MondayHW

Section 1

Data Processing

Equation One

\[\text{price} = \beta_0 + \beta_1 (\text{carat}) + \varepsilon\]

Below is the slope using bivariate regression and the coef command:

(Intercept)       carat 
  -2256.361    7756.426 

Here is the slope using

\(\frac{\text{Cov}(\text{carat}, \text{price})}{\text{Var}(\text{carat})}\):

[1] 7756.426

Equation Two

\[\text{price} = \beta_0 + \beta_1 (\text{carat}) + \beta_2 (\text{depth}) + \varepsilon\]

(Intercept)       carat       depth 
  4045.3332   7765.1407   -102.1653 

The equation \(\frac{\text{Cov}(\text{carat}, \text{price})}{\text{Var}(\text{carat})}\) does not hold for a multivariate regression. Normally, it works becuase \(X_1\) is the only variable in the model, so its slope absorbs all correlation with \(Y\). In multivariate regression, both x values are usually correlated somehow, meaning that the formula fails.

Predictions

We can see that the Apple stock will most likely follow the drift pattern and continue increasing. The Drift model follows the slope from x = 0 to the end of the training data. The postivity of the slope best represents the increasing trend for the Apple stock. However recently, the stock has been fluctuating similar to a SNaive pattern, meaning that it could begin following a cycle. In real life however, the stock continued to soar, outdoing all of the predictions from the three model types.

I believe we should HOLD the stock because simple baseline models like Naive, SNaive, and Drift fail to reliably capture real-world market dynamics. While the long-term trend shows strong historical growth, these basic models either assume a completely flat price, force unrealistic annual seasonality, or project a rigid linear line with extreme uncertainty. Therefore, buying based on these simple forecasts carries too much financial risk, making holding the position the wisest choice.