Presidential Probit Model: A Working Paper

by Graham Giller February 23, 2012 18:30
I wrote up the Presidential Probit Model work as a working paper that is available from the Social Science Research Network.  

Towards Non-Linear Models on Quantized Data: A Support Vector Machine for the Presidential Election Data

by Graham Giller August 03, 2010 23:55

To finalize our analysis of the Presidential election data, I will quickly compare the usage of a Support Vector Machine, which is a modern classification engine based on research by Vapnik et al., with that of the probit model. The SVM is implemented in several packages for R; in the following we will use that from the kernlab package. The code fragments below illustrate how to compute the probit and SVM models on our data.

pmodel<-glm(PRES~HEIGHT+CHANGE,family=binomial(link='probit'),data=presdata)
presdata$PROBIT<-predict(pmodel,type='response')
library(kernlab)
kmodel<-ksvm(PRES~HEIGHT+CHANGE,type='C-svc',data=presdata)
presdata$SVM<-predict(kmodel)

The table below illustrates the output of both models:

YEAR PROBIT SVM PRES
1896 0.15 0 1
1900 0.60 1 1
1904 0.24 0 1
1908 0.69 1 1
1912 0.34 0 0
1916 0.45 0 0
1920 1.00 1 1
1924 0.71 1 1
1928 0.74 1 1
1932 0.15 0 0
1936 0.03 0 0
1940 0.80 1 0
1944 0.25 0 0
1948 0.80 1 0
1952 0.90 1 1
1956 0.90 1 1
1960 0.39 0 0
1964 0.15 0 0
1968 0.90 1 1
1972 0.84 1 1
1976 0.34 0 0
1980 0.83 1 1
1984 0.96 1 1
1988 0.96 1 1
1992 0.45 0 0
1996 0.24 0 0
2000 0.80 1 1
2004 0.48 0 1
2008 0.02 0 0

We see from the above table that the SVM is categorizing the data quite accurately, but it can tell us nothing about the likelihood of a particular outcome — because that is not what it has been asked to do. We also built our SVM using the parameter set indicated by our probit analysis, because that system gave us tools to investigate those choices. However, the SVM did accomplish something that we only implicitly asked of the probit analysis, which is to classify the data. Although the probit probabilities are useful out-of-sample in a Bayesian sense, in-sample they do not help us quantify the success of the method — we have to overlay an ad hoc classification scheme to permit that interpretation.

 

Towards Non-Linear Models on Quantized Data: A Probit Analysis of Presidential Elections Part II

by Graham Giller August 03, 2010 15:55

To finish off our “toy” example of using Discrete Dependent Variables analysis, following on from the discussion of our data, we will use the probit link function and the NY Times data to test each of our variables for inclusing in a model forecasting the probability to a Republican President. The predictors are:

  • height difference (Republican candidate height in feet minus Democrat candidate height in feet);
  • weight difference (Republican candidate weight in pounds minus Democrate candidate weight in pounds);
  • incumbency i.e. the prior value of the Presidential Party state variable; and,
  • change i.e. the value of the Presidential Party state variable at two lags.

I am going to perform this analysis in RATS, but to continue from the prior post I will also illustrate the command used in R to fit such models.

summary(glm(PRES ~ HEIGHT + WEIGHT + INCUMBENCY + CHANGE,
    family=binomial(link="probit"), data=presdata))

The results are:

Variable Likelihood Ratio Parameter
Estimate
λ α
HEIGHT 6.422 0.01127461 3.1360 ± 1.4517
HEIGHT+WEIGHT 0.487 0.48538310 2.6919 ± 1.6049
HEIGHT+INCUMBENCY 0.761 0.38315424 3.0773 ± 1.4722
HEIGHT+CHANGE 5.273 0.02165453 3.6310 ± 1.5546

This analysis performed on all data up to and including 2004 — replicating the original out-of-sample nature of the 2008 election. With over 98% confidence we reject the null hypothesis and add the HEIGHT variable to the model. We then test the remaining variables and will over 97% confidence add CHANGE to the model including HEIGHT

Time series of the Probability of a Republican President Estimated using a Probit Model

The above chart illustrates the implied probabilities extracted from the estimated model. (The shading illustrates the actual party selection.) Out of sample, this exhibits a very strong prediction that President Barack Obama would have been elected and, given the unknown stature of his coming opponent, a marginal probability favouring his relection. Finally, after entertaining ourselves with this “toy” model, we will return to applying this methodology to market data in a coming post.

