The Trump Rally, Really?

Today, The Dow Jones Industrial Average (DJIA) surpassed the 20,000 mark for the first time in history. At the time of the writing of this posting (12:31 PM on January 25), it is actually 20,058.29, so, I am not sure if it will close above 20,000 points, but, nevertheless, a lot of people are crediting this to Trump’s presidency, but I’m not so sure you can do that. First, the point must be made, that it is really the Obama economic policies that set the stage for this. On January 20, 2009, when Obama was sworn in, the Dow closed at 7949.089844 points. On November 8, 2016, when Trump won the election, the Dow closed at 18332.74023. So, during the Obama administration, the Dow increased by approximately 130.63%. I just wanted to make that point.

Now, the question that I wanted to investigate was would the Dow have closed past 20,000 points had Trump not been elected president. That is, assuming that the Obama administration policies and subsequent effects on the Dow were allowed to continue, would the Dow have surpassed 20,000 points.

For this, I looked at the DJIA data from January 20, 2009 (Obama’s first inauguration) to November 08, 2016 (Trump’s election). I specifically calculated the daily returns and discovered that they are approximately normally distributed using a kernel density method:

obamadowpdf

Importantly, one can calculate that the mean daily returns, \mu = 0.00045497596503813, while the volatility in daily returns, \sigma = 0.0100872666938282. Indeed, the volatility in daily returns for the DJIA was found to be relatively high during this period. Finally, the DJIA closed at 18332.74023 points on election night, November 08, 2016, which was 53 business days ago.

The daily dynamics of the DJIA can be modelled by the following stochastic differential equation:

S_{t} = S_{t-1} + \mu S_{t-1} dt + \sigma S_{t-1} dW,

where dW denotes a Wiener/Brownian motion process. Simulating this on computer, I ran 2,000,000 Monte Carlo simulations to simulate the DJIA closing price 53 business days from November 08, 2016, that is, January 25, 2017. The results of some of these simulations are shown below:

djiaclosingvaluesims

We concluded the following from our simulation. At the end of January 25, 2017, the DJIA was predicted to close at:

18778.51676 \pm 1380.42445

That is, the DJIA would be expected to close anywhere between 17398.0923062336 and 20158.94121. This range, albeit wide, is due to the high volatility of the daily returns in the DJIA, but, as you can see, it is perfectly feasible that the DJIA would have surpassed 20,000 points if Trump would not have been elected president.

Further, perhaps what is of more importance is the probability that the DJIA would surpass 20,000 points at any time during this 54-day period. We found the following:

probofexceeding

One sees that there is an almost 20% (more precisely, 18.53%) probability that the DJIA would close above 20,000 points on January 25, 2017 had Trump not been elected president. Since, by all accounts, the DJIA exceeding 20,000 points is considered to be an extremely rare/historic event, the fact that the probability is found to be almost 20% is actually quite significant, and shows, that it is quite likely that a Trump administration actually has little to do with the DJIA exceeding 20,000 points.

Although, this simulation was just for 53 working days from Nov 08, 2016, one can see that the probability of the DJIA exceeding 20,000 at closing day is monotonically increasing with every passing day. It is therefore quite feasible to conclude that Trump being president actually has little to do with the DJIA exceeding 20,000 points, rather, one can really attribute it to the day-to-day volatility of the DJIA!

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Optimal Strategies for the Clinton/Trump Debate

Consider modelling the Clinton/Trump debate via a static game in which each candidate can choose between two strategies: \{A,P\}, where A denotes predominantly “attacking” the other candidate, while P denotes predominantly discussing policy positions.

Further, let us consider the mixed strategies \sigma_1 = (p,1-p) for Clinton, and \sigma_2 = (q,1-q) for Trump. That is, Clinton predominantly attacks Trump with probability p, and Trump predominantly attacks Clinton with probability q.

Let us first deal with the general case of arbitrary payoffs, thus, generating the following payoff matrix:

\left( \begin{array}{cc} \{a,b\} & \{c,d\} \\ \{e,f\} & \{g,h\} \\ \end{array} \right)

That is, if Clinton attacks Trump and Trump attacks Clinton, the payoff to Clinton is a, while the payoff to Trump is b. If Clinton attacks Trump, and Trump ignores and discusses policy positions instead, the payoff to Clinton is c, while the payoff to trump is d. If Clinton discusses policy positions while Trump attacks, the payoff to Clinton is e, while the payoff to Trump is f, and if both candidates discuss policy positions instead of attacking each other, the payoff to them both will be g and h respectively.

With this information in hand, we can calculate the payoff to Clinton as:

\pi_c(\sigma_1, \sigma_2) = a p q+c p (1-q)+e (1-p) q+g (1-p) (1-q)

while the payoff to Trump is:

\pi_t(\sigma_1,\sigma_2) = b p q+d p (1-q)+f (1-p) q+h (1-p) (1-q)

With these payoff functions, we can compute each candidate’s best response to the other candidate by solving the following equations:

\hat{\sigma}_1 \in \text{argmax}_{\sigma_1} \pi_1(\sigma_1,\sigma_2)

\hat{\sigma}_{2} \in \text{argmax}_{\sigma_2} \pi_2(\sigma_1,\sigma_2)

where \hat{\sigma}_{1,2} indicates the best response strategy to a fixed strategy for the other player.

