Wikipedia:Reference desk/Archives/Mathematics/2007 June 9

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June 9

ARCH and GARCH

Hi. I am a research scholar of a university and currently I am doing a simple study on volatility of stock market. While browsing for literature I found that virtually all the studies in volatility of stock market use use tool called GARCH to model volatility clustering and persistence. Fascinated by the tool, I also used it in my study and discussed with one of my research guides. All I knew about the tools is that they model time-varying volatility and find whether past variances and disturbance terms affect present volatility or not. Like all other the stock markets in the world, it was significant in case of the stock market of my study as well. My research guide asked me what is the meaning of the results of ARCH and GARCH models.

I said "it means the stock market shows the existence of of volatility clustering and persistence and implies that past volatility significantly affects present volatility of returns. Since the sum of ARCH and GARH coefficient is less than one, the volatility shock is time-decaying and mean-reverting which means any volatility shock does not persist for infinity and it gradually comes down to its long run mean-level at a rate suggested by the sum of ARCH and GARH coefficients."

Now he said to me "It sounds all right but what does it tell you about the stock market behavior and what conclusions can you draw from the results of ARCH and GARCH so that it will be useful for the readers, the investors and the policy makers? You should put your results in words in such a way that it is understandable by everybody"

Now I need to know what unique results about my stock returns can I infer from the GARCH tool and interpret it so that it is useful to every reader who are not much familiar with econometrics like myself. I KNOW THE QUESTION IS VERY DEMANDING. A comprehensive and non-technical answer would be appreciated. Thanks in advance.--202.79.62.21 08:35, 9 June 2007 (UTC)

This is not really a question about mathematics. The Google search results for "arch garch volatility"[1] may contain something you can use.  --Lambiam^Talk 10:20, 9 June 2007 (UTC)

Appropriate wikipedia articles include Autoregressive conditional heteroskedasticity, and Volatility. Volatility measure how changable a stock price is in a given time period. volatility clustering and persistence mean that highly changable stocks will continue to be highly changable, the volatility shock is time-decaying and mean-reverting - peroids of high changability will not last forever, in the long term thing quieten down. --Salix alba (talk) 14:07, 9 June 2007 (UTC)

Thanks. In fact I have gone through several articles including the wikipedia articles regarding this. But the problem is their interpretation is always statistical and conclusions are always technical. However, the widespread use of GARCH indicates that its results must tell something important about the financial time series (e.g. stock returns) behavior. Studies show every market shows that volatility shock is time-decaying and mean-reverting. What can I infer something "unique" for the market I'm studying about. In other words, to a layperson (a market participant), how to interpret the results of ARCH and GARCH that makes sense to him/her ? As per my understanding it has something to do with the coefficients of alpha and beta (also known as ARCH and GARCH coefficients) for the widely used GARCH (1, 1) model--202.79.62.21 15:38, 9 June 2007 (UTC)

You have two problems. The first is that you, yourself, must decide what is unique about the market of interest, as measured. The second is that you must remember how to speak plain English, devoid of technical jargon. I'm not sure which is the greater challenge. And the two together may pull your brain in different directions. :-) --KSmrq^T 15:52, 9 June 2007 (UTC)

Fokker periodicity blocks

What's a basic or easier way to explain what Fokker periodicity blocks are ? Guroadrunner 17:43, 9 June 2007 (UTC)

I'm amazed the Fokker article does not explicitly mention the 31-tone organ he had constructed (still in working condition) and the paper he wrote about it, but only refers indirectly to those through links. Are you interested in this stuff? Perhaps you can help to edit this article into something approximating intelligibility; it ought to be understandable to musicians who have no difficulty grasping equal temperament. After much puzzling I understand the following. I'll restrict myself to the n = 3 case, but it should be easy to infer the general case from that. For integer values p, q and r (no need to restrict this to natural numbers), define C(p,q,r) = 2^−k3^p5^q7^r, where k is the largest number such that 2^k ≤ 3^p5^q7^r. It follows that 1 ≤ C(p,q,r) < 2. (Not coincidentally, 3, 5 and 7 are the first three primes after 2). For example, C(1,5,1) = 65625/65536 = 1.00135... and C(1,2,−4) = 4800/2401 = 1.99916... . These numbers are close to 1 and 2, respectively, which is what we're looking for. The values of p, q and r are, moreover, not very large, which is also desirable. So (1,5,1) and (1,2,−4) could be two of the three unison vectors we need for constructing a block. I have to stop here for now.  --Lambiam^Talk 22:54, 9 June 2007 (UTC)

I

I know that the square root of -1 is I, but how can you contemplate this imaginary number? Is it within the bounds of the human brain? Or does it just take a lot of (understatement) thinking outside of the box? I'm 11 and interested about this stuff. Thanks! Gbgg89 17:50, 9 June 2007 (UTC)

First, let's get the formalities out of the way: The usual notation for ${\sqrt {-1}}$ is i (small letter), and for information regarding it you may want to take a look at imaginary unit.

