![]() Next, let’s find the value of the sum of □ squared. We have two more things we need to calculate. This means we’ve so far managed to find the values of □, the sum of □, and the sum of □. We get that the sum of □ is equal to 121. So in this case the sum of □ is going to be 25 plus 18 plus 24 plus 25 plus 12 plus 17. We just want to add all of the values of □ in our table together. We see that it’s equal to 104.Īnd we can then do exactly the same thing to find the sum of □. So in this case, the sum of □ is 10 plus 22 plus 22 plus 13 plus 16 plus 21. We just need to add all of the □-values in our table together. We can also find the sum of □ and the sum of □ from our table. We can see that there are only six data points in this example. We can actually see this directly from our table. Once we’ve found these five values, we just need to substitute these into our formulae to find the values of □ and □. We need to find the value of □, the sum of □, the sum of □, the sum of □ times □, and the sum of □ squared. First, although this seems very complicated, there are only five things we need to find. However, there’s a lot of things we need to take in. We’re now ready to start finding the equation of our regression line. Similarly, the mean value of □ will be the sum of □ over □. The mean value of □ will be the total of all of our data points of □ divided by the number of data points. Similarly, we know how to find the average value of □ and □. We also need the formula for □ sub □□, □ sub □□, and □ bar and □ bar.įirst, we recall □ sub □□ is equal to the sum of □ squared minus the sum of □ all squared over □ and □ sub □□ is the sum of □ times □ minus the sum of □ times the sum of □ over □. Of course, this alone is not quite enough to find the values of □ and □. And our value of □ is going to be equal to □ bar minus □ times □ bar, where □ bar is the mean □-value and □ bar is the mean □-value. □ will be equal to □ sub □□ divided by □ sub □□, where □ sub □□ is a measure between the covariance of □ and □ and □ sub □□ is a measure of the variance of □. We recall to find the least squares regression line between two variables □ and □, we can use the following formula. To answer this question, let’s start by recalling how we find the least squares regression line linking two variables □ and □. We’re also told we only need to approximate the value of □ and □ to three decimal places. We’re told to give our answer in the form □ hat is equal to □ plus □□. We need to use this table to find the equation of the regression line linking □ and □. In this question, we’re given a table of data points which show a relationship between two variables, the variable □ and the variable □. Approximate □ and □ to three decimal places. Find the equation of the regression line in the form □ hat is equal to □ plus □□. But it seems that you application is really about "predictive causality," which is exactly what the Granger causality approach is meant for.The table shows the relation between the variables □ and □. Granger causality has been critiqued for not actually establishing causality (in some cases). Then do the same the reverse (so, now regress the past values of X and Y on X(t)) and see which regressions have significant F-values.Ī very straightforward example, with R code, is found here. Check the significance of the F-statistics for each regression. regress X(t-1), X(t-2), X(t-3), Y(t-1), Y(t-2), Y(t-3) on Y(t)Ĭontinue for whatever history length might be reasonable.Also, considering that you are dealing with time series data, it tells you how much of the history of X counts towards the prediction of Y (or vice versa).Ī time series X is said to Granger-cause Y if it can be shown, usually through a series of t-tests and F-tests on lagged values of X (and with lagged values of Y also included), that those X values provide statistically significant information about future values of Y. In other words, it tells you whether beta or gamma is the thing to take more seriously. This would help you to assess whether X is a good predictor of Y or whether X is a better of Y. Perhaps the approach of "Granger causality" might help. $\dagger$ This is a simplification but it suits the purpose of explaining how one can, and should, perform the analysis to find an optimal ratio without a regression line. Also you could incorporate the time in a more sophisticated model to make better predictions of future values/distributions for the pair $X,Y$. Improvements of the model can be made by using different distributions than multivariate normal. In a situation with more than two variables/stocks/bonds you might generalize this to the last (smallest eigenvalue) principle component. To see the connection between both representations, take a bivariate Normal vector:
0 Comments
Leave a Reply. |
AuthorWrite something about yourself. No need to be fancy, just an overview. ArchivesCategories |