Monte Carlo Simulations and PcNaive Heino Bohn Nielsen 1 of 21 - PDF document

Econometrics 2 — Fall 2005 Monte Carlo Simulations and PcNaive Heino Bohn Nielsen 1 of 21 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 di ffi cult statistical theory. (1) Many asymptotic results are technically demanding. Sometimes also di ffi cult to fi rmly understand. → Use MC simulations to obtain intuition. (2) The fi nite sample properties are often analytically intractable. → Analyze fi nite sample properties. 2 of 21

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 PcNaive. (5) Example 3: Consistency and unbiasedness of OLS in a cross-sectional regression. Genereal-to-Speci fi c or Speci fi c-to-General? 3 of 21 The Monte Carlo Idea The basic idea of the Monte Carlo method: “Replace a di ffi cult 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 (LLN): sample moments converge to population moments. 4 of 21

Example • Consider a regression y i = x 0 i β + � i , i = 1 , 2 , ..., N, ( ∗ ) where x i and � i have some speci fi ed properties; and the OLS estimator à N ! − 1 à N ! X X b x i x 0 β = x i y i . i i =1 i =1 We are often interested in E [ b β ] to check for bias. This is di ffi cult in most situations. • But if we could draw realizations of b β , we could estimate E [ b β ] . MC simulation: (1) Construct M arti fi cial data sets from the model ( ∗ ). (2) Find the estimator, b β m , for each data set, m = 1 , 2 , ..., M . Then from the LLN: X M β m → E [ b b M − 1 β ] M → ∞ . for m =1 5 of 21 Note 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 speci fi c 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. 6 of 21

Example 1: Mean of IID Normals • Consider a model where we know the fi nite sample properties: � i ∼ N (0 , η 2 ) , y i = µ + � i , i = 1 , 2 , ..., N. ( ∗∗ ) • The OLS estimator b µ of µ is the sample mean X N µ = N − 1 b y i . i =1 Note, that b µ is consistent, unbiased and (exactly) normally distributed µ ∼ N ( µ, N − 1 η 2 ) . b • The standard deviation of the estimate, in PcNaive called the estimated standard error, can be calculated as v u q u X N η 2 = t N − 2 µ ) 2 . ESE ( b N − 1 b ( y i − b µ ) = i =1 7 of 21 Ex. 1 (cont.): Illustration by Simulation We can illustrate the results, if we can generate data from ( ∗∗ ). We need: (1) A fully speci fi ed Data Generating Process (DGP), e.g. � i ∼ N (0 , η 2 ) , y i = µ + � i , i = 1 , 2 , ..., N ( # ) µ = 5 η 2 = 1 . An algorithm for drawing random numbers from N ( · , · ) . Specify a sample length, e.g. N = 50 or N ∈ { 10 , 20 , 30 , ..., 100 } . (2) An estimation model for y i and an estimator. Consider OLS in y i = β + u i . ( ## ) Note that the statistical model ( # ) and the DGP ( ## ) need not coincide. 8 of 21

Ex. 1 (cont.): Four Realizations • Suppose we draw � 1 , ..., � 50 from N (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, b ESE ( b β = 4 . 98013 , β ) = 0 . 1477 . • We can look at more realizations b ESE ( b Realization, m β m β m ) 1 4.98013 0.1477 2 5.04104 0.1320 3 4.99815 0.1479 4 4.82347 0.1504 Mean 4.96070 0.1445 9 of 21 Four Realization First realization, Mean=4.98013 Second realization, Mean=5.04104 6 6 4 4 Third realization, Mean=4.99815 Fourth realization, Mean=4.82347 7.5 7.5 5.0 5.0 2.5 2.5 10 of 21

Ex. 1 (cont.): Formalization Now suppose we generate data from ( # ) M times, y m 1 , ..., y m 50 , m = 1 , 2 , ..., M. For each m we obtain a sample mean b β m . • We look at the mean estimate and the Monte Carlo standard deviation: M X MEAN ( b b β ) = M − 1 β m m =1 v u ³ ´ 2 u M X t M − 1 MCSD ( b β m − MEAN ( b b β ) = β ) m =1 • For large M we expect: MEAN ( b β ) to be close to the true µ (LLN). The small sample bias is BIAS = MEAN ( b β ) − µ. 11 of 21 Ex. 1 (cont.): Measures of Uncertainty Note, that MEAN ( b β ) is also an estimator (stochastic variable). The standard deviation of MEAN ( b β ) is the Monte Carlo standard error MCSE ( b 2 MCSD ( b β ) = M − 1 β ) . Note the di ff erence • MCSD ( b β ) measures the uncertainty of b ( ≈ ESE ( b β β m ) ). • MCSE ( b β ) measures the uncertainty of MEAN ( b β ) in the simulation. MCSE ( b β ) → 0 for M → ∞ . 12 of 21

