... shapes The RCS of real aircraft must be measured It varies significantly depending upon the direction ofthe illuminating radar 4-11.2 Figure shows a typical RCS plot of a jet aircraft The plot ... aspect (increasing reflectivity) The next highest RCS area is the nose/tail area, largely because of reflections off the engines or propellers Most self-protection jammers cover a field of view of ... example is 100 m2 inthe beam, and the weakest is slightly more than m2 inthe 135E/225E positions These RCS values can be very misleading because other factors may affect the results For example,...
... denote the position ofthe centroid ofthe plane lamina of Figure 8.2 At the centroid the moment of area is zero, so thatthe following equations apply Z x dA = Z y d A = dA = elemental area ofthe ... as theperpendicular axes theorem which states thatthe sum ofthe second moments of area of two mutually perpendicular axes of a lamina is equal to the polar second moment of area about a point ... second moment of area about x-x In = the second moment of area about X-X The importance ofthe parallel axes theorem is that it is useful for calculating second moments of area of sections of RSJs,...
... denotes the length ofthe sliding window and it is the sum ofthe number of detections (N), and the number of misses (M) inthe window Note thatthe function (14) corresponds to an average ofthe ... number of detections inthe window On the contrary, the window size of ML(N = n) can change to retain n detections inthe window The average SNR, SNR, was set to a constant having a value of 8, ... estimates of σ that are obtained based on the observations ofthe detections and misses inthe window To be more specific, (zi − 1)/αi inthe first term of (14) is an (one-sample) unbiased estimate of...
... respectively, R0 is the intersection point ofthe circles and X-axes The motion ofthe charged particle inthe plane of cross section is localized inthe dashed area as it is shown in Fig 1b Inthe strong ... by the fact thatthe Taylor expansion of potential (17) for small values ofthe x-coordinate is parabolic as it is the case for (12) also On the other hand, at higher values ofthe ¨ x-coordinate ... quantum number When the coordinate ofthe ‘‘slow’’ subsystem, x, is fixed, the motion ofthe electron is localized inthe onedimensional effective potential well, having the following spatial profile:...
... described inthe text; others cover new ideas that can be analyzed using the tools presented inthe current and previous chapters Several ofthe problems require using the data sets that are included ... basis The book contains proofs or outlines the proofs of many assertions, focusing on the role played by the assumptions with economic content while downplaying or ignoring regularity conditions The ... Models Introduction The Linear Probability Model for Binary Response Index Models for Binary Response: Probit and Logit Maximum Likelihood Estimation of Binary Response Index Models Testing in Binary...
... studying the e¤ect of w on the expected value of y The reason we control for these variables is that we think w is correlated with other factorsthat also in uence y If w is continuous, interest ... regardless of how they are related to the xi Assuming thatthe population mean ofthe error is zero is without loss of generality when an intercept is included inthe model Thus, the statement ‘ The ... binary variable indicating marital status The variable u, called the error term or disturbance, contains unobserved factorsthat a¤ect the wage o¤er Interest lies inthe unknown parameters, the...
... treated inthe same framework because they are linear inthe parameters bj The fact that equation (2.3) is nonlinear in x has important implications for interpreting the bj , but not for estimating ... density of y given x by integration, summation, or a combination ofthe two (depending on the nature of y) It follows thatthe conditional expectation operator has the same linearity properties as the ... for relating the two expectations Obtaining E½m1 ðx; zÞ j x generally requires integrating (or summing) m1 ðx; zÞ against the conditional density of z given x, but in many cases the form of Eðy...
... find the limiting distribution of many interesting functions of xN This is especially useful for determining the asymptotic distribution of test statistics once the limiting distribution of an ... continuous function, then gðxN Þ ! gðxÞ The usefulness of Lemma 3.6, which is called the continuous mapping theorem, cannot be overstated It tells us that once we know the limiting distribution of ... continuous at plim xN Slutsky’s theorem is perhaps the most useful feature ofthe plim operator: it shows thatthe plim passes through nonlinear functions, provided they are continuous The expectations...
... e¤ects ofthe observed explanatory variables holding the other explanatory variables constant, including the unobservable q Inthe context of this additive model, there is no point in allowing for ... context of models that are linear inthe parameters under random sampling, identi- The Single-Equation Linear Model and OLS Estimation 53 fication of b simply means that b can be written in terms of ... correlated Of course, these are just the findings from one sample Adding IQ explains only one percentage point more ofthe variation in logðwageÞ, and the equation predicts that 15 more IQ points (one...
... define the linear combination of interest, say y a1 b1 þ a2 b2 þ Á Á Á þ aK bK , and then to write one ofthe bj in terms of y and the other elements of b Then, substitute into the equation of interest ... class of all instrumental variables estimators using instruments linear in z ^ ~ Proof: Let b be the 2SLS estimator, and let b be any other IV estimator using ~ ~ instruments linear in z Let the instruments ... violated (This reasoning assumes that we are interested in estimating the pure e¤ect of serving in Vietnam, as opposed to including indirect e¤ects such as reduced job training.) Instrumental Variables...
