Multiple Regression on the Example of the Presidential Elections in the United States during the Period from 1916 to 2000

Multiple regressions are studied in Russia and abroad. So Chetyrkin [1] describe a method of constructing it manually. Winston [2,3] described the quickest way to build it in the program Microsoft Office Excel, and taken them an example of submission of multiple regression, as well as details of its interpretation of some of the best works on the statistics. Sachs [4] not a lot of concerns of multiple regression, but look at how he described the manual methods of investigation regression really understand what the German precision [5].


Introduction
Multiple regressions are studied in Russia and abroad. So Chetyrkin [1] describe a method of constructing it manually. Winston [2,3] described the quickest way to build it in the program Microsoft Office Excel, and taken them an example of submission of multiple regression, as well as details of its interpretation of some of the best works on the statistics. Sachs [4] not a lot of concerns of multiple regression, but look at how he described the manual methods of investigation regression really understand what the German precision [5].
But outside attention the authors were: 1. Systematization of knowledge about the multiple regressions; 2. Construction of the system of normal equations for mathematical models with more than two explanatory variables; conclusion of these equations is considered redundant because, firstly, the need to build an auxiliary table, and secondly, the solution of these equations requires a lot of work if it is to perform manually. But if you examine the sum of squared errors, standard errors of prediction and covariance, and you'll need. A solution of large systems is now easier MATLAB. Plus the consideration of such a system shows the interaction of variables;

Conclusion of normal equations of the functional equation by differentiation;
4. The decision referred to in MATLAB as the normal equations, and direct-matrix method; 5. The decision of using a calculator and matrix "matrix arithmetic"; 6. Graphical analysis of residues; 7. Investigation of multiple regressions with z -statistics and graphs standard normal probability. The regression statistics -the most difficult section of statistics. In turn, the multiple regression -one of the most difficult in the regression statistics. The formula is as follows: Chetyrkin [1] wrote: "If selected as the independent variable is the dominant factor, respectively, the corresponding pair regression adequately describes the mechanism of causation. The most common change of y due to the influence of several factors (sometimes acting in opposite directions). In these cases, naturally the desire to enter some explanatory variables. This is called multiple regressions. Multiple regression equation to better to explain the behavior of the dependent variable than steam regression, in addition, it makes it possible to compare the effectiveness of various factors".

Multiple Regressions
Wayne Winston -Indiana University professor who advises the company Ford Motor corporation General Motors, Intel, Microsoft, Proctor and Gamble, the US Army, US Department of Defense and other organizations, a graduate of Yale University with a Ph.D. and the Faculty of Mathematics at MIT, who among universities in the world by Times Higher Education World Reputation Rankings 2015 took 4th place, refers to a book economist Roy Fair [6] that the economy has a major impact on the results of the presidential elections. How can I predict U.S. presidential elections? Presidential advisor James Carville said "It's the economy" when asked about which factors drive presidential elections. Yale economist Roy Fair showed that Carville was right in thinking that the state of the economy has a large influence on the results of presidential elections. Fair's dependent variable for each election  was the % of the two party vote (ignoring votes received by third party candidates) that went to the incumbent party. He tried to predict the incumbent party's % of the two party votes by using the following independent variables: 1) Party in power. In our data, we use a 1 to denote that the Republican Party is in power and a 0 to denote that the Democratic Party is in power.
2) % growth in GNP during the first nine months of the election year.
3) Absolute value of the inflation rate during the first nine months of the election year. We use the absolute value because either a positive or a negative inflation rate is bad. 4) Number of quarters during the last four years in which economic growth has been strong. Strong economic growth is defined as growth at an annual level of 3.2% or more. 5) Time incumbent party has been in office. Fair used 0 to denote one term in office, 1 for two terms, 1.25 for three terms, 1.5 for four terms, and 1.75 for at least five terms. This definition implies that each term after the first term in office has less influence on the election results than the first term in office.
Is the election during wartime? The elections in 1920 (World War I), 1944 (World War II), and 1948 (World War II was still underway in 1945) were defined as wartime elections. Elections held during the Vietnam War were not considered wartime elections. During wartime years, the variables related to quarters of good growth and inflation was deemed irrelevant and was set to 0.
6) Is the current president running for re-election? If so, this variable is set to 1; otherwise, this variable is set to 0. In 1976, Gerald Ford was not considered a president running for re-election because he was not elected as either president or vice-president.
I've attempted to use the data from the elections in 1916 through 1996 to develop a multiple regression equation that can be used to forecast future presidential elections. I saved the infamous 2000 election as a «validation point».
In Table 1, you can see that the p-value for each independent variable is much less than.15, which indicates that each of our independent variables is helpful in predicting presidential elections. We can predict elections using an equation…: Projected percentage of votes in the presidential election = 45,53 + 0,70 growth GNP -0,71 abs.inf. + 0,90 quarter growth GNP -3,33 terms of power + 5,66 Republicans + 4,71 war + 3,99 President of a new term The coefficients of the independent variables can be interpreted as follows (After adjusting for all other independent variables used in equation 2): 1. A 1% increase in the annual GNP growth rate during an election year is worth. 7% to the incumbent party.

