it will also converge to the to different smoothness.
So you would imagine the two smoothness are kind of related. Why? Because at the highest frequency the smoothness is slightly different from the smoothness as a causer frequency, because you just keep one you you skip one you know the the sample pause become either more smooth or less smoother depending on your regional
smoothness that in terms of it turns out, if you take the ratio of the two second, all the difference the functionals of the two second, all the difference.
In this two papers long, and the rule is bound of Nelson and all they show that if I use that as a moment condition, then I can we can propose and method moment estimator for H.
Which is a half times log. Two, two is the base for the logarithm of the change of frequency.
So that's the the first estimator. This estimator basically require one line of code to implement it.
So you give me data. I use one nine of our code to implement this estimation method very simple.
To ask me.
So that's how we estimate H. In fraction of boundary motion. That's the only parameter we care, because there's only one parameter in fraction upon a motion.
啊if。
The if. If the data comes from fractional Gaussian noise, then what we need to adjust is, we know, fraction of thousand noise is already the increment of a fraction of bond emotion.
Therefore you don't need to do a second all the difference anymore. You just do first, all the difference. So this will become Xi Delta minus x one x. I.
Minus one delta in the numerator. In the denominator, numerator will be X.
I plus two delta minus x i delta in the in the numerator. So you need to just the formula based on the first.
All the difference at the highest frequency, and then at the frequency, by skipping one observation.
Okay. So that's again one nine of a code, well, you know, will be good enough.
What about the asymptotic theory? It turns out. Suppose I use these three notations, and is a total number of observations.
Is the sampling individual T is a tom span, so t equals n times delta.
So you sample one observation every time you sample and terms. That's why the Tom span is T.
So, in order to establish the as a methodical distribution for H. Hat, C. O. F.
Estimator. You can do two ways.
You can fixt. Fix Tom Span, and then shirring, sampling in the. If you do that, then you have a more and more observation that's called in fear of some topics.
So over a fixed time in the row. You have a increasing sample size the way you increase your sample size is because you sample more and more often.
That's one way the other way is you fix their out. But you sample over long, and the longer time span that's called long span, as some products.
In economics, traditional geometrics correspond to long Spanish and products. But you can see here in finance.
We can also do info some products in this particular setup. Whether or not is a long span, and the info you get the same as the target distribution for H.
Hat to, so we can show that H. Hat is a consistent estimate of H. And the convulsions rate is the square root of N, which is a sample size.
The limiting distribution is normal. The variance the asymptotic variance is given by this guy.
Well, sigma, one, one sigma, two, two sigma, one, two are given by equation, thirteen.
And the row here row has nothing to do with the auto-relation function. Okay. The role here is given by sixteen.
Okay, for instance, if H. Is half, then the coof estimator will give this asymptotic distribution, where the asymptotic variance is seven, divided by eight Tom is a log two square yeah.
Coof, estimator, and.
Coof estimate is based on one moment you would expect this estimator, although it's extremely simple to implement.
It will not be efficient, because you can know, are the moment conditions you only use one moment.
A more efficient estimation. Method would be based on the maximum likelihood. So again, let me collect the data.
X条。
You collect this vector of observations you denoted as a X. So x is my data.
We know by the model, imply that x is normally distributed with mean zero variance.
Sigma x.
Sigma x is a variance covariance, matrix of x. So the element of a sigma x can be found so element of a variance covariance.
Matrix must be covariance. Right? So the covariance of fraction of bonding motion was given earlier covariance of fraction, of Gaussian noise was given earlier.
In particular, the expression of the Covariance for fraction upon the motion you question two.
Give up expression for the covariance of fractional Gaussian noise.
Therefore, you know the full distribution of the data. Therefore you can establish the likelihood function, and then take a log.
You can log like to the function which is proportional to this guy. So it's a normal, with a variance covariance, sigma X.
为什么？
So that's the log likelihood function is our Tom Domain logarithmic function.
So that's why I call td means time domain. So since I have a lot like the function, I can numerically optimize it.
Lead to maximum likelihood, estimator of H. And the Sigma Square. This method is call Tom domain maximum likelihood.
TDML?
To implement this method. You need to numerically calculate the determinant. Of a variance covariance matrix, and the integration of a variance combination.
Matrix if sample size, and is very, very large, say many thousand. Then. Such calculation may not be easy.
Integration of, say, hundred thousand observation.
Covariance. Matrix of a hundred thousand dimension. Is not that easy?
But if n is not so big, then this is okay.
So that's the second method. Tom Domain method.
There is another maximum likelihood method which is call, we told, maximum likelihood.
We, to maximum likelihoods requires. Two have.
Spectral density. It basically use a spectral density as a tool to approximate.
To approximate the in the determinant and the inverse of the various covariance function.
So, in order to avoid the calculation of a determinant and the universe of various covariance function, we use spectral density.
So this this way. This method was originally proposed by wheel. We to said that.
We can approximate the Tom domain, not good function, using spectral density so clearly in order to implement.
We told maximum likelihood. You need to know how to calculate the spectral density of a fractional Gaussian noise, but we already mentioned that.
第一。
Density of fractional gals and noise.
Involve infinite sum. Right? So you may argue that we can calculate this infinite sum by truncation, method.
For instance, you can truncate K from minus one thousand to one thousand. You just you know.
的，的。
One thousand, or less than minus one thousand. You know what I mean so you, instead of calculating this infinite sum, you basically calculate the sum of this guy.
One case is running from minus one thousand to one thousand.
That appears to be reasonable if age take a big value, but if h take a small value, suppose H. Take a value very close to zero, what you are going to see is.
This guy behave like one over K. Why? Because when K. Is large, K.
Two pi k domain number. So this is like a two pi K to the power of minus one minus two h.
But two H. Is close to zero so the power is more like a minus one, so that's why I say that you are more like dealing with series.
Which is a summation of one over K. One over K. Converge to zero. Very, very slowly.
You can not. You can know a high value of K.
So that's the problem you cannot use a simple truncation method to approximate the equation.
Three in particular, the infinite sum in equation three, one age take a small value.
In practice. We don't know how Big H. Is right, so that's why you should not use truncation matter.
Okay?
I already mentioned to you in practice. H. Take a value of point one that's a very close to zero.
So this basically say that you should be extremely careful with using truncation method.
Okay, so instead of using truncation method to calculate spectral density in a recent paper, still not published, yet.
啊，是you and yang。
Obtain an analytical expression for the we to, for the spectral density, the spectral density.
Is now not based on the infinite sum is obtained by the data function which can be computed by many standard software.
So they can more or less treat this as a analytical function, so you don't need to use chuncation method to approximate spectral density.
You can use use a data function to compute the spectrum spectrum density. Exactly, and then you plug that spectrum density in the wheat or likelihood function, and then you can you know do the we to maximum likelihood.
Later I will show you what's the expression for? The veto likelihood, but I think it's a time to stop for a lunch.
Break. Thank you so much.