計量方法的演變 Specification Y i =β 1 +β 2 X 2i +…+ β k X ki + e i,i = 1,…,N linear model nonlinear model nonparametrics least squares (least absolute deviation), QMLE, GMM, computationally intensive methods, etc. single equation multiple equations (simultaneous-equation model, VAR) Method Structure
Does the era of linearity and least-squares come to an end? Legendre (1798): “balance the errors in such a way that they are borne equally by all equations” Legendre (1805): “distributing the errors among the equations” Minimizing sum of squared errors Least squares: Euler Laplace Legendre Key: “averaging” the errors Least absolute deviation:Boscovich, 1755 Minimizing sum of absolute errors
When a random structure is imposed We are interested in the random behavior of Y, given the information contained in X 2,…,X k. Least squares: β 1 +β 2 X 2 +…+ β k X k as the conditional mean function. Least absolute deviation: β 1 +β 2 X 2 +…+ β k X k as the conditional median function. Both describe the “averaging” behavior of Y
Other Aspects of Conditional Distribution Conditional variance: volatility models Conditional higher moments Conditional quantiles ( 分位數, 分量 )
Quantile Regression ( 分量迴歸 ) Koenker and Bassett (1978) For 0 < θ < 1, minimize weighted sum of absolute errors: β 1 +β 2 X 2 +…+ β k X k now can characterize the tail behaviors of Y. β 1 …,β k vary with θ ; compare the behaviors with different θ (e.g. θ =0.05 vs. θ =0.95).
Example 1: (Koenker and Hallock, 2001) A study of infant birth weights For a girl born to an unmarried, white mother with below-high-school education Least squares: 3350g 5% quantile: 2500g;95% quantile 4100g Difference between boys and girls Least squares: 100g 5% quantile: 45g;95% quantile: 130g Difference between black and white mothers Least squares: -200g 5% quantile: -330g;95% quantile: -175g
Other Applications Economic growth of bad years Value at Risk ( 涉險值 ) Wage differentials Demand analysis High risk group of an insurance policy Productivity of firms Schooling years
Example 2: Index of Financial Crisis Eichengreen et al. (1996) EMP t = e t + i t - r t Crisis t = 1,if EMP t > + n Quantile Regression EMP on important financial and economic variables Focusing on the quantiles of the right tail
A Summary Least squares and linear models will still prevail. After almost 200 years, we may start paying more attention to other methods. We may now go beyond the models for conditional mean (median) and study aberrant behaviors.
Do we really need a large system of equations? From one variable to many variables Single equation model Simultaneous-equation model (Cowles Commission since 50’s ) Univariate time-series model multivariate model (VAR; Sims, 1980)
Drawbacks of a large system Simultaneous-equation model Model identification Model estimation: Estimating each equation by LS is meaningless VAR Too many parameters Difficult to impose prior restrictions Others Data must be of the same frequency (e.g. quarterly) Model specification error
A large information set but not necessary a large system Include as many economic variables as possible. Extract information from these variables without specifying many models. Adopt data dimension reduction techniques (e.g. principal component analysis).
Example 1: Diffusion Index (Stock and Watson, 1998) NBER Diffusion indexes as dynamic factors Reduce a large information set to a small number of factors. No parametric model is specified. Can handle mixed-frequency data (e.g. monthly and quarterly).