Now, as you no doubt have gathered, this is a very difficult question to answer, and my explanation may not be very good (or even correct). Think of this, however: What is ${\sqrt {2}}$? Did you ever pay ${\sqrt {2}}$ dollars at a store, or have you ever measured, using a ruler, a length of exactly ${\sqrt {2}}$ centimeters (as opposed to, say, around 1.41)? I would think not, and in fact humanity may never even know if ${\sqrt {2}}$ has any physical manifestation (if the universe is discrete, then it arguably doesn't). But for mathematicians, and people who use mathematics, it is convenient to assume that there is a number whose square is 2, and they even may have an explanation why such an entity should exist. But it is still just a mathematical abstraction introduced because of its usefulness. The fact that you did not ask the exact same question with 2 instead of -1, is evidence that its usefulness is such that it seems absurd to even question its existence.

Having this in mind, you will see that there is no reason why ${\sqrt {-1}}$ should be any different. It, too, is a mathematical abstraction introduced due to its usefulness (which is just as great as that of ${\sqrt {2}}$). The motivation for its plausibility may be slightly different, but that doesn't make it "harder to grasp" or on the other side of the bounds of the human brain (as long as you block out the perpetuated brainwash, comitted by mathematics teachers for younger ages, that no number squared can ever be negative). -- Meni Rosenfeld (talk) 18:16, 9 June 2007 (UTC)

Yes, we have a simple explanation for a number, denoted by i or i (or sometimes j), that squares to −1. But first let's dispel the mystique unnecessarily introduced for √2. Draw a square with side length one. The length of its diagonal is √2, by the Pythagorean theorem. When the ancient Greeks realized this number could not be written as a ratio of two integers, like ²⁹⁄₇₀, it troubled them deeply. But it is hardly a mere abstraction!

The need for a number that squares to −1 also arises naturally. Historically, one of the circumstances that forced the issue was solution of a general cubic equation. Like virtual particles in quantum field theory, complex numbers could appear in the midst of a solution only to disappear at the end.

The simple explanation, like that for √2, is geometric. Stretch out the usual real numbers along a line. With zero and one marked off, we can measure any amount, including √2 and π and their negatives, and so on. Multiplication by −1 has the effect of reversing the line, of rotating it by a half turn (180°) around zero. Multiplication by a square root of −1 works out nicely as a quarter turn (90°), and we can place i one unit above zero, rather than left or right. Thus every complex number has a place, not on a line, but in the complex plane. --KSmrq^T 19:34, 9 June 2007 (UTC)

"Square", "diagonal" etc. are also abstractions, in the sense that if you attempt to actually draw a square, you will obtain a figure with sides more or less equal, with diagonal length of around 1.414 times the "side length".

Of course, I had no intention to confuse the OP. I was merely trying to emphasize that one should not be perplexed by a new mathematical construct, just because it does not have any immediately obvious physical interpretation. -- Meni Rosenfeld (talk) 19:46, 9 June 2007 (UTC)

I understand what the original poster is saying, that they need to be able to physically visualize something to understand it. Many people learn in this way. Unfortunately, higher mathematics often can't be visualized physically, making the subject difficult to understand for many. The same is also true of advanced particle physics and many other fields. In some cases a "conventionalization" (simplified model) can help, such as the model of electrons in circular, coplanar orbits about a nucleus, versus the reality of wave-probability functions defining the electron shells. In the case of imaginary numbers, thinking of the complex plane versus a number line may help. StuRat 17:54, 11 June 2007 (UTC)

(edit conflict) If multiply them this way, it may be easier to express them as re^θi where r is the distance from zero, e is Euler's number, θ is the angle in radians (there are 180/π degrees in a radian), and i is the square root of negative one. There's a four dimentional extention of complex numbers called quaternions, and the number of dimentions can be doubled any number of times quite simply. Quaternions aren't used often partially because multiplication isn't cumulative with it (a*b doesn't necessarily equal b*a). I'd also like to add that no form of numbers is strictly necessary, and any sufficiently complex mathematical system can be used to represent any other, and therefore represent reality just as well, though one system might make it easier than another.— Daniel 20:07, 9 June 2007 (UTC)

