#ec2020

Earth Day, Oh Happy Day!

In honor of Earth Day, we are sharing this beautiful photo, taken in the Sierra Nevadas; it reminds Susan of some of her happiest moments being in the mountains hiking and backpacking with friends.

Every year on Earth Day people around the globe volunteer and share messages about improving care for our local and global environment. In addition to being essential for sustaining our lives, this beautiful planet is key to our happiness and health. Earth Day is a great opportunity for each of us as individuals to consider our personal daily habits in connection to the earth, our health and happiness.

The cover of this month’s Outside magazine states: “Science’s Newest Miracle Drug is FREE”. The article goes on to describe how doctors are beginning to prescribe time outside to help individuals improve their wellness and manage depression and anxiety. Helping others on a regular basis also helps improve one’s health and happiness because of the natural hormones our body releases when we do. The platform we are building will help people develop this healthy habit! We hope you will be one of the first people experience the “Joy of Giving” using our system by signing up for our email list to be notified when it is available.

Back to Earth Day, daily life practices done by billions of people have an enormous impact on the earth’s environment. Today, people are making personal choices such as eliminating their use of single use plastic, taking the Swear Jar Challenge, recycling, consuming food that is locally and sustainably produced, purchasing used consumer goods when possible and sustainably produced goods when buying new, eliminating the use of acrylic clothing in your wardrobe to prevent micro plastics from being created and entering our water systems when you wash them. What personal choices can you make?

Baku is hosting the 2020 European Championships for Men’s gymnastics and Armenia is boycotting - sort of. The Armenian federation head said he will not send any athletes to compete in Azerbaijan. This is even when it risks their athletes chances of qualifying to the Olympics and being granted deferrals for compulsory military service.

The 2020 European Championships are being hosted by two cities for the women’s and men’s competition. The women are going to be competing in Paris at the Bercy - it is where that major Paris event is held. While the men are going to be competing in Baku at the European Games 2015 venue. The women in early May and the men in late May. It is that usual senior and junior format with team over all-around format. This format is a return to the old format of hosting from the previous one which was hosted solely by Bern in 2016. I prefer the one city format but it does allow for more cities/countries getting hosting duties - if they so please.

Purpose: Test for heteroscedasticity

Park suggests that the unknown variance \(\sigma_{i}^2\) is some function of the explanatory variable \(X_{i}\).

The suggested form:

\[\sigma_{i}^{2} = \sigma^{2} X_{i}^{\beta} e^{\nu i }\]

\[ln\sigma_{i}^{2} = ln \sigma^{2} + \beta ln X_{i} + ln \nu i \]

Since \(\sigma_{i}\) is generally unknown, Park suggests using \(\hat{u_{i}}^{2}\) as a proxy and running the following regression:

\[ln\hat{u_{i}}^{2} = ln \sigma^{2} + \beta ln X_{i} + ln \nu i \]

Test

\(H_{0} : \beta = 0\) Homoscedastic \(H_{1} : \beta \neq 0 \) Heteroscedastic

use t-test to find out if \(\beta\) is significant. 

If \(H_{0}\) were rejected, then we can conclude that there is heteroscedasticity in the model.

Purpose: Test for functional misspecification

Suppose we want to test whether the model includes non linear explanatory terms.

Consider the model

\[Y_{i} = \beta_{1} + \beta_{2}\: X_{2i} + ... + \beta_{k}\:X_{ki} + u_{i}\]

So the estimated model is 

\[Y_{i} = \widehat{\beta_{1}} + \widehat{\beta_{2}}\: X_{2i} + ... + \widehat{\beta_{k}}\: X_{ki} \]

Now, we introduce the square of the estimated values of Y, \(y_{i}^{2}\), which measures the effect of the nonlinear explanatory variables and their interaction.

\[Y_{i} = \beta_{1} + \beta_{2}\: X_{2i} + ... + \beta_{k}\:X_{ki} +\beta_{k+1}\widehat{y_{i}}^{2}+ u_{i}\]

Note: if the \(\beta_{k+1} = 0\), the model would be linear

Test

\(H_{0}: \beta_{k+1} =\:0 \) \(H_{1}: \beta_{k+1} \neq \: 0\)

Purpose: To test significance of a single or a set of MLE estimators.

Test: 

\(H_{0} = \theta = \theta_{0}\) \(H_{1} = \theta \neq \theta_{0}\)

let \(\hat{\theta}\) be the Maximum Likelihood Estimator (MLE).

Test statistic 

\[\frac{(\hat{\theta} - \theta_{0})^{2}}{\hat{\sigma_{\hat{\theta}}}^{2}} \sim \chi^{2}_{k}\]

where k is the number of restrictions

Rejection Rule

Purpose: Test the validity of the model specification.

Consider the models,

$$ \begin{eqnarray}  Y_{t} &=& \lambda_{1} + \lambda_{2}Y_{t-1} + \lambda_{3}X_{t} + \lambda_{4}X_{t-1} + \epsilon_{t} & unrestricted \:\:ADL ^{1} &\\\ Y_{t} &=& \beta_{1} (1 - \rho) + \rho Y_{t-1} + \beta_{2} X_{t} - \beta_{2} \rho X_{t-1} + \epsilon_{t} & restricted\:\: AR(1) &\\\ \end{eqnarray} $$

The restriction is \(\lambda_{4} = - \lambda_{2}\lambda_{3}\)

Test:

\(H_{0}\): The restriction is valid (ie. no model misspecification) \(H_{1}\): The restriction is not valid (ie. there is model misspecification)

Test statistics:

\[n\: log\:(\frac{RSS_{R}}{RSS_{U}}) \:\sim \:\chi^{2}_{k}\]

where k is the degrees of freedom which is equal to the number of explanatory variables in the model.

Rejection rule

We would reject \(H_{0}\) if the test statistic is calculated to be more than the critical value at a particular significant level and conclude that there is misspecification in the model and we will be with the unrestricted one.