General

Re: Short essay on the coming collapse

Solredime — Thu, 08 Apr 2010 09:25:24 GMT

abskebabs:

Looking at certain price charts I may find a linear dependence on demand for a set of goods, but in another set I may find a completely different dependence. Even though other variables could be accounted for in the different cases(I guess this is the entire aim of econometrics), it puzzles me as to how this could be achieved with unambiguous success. Regardless of that point, many of the factors affecting the formation of economic data are entirely qualitative and difficult or impossible to express in numbers(like ideology), that it seems strange how econmetrics could be seen as the "magic bullet", to allow economics to assume the form of a natural science.

Well, any individual explanatory variable can be given its own functional form, linear, logarithimic, quadratic, whatever.

Also, qualitative information can be expressed using dummy variables. For example, gender can assume a 1 or 0. Whether they have a degree can assume a 1 or 0, whether they graduated with a first, second, third class. All of these things could technically be accounted for, assuming sufficient and accurate data. Of course, there are lots of problems too.

Re: Short essay on the coming collapse

abskebabs — Wed, 07 Apr 2010 18:22:26 GMT

Fred Furash:

abskebabs:

the equation has still had to be specified by constants to the extent that the powers of the variables have been set to zero, and hence the expression is linear as opposed to nonlinear.

Do you mean powers set to 1?

Linearity in variables or linearity in parameters is introduced in like, chapter 2 of my econometrics book.

And I think regression analysis pretty much assumes the kind of constancy you are talking about for the models to be valid. As an example, in time series, it is assumed that variances don't change over time, which they probably do. If variances change over time, then you have heteroscedasticity, There are also ways to try to correct for this.

Don't think that these issues are just being ignored, there is a whole discipline out there that tries to deal with them. How successful or not I do not know, since I'm still a noob at econometrics.

Ah, yes, I meant 1. Thinking about it again, beyond simply being able to establish the constants, one needs to show that the same uniform relationship between the variables holds in multiple repeated experiments in which other variables are controlled for. I confess I'm probably even more ignorant of econometrics not being an economics undergraduate, but I think it's strange to assume a constancy in the nature of the relation between economic variables, since for example:

Looking at certain price charts I may find a linear dependence on demand for a set of goods, but in another set I may find a completely different dependence. Even though other variables could be accounted for in the different cases(I guess this is the entire aim of econometrics), it puzzles me as to how this could be achieved with unambiguous success. Regardless of that point, many of the factors affecting the formation of economic data are entirely qualitative and difficult or impossible to express in numbers(like ideology), that it seems strange how econmetrics could be seen as the "magic bullet", to allow economics to assume the form of a natural science.

Re: Short essay on the coming collapse

Solredime — Wed, 07 Apr 2010 15:26:51 GMT

abskebabs:

the equation has still had to be specified by constants to the extent that the powers of the variables have been set to zero, and hence the expression is linear as opposed to nonlinear.

Do you mean powers set to 1?

Linearity in variables or linearity in parameters is introduced in like, chapter 2 of my econometrics book.

And I think regression analysis pretty much assumes the kind of constancy you are talking about for the models to be valid. As an example, in time series, it is assumed that variances don't change over time, which they probably do. If variances change over time, then you have heteroscedasticity, There are also ways to try to correct for this.

Don't think that these issues are just being ignored, there is a whole discipline out there that tries to deal with them. How successful or not I do not know, since I'm still a noob at econometrics.

Re: Short essay on the coming collapse

abskebabs — Wed, 07 Apr 2010 15:02:59 GMT

Smiling Dave:

You can make the power a letter also. In fact, its done routinely.

I don't now how to make summation signs and subscripts, so I'll copy one from wikipedia on the fundamental theorem of algebra:

Indeed, good point again, what you've written is basically the definition of a power series which can be extended to describe functions f(z) with complex domains and negative powers.

This is just the general definition of a power/laurent series however, and could be used to describe any analytic function. Of course any theorems that apply generally in complex analysis would apply to these too, though again just having this definition doesn't aid any application. That's got me thinking, do economists make any use of complex analysis?

