Monte Carlo Simulations and PcNaive

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1 Econometrics 2 Fall 2005 Monte Carlo Simulations and Pcaive Heino Bohn ielsen 1of21 Monte Carlo Simulations MC simulations were introduced in Econometrics 1. Formalizing the thought experiment underlying the data sampling. In this course we will frequently use MC simulations. Standard tool in econometrics. Underlying the econometric results is a layer of difficult statistical theory. (1) Many asymptotic results are technically demanding. Sometimes also difficult to firmly understand. Use MC simulations to obtain intuition. (2) The finite sample properties are often analytically intractable. Analyze finite sample properties. 2of21

2 Outline of the Lecture (1) The basic idea in Monte Carlo simulations. (2) Example 1: Sample mean(ols)of IID normals. (3) Example 2: Illustration of a Central Limit Theorem. (4) Introduction to Pcaive. (5) Example 3: Consistency and unbiasedness of OLS in a cross-sectional regression. Genereal-to-Specific orspecific-to-general? 3of21 The basic idea of the Monte Carlo method: The Monte Carlo Idea Replace a difficult deterministic problem with a stochastic problem with the same solution. If we can solve the stochastic problem by simulations, labour intensive work can be replaced by cheap capital intensive simulations. Problem: How can we be sure that the deterministic and stochastic problem have the same solution? General answer is the law of large numbers (LL): sample moments converge to population moments. 4of21

3 Consider a regression Example y i = x 0 iβ + i, i =1, 2,...,, ( ) where x i and i have some specified properties; and the OLS estimator Ã! 1 Ã X! X bβ = x i x 0 i x i y i. We are often interested in E[ b β] to check for bias. This is difficult in most situations. But if we could draw realizations of b β, we could estimate E[ b β]. MC simulation: (1) Construct M artificial data sets from the model ( ). (2) Find the estimator, b β m, for each data set, m =1, 2,...,M. Then from the LL: M 1 M X m=1 bβ m E[ b β] for M. 5of21 ote of Caution The Monte Carlo method is a useful tool in econometrics. BUT: (1) Simulations do not replace theory. Simulation can illustrate but not prove theorems. (2) Simulations results are not general. Results are specific to the chosen setup. We have to totally specify the model. (3) Work like good examples. In this course we hope to give you a Monte Carlo intuition. 6of21

4 Example 1: Mean of IID ormals Consider a model where we know the finite sample properties: y i = µ + i, i (0,η 2 ), i =1, 2,...,. ( ) The OLS estimator bµ of µ isthesamplemean X bµ = 1 y i. ote, that bµ is consistent, unbiased and (exactly) normally distributed bµ (µ, 1 η 2 ). The standard deviation of the estimate, in Pcaive called the estimated standard error, can be calculated as v q u X ESE(bµ) = 1 bη 2 = t 2 (y i bµ) 2. 7of21 Ex. 1 (cont.): Illustration by Simulation We can illustrate the results, if we can generate data from ( ). We need: (1) A fully specified Data Generating Process (DGP), e.g. y i = µ + i, i (0,η 2 ), i =1, 2,..., (#) µ = 5 η 2 = 1. An algorithm for drawing random numbers from (, ). Specify a sample length, e.g. =50or {10, 20, 30,..., 100}. (2) An estimation model for y i andanestimator.considerolsin y i = β + u i. (##) ote that the statistical model (#) and the DGP(##) need not coincide. 8of21

5 Ex. 1 (cont.): Four Realizations Supposewedraw 1,..., 50 from (0, 1) and construct a data set, y 1,..., y 50. We then apply OLS to the regression model y i = β + u i, to obtain the sample mean and the standard deviation in one realization, bβ = , ESE( b β)= Wecanlookatmorerealizations Realization, m βm b ESE( β b m ) Mean of21 Four Realization First realization, Mean= Second realization, Mean= Third realization, Mean= Fourth realization, Mean= of 21

6 Ex. 1 (cont.): Formalization ow suppose we generate data from (#) M times, y1 m,..., y50, m m =1, 2,..., M. For each m we obtain a sample mean β b m. We look at the mean estimate and the Monte Carlo standard deviation: MEA( β) b X M = M 1 bβ m MCSD( b β) = m=1 v u t M 1 MX ³ bβm MEA( β) 2 b m=1 For large M we expect: MEA( b β) to be close to the true µ (LL). The small sample bias is BIAS = MEA( b β) µ. 11 of 21 Ex. 1 (cont.): Measures of Uncertainty ote, that MEA( b β) is also an estimator (stochastic variable). The standard deviation of MEA( b β) is the Monte Carlo standard error MCSE( b β)=m 1 2 MCSD( b β). ote the difference MCSD( b β) measures the uncertainty of b β ( ESE( b β m )). MCSE( b β) measures the uncertainty of MEA( b β) in the simulation. MCSE( b β) 0 for M. 12 of 21

7 Ex. 1 (cont.): Results Consider the results for =50, M = 5000 : bβ m ESE( β b m ) MEA( β)= b MEA(ESE)= MCSD( β)= b MCSE( b β)= of 21 Ex. 1 (cont.): Results for Different 1.0 Density, T=5 Density, T= Density, T= Estimates, different T of 21

8 Example 2: A Central Limit Theorem (CLT) Recall the idea of a CLT (Lindeberg-Levy): Let z 1,..., z be IID with E [z i ]=µ and V [z i ]=σ 2.Then 1 X z i µ σ (0, 1) for. This can be extended to Heterogeneous processes. (Limited) time dependence. We will illustrate this for two examples Uniform distribution. Exponential distribution. 15 of 21 Consider as an example Ex. 2 (cont.): Uniform Distribution z i Uniform (0 : 1), i =1, 2,...,. It holds that E [z i ] = 1 2 V [z i ] = (1 0)2 = We look at the estimated distribution of 1 X z i µ σ based on M = replications. = 1 X µ 12 z i 1, 2 16 of 21

9 Ex. 2 (cont.): Uniform Distribution =1 = =5 = of 21 Ex. 2 (cont.): Exponential Distribution Consider as a second example z i Exp (1), i =1, 2,...,. It holds that We look at the estimated distribution of E [z i ] = 1 V [z i ] = 1 2 =1. 1 X based on M = replications. z i µ σ = 1 X (z i 1), 18 of 21

10 Ex. 2 (cont.): Exponential Distribution =1 = = = of 21 Pcaive Pcaive is a menu-driven module in GiveWin. Technically, Pcaive generates Ox code, which is then executed by Ox. Output is returned in GiveWin. Outline: (1) SetuptheDGP. AR(1) Static Pcaive General (2) Specify the estimation model. (3) Choose estimators and test statistics to analyze. (4) Set specifications: M, etc. (5) Select output to generate. (6) Save and run. 20 of 21

11 Example 3: Pcaive Static DGP DGP: for i =1, 2,...,. µ x1i x 2i y i = α 1 x 1i + α 2 x 2i + i, i (0, 1) µ 0 0 µ 1 c, c 1 Estimation model: Apply OLS to the linear regression model Example: y i = β 0 + β 1 x 1i + β 2 x 2i + u i. (1) Unbiasedness and consistency of OLS in this setting. (2) Effect of including a redundant regressor. (3) Effect of excluding a relevant regressor. 21 of 21

Monte Carlo Simulations and the PcNaive Software

Econometrics 2 Monte Carlo Simulations and the PcNaive Software Heino Bohn Nielsen 1of21 Monte Carlo Simulations MC simulations were introduced in Econometrics 1. Formalizing the thought experiment underlying