Because sometimes its more than correlation; Its causationRaquel Acosta RA6977430Raquel Acosta RA6977430 Lisa Kustina RA6977197Lisa Kustina RA6977197?Used to determine if one time series is useful in forecasting another?Claims to move beyond correlation to test for causation?Can only be applied to pairs of variables?May be misleading if true relationship involves three or more variablesWhat is causality?What do you think?What do you think?Sir Clive William John GrangerSir Clive William John GrangerBorn 1943Studied applied mathematics and statisticsWon Nobel Prize in Economics 2003Published a series of articles and two books analyzing the relationship between time seriesThe topic of how to define causality has kept philosophers busy for over two thousand years and has yet to be resolved. It is a deep convoluted question with many possible answers which do not satisfy everyone, and yet it remains of some importance. Investigators would like to think that they have found a “cause”, which is a deep fundamental relationship and possibly potentially usefulSir Clive William John GrangerSir Clive William John GrangerBorn in 1943Studied applied mathematics and statisticsWon Nobel Prize in Economics 2003Published a series of articles and two books analyzing the relationship between time seriesWhat is causality?is all the information in the universe at time T?F(A|B) is the conditional distribution of A given B?and are two time seriesIf this condition holds, then does not cause . If this condition does not hold, then can be said to cause because it contains special information that is not available elsewhere.1.Run a regression of X 2.Determine a statistically significant time lag interval3.Run multiple regressions1.Make sure additional regressions are also significant2.Include regressions that add explanatory power 4.Can be repeated for multiple Ys5.Hopefully, find a clear relationship, such as Y granger-causes X?Do a series of F-tests on lagged values of Y to see if those values provide statistically significant information about future values of X?Can involve more variables if vector autoregression is applied to capture interdependenciesLinear Regression models?Used to describe the dependence of the mean value of one variable on one or more other variables.?E(Y | x) the regression of Y on x. ?x is the explanatory variable, or regressor. Y the mean of variable y. i.e. (y is the occurrences of embezzlement and x is the individuals yearly income)?Since its linear E(Y|x) 0+1x are our regression coefficents. Y is of unknown variance 2 Given a sample of n independent pairs of observations (xi, yi) the best estimators of 0 +1is given by the method of Least Squares.?i.e. minimize (yi - 0- 1x1)?Gauss-Markov theorem states that the least squares estimator gives the unbiased* estimator of a parameter having minimum variance.?Its good to note the least-squares regression of x on y is not in general the same line as the least- squares regression of y on x. i.e. what is meant by the line of best fit to a data set depends on what assumptions are made about the nature of any deviations from a fitted line?Goal select parameters so as to minimize the sum of the squared residuals?OLS Ordinary least squares estimationBest linear unbiased estimates if and only if the Gauss-Markov assumptions are satisfied.Time series models?Standard regression cant be applied?Common forms?Autoregressive models?Moving average models?Box-Jenkins (Combines these 2)?ARCH - Autoregressive conditional heteroskedasticity?GARCH Generalized ARCHFrequently used for financial time series?VAR Vector autoregressionUsed to understand inter-relationships of variables represented by systems of equations?Estimates difference equations containing stochastic componentsAutoregressive modelXt = c + Xt-1 + twhere tis white noise with zero mean and variance 2 If = 1 then Xtexhibits a unit root and can be considered a random walk.For AR of order 1, processes where its parameters , are stationary if 1.?Let be the mean. Then E(Xt) = E(c) + E(Xt-1 ) + E(t) ; so = c + +0?So var(Xt) = E(Xt2) 2= 2 /(1- 2)?Autocovariance is given by E(Xt+n| Xt) - 2= (2 /(1- 2) |n|Tools?Correlation coefficient ?Product moment CC Pearson CCFor straight line relationship is plus or minus oneIf = 0, X and Y are uncorrelated?Doesnt imply independence unless X and Y have a bivariate distributioni.e. for n paired observations the sample correlation coefficient is r = Sxy /(SxxSyy)1/2?Sxyis the sum of the products of deviations of xiand yifrom their means?Sxxand Syyare the sums of squares of deviations from their means?r =1 implies a direct relationship, r = -1 implies an inverse one?r = 0 implies virtually no linear relationship, but there may be some other association - i.e. Points scattered around the circumference of a circle?Spearman CC2 sets of pa