Beginning Work in LaTeX

This week I finished breaking down the simplest form of the panel data model and typed it up in LaTeX. At first I wanted to write it all out without using any variables other than x’s and y’s. However, doing so made matrices that were too large to fit on a page. Thus I wrote out each variable so that it could be understood what was going into them. The results can be seen in the attached PDF.

Capstone

Capstone Update 2

Last weekend I participated in the Mathematical Competition in Modeling (MCM). This event took place over a 96-hour period starting on Thursday night at 5 Pm and ending Monday night at 5 PM. Having dedicated my entire weekend (and then some) to this endeavor, I was thus forced to spend the rest of the week playing catch up on my homework. Consequently, very little progress was made on my capstone project.

Over the next week I hope to finish constructing the basic forms of the panel data model at their simplest levels. I then will type them using LaTeX so they are easier to read and understand.

Capstone Update

Over winter break and J-Term I thought of my capstone project seldom and worked on it even less. However, towards the end of J-Term I did make a trip to PLU’s library. I checked out three books on data analysis and panel data modeling, and began breaking the panel data model down to its simplest form. Ironically, its “simplest” form was too big to fit on a single piece of paper, and involved several matrices filled with summations.

One interesting thing that I learned was that the basic form of the panel data model actually has four variations, each based on its own set of assumptions. Needless to say I have a lot of work ahead of me.

I also met with Professor Munson last Wednesday to discuss the error term of the basic form(s) of the panel data model. We agreed we needed to do some more research and meet again at a later time.

Human Knots

I read a post from The Math Less Traveled entitled, The Mathematics of Human Knots. In this post, the author discusses the game “human knot”, in which every player stands in a circle and grabs the hands of someone else in the circle so that everyone is seemingly tangled up. The goal of the game is then to untangle the knot without any pair of hands letting go of each other. Many people believe that untangling is always possible, however after Dr. Heath’s lecture on knots (and reading this article), I know that this is “knot” true (I crack myself up).

The reason that accomplishing the goal of “human knot” is not always possible is because certain knots exist which cannot be untangled. The blog post provides the example of the figure eight knot:

Just from looking at this knot one can see that the knot couldn’t be untangled without going through itself.

This spring Professor Heath is teaching a course on knot theory. I look forward to taking it.

Modeling Crime

A recent article from Math in the News titled Preventing Crime with Math discusses the work of University of California-Santa Clara Professor of Mathematics George Mohler. Mohler has applied an algorithm used in modeling seismic activity to crime in Los Angeles. The model developed by Mohler models burglaries and gang violence.

According to the article, crime is similar to seismic activity in that it is initially difficult to predict, but is usually followed by smaller “afterschocks”. The model uses the algorithm to compute which houses are likely to be struck by a burglar given their initial crime. This model has been considered so effective that the Santa Cruz Police Department is working with Mohler to adapt the model for their use.

Random Effects

My last post was on the fixed effects model. I established that the fixed effects model assumes that variables not included in the regression are correlated with the variables included in the regression, and thus the results of the regression cannot be used to assess the effects of unobserved variables. The random effects model, on the other hand, assumes that unobserved variables are not correlated with observed variables, and allows the regression to be used to investigate the effects of variables not included in the regression.

In my past posts on the panel data model and its specific variations, I explained that the general form of the panel data model is $y_{it} = \alpha_i + \beta'x_{it} + \epsilon_{it}$, and the general form of the fixed effects model is $y_{it} = D \alpha_i + \beta'x_{it} + \epsilon_{it}$. With the random effects model, the general form is $y_{it} = \alpha + \beta'x_{it} + u_i + \epsilon_{it}$. In this model, $\alpha$ is taken to be constant, and $u_i$ is a measurement of random disturbance for each cross-sectional unit.

In choosing whether to use a fixed effects model or a random effects model, one must first test to see if individual effects exist. This is done using a Langrange Multiplier (LM) test. If they do indeed exist, then a Hausman test can be used. The Hausman test uses a hypothesis test to determine whether or not the fixed effects model and the random effects model have the same variance. If their variances are the same, then a random effects model may be used. If not, the more restricting fixed effects model must be used.

Fixed Effects

In one of my earlier posts, I discussed panel data modeling and my intentions to use it as my capstone. A panel data model is rather general, however, and thus I want to consider specific variations of it. The first variation that I have decided to take a closer look at is the fixed effects model.

The fixed effects model operates under the assumption that unobserved variables are correlated with variables included in the regression. Consequently, studies conducted using this model can only be used to describe the effects of the included independent variables on the dependent variable, and thus cannot extend their results to explain the effects of other variables on the dependent variable.

Recall the general form of a panel data model: $y_{it} = \alpha_i + \beta'x_{it} + \epsilon_{it}$. The fixed effects model assumes that the individual effects coefficients ($\alpha_i$) vary across each cross-sectional unit, while the $\beta'$ coefficient is held constant.  As a result of the variability in the individual effects coefficients, it becomes necessary to use dummy variables representing each cross-sectional unit in order to properly estimate the regression. The resulting equation is $y_{it} = D \alpha_i + \beta'x_{it} + \epsilon_{it}$ where $D$ is the set of dummy variables.

Equal Understanding

According to an article from Science News titled, Students’ Understanding of the Equal Sign Not Equal, Professor Says, one of the reasons students in the United States are so far behind their foreign peers in mathematics is that they have a lesser understanding of the equal sign. Research conducted at Texas A & M university suggests that roughly 70% of students in middle grades in the U.S. have misconceptions about what the equal sign means. This is a staggeringly large proportion considering that students in some Asian countries have almost no misconceptions about the equal sign.

A large part of the problem is believed to come from students memorizing procedures rather than learning how to interpret the symbols. An example given in the article is 4+3+2=__+2 where students are asked to fill in the blank. Many U.S. middle grade students would solve this by recognizing that 4+3+2=9, and thus filling in the blank with 9. The source of this problem is thought to be the textbooks that students use. In comparing textbooks from China with those from the U.S., researchers noticed that the Chinese texts offered examples explaining things such as the equal sign, while those in the U.S. did not.

Always Check Your Work

How many times have you had an exam returned and thought to yourself, “Well, that was stupid of me.” Many times re-reading the directions given on an exam or double-checking your work can make a substantial difference in your grade. In the “real world”, the consequences of mistakes can be more severe. This is exemplified in a comic strip titled Los Alamos from XKCD.

In the strip, the characters discuss the important implications of the test they are about to conduct, and the dire consequences of the test if their math is incorrect. One character then asks, “Is it ‘SOH CAH TOA’ or ‘COH SAH TOA’?” Always remember to check your work.

Math Games

Many of us have fallen victim to the distraction of online games. Whether playing solitaire or harvesting crops on FarmVille (which I can proudly say I have never played), there is probably something else that you should be doing. Well, tonight while I was playing online math games, I decided to turn procrastination into productivity by blogging about it.

The web site that stood out to me was The Truth Tree’s Math and Logic Board. Story problems are the main focus of this particular website (if you had told me I’d be doing story problems for fun 13 years ago I would have said you were crazy). Unlike many “math game” web sites, this one is fairly advanced, with problems ranging from logical riddles, volumes of spheres, counting, topology and more. The only downside is that the answers are not provided. There is, however, an index of similar sites on the page.