(note:- I can't seem to attach Latex images from texify on this forum, so I had to copy that image from wikipedia too)

Re: Short essay on the coming collapse

Smiling Dave — Wed, 07 Apr 2010 14:41:29 GMT

abskebabs:

Indeed, going a little more abstract again, even with an equation like v=ax+by+cz , even if we consider the constants unknown and variable, the equation has still had to be specified by constants to the extent that the powers of the variables have been set to zero, and hence the expression is linear as opposed to nonlinear. Hence, to get some kind of determination you still can't have absolutely no constants...

You can make the power a letter also. In fact, its done routinely.

I don't now how to make summation signs and subscripts, so I'll copy one from wikipedia on the fundamental theorem of algebra:

Re: Short essay on the coming collapse

abskebabs — Wed, 07 Apr 2010 14:12:32 GMT

Good points.

I definitely agree, to actually apply mathematics in your field of study one needs to have established what the constants in your equations are, and in the same sense established there are constant relations. This is the same in physics and other areas where mathematics is applied, one may use shorthand notation for constants but ultimately they are defined as some number according to a certain scale or metric.

Indeed, going a little more abstract again, even with an equation like v=ax+by+cz , even if we consider the constants unknown and variable, the equation has still had to be specified by constants to the extent that the powers of the variables have been set to 1, and hence the expression is linear as opposed to nonlinear. Hence, to get some kind of determination you still can't have absolutely no constants...

Re: Short essay on the coming collapse

Smiling Dave — Wed, 07 Apr 2010 05:37:06 GMT

abskebabs:

Smiling Dave:

2. My q was not about math for use in economics, but about math. Esuric said constants are needed in math, and Im wondering which constants are needed in those fields.

Sorry to beat a dead horse, but I was thinking about this the other day and had a Eureka moment. To express any remotely meaningful equation we require constants, whether explicit or implicit. So, if you know v has a dependence on x, y and z, without knowledge of constants one could not write any more meaningful equation than v=v(x,y,z). For any meaningful equation one needs constants whether explicit, so v=x+2y+3z or implicitly in v=sin(z), in which the constants that specify the dependency can be discovered from the function's Taylor series.

Perhaps this is a trivial point, but I thought I'd give you my 10 cents.

1. All very true as far as it goes.

But huge tracts of math dont need numbers at all. Look at the list in an earlier post. And the ones that do need constants are usually studied using variables for constants, like v=ax+by +cz, where a, b and c are assumed to be constants.

Which is a bit of a subtle point. How can a letter of the alphabet be a constant? After all you are studying many equations at once, letting a, b, and c vary over the whole range of numbers. So they are not constants like pi or euler's number is a constant.

I'm under the influence of a few beers, and it's been a while since I gave these things any thought. Not sure how relevant it is anyway.

At any rate, the time has come to let the cat out of the bag:

I think Esuric was echoing Rothbard or someone like that whose writings I have seen in passing. Here is a link that turned up when I searched.

Their point is that TO APPLY a mathematical theory to a real world situation, one needs constants to work with. [In other words,one has to be able to replace the a, b and c with 3, 7 and 4 for example]. I suspect that's what Esuric meant as well.

But math itself can get along fine in many areas without constants.

2. Going into it a bit more, math does not "need" the rock solid constants such as pi to get going. It starts out with nothing almost, the very abstarct axioms of set theory which dont mention numbers at all, and takes it from there. Pi and Euler's number and such like make their appearance on the stage of Math as EXISTENCE THEOREMS.

They are RESULTS of studying math from scratch. They are not needed for math. Had math discovered, for example, that the ratio of the diameter to the circumference of a circle depends on the size of the circle and is not the constant number pi, math would not have come to an end. It would contnue on, happily studying the conclusions of that revelation.

Re: Short essay on the coming collapse

abskebabs — Wed, 07 Apr 2010 03:42:10 GMT

Smiling Dave:

2. My q was not about math for use in economics, but about math. Esuric said constants are needed in math, and Im wondering which constants are needed in those fields.

