Empirical Finance:Analysis of non-stationary processes II: long-run relationships in empirical finance [3]
论文作者:留学生论文论文属性:案例分析 Case Study登出时间:2011-02-17编辑:anterran点击率:9214
论文字数:2412论文编号:org201102170919286578语种:英语 English地区:英国价格:免费论文
附件:20110217091928703.pdf
关键词:Empirical FinanceAnalysisnon-stationary processeslong-run relationshipsempirical finance
mator converges on the population coefficients much fasterthan in the stationary case.Why? OLS chooses parameter values which minimize the residual variance: –Only the cointegratingrelationship will have a finite variance: ε~I(0).–All other linear combinations are associated with an infinite residual variance: ε~I(1).Therefore OLS is very efficientat finding the cointegratingrelationships (if they exist) from amongst all the other (non-stationary) linear combinations.ttnnttXbXbbYε++++=,,221KHere we’ve just normalized the cointegratingrelationship on one of the variables (X1) which we’ve then made the dependent variable (Y). The error term is the same as z in the previousformulation. Cointegrationimplies that the error termin a regression with I(1) variables is I(0).Warwick Business School 11
Simulation evidence on the properties of the OLS estimator: sampling distributionsof the OLS estimator from a regression of Y on X in two instances
()1,0~000,10,...,1 ,5.0NIDtXYttttεε=+=0400080001200016000200002400028000320000.497500.498750.500000.50125Series: SLOPES_NSSample 1 100000Observations 100000Mean 0.499999Median 0.499999Maximum 0.502237Minimum 0.496996Std. Dev. 0.000332Skewness -0.017694Kurtosis 4.905678Jarque-Bera 15136.92Probability 0.0000000200040006000800010000120000.4750.5000.525Series: SLOPES_SSample 1 100000Observations 100000Mean 0.499972Median 0.500006Maximum 0.542554Minimum 0.457939Std. Dev. 0.010025Skewness -0.002035Kurtosis 2.982246Jarque-Bera 1.382321Probability 0.500994Y is a simulated series with the followingDGP:100,000 samples of Y and X were obtained (X is either a random walk or an I(0) process –see below). The model was estimated by OLS 100,000 times to estimate the sampling distribution of . The simulation was conducted in two instances:1.Y,X and εare I(0) (a classical stationarymodel).2. Y and X are I(1); εis I(0)(a cointegratingmodel).βˆStationary modelCointegratingmodelWarwick Business School 12
The simulation evidence highlights…
Both the sampling distributions are centred very close to 0.5 (the true parameter value). Recall, the sample size is large: T=10,000. However the mean is closer to 0.5in the cointegratingmodel.–The relative erroris 0.0056% in the stationary model versus 0.0002% in the cointegratingmodel.Also the estimates are less dispersedabout the mean in the cointegratingmodel:–The std dev is 0.01 in the stationary model versus 0.0003 in thecointegratingmodel.–The range (max value-min value) is 0.085 in the stationary model versus 0.005 in the cointegratingmodel.This highlights that the OLS estimator is collapsing on the truevalue fasterfor the cointegratingmodel than the stationary model ⇒OLS is SUPER-CONSISTENT.–In fact the sampling distribution is collapsing on βat the rate T for the cointegratingmodel versus √Tfor the stationary model.However the sampling distribution in the cointegratingmodel is highly non-normal(unlike the distribution in the stationary model which is normal).This highlights again that classical inferences do not apply with non-stationary models(see also the DF distribution in lecture 7).100ˆerror Relative×−βββWarwick Business School 13
Examples of cointegrationin empirical finance
Relationship between spot and future pricesSpot (s) and forward (f) prices are I(1).However for a given asset we would expect s and f to be driven by the same fundamentals (share a common stochastic trend).In that case there should exist a cointegratingrelationshi
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