5 Yi =β1+β2 X2i+…+ βk Xki+ ei , i =1,…,N 計量方法的演變Yi =β1+β2 X2i+…+ βk Xki+ ei , i =1,…,NSpecificationlinear model nonlinear model nonparametricsMethodleast squares(least absolute deviation),QMLE,GMM,Beta_1 = Arial+fontsize18+ 下標_I = monotype corsiva+ fontsize18+ 下標computationally intensive methods, etc.Structuresingle equation multiple equations(simultaneous-equation model,VAR)
6 Does the era of linearity and least-squares come to an end? Euler Laplace LegendreLegendre (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 errorsLeast absolute deviation:Boscovich, 1755Minimizing sum of absolute errorsKey: “averaging” the errors
7 Legendre Boscovich http://www-groups.dcs.st-andrews.ac.uk/ The MacTutor History of Mathematics archive | Biographic Index | --search by alphabet for Legendre and Boscovich
8 When a random structure is imposed We are interested in the random behavior of Y, given the information contained in X2,…,Xk.Least squares: β1+β2 X2+…+ βk Xk as the conditional mean function.Least absolute deviation: β1+β2 X2+…+ βk Xk as the conditional median function.Both describe the “averaging” behavior of Y“…“ -- MT Extra
9 Other Aspects of Conditional Distribution Conditional variance: volatility modelsConditional higher momentsConditional quantiles (分位數, 分量)
10 Quantile Regression (分量迴歸) Koenker and Bassett (1978)For 0 < θ < 1, minimize weighted sum of absolute errors:β1+β2 X2+…+ βk Xk now can characterize the tail behaviors of Y.β1…,βk vary with θ; compare the behaviors with different θ (e.g. θ=0.05 vs. θ=0.95).
11 Example 1: (Koenker and Hallock, 2001) A study of infant birth weightsFor a girl born to an unmarried, white mother with below-high-school educationLeast squares: 3350g % quantile: 2500g; 95% quantile 4100gDifference between boys and girlsLeast squares: 100g % quantile: 45g; 95% quantile: 130gDifference between black and white mothersLeast squares: -200g % quantile: -330g; 95% quantile: -175g
12 Other Applications Economic growth of bad years Value at Risk (涉險值) Wage differentialsDemand analysisHigh risk group of an insurance policyProductivity of firmsSchooling years
13 Example 2: Index of Financial Crisis Eichengreen et al. (1996)EMPt = e t + it - rtCrisist = 1, if EMPt > + n Quantile RegressionEMP on important financial and economic variablesFocusing on the quantiles of the right tail
14 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.
15 Do we really need a large system of equations? From one variable to many variablesSingle equation model Simultaneous-equation model (Cowles Commission since 50’s )Univariate time-series model multivariate model (VAR; Sims, 1980)
16 Drawbacks of a large system Simultaneous-equation modelModel identificationModel estimation: Estimating each equation by LS is meaninglessVARToo many parametersDifficult to impose prior restrictionsOthersData must be of the same frequency (e.g. quarterly)Model specification error
17 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).
18 Example 1: Diffusion Index (Stock and Watson, 1998) NBERDiffusion indexes as dynamic factorsReduce 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).