Variance reduction techniques (vrt)

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De La Salle University-Manila

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Variance Reduction Techniques (VRT) (Law and Kelton)

VRTs ,[object Object],[object Object],[object Object],[object Object],[object Object]

Common Random Numbers (CRN) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Antithetic Variates (AV) ,[object Object],[object Object],[object Object],[object Object],[object Object]

Control Variates (CV) ,[object Object],[object Object],[object Object]

Indirect Estimation ,[object Object],[object Object]

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Distribution of Estimates Linear Regression Model Assume (yt, xt) are independent and identically distributed and E(xtet) = 0 Estimation Consistency The estimates approach the true values as the sample size increases. Estimation variance decreases as the sample size increases. Illustration of Consistency Take a random sample of U.S. men Estimate a linear regression of log(wages) on education Total sample = 9089 Start with 100 observations, and sequentially increase sample size until in the final regression use the whole 9089. Sequence of Slope Coefficients Asymptotic Normality 4 Illustration of Asymptotic Normality Time Series Do these results apply to time-series data? Consistency Asymptotic Normality Variance Formula Time-series models AR models, i.e., xt = yt-1 Trend and seasonal models One-step and multi-step forecasting Derivation of Variance Formula For simplicity Assume the variables have zero mean The regression has no intercept Model with no intercept: Model with no intercept OLS minimizes the sum of squares The first-order condition is Solution Now substitute We have The denominator is the sample variance (when x has mean zero), so 10 Then Where Since Then From the covariance formula When the observations are independent, the covariances are zero. And since We obtain We have found As stated at the beginning. Extension to Time-Series The only place in this argument where we used the assumption of the independence of observations was to show that vt = xtet has zero covariance with vj = xjej. This is saying that vt is not autocorrelated. Unforecastable one-step errors In one-step-ahead forecasting, if the regression error is unforecastable, then vt is not autocorrelated. In this case, the variance formula for the least-squares estimate is Why is this true? The error is unforecastable if For simplicity, suppose that xt = 1. Then for Summary In one-step-ahead time-series models, if the error is unforecastable, then least-squares estimates satisfy the asymptotic (approximate) distribution As the sample size T is in the denominator, the variance decreases as the sample size increases. This means that least-squares is consistent. Variance Formula The variance formula for the least-squares estimate takes the form This formula is valid in time-series regression when the error is unforecastable. Classical Variance Formula If we make the simplifying assumption Then Homoskedasticity The variance simplification is valid under “conditional homoskedasticity” This is a simplifying assumption made to make calculations easier, and is a conventional assumption in introductory econometrics courses. It is not used in serious econometrics. Variance Formula: AR(1) Model Take the AR(1) model with unforecastable homoscedastic errors Then the variance of the OLS estimate is Since in this model AR(1) Asymptotic Variance We know that So The asymp ...

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