Solving these equations, we obtain the following:

If

latex-image-32
then,

Clinton’s best response is to choose p = 1/2.

If

latex-image-33

then,

Clinton’s best response is to choose  p = 1.

Otherwise, her best response is to choose p = 0.

 

While for Trump, the best responses are computed as follows:

If

latex-image-34

Trump’s best response is to choose q = 1/2.

If

latex-image-35

Trump’s best response is to choose q = 1.

Otherwise, Trump’s best response is to choose q = 0.

To demonstrate this, let us work out an example. Assume (for this example) that the payoffs for each candidate are to sway independent voters / voters that have not made up their minds. Further, let us assume that these voters are more interested in policy positions, and will take attacks negatively. Obviously, this is not necessarily true, and we have solved the general case above. We are just using the following payoff matrix for demonstration purposes:

\left( \begin{array}{cc} \{-1,-1\} & \{-1,1\} \\ \{1,-1\} & \{1,1\} \\ \end{array} \right)

 

Using the above equations, we see that if 0 \leq q \leq 1, Clinton’s best response is to choose p=0. While, if 0 \leq p \leq 1, Trump’s best response is to choose q =0. That is, no matter what Trump’s strategy is, it is always Clinton’s best response to discuss policy positions. No matter what Clinton’s strategy is, it is always Trump’s best response to discuss policy positions as well. The two candidates’ payoff functions take the following form:

payofffuncs

What this shows for example is that there is a Nash equilibrium of:

(\sigma_1^{*}, \sigma_{2}^{*}) = (0,0).

The expected payoffs for each candidate are evidently

\pi_c = \pi_t = 1.

Let us work out an another example. This time, assume that if Clinton attacks Trump, she receives a payoff of +1, while if Trump attacks Clinton, he receives a payoff of -1. While, if Clinton discusses policy, while being attacked by Trump, she receives a payoff of +1, while Trump receives a payoff of -1. On the other hand, if Trump discusses policy while being attacked by Clinton, he receives a payoff +1, while Clinton receives a payoff of -1. If Clinton discusses policy, while Trump discusses policy, she receives a payoff of +1, while Trump receives a payoff of -1. The payoff matrix is evidently:

\left( \begin{array}{cc} \{1,-1\} & \{1,-1\} \\ \{1,-1\} & \{1,-1\} \\ \end{array} \right)

In this case, if 0 \leq q \leq 1, then Clinton’s best response is to choose p = 1/2. While, if 0 \leq p \leq 1, then Trump’s best response is to choose q = 1/2. The Nash equilibrium is evidently

(\sigma_1^{*}, \sigma_{2}^{*}) = (1/2,1/2).

The expected payoffs for each candidate are evidently

\pi_c = 1, \pi_t = -1.

In this example,  even though it is the optimal strategy for each candidate to play a mixed strategy of 50% attack, 50% discuss policy, Clinton is expected to benefit, while Trump is expected to lose.

Let us also consider an example of where the audience is biased towards Trump. So, every time Trump attacks Clinton, he gains an additional point. Every time Trump discusses policy, while Clinton does the same he gains an additional point. While, if Clinton attacks while Trump discusses policy positions, she will lose a point, and he gains a point. Such a payoff matrix can be given by:

\left( \begin{array}{cc} \{1,2\} & \{-1,1\} \\ \{0,1\} & \{0,1\} \\ \end{array} \right)

Solving the equations above, we find that if q = 1/2, Clinton’s best response is to choose p =1/2. If 1/2 < q \leq 1, Clinton’s best response is to choose p = 1. Otherwise, her best response is to choose p = 0. On the other hand, if p = 0, Trump’s best response is to choose q = 1/2. While, if 0 < p \leq 1, Trump’s best response is to choose q = 1. Evidently, there is a single Nash equilibrium (as long as 1/2 < p \leq 1):

 (\sigma_1^{*}, \sigma_{2}^{*}) = (1,1).

Therefore, in this situation, it is each candidate’s best strategy to attack one another. It is interesting that even in an audience that is heavily biased towards Trump, Clinton’s best strategy is still to attack 100% of the time.

The interested reader is invited to experiment with different scenarios using the general results derived above.

2016 Real-Time Election Predictions

Further to my original post on using physics to predict the outcome of the 2016 US Presidential elections, I have now written a cloud-based app using the powerful Wolfram Cloud to pull the most recent polling data on the web from The HuffPost Pollster, which “tracks thousands of public polls to give you the latest data on elections, political opinions and more”.  This app works in real-time and applies my PDE-solver / machine learning based algorithm to predict the probability of a candidate winning a state assuming the election is held tomorrow.

