Quantitative Methods in Finance

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As we plug the coefficients into our model, we obtain the excess returns from our stock.

[pic 2]

Once the coefficients plugged in, we assume that the errors are eliminated.[a] We noticed that our regression is significant as its significance F (1.11*10-19) is far below 0.10.

We formulate the following null hypothesis: [b][pic 3]

Moreover, R2 is equal to 32%, which means that our model explains 32% of Amazon’s stock variations by a change in one of the four explanatory variables. While this is not as high as we could expect it, the R2 is only of 30% using the Fama-French three-factor model, while the CAPM is as low as 25%.

Let’s now check the significance of our coefficients by testing our null hypothesis, which states that: with[pic 4][pic 5]

To do that, we calculate the t-stat of our coefficient. We then compare our t-stat to the critical level at a 5% significance level, which is distributed as a t-distribution with 252 observations and 5 parameters[c].[pic 6]

By comparing our p-values[d] to the critical value (1.9696), we observe that each of our p-value [e]is lower than the critical value, which means that all of our parameters, and in that matter our overall regression, are significant.

Our results are based on the fact that the following assumptions are true:[f]

Now that we have built our model and tested its significance, we need to check its coefficients. Our estimations of the coefficients will be unbiased, consistent, and effective only if the following five assumptions hold true:

- Our model is logical and well specified.

- The errors have a zero mean.

- Variance of the errors is homoscedastic.

- There is no autocorrelation among the errors.

- The errors are normally distributed, allowing us to infer about the population parameters from our sample parameters.

In this case, we could say that OLS-estimators are unbiased, consistent, and effective. Otherwise, we can not completely rely on OLS-estimators.[g]

Let’s now tackle each of these assumptions to check whether or not our model and the calculated coefficients are unbiased, consistent, and effective.

- Logic and good specification

The Carhart model is an extension of the famous Fama-French three-factor model. As a result, it relies on a model that has been praised and used worldwide for over twenty years. The Carhart model itself adds a logical element aimed at narrowing down the unexplained abnormal errors. The additional parameter can be explained in financial terms and adds value to the basic model. On top of that, it has been in use for over fifteen year.

Overall, we can safely assume that the Carhart four-factor model is both logical and well specified.

- Zero mean of the errors

To check the zero mean of the errors, we computed the average of our residuals and calculated a mean of 1.34*10-19.

The mean is so close to zero that it can safely be assumed to be zero, meaning that the second assumption holds true for the Carhart model.

- Homoscedasticity of the variance

Let’s now have a look at the third assumption, which is the homoscedasticity of the variance so that Var() =