Ex. 1 (cont.): Results Consider the results for N = 50 , M = 5000 : b ESE ( b β m β m ) 1 4.98013 0.1477 2 5.04104 0.1320 . . . . . . . . . 5000 4.92140 0.1254 MEAN ( b β ) =4.9985 MEAN ( ESE ) =0.14083 MCSD ( b β ) =0.14061 MCSE ( b β ) =0.0019886 13 of 21 Ex. 1 (cont.): Results for Di ff erent N Density, T=5 Density, T=10 1.0 1.0 0.5 0.5 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 Density, T=50 Estimates, different T 3 5.5 2 5.0 1 4.5 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 0 50 100 150 200 250 14 of 21

Example 2: A Central Limit Theorem (CLT) Recall the idea of a CLT (Lindeberg-Levy): • Let z 1 , ..., z N be IID with E [ z i ] = µ and V [ z i ] = σ 2 . Then N X 1 z i − µ √ → N (0 , 1) for N → ∞ . σ N i =1 This can be extended to • Heterogeneous processes. • (Limited) time dependence. We will illustrate this for two examples • Uniform distribution. • Exponential distribution. 15 of 21 Ex. 2 (cont.): Uniform Distribution Consider as an example z i ∼ Uniform (0 : 1) , i = 1 , 2 , ..., N. It holds that 1 E [ z i ] = 2 (1 − 0) 2 = 1 V [ z i ] = 12 . 12 We look at the estimated distribution of µ ¶ X N X N √ 1 z i − µ 1 z i − 1 √ √ = 12 · , σ 2 N N i =1 i =1 based on M = 20000 replications. 16 of 21

Ex. 2 (cont.): Uniform Distribution N=1 N=2 1.0 0.4 0.5 0.2 -4 -2 0 2 4 -4 -2 0 2 4 N=5 N=10 0.4 0.4 0.2 0.2 -4 -2 0 2 4 -4 -2 0 2 4 17 of 21 Ex. 2 (cont.): Exponential Distribution Consider as a second example z i ∼ Exp (1) , i = 1 , 2 , ..., N. It holds that E [ z i ] = 1 V [ z i ] = 1 2 = 1 . We look at the estimated distribution of X N X N 1 z i − µ 1 √ √ = ( z i − 1) , σ N N i =1 i =1 based on M = 20000 replications. 18 of 21

Ex. 2 (cont.): Exponential Distribution N=1 N=2 0.75 0.4 0.50 0.2 0.25 -2.5 0.0 2.5 5.0 7.5 -2.5 0.0 2.5 5.0 N=5 N=50 0.4 0.4 0.2 0.2 -4 -2 0 2 4 6 -4 -2 0 2 4 19 of 21 PcNaive • PcNaive is a menu-driven module in GiveWin. • Technically, PcNaive generates Ox code, which is then executed by Ox. Output is returned in GiveWin. • Outline: (1) Set up the DGP. — AR(1) — Static — PcNaive — General (2) Specify the estimation model. (3) Choose estimators and test statistics to analyze. (4) Set speci fi cations: M , N etc. (5) Select output to generate. (6) Save and run. 20 of 21

Example 3: PcNaive Static DGP DGP: y i = α 1 · x 1 i + α 2 · x 2 i + � i , � i ∼ N (0 , 1) µ ¶ ∙ µ ¶ µ ¶¸ x 1 i 0 1 c ∼ N , x 2 i 0 c 1 for i = 1 , 2 , ..., N . Estimation model: Apply OLS to the linear regression model y i = β 0 + β 1 · x 1 i + β 2 · x 2 i + u i . Example: (1) Unbiasedness and consistency of OLS in this setting. (2) E ff ect of including a redundant regressor. (3) E ff ect of excluding a relevant regressor. 21 of 21