The app can be accessed by clicking the image below: (Note: If you obtain some type of server error, it means Wolfram’s server is busy, a refresh usually works. Also, results are only computed for states for which there exists reliable polling data. )

 

Some Thoughts on The US GDP

Here are some thoughts on the US GDP based on some data I’ve been looking at recently, mostly motivated by some Donald Trump supporters that have been criticizing President Obama’s record on the GDP and the economy. 

First, analyzing the real GDP’s average growth per year, we obtain that (based on a least squares regression analysis)

According to these calculations, President Clinton’s economic policies led to the best average GDP growth rate at $436 Billion / year. President Reagan and President Obama have almost identical average GDP growth rates in the neighbourhood of $320 Billion / year. However, an obvious caveat is that President Obama’s GDP record is still missing two years of data, so I will need to revisit these calculations in two years! Also, it should be noted that, historically, the US GDP has grown at an average of about $184 Billion / year. 

The second point I wanted to address is several Trump supporters who keep comparing the average real GDP annual percentage change between President Reagan and President Obama. Although they are citing the averages, they are not mentioning the standard deviations! Computing these we find that:


Looking at these calculations, we find that Presidents Clinton and Obama had the most stable growth in year-to-year real GDP %. Presidents Bush and Reagan had highly unstable GDP growth, with President Bush’s being far worse than President Reagan’s. Further, Trump supporters and most Republicans seem quick to point out the mean of 3.637% figure associated with President Reagan, but the point is this is +/- 2.55%, which indicates high volatility in the GDP under President Reagan, which has not been the case under President Obama. 

Another observation I would like to point out is that very few people have been mentioning the fact that the annual real US GDP % is in fact correlated to that of other countries. Based on data from the World Bank, one can compute the following correlations: 


One sees that the correlation between the annual growth % of the US real GDP and Canada is 0.826, while for Estonia and The UK is roughly close to 0.7. Therefore, evidently, any President that claims that his policies will increase the GDP, is not being truthful, since, it is quite likely that these numbers also depend on those for other countries, which, I am not entirely  convinced a US President has complete control over!

My final observation is with respect to the quarterly GDP numbers. There are some articles that I have seen in recent days in addition to several television segments in which Trump supporters are continuously citing how better Reagan’s quarterly GDP numbers were compared to Obama’s. We now show that in actuality this is not the case. 

The problem is that most of the “analysts” are just looking at the raw data, which on its face value actually doesn’t tell you much, since, as expected, fluctuates. Below, we analyze the quarterly GDP% data during the tenure of both Presidents Reagan and Obama, from 1982-1988 and 2010-2016 respectively, comparing data from the same length of time. 

For Reagan, we obtain: 


For Obama, we obtain:


The only way to reasonably compare these two data sets is to analyze the rate at which the GDP % has increased in time. Since the data is nonlinear in time, this means we must calculate the derivatives at instants of time / each quarter. We first performed cubic spline interpolation to fit curves to these data sets, which gave extremely good results: 


We then numerically computed the derivative of these curves at each quarter and obtained: 

The dashed curves in the above plot are plots of the derivatives of each curve at each quarter. In terms of numbers, these were found to be: 


Summarizing the table above in graphical format, we obtain: 


As can be calculated easily, Obama has higher GDP quarterly growth numbers for 15/26 (57.69%) quarters. Therefore, even looking at the quarterly real GDP numbers, overall, President Obama outperforms President Reagan. 

Thanks to Hargun Singh Kohli, B.A. Honours, LL.B. for the data collection and processing part of this analysis. 

2016 Michigan Primary Predictions

Using the Monte Carlo techniques I have described in earlier posts, I ran several simulations today to try to predict who will win the 2016 Michigan primaries. Here is what I found:

For the Republican primaries, I predict:

Trump: 89.64% chance of winning

Cruz: 5.01% chance of winning

Kasich: 3.29% chance of winning

Rubio: 2.06% chance of winning

The following plot is a histogram of the simulations:

trumpRepubs

 

 

The Effect of Individual State Election Results on The National Election

A short post by me today. I wanted to look at the which states are important in winning the national election. Looking at the last 14 presidential elections, I generated the following correlation plot:

  
For those not familiar with how correlation plots work, the number bar on the right-hand-side of the graph indicates the correlation between a state on the left side with a state at the top, with the last row and column respectively indicating the national presidential election winner. Dark blue circles representing a correlation close to 1, indicate a strong relationship between the two variables, while orange-to-red circles representing a correlation close to -1 indicate a strong anti-correlation between the two variables, while almost white circles indicate no correlation between the two variables.

For example, one can see there is a very strong correlation between who wins Nevada and the winner of the national election. Indeed, Nevada has picked the last 13 of 14 U.S. Presidents. Darker blue circles indicate a strong correlation, while lighter orange-red circles indicate a weak correlation. This also shows the correlation between winning states. For example, from the plot above, candidates who win Alabama have a good chance of winning Mississippi or Wyoming, but virtually no chance of winning California.

This could serve as a potential guide in determining which states are extremely important to win during the election season!