Sorry to beat a dead horse, but I was thinking about this the other day and had a Eureka moment. To express any remotely meaningful equation we require constants, whether explicit or implicit. So, if you know v has a dependence on x, y and z, without knowledge of constants one could not write any more meaningful equation than v=v(x,y,z). For any meaningful equation one needs constants whether explicit, so v=x+2y+3z or implicitly in v=sin(z), in which the constants that specify the dependency can be discovered from the function's Taylor series.

Perhaps this is a trivial point, but I thought I'd give you my 10 cents.

Re: Short essay on the coming collapse

Solredime — Mon, 05 Apr 2010 11:49:44 GMT

mash:

Out of curiosity: how would accounting sit as a complimentary methodology for economics, at its heart accounting a mathematical science with an identity? What happens when economic theory contradicts accounting - is the accounting at fault, or the economic theory?

Accounting doesn't attempt to describe, explain, or predict human action. It is simply a record of transactions. Just like the GDP equation is technically an identity, not a theory. All it says is that total spending in an economy is the spending of consumers, firms, and the government.

It is the theory behind the equations that may be dubious; such as saying that government spending, G, is similar in nature to C or I. Such overaggregation leads some economists to believe that an output gap caused by a fall in C and I, can be sustainably rectified by spending the difference through G.

Re: Short essay on the coming collapse

mash — Mon, 05 Apr 2010 11:26:22 GMT

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Out of curiosity: how would accounting sit as a complimentary methodology for economics, at its heart accounting a mathematical science with an identity? What happens when economic theory contradicts accounting - is the accounting at fault, or the economic theory?

Re: Short essay on the coming collapse

czelaya — Sun, 04 Apr 2010 21:13:05 GMT

Smiling Dave:

abskebas and czelay, you came in the middle

1. The OP said he's done with the thread and opened it for any discussion.

2. My q was not about math for use in economics, but about math. Esuric said constants are needed in math, and Im wondering which constants are needed in those fields.

3. Oh, may as well toss in Euclid's geometry, what constant values did he use in his axioms or deduce in his theorems?

Honestly, all I can tell you is how and why constants exist in equations.

1.) Constants usually come in two forms: dimensional and dimensionless (ratio or product forms). You are already familiar with dimensional constants such as c, the speed of light [3.0 x 10^8 m/s]. It has the dimensions of meters per second, which relates to speed (SmilingDave, I hope I'm not insulting your intelligence or anybody else’s on this forum. It's just for the sake of clarity). Dimensional constants are observed and don't have any deeper meaning than what they are intended to express. Dimensionless constants are a little more complicated. They are unit less. They are derived when constructing equations using a number of metrics such as length & charge. The constants are derived from some proportionality ( or products) of variables and or other constants. For instance, the gravitational coupling constant g which represents the gravitational attraction between two charged particles is derived as:

g=(Me/Mp)^2=1.7518x10^-45 (where Me=mass of the electron, and Mp=Planck mass)

The two units that measure mass cancel each other and a unit less number is derived.

2.) Mathematical constants exist naturally because of "pure" mathematical observation (meaning they don't need physical observation to be calculated. However, it must be noted they are essential in a number of physical equations). Examples include pi and the exponential constant e. Pi is simply a constant that is derived by dividing the circumference of a circle by its diameter. e is a little more difficult to derive but it's found everywhere in mathematics from the derivatives of logarithmic functions to geometric growth.

3.) I have a very little understanding of the historical fondation of mathematics but I was just reading on Euclid's geometry at Wikipedia. I'm sure this can help:

http://en.wikipedia.org/wiki/Euclidean_geometry

Basically, a number of constants arise naturally that are easy to understand because it deals with Eucladian space (systems where the basis are just straight lines) or if you incorporate time into the basis-Minkowski Space.

Re: Short essay on the coming collapse

Smiling Dave — Sun, 04 Apr 2010 18:32:39 GMT

abskebas and czelay, you came in the middle

1. The OP said he's done with the thread and opened it for any discussion.

2. My q was not about math for use in economics, but about math. Esuric said constants are needed in math, and Im wondering which constants are needed in those fields.