... industry, and type of injury These allow for the fact thatthe kind of people and type of injuries di¤er systematically inthe two years Perhaps not surprisingly, controlling for these factors has little ... from the incinerator Is this regression appropriate for determining the causal e¤ects of incinerator on housing prices? Explain b Pooling the two years of data, consider the model Additional Single-Equation ... generated instruments The tools needed to make the proof rigorous are introduced in Chapter 12, but the key components ofthe proof can be given here inthe context ofthe linear model Write the model...
... use a SUR software routine in obtaining the estimates: we may be interested in testing joint hypotheses involving parameters ^ in di¤erent equations In order to so we need to estimate the variance ... important consequences The most important of these is that, for performing asymptotic inference about b using ^ ^ b , we not have to worry that W is an estimator of W Of course, whether the asymptotic ... tending to in nity Inthe household demand example, we are interested in a set of three demand functions, and the unit of obser- Estimating Systems of Equations by OLS and GLS 145 vation is the...
... given in expression (8.26) The question is, Can we say that using the full set of instruments (with the optimal weighting matrix) is better than using the reduced set of instruments (with the optimal ... means that z and nonlinear functions of z are valid instruments in every equation 206 Chapter a Suppose that Eðxg j zÞ is linear in z for all g Show that adding nonlinear functions of z to the instrument ... obtained—by estimating the linear projection of xiK on zi inthe first stage—it makes perfectly good sense that zi2 can be omitted under condition (8.54) without a¤ecting e‰ciency of 2SLS In the...
... estimation In plugging fitted values into equation (9.61), our mistake is in thinking thatthe linear projection ofthe square is the square ofthe linear projection What the 2SLS estimator does inthe ... using the usual rank condition Adding equation (9.58) to the original system and then studying the rank condition ofthe first two equations is equivalent to studying the rank condition inthe ... by 2SLS The bottom line is thatthe methods studied in Chapters and are directly applicable All ofthe tests we have covered apply, including the tests of overidentifying restrictions in Chapters...
... appearing inthe structural equation to obtain the estimating equation, including any binary indicators indicating participation inthe program The estimates should be interpreted inthe orginal ... Estimation ofthe E¤ects of Job Training Grants): We now use the data in JTRAIN1.RAW to estimate the e¤ect of job training grants on firm scrap rates, using a random e¤ects analysis There are 54 firms that ... kinds of serial correlation inthe composite error vit , and so a rejection ofthe null should not be interpreted as implying thatthe random e¤ects error structure must be true Finding that the...
... e¤ect ci contains cityspecific unobserved factorsthat can a¤ect sexual conduct, as well as the incidence of HIV Equation (11.8) is one way of capturing the fact thatthe spread of HIV is in uenced ... heteroskedasticity Inthe pure AR(1) model, using lags of yit as an instrument for Dyi; tÀ1 means that we are assuming the AR(1) model captures all ofthe dynamics If further lags of yit are added to the structural ... =sx à being important in determining the size ofthe measurement error bias, the à ratio ð1 À rr Þ=ð1 À rx Ã Þ is also important As the autocorrelation in xit increases rel^ increases In fact,...
... We introduce the subscript ‘‘o’’ on theta to distinguish the parameter vector appearing in Eðy j xÞ from other candidates for that vector (Often, the value yo is called ‘ the true value of theta,’’ ... assume that lo is inthe interior of its parameter space, L, under H0 We also assume that d is twice continuously di¤erentiable on the interior of L ~ Let l be the solution to the constrained minimization ... results The general derivation ofthe LM statistic also assumed that y o A intðYÞ under H0 Nevertheless, for certain applications ofthe LM test we can drop the requirement that y o is inthe interior...
... to assume the regularity conditions there; in particular, we assume that yo is inthe interior of Y, and li ðy Þ is twice continuously di¤erentiable on the interior of Y The score ofthe log likelihood ... nðdzÞ Then ð pðy j xÞ ¼ pðy j x; zÞpðz j xÞnðdzÞ Z In other words, we can obtain the density of y given x by integrating the density of y given the larger conditioning set, ðx; zÞ, against the ... elements of any ofthe matrices We discussed each of these estimators inthe general M-estimator case in Chapter 12, but a brief review is in order The first estimator, based on the Hessian of the...
... McFadden (1994, Theorem 3.4): theorem 14.2 (Asymptotic Normality of GMM): In addition to the assumptions in Theorem 14.1, assume that (a) yo is inthe interior of Y; (b) gðw; ÁÞ is continuously di¤erentiable ... has a limit2 ing wQ distribution 14.3 Systems of Nonlinear Equations A leading application ofthe results in Section 14.2 is to estimation ofthe parameters in an implicit set of nonlinear equations, ... asymptotic) version ofthe Gauss-Markov theorem from linear regression analysis: while the Gauss-Markov theorem states that OLS has the smallest variance inthe class of linear, unbiased estimators,...
... issue of fitted values being outside the unit interval, the LPM implies that a ceteris paribus unit increase in xj always changes Pð y ¼ j xÞ by the same amount, regardless ofthe initial value of ... means that if nonwife income increases by 10 ($10,000), the probability of being inthe labor force is predicted to fall by 034 This is a small e¤ect given that an increase in income by $10,000 in ... xj If x contains nonlinear functions of some explanatory variables, such as natural logs or quadratics, there is the issue of using the log ofthe average versus the average ofthe log (and similarly...