I have to take issue with your statement that we can double the number of dimensions "simply" as many times as we like. You did note that the quaternions are not commutative; if we double it again, the most likely candidate is the octonions, which are not only non-commutative but not even associative! And what next? All these are normed division algebras (also called composition algebras), and by Hurwitz's theorem there is no real composition algebra of dimension 16. Sure, we can continue the Cayley-Dickson construction arbitrarily, but you start losing properties left and right -- the octonions are the last algebra before you start getting zero divisors. Tesseran 23:10, 11 June 2007 (UTC)

This is not any more mysterious than the number −1 itself. If all you're used to is numbers for counting sheep, you'll have a hard time grasping −1 sheep, and you may wonder, should you overhear some mathematicians passing by while herding your sheep: How can adding a sheep to some number of sheep result in no sheep? Is it some kind of antisheep? But how can that be? Is −1 within the bounds of the human brain? Well, yes, it is, and so is i. Here is yet another way of thinking about it. It may be a bit above your head, but hopefully you'll get the drift. First, consider the way the integers are constructed from the natural numbers, the rational numbers from the integers, and the reals from Cauchy sequences of rationals (see, respectively, Integer#Construction, Rational number#Formal construction, and Construction of real numbers#Construction from Cauchy sequences). These constructions all follow the same pattern. First, a concrete kind of mathematical objects is constructed, such as pairs of integers. These new objects capture some notion we have that allows us to solve more equations than before, and they provide a concrete representation for that. For example, in natural numbers you cannot solve the equation x + 3 = 1, in integers you cannot solve the equation x × 3 = 2, and in rational numbers you cannot solve the equation x³ = 2. However, because of the specifics of the representation, "too much" is captured; for example, viewed as pairs of integers, ²/₃ and ¹⁴/₂₁ are different, but they represent the same idea: a solution of x × 3 = 2. So they are equivalent, as far as we are concerned. What to do? Well, just form yet another concrete kind of objects, each of which consists of a whole class of the objects we had before. In doing so, all objects that are in some (precisely defined!) way equivalent are lumped together in the same class. These are called equivalence classes, and they are the things we actually want. The technical term for this is quotient set. If the relevant equivalence relation is chosen the right way, it is a congruence relation for the relevant mathematical operations, which means that they are "automatically" also defined on the new quotient set. So far, so good. The complex numbers can be formed in just the same way. Take the polynomials with real-valued coefficients in one variable X. This gives us a ring. Now we consider two polynomials equivalent if, on polynomial division by the polynomial X²+1, they give the same remainder. We can then form the quotient set, which in this case is again a ring: the quotient ring. Miraculously, it is even a field. We embed the real numbers as usual: [r] = r. Now consider the class having X as a representative. Let us give it a name: i. What is i²? Well, i² = [X²] = [X² − (X²+1)] = [−1] = −1. Voilà, we have constructed a concrete kind of mathematical objects, forming a field with the real numbers embedded, that contains an element i such that i² = −1.  --Lambiam^Talk 21:58, 9 June 2007 (UTC)

Well, this work, but don't forget that the OP is 11, after all. Cthulhu.mythos 09:48, 12 June 2007 (UTC)

Kids are very smart these days :) -- Meni Rosenfeld (talk) 15:50, 12 June 2007 (UTC)

Just to see if I can clarify what I think the original poster might have meant, positive real numbers can be easily visualized as lengths of line segments (like the length of a diagnol as described above). A negative number can be visualized as a directed ray segment the same length as a positive segment in the opposite direction. The analogy of "imaginary" rays and line segments, though, is harder to visualize. The complex plane partially works, but mainly as showing that imaginary components of complex numbers are akin to a second component of a binary vector. It doesn't visually explain however why squaring an imaginary number results in a real number with negative sign. So it's the visualization of lengths and areas and directed vectors that seem to have a hard time explaining what imaginary numbers "look like". I'm thinking that maybe what the poster wants, therefore, is a physical setting using both imaginary and real numbers that reflects how imaginary numbers squared become negative real numbers.Dugwiki 16:43, 11 June 2007 (UTC)

Thanks for all your help! It really cleared things up! Gbgg89 02:24, 13 June 2007 (UTC)