Generalized Extreme-Value Model
McFadden (1978) introduced the generalized extreme-value (GEV) distribution defined by F(€ue2,. . . ,ej (9.3.67) = exp {—(/[exp (-eO, exp (-e2), . . . , exp ( Cm)]), where G …
Heteroscedasticity
Hurd (1979) evaluated the probability limit of the truncated T obit MLE when a certain type of heteroscedasticity is present in two simple truncated Tobit models: (1) the i. i.d. …
Distribution Theory
The theorems listed in this appendix, as well as many other results concerning the distribution of a univariate continuous random variable, can be found in Johnson and Kotz (1970a, b). …
Box-Cox Transformation
Box and Cox (1964) proposed the model3z,(A) = x# + u„ where, forj>, > 0, — 1 Z,(A)== )T if ХФ0 = log y, if A = 0. Note that …
Results of Cosslett: Part I
Cosslett (1981a) proved the consistency and the asymptotic normality of CBMLE in the model where both /and Q are unknown and also proved that CBMLE asymptotically attains the Cramer-Rao lower …
Amemiya’s Least Squares and Generalized Least Squares Estimators
Amemiya (1978c, 1979) proposed a general method of obtaining the estimates of the structural parameters from given reduced-form parameter estimates in general Tobit-type models and derived the asymptotic distribution. Suppose …
Three Error Components Models
6.6.1 Three error components models are defined by = л. i= 1, 2,. . . ,N, (6.6.1) t= 1,2,. . . ,Г, and ий = а + Л, + €й, …
Berkson’s Minimum Chi-Square Method
There are many variations of the minimum chi-square (MIN x1) method, one of which is Berkson’s method. For example, the feasible generalized least squares (FGLS) estimator defined in Section 6.2 …
Standard Tobit Model (Type 1 Tobit Model)
Tobin (1958) noted that the observed relationship between household expenditures on a durable good and household incomes looks like Figure 10.1, Income Figure 10.1 An example of censored data where …
Markov Chain Models
11.1.1 Basic Theory Define a sequence of binary random variables yjit) = 1 if /th person is in state j at time t (11.1. l) = 0 otherwise, /=1,2,. . …
Full Information Maximum Likelihood Estimator
In this section we shall define the maximum likelihood estimator of the parameters of model (7.1.1) obtained by assuming the normality of U, and we shall derive its asymptotic properties …
Universal Logit Model
In Section 9.2.1 we stated that for a binary QR model a given probability function G(xf, в) can be approximated by F[H(xf, в)] by choosing an appropriate H(xf, в) for …
Serial Correlation
Robinson (1982a) proved the strong consistency and the asymptotic normality of the Tobit MLE under very general assumptions about ut (normality is presupposed) and obtained its asymptotic variance-covariance matrix, which …
Classical Least Squares Theory
1. This statement is true only when xf is a scalar. If y, x,, and x2 are scalar dichotomous random variables, we have E{y|x,, x2) = /?0 + fiixi + …
The F Test
In this section we shall consider the test of the null hypothesis Q'fi = c against the alternative hypothesis Q'fi Ф c when it involves more than one constraint (that …
Nonlinear Least Squares Estimator
4.3.1 Definition We shall first present the nonlinear regression model, which is a nonlinear generalization of Model 1 of Chapter 1. The assumptions we shall make are also similar to …
A Useful Transformation for the Calculation of the Feasible Generalized Least Squares Estimator
Let Rt be the matrix obtained by inserting p into the right-hand side of (5.2.10) . Then we have by (5.2.14) fip = (X'ft'&X^X'RJRj. Thus fip can be obtained by …
Canonical Model
Let H be an orthogonal matrix that diagonalizes the matrix X'X, that is, H'H = I and H'X'XH = Л, where Л is the diagonal matrix consisting of the characteristic …
Asymptotic Normality of the Median
Let {Yt), t = 1, 2,. . . , T, be a sequence of i. i.d. random variables with common distribution function Fand density function f. The population median M …
Least Squares as Best Linear Unbiased Estimator (BLUE)
The class of linear estimators of 0 can be defined as those estimators of the form С' у for any TX К constant matrix C. We can further restrict the …
Distribution Function
Definition 3.1.5. The distribution function F(x) of a random variable X(a>) is defined by F(x) = P{o)Х(оз) < x). Note that the distribution function can be defined for any random …
The Almon Lag
Almon (1965) proposed a distributed-lag model yl=='ZPjX,+i-j+v» 1-І in which 0X. . , 0N lie on the curve of a gth-order polynomial; that is, /«, j=l,2,...,N. (5.6.6) ,Y and S=(S0,Sl,. …
A Test of Structural Change when Variances Are Equal Suppose we have two regression regimes
Уі = X] A + u, (1.5.23) and y2 — X202 + u2, where the vectors and matrices in (1.5.23) have T{rows and those in (1.5.24) T2 rows, X, is …
Bootstrap and Jacknife Methods
In this subsection, we shall consider briefly two methods of approximating the distribution of the nonlinear least squares estimator, these methods are called the bootstrap and the jackknife methods (see …
Durbin-Watson Test
In this subsection we shall consider the test of the hypothesis p = 0 in Model 6 with {u,} following AR(1). The Durbin-Watson test statistic, proposed by Durbin and Watson …
Multicollinearity and Principal Components
In Model 1 we assumed that X is of full rank [that is, rank(X) = КШТ], or, equivalently, that X'X is nonsingular. If it is not, X'X cannot be inverted …
Consistency of Least Absolute Deviations Estimator
Consider a classical regression model у = ХД, + u, (4.6.17) where X is a TX К matrix ofbounded constants such that limr_«, Г_1Х'Х is a finite positive-definite matrix and …
Model 1 with Normality
In this section we shall consider Model 1 with the added assumption of the joint normality of u. Because no correlation implies independence under normality, the {w,} are now assumed …
Various Modes of Convergence
In this section, we shall define four modes of convergence for a sequence of random variables and shall state relationships among them in the form of several theorems. Definition 3.2.1 …
Generalized Least Squares Theory
One of the important assumptions of the standard regression model (Model 1) is the assumption that the covariance matrix of the error terms is a scalar times the identity matrix. …
A Test of Structural Change when Variances Are Unequal
In this section we shall remove the assumption <r? = a and shall study how to test the equality of jj, and fi2. The problem is considerably more difficult than …
Tests of Hypotheses
In the process of proving Theorem 4.3.2, we have in effect shown that asymptotically A—A*(G'G)rlG'u, (4.3.28) where we have put G = (df/d/Пд,. Note that (4.3.28) exactly holds in the …
Joint Presence of Lagged Endogenous Variables and Serial Correlation
In this last subsection we shall depart from Model 6 and consider briefly a problem that arises when X in (6.1.1) contains lagged values of y. An example of such …
Advanced Econometrics Takeshi Amemiya
This book is intended both as a reference book for professional econometricians and as a graduate textbook. If it is used as a textbook, the material contained in the book …
Stein’s Estimator: Homoscedastic Case
Let us consider a special case of the canonical model (2.2.5) in which Л = I. James and Stein (1961) showed that £l|[l — c(d'd)_1]d — all2 is minimized for …
