Investment

二项分布概率The distribution of random errors, ut, is not normal, 1Logit Model ,[pic 1]odds ratio Probit Model, [pic 2][pic 3][pic 4]cumulative distribution function (cdf) for the standard normal distribution, where (z) is the standard normal probability density function (pdf), Estimation of Binary Choice Models: Weighted Least Squares (WLS) [pic 5][pic 6]

 and Maximum Likelihood (ML) Estimation Consistent. plim = Asymptotically normally distributed.   asymp ~ N Asymptotically efficient.It has the smallest variance among the group of consistent and asymptotically normally (CAN) distributed estimators. Goodness-of-fit McFadden’s R2 (likelihood ratio index)  = 1 - [ logL(^) / logL(0) ] logL(0) the value of the maximised log likelihood function with an intercept term only logL(^)  all independent variables+ intercept (R^2 = 1) never happen.  [pic 13][pic 14][pic 15][pic 16][pic 7][pic 8][pic 9][pic 10][pic 11][pic 12]

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If the transformation function F(It) is a proper cdf, then even with many regressors,the model cannot fit perfectly unless It explodes to + or -. n1=n11,y=1,y=1,n10,y=1,predictedy=0, Likelihood Ratio (LR) test 时间[pic 23]ARDL(1,1) model can be rewritten as a DL(∞).Deterministic processyt = δ0 + δ1t Stationarity of a stochastic process Yt = β0 + β1Yt−1 + εtNonstationary ProcessesTrend Stationary: Yt = 10 + 0.3t + et (linear deterministic trend) Yt = 10 + 0.001t + 0.005t2 + et (quadratic deterministic trend)(nonstationary) Difference Stationary (stochastic trend):All processes, Yt, listed below are integrated of order 1, or simply I(1)(meaning that they have one unit root). Random walk: Yt = Yt−1 + et Breusch‐Godfrey (LM) test for autocorrelation 一.Regression model: yt = 1 +2)xt + 3)zt + et, 二When autocorrelation of order p is suspected et = 1)et−1 + 2)et−2 + ... +et−p + vt 三The auxiliary regression et^ = a1 + a2xt + a3zt + 1)et−1^+ 2)et−2^ + ... +et−p^ + vt 四The hypotheses H0: 1 = 2 = ... = = 0 (no autocorrelation) H1: At least one of the p ’s ≠ 0 (autocorrelation of order up to p) 五The test statistic and its distribution under H0:T× asymptotically ~ χ2(p)when H0 is true. The DurbinWatson (DW) test for a finite sample test H0:= 0 (no autocorreation) H1: > 0 (positive aut.) {or < 0 (negative aut.)} where  is the coefficient in the first‐order autocorrelation model: et =et−1 + vt, d ≈ 2(1 − ^) where^ is the OLS estimate of . [pic 46] Limitations DW Invalid if Use Durbin h test if the stochastic variable is yt−1, or use BG test.) Cannot test higher order autocorrelation. (Use BG test.)Invalid if the regression model does not have an intercept term. (Use BG test.) The Durbin h test limitation The regression model may have multiple lagged dependent variables (e.g., yt−1, yt−2, etc) on the right hand side, but only the variance of the OLS estimator for the coefficient of yt−1 is used in h. If n times V(b2) is greater than 1, the test will not be applicable. Valid in large samples. It has lower power than the BG test in both large and small samples 补救The Cochrane Orcutt iterative estimator [pic 47][pic 48][pic 49][pic 50][pic 51][pic 52][pic 53][pic 54][pic 55][pic 56][pic 20][pic 21][pic 22][pic 24][pic 25][pic 26][pic 27][pic 28][pic 29][pic 30][pic 31][pic 32][pic 33][pic 34][pic 35][pic 36][pic 37][pic 38][pic 39][pic 40][pic 41][pic 42][pic 43][pic 44][pic 45]

Difference‐stationary variables If all variables are I(1), then the estimation results are either (i) spurious and a differenced model is appropriate: , DICKEY-FULLER (DF) FOR UNIT ROOT Test equations: [pic 57][pic 58][pic 59][pic 60][pic 61][pic 62]

The Hypotheses: H0 : a1 = 0 (unit root → difference stationary) H1 : a1 < 0 (no unit root → stationary [or trend‐stationary]) Test Statistic: If the absolute value of t ratio for the a1 estimate is greater than the critical value (i.e., the t ratio is sufficiently negative), then the process is implied to be “stationary”. The observed value of the test n=多少COINTEGRATION interest rates on assets of different maturities; The Engle-Granger (EG) Approach: single static equation model (The residual-based approach to cointegration) Consider, for simplicity, a case of two I(1) variables y and x. If y and x are cointegrated, there exists a linear combination of y and x (plus some other deterministic terms such as t) is I(0). et = yt-a-xt; yt = a + xt + et Estimate the cointegrating relationship by OLS and obtain the residuals, et.    [pic 66][pic 67][pic 63][pic 64][pic 65]