MonteCarlo蒙特卡洛方法

1、l：NIVLRSITY OF COPKN H GI!ND I; F A R T M E N T OF ECONOMICSEconometrics CMonte Carlo SimulationsPrinciple and Examplesl： N I V LR S I T V or COPI N M %GI!ND I; F A R T M £ N T OF ECONOMICSMonte Carlo Simulations MC simulations were introduced in Econometrics A.Formalizing the thought experimen

2、t underlying th© data sampling.Mimic the data generation and study the behaviour. In this course we will frequently use MC simulations. Standard tool in econometrics. Underlying the econometric results Is a layer of difficult statistical theory. Many asymptotic results are technically demanding

3、.Sometimes also drfftcult to firmly understand. Use MC simuiatlons to obtain intuition. The finite sample properties are otten analytically intractable. Analyze finrte sample propertiesUNIVERSITY OF COPi N HDI1FARTMENT OF tf C O N O M IC SThe Monte Carlo IdeaThe basic idea of the Monte Carlo method:

4、Replace a difficult deterministic problemwith a stochastic problem with the same solution.If we can solve the stochastic problem by simulations. laDour Intonsive work can be replaced by cheap capital Intensive simulations. How can we be sure that deterministk: and stochastic problems have same solut

5、ion?General answer is the law of large numbers (LLN). As an example, consider a stochastic variaWe x /(x).Calculation of the mean Is a (potentially difficult) deterministic problem:£(*) = Ixf(x)dxIf we can draw realizations x仁 &“xM from f(x). we can useE(x) = M"工* £ E(x) for M t x

6、.l：NIVtKSITY or COPKN H %GtlhC O N O M IC SD I; F A R T M E N T OFExample 1: Un biased ness Con引der a regressionYt = Xf3 f. t = 1.2T.where xt and “ have some specified properties: and the OLS estimator了 =（力衲寸（力Xr".We are often Interested In W( 7) to check for bias This Is difficult In most situ

7、ations、 But If we could draw realizatlons of 氏 then we could estimate E( J).MC simulation:0 Construct M artificial data sets from the model () Find the estimate. lor each data set. m 12 MThen from the LLN:MM八力益-E(7) tor MtxwExmp隱 2 A Central Limit Theorem (CLT)Recall the idea of a CLT (Undeberg-Lwy

8、L創(chuàng) z, . £? t)e IID with E(eJ 卜 and V(Zj “ and T r / z： be the empirtca< mean of zT ThenUNIVERSITY OF COPENHAGEN DEPARTMENT OF ECONOMICS Example 2: lllustrati on of CLT Con sider as an example zt Uni form (0, 1 , It holds that E (zt V (zt =1 2 (1 - 02 1 = . 12 12V T V 12, t = 1,2, ., T . We l

9、ook at the simulated distribution ofVT z -卩(T = z- 1 2 based on M = 10000 replicatio ns. Monte Carlo Simulatio nSlideUNIVERSITY OF COPENHAGEN T=1 0.4 DEPARTMENT OF ECONOMICST=2 0.30 De nsity 0.20 De nsity -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 0.10 0.00 0.0 0.1 0.2 0.3 -2 -1 0 1 2 T=5 0.4 0.4 T = 100 0.3 De

10、n sity Den sity -3 -2 -1 0 1 2 3 4 0.2 0.1 0.0 0.0 -4 0.1 0.2 0.3 -2 0 2 Mon te Carlo Simulatio nsSlide 7 UNIVERSITY OF COPENHAGEN DEPARTMENT OF ECONOMICS Example 3:Mon te Carlo to Check Con siste ncy Con sider the followi ng data gen erati ng processes: x1tN (0, 1 x2t N (0, 1 p = Corr (x1t , x2t b0 = 0 b1 = 1 b2 = 1 Case 1 2 3 4 5 DataGen erati ng Process yt = b0 + b1 x1t +£ t yt = b0 + b1 x1t + b2 x2t +£ t yt = b0 + b1b2 x2t +£ t yt = b1 yt -1 +£ t yt = b1 yt -1 +£ t p =N (0,p1 N6£ t £ t £ tB 1 be con sis(0, 1 N (0, 1 N (0, 1 = 0.5£