2.
A 1% deviation from the ideal (0% inflation) costs the incumbent party. 71% of the vote.
3. Every good quarter of growth during an incumbent's term increases his (maybe her someday soon!) vote by .90%. 4. Relative to having one term in office, the second term in office decreases the incumbent's vote by 3.33%, and each later term decreases the incumbent's vote by .25*(3.33% ) = .83%. 5. A Republican has a 5.66% edge over a Democrat.

Applicants
Year We fix a, b, d, e, h, p and q and differentiate Е (а, b, с, d, e, h, p, q) by c. Receive   We find that 94% of the variation in the % received by an incumbent in a presidential election is explained by our independent variables. We have made no mention whether the candidates are "good or bad" candidates ( Table 1).
The author of the article draws attention to the fact that the fall in the growth rate of GDP in the election year, almost always leads to a change of the ruling party (Table 1).
First, we decide through the normal equations.
Finally got a model that reflects the policy of the United States for nearly a century:

T T T T T T T T i Q e e e y Xa y Xa y y a X y y Xa a X Xa
Here and below T denotes the transpose of a vector or matrix. So how a T X T y = y T Xa, then 2 . We equate the result to zero. After that, we can easily find the system of normal equations that in matrix form is written as 1 (

T T T T T Q y y a X y a X Xa
) .
T T T T X y X Xa отсюда a X X X y − = = × Formula solutions matrix method takes the form: .
.  3.0000 0 10.0000 7.0000 4.0000 9.0000 8.0000 0 0 6.0000 5.0000 5.0000 10.0000 7.0000 4.0000 4.0000 5.0000 7.0000 6.0000 1. We can solve the problem and using the calculator matrices [7] and "matrix arithmetic". Enter the original data into the calculator ( Figure  1). We find the transposed matrix ( Figure 2). Go to the window "matrix arithmetic" (Figure 3). We get work and transpose of the original matrix ( Figure 4). We find the inverse matrix (Figures 5 and 6). In the same window, find the product of the transposed matrix of the vector and the column y. And by multiplying the inverse matrix that happened, we obtain a matrix of unknown coefficients (Figures 7 and 8                       Russian roulette with one bullet in the drum. Do not we would like to see the drum, for example, 1,000 or a million rounds of ammunition? Where is the limit in medicine temperature 37°C? Almost all authors agree on the fact that 95% -a statistical threshold beyond which -a disease. And, secondly, if the counting carried out with the reverse end, the error is increased from 5% to about 7% and 15% (three times). That is, at 95% reliability a failure on 20 attempts, and at 93.0503% reliability a failure already at 14.38911032. When reliability of 85%, which is offered by Professor Winston, single failure accounted for 6.67 attempts. In a game of Russian roulette we get 20 and 14 and the 6-7 Chargers revolvers charged 1 cartridge. Even in the first case, we cannot risk it. Sachs [4] points out: "In special cases, especially when the test poses a risk to human life, you should take less than α = 0,001 probability of errors." In the second case, the risk has increased. In the third your chances are dangerous. The offender Gabbar Singh has played in Russian roulette, spinning drum six-shooter revolver. Thus, Mr. Gabbar would fulfill the conditions of a statistical reliability of 85%, if he had a 6.67 charging revolver with one cartridge (Figure 11).
We have no right to risk, said Holmes. Ten to one that they go down to the river, rather than up. And yet, we must take into account In Tables 3-7 we have made the correction, whose outcome in  Table 12. The last two columns are not the same way. In this way Table  3 contains, first, a mathematical error instead of the 95% reliability should indicate the reliability of 93.0502%. Second -under certain conditions and statistical. What conditions? If you are willing to risk -the coefficient of reliability of 93.0502% and 85% is statistically true. But if the risk is not acceptable -the coefficient of 4.714529 must be zero. Thus Tables 3 and 6 after the both patches reconstructed in Table  8. In support of Sax in [4] causes test the significance of the regression coefficient: "Checks null hypothesis H 0 : β yx = 0, i.e checks whether significantly different estimates of regression coefficients from zero. The border is set on the basis of the significance of the t-distribution [8].
С (n -2) degrees of freedom. If statistics are greater than or equal to the limit of the significance of it, то β yx significantly different from zero".
In addition, Sachs [4] points out: Let's device consists of 300 complex elements. If these elements, for example, 284 completely smoothly, 12 have the reliability of 99%, and 4 -98% reliability, the reliability of the device, provided independence is reliable elements.  (8) Finally, Professor Winston in his work is not conducted analysis of residues (Figures 12-19) (Tables 12-17).         (7) calculator, the total reliability for the purchased unit.

Surveillance
The predicted share in the governments   More precisely will -to 98.2269% reliability (Figures 20-31).
Thus, having considered the multivariate regression model of