3. Oh, may as well toss in Euclid's geometry, what constant values did he use in his axioms or deduce in his theorems?

Re: Short essay on the coming collapse

czelaya — Sun, 04 Apr 2010 15:54:32 GMT

abskebabs:

Smiling Dave:

Esuric:
there are no constants in economics (which is needed for mathematics

1. What constants are required in these branches of mathematics:

set theory. group theory. algebra. linear algebra. lattice theory. probability theory. measure theory. differential equations. model theory. calculus. functional analysis.

2. Please name one branch of math that needs constants, and what those constants are.

The issue is epistemic. One cannot say one has found a quantitative relations in Economics because one cannot conduct universally applicable experiments that control for unknowns reasonably and hold all other variables constant. Hence recording the variations of the prices of potatoes with quantity demanded in 1905, does not mean you've measured "elasticity" of demand. This is in contrast to equations from physics e.g Newton's gravitational equation from which we can measure constants once we control for mass and distance variables using the Galilean equivalance principle to equate Newton's 2nd law with it. This allows us to find G, a universal constant.

To even state an equality( a constant relation between a set of variables), one needs to have rigorously established that it universally holds, or at least have established the bounds within which it does. I think Mises' objection to the use of mathematics also relates to the concept of ordinal utility, the main driving factor that cannot be cardinalised since prices for example are not equalities but exchange ratios, killing the possibillity of designing a form of cardinal theory, utilising utillity functions etc.

Well stated. This seems to be the central issue with mathematics in economics. If you can't reproduce the conditions and hold variables fixed, then you do not understand the physical conditions of your equations. In addition, your comment on boundary conditions for differential & partial differential equations is the MOST significant. How can any individual have faith in equations without first knowing the state of the system to begin with? I'm more curious with the functions they are using? Are they real or complex and why would any economist employ differential geometry in economics? What does a manifold have to do with economics? As soon as I'm finished with Ph.D, I want to start a serious study in econometrics and quant. I want to know how economist magically overcome the complexities and rigor in mathematics without adding dubious and whacked assumptions. Next, I'm sure were going to hear that economist are using Fourier and Laplace transforms in their analysis, and finding new novel methods of information.

So sorry, didn't mean to hi jack this thread either.

Re: Short essay on the coming collapse

abskebabs — Sun, 04 Apr 2010 13:13:26 GMT

Smiling Dave:

Esuric:
there are no constants in economics (which is needed for mathematics

1. What constants are required in these branches of mathematics:

set theory. group theory. algebra. linear algebra. lattice theory. probability theory. measure theory. differential equations. model theory. calculus. functional analysis.

2. Please name one branch of math that needs constants, and what those constants are.

The issue is epistemic. One cannot say one has found a quantitative relations in Economics because one cannot conduct universally applicable experiments that control for unknowns reasonably and hold all other variables constant. Hence recording the variations of the prices of potatoes with quantity demanded in 1905, does not mean you've measured "elasticity" of demand. This is in contrast to equations from physics e.g Newton's gravitational equation from which we can measure constants once we control for mass and distance variables using the Galilean equivalance principle to equate Newton's 2nd law with it. This allows us to find G, a universal constant.

To even state an equality( a constant relation between a set of variables), one needs to have rigorously established that it universally holds, or at least have established the bounds within which it does. I think Mises' objection to the use of mathematics also relates to the concept of ordinal utility, the main driving factor that cannot be cardinalised since prices for example are not equalities but exchange ratios, killing the possibillity of designing a form of cardinal theory, utilising utillity functions etc.

Re: Short essay on the coming collapse

Smiling Dave — Sun, 04 Apr 2010 10:54:09 GMT

Esuric:
there are no constants in economics (which is needed for mathematics

1. What constants are required in these branches of mathematics:

set theory. group theory. algebra. linear algebra. lattice theory. probability theory. measure theory. differential equations. model theory. calculus. functional analysis.

2. Please name one branch of math that needs constants, and what those constants are.