 

Towards Non-Linear Models on Quantized Data: A Probit Analysis of Presidential Elections Part I

by Graham Giller August 02, 2010 09:00

This post is concerned with an analytical method that could be thought of as a half-way house between the standard linear models that work at lower frequencies and the non-linear models that we are hypothesizing are necessary to deal with highly quantized high frequency data. In common with our prior post, on computing bias in Presidential approval rating opinion polling, it is also using a political not financial data set. However, that merely changes the meaning of the analysis and not the method of analysis. (It actually represents a piece of work I did prior to the recent Presidential election, which happened to use this method, so I thought I'd include it here as it is entertaining.)

What I'm discussing are methods to deal with what are now known as discrete dependent variables in statistics. To my knowledge this methodology was pioneered by Chester Bliss in his analysis on the Probability of Insect Toxicity or Probit Models. Faced with a binary outcome (either the insects die, or they don't), a standard linear regression was unsuitable for estimating this quantity. Instead, Bliss decided to estimate the probability of the outcome via what we now call a link function.

LaTeX Rendered by www.forkosh.com/mathtex.html

Here Φ is the c.d.f. for the Normal (or other suitable) distribution. We note that this formalism is identical to that of the Perceptron in which the discriminant output of the system is a sigmoid (i.e. “S” shaped) function of an linear function of the data. However, unlike a typical Neural Net, which generally just describes a functional relationship of some kind, we have a specific probabilistic interpretation of the output of our system. Thus we can adopt the entire toolkit developed for maximum likelihood analysis to this problem. Specifically, if our models do not involve degenerate specifications, we can use the maximum likelihood ratio test to evaluate the utility of various composite hypotheses.

At this point I'm going to describe our data briefly, and then present our analysis in a later post. The data comes from the 6th. October, 2008, edition of the New York Times. Specifically an article entitled The Measure of a President (the currently linked to document contains corrections printed the next day). I have tabulated this data and it is available from this website. I have also included the outcome of the 2008 election, which was unknown at the the time the article was printed (and when my analysis was orignally done). You can download it and read it the analysis programme of your choice. For example, the following will load the data directly into R and then create the needed auxiliary variables.

presdata<-
    read.table("http://www.gillerinvestments.com/Downloader/Files/Election.txt",
    skip=6,col.names=c("YEAR","D_FT","D_IN","D_LB","R_FT","R_IN","R_LB","PRES"))
presdata$D_HT<-presdata$D_FT+presdata$D_IN/12.
presdata$R_HT<-presdata$R_FT+presdata$R_IN/12.
presdata$HEIGHT<-presdata$R_HT-presdata$D_HT
presdata$WEIGHT<-presdata$R_LB-presdata$D_LB
presdata$INCUMBENCY<-c(0,presdata$PRES[1:(length(presdata$PRES)-1)])
presdata$CHANGE<-c(1,0,presdata$PRES[1:(length(presdata$PRES)-2)])

This data presents the heights and weights of Presidential candidates since 1896, together with the binary outcome (“0” for a Democrat and “1” for a Republican — and this data is organized alphabetically not by any kind of hidden preference). The light hearted goal of the article was to investigate whether the public prefers the lighter, taller candidate over the shorter, heavier alternate. We are going to augment this data by computing the Body Mass Index, to ask whether the data public preferred the “healthier” candidate and also the first two lags of the Presidential party indicator, which we will call incumbency and change respectively. We will therefore seek to estimate the probability of a Republican President from this data via Probit Analysis. Results to follow.

 

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About the Author

Graham Giller - Headshot GRAHAM GILLER
Dr. Giller holds a doctorate from Oxford University in experimental elementary particle physics. His field of research was statistical astronomy using high energy cosmic rays. After leaving Oxford, he worked in the Process Driven Trading Group at Morgan Stanley, as a strategy researcher and portfolio manager. He then ran a CTA/CPO firm which concentrated on trading eurodollar futures using statistical models. From 2004, he has managed a private family investment office. In 2009, he joined a California based hedge fund startup, concentrating on high frequency alpha and volatility forecasting. My updated resume is on LinkedIn.

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