A72 Idempotent Quadratic Forms
Quadratic forms in idempotent matrices play an important role in the distributions of many test statistics. As such, we shall encounter them fairly often. Two central results are of interest. Every symmetric idempotent matrix is nonnegative definite. A-116 Proof All roots are one or zero hence, the matrix is nonnegative definite by definition. Combining this with some earlier results yields a result used in determining the sampling distribution of most of the standard test statistics. If A is...
. Compute The Multiple Regression Of Per Capita Consumption Of Gasoline G Pop
The true model underlying these data is y xi x2 X3 e. a. Compute the simple correlations among the regressors. b. Compute the ordinary least squares coefficients in the regression of y on a constant xi,x2, and X3. c. Compute the ordinary least squares coefficients in the regression of y on a constant x and x2, on a constant x and x , and on a constant x2 and x . d. Compute the variance inflation factor associated with each variable. e. The regressors are obviously collinear. Which is the...
Summary Of Testing Procedures
The preceding has examined several testing procedures for locating autocorrelation in the disturbances. In all cases, the procedure examines the least squares residuals. We can summarize the procedures as follows LM Test LM TR2 in a regression of the least squares residuals on x , et- , lt ',- gt . Reject Hq if LM gt xl P - This test examines the covariance of the residuals with lagged values, controlling for the intervening effect of the independent variables. Q Test Q T T - 2 Yfj i r T - ' ....
B114 Quadratic Forms In A Standard Normal Vector
The earlier discussion of the chi-squared distribution gives the distribution of x'x if x has a standard normal distribution. It follows from A-36 that x'x Y xf Y Xi f 2 n 2' B-104 We know from B-32 that x'x has a chi-squared distribution. It seems natural, therefore, to invoke B-34 for the two parts on the right-hand side of B-104 . It is not yet obvious, however, that either of the two terms has a chi-squared distribution or that the two terms are independent, as required. To show these...
E52 The Standard Normal Cumulative Distribution Function
The standard normal cumulative distribution function cdf is ubiquitous in econometric models. Yet this most homely of applications must be computed by approximation. There are a number of ways to do so.11 Recall that what we desire is 4Hi dt, where lt p t e'2 2. oo V2jt One way to proceed is to use a Taylor series The normal cdf has some advantages for this approach. First, the derivatives are simple and not integrals. Second, the function is analytic as M gt oo, the approximation converges to...
B111 Marginal And Conditional Normal Distributions
Let Xi be any subset of the variables, including a single variable, and let x2 be the remaining variables. Partition l and X likewise so that Then the marginal distributions are also normal. In particular, we have the following theorem. THEOREM B.7 Marginal and Conditional Normal Distributions If xi, x2 have a joint multivariate normal distribution, then the marginal distributions are 8This result is obtained by constructing A, the diagonal matrix with cr, as its ith diagonal element. Then, R...
Partitioned Regression And Partial Regression
It is common to specify a multiple regression model when, in fact, interest centers on only one or a subset of the full set of variables. Consider the earnings equation discussed in Example 2.2. Although we are primarily interested in the association of earnings and education, age is, of necessity, included in the model. The question we consider here is what computations are involved in obtaining, in isolation, the coefficients of a subset of the variables in a multiple regression for example,...
Identification Through Covariance
RESTRICTIONS THE FULLY RECURSIVE MODEL The observant reader will have noticed that no mention of X is made in the preceding discussion. To this point, all the information provided by it is used in the estimation of X for given T, the relationship between S2 and X is one-to-one. Recall that X T'ftr. But if restrictions are placed on X, then there is more information in il than is needed for estimation of X. The excess information can be used instead to help infer the elements l3The analysis is...
Info Mwx
a. Compute the FGLS estimates of fa and fa. b. Test the hypothesis that fa fa. c. Compute the maximum likelihood estimates of the model parameters. d. Use the likelihood ratio test to test the hypothesis in part b. generalized least squares is equivalent to equation-by-equation ordinary least squares if Xi X2. Does your result hold if it is also known that Pi J2 5. Consider the two-equation system Assume that the disturbance variances and covariance are known. Now suppose that the analyst of...
Limited Information Maximum Likelihood And The K Class Of Estimators
The limited information maximum likelihood LIML estimator is based on a single equation under the assumption of normally distributed disturbances LIML is efficient among single-equation estimators. A full lengthy derivation of the log-likelihood is provided in Theil 1971 and Davidson and MacKinnon 1993 . We will proceed to the practical aspects of this estimator and refer the reader to these sources for the background formalities. A result that emerges from the derivation is that the LIML...
Info Gwl
x w hy,J v H'iHv,J v H'y,J. 13-41 The estimator of the asymptotic covariance matrix for S is the inverse matrix in brackets. 30In some treatments e.g., Blundell and Bond 1998 , an additional condition is assumed for the initial value, y,-0, namely E y,o exogenous data o- This would add a row at the top of the matrix in 13-38 containing gt 0-mo , 0,0 , The remaining loose end is how to obtain the consistent estimator of S to compute I. Recall that the GMM estimator is consistent with any...
Full Rank
Assumption 2 is that there are no exact linear relationships among the variables. Assumption X is an n x K matrix with rank K. 2-5 Hence, X has full column rank the columns of X are linearly independent and there are at least K observations. See A-42 and the surrounding text. This assumption is known as an identification condition. To see the need for this assumption, consider an example. Example 2.5 Short Rank Suppose that a cross-section model specifies C fa 1S2 nonlabor income jS3 salary fa...
Summary And Conclusions Olt
This chapter has discussed the functional form of the regression model. We examined the use of dummy variables and other transformations to build nonlinearity into the model. We then considered other nonlinear models in which the parameters of the nonlinear model could be recovered from estimates obtained for a linear regression. The final sections of the chapter described hypothesis tests designed to reveal whether the assumed model had changed during the sample period, or was different for...
The Implications Of Stochastic Regressors
The preceding analysis is done conditionally on the observed data. A convenient method of obtaining the unconditional statistical properties of b is to obtain the desired results conditioned on X first, then find the unconditional result by averaging e.g., by integrating over the conditional distributions. The crux of the argument is that if we can establish unbiasedness conditionally on an arbitrary X. then we can average over X's to obtain an unconditional result. We have already used this...
Spherical Disturbances
The fourth assumption concerns the variances and covariances of the disturbances Var 1X a2, for all i 1, , n, C.ov e,, s j X 0, for all i j. Constant variance is labeled homoscedasticity. Consider a model that describes the profits of firms in an industry as a function of, say, size. Even accounting for size, measured in dollar terms, the profits of large firms will exhibit greater variation than those of smaller firms. The homoscedasticity assumption would be inappropriate here. Also, survey...