The Regression Equation

Chapter 16 Markov processes and the Kolmogorov equations 16.1 Stochastic Differential Equations Consider the stochastic differential equation: dX t= at; X t dt +  t; X t dB t: (SDE) Here at; x and  t; x are given functions, usually assumed to be continuous in t; x and Lips- chitz continuous in x ,i.e., there is a constant L such that jat; x , at; y j Ljx,yj; jt; x , t; y j Ljx,yj for all t; x; y . Let t 0 ;x be given. A solution to (SDE) with the initial condition t 0 ;x is a process fX tg tt 0 satisfying X t 0 =x; X t= Xt 0 + t Z t 0 as; X s ds + t Z t 0 s; X s dB s; t  t 0 The solution process fX tg tt 0 will be adapted to the filtration fF tg t0 generated by the Brow- nian motion. If you know the path of the Brownian motion up to time t , then you can evaluate X t . Example 16.1 (Drifted Brownian motion) Let a be a constant and  =1 ,so dX t=adt+dB t: If t 0 ;x is given and we start with the initial condition X t 0 =x; 177 178 then X t=x+at,t 0 +Bt,Bt 0 ; t  t 0 : To compute the differential w.r.t. t , treat t 0 and B t 0  as constants: dX t=adt+dB t: Example 16.2 (Geometric Brownian motion) Let r and  be constants. Consider dX t=rX t dt + X t dB t: Given the initial condition X t 0 =x; the solution is X t=xexp   B t , B t 0  + r , 1 2  2 t , t 0   : Again, to compute the differential w.r.t. t , treat t 0 and B t 0  as constants: dX t=r, 1 2  2 Xtdt + Xt dB t+ 1 2  2 Xt dt = rX t dt + X t dB t: 16.2 Markov Property Let 0  t 0 t 1 be given and let hy  be a function. Denote by IE t 0 ;x hX t 1  the expectation of hX t 1  , given that X t 0 =x .Nowlet  2 IR be given, and start with initial condition X 0 = : We have the Markov property IE 0;  hX t 1      F t 0   = IE t 0 ;X t 0  hX t 1 : In other words, if you observe the path of the driving Brownian motion from time 0 to time t 0 ,and based on this information, you want to estimate hX t 1  , the only relevant information is the value of X t 0  . You imagine starting the SDE at time t 0 at value X t 0  , and compute the expected value of hX t 1  . CHAPTER 16. Markov processes and the Kolmogorov equations 179 16.3 Transition density Denote by pt 0 ;t 1 ; x; y  the density (in the y variable) of X t 1  , conditioned on X t 0 =x .Inotherwords, IE t 0 ;x hX t 1  = Z IR hy pt 0 ;t 1 ; x; y  dy : The Markov property says that for 0  t 0  t 1 and for every  , IE 0;  hX t 1      F t 0   = Z IR hy pt 0 ;t 1 ; Xt 0 ;y dy : Example 16.3 (Drifted Brownian motion) Consider the SDE dX t=adt+dB t: Conditioned on X t 0 =x , the random variable X t 1  is normal with mean x + at 1 , t 0  and variance t 1 , t 0  , i.e., pt 0 ;t 1 ; x; y= 1 p 2t 1 ,t 0  exp  , y , x + at 1 , t 0  2 2t 1 , t 0   : Note that p depends on t 0 and t 1 only through their difference t 1 , t 0 . This is always the case when at; x and  t; x don’t depend on t . Example 16.4 (Geometric Brownian motion) Recall that the solution to the SDE dX t=rX t dt + X t dB t; with initial condition X t 0 =x , is Geometric Brownian motion: X t 1 =xexp  B t 1  , B t 0  + r , 1 2  2 t 1 , t 0   : The random variable B t 1  , B t 0  has density IP fB t 1  , B t 0  2 dbg = 1 p 2t 1 , t 0  exp  , b 2 2t 1 , t 0   db; and we are making the change of variable y = x exp  b +r, 1 2  2 t 1 , t 0   or equivalently, b = 1  The Regression Equation The Regression Equation By: OpenStaxCollege Data rarely fit a straight line exactly Usually, you must be satisfied with rough predictions Typically, you have a set of data whose scatter plot appears to "fit" a straight line This is called a Line of Best Fit or Least-Squares Line Collaborative Exercise If you know a person's pinky (smallest) finger length, you think you could predict that person's height? Collect data from your class (pinky finger length, in inches) The independent variable, x, is pinky finger length and the dependent variable, y, is height For each set of data, plot the points on graph paper Make your graph big enough and use a ruler Then "by eye" draw a line that appears to "fit" the data For your line, pick two convenient points and use them to find the slope of the line Find the y-intercept of the line by extending your line so it crosses the y-axis Using the slopes and the yintercepts, write your equation of "best fit." Do you think everyone will have the same equation? Why or why not? According to your equation, what is the predicted height for a pinky length of 2.5 inches? A random sample of 11 statistics students produced the following data, where x is the third exam score out of 80, and y is the final exam score out of 200 Can you predict the final exam score of a random student if you know the third exam score? x (third exam score) y (final exam score) 65 175 67 133 71 185 71 163 66 126 75 198 67 153 1/11 The Regression Equation x (third exam score) y (final exam score) 70 163 71 159 69 151 69 159 Table showing the scores on the final exam based on scores from the third exam Scatter plot showing the scores on the final exam based on scores from the third exam Try It SCUBA divers have maximum dive times they cannot exceed when going to different depths The data in [link] show different depths with the maximum dive times in minutes Use your calculator to find the least squares regression line and predict the maximum dive time for 110 feet X (depth in feet) Y (maximum dive time) 50 80 60 55 70 45 80 35 90 25 100 22 ŷ = 127.24 – 1.11x 2/11 The Regression Equation At 110 feet, a diver could dive for only five minutes The third exam score, x, is the independent variable and the final exam score, y, is the dependent variable We will plot a regression line that best "fits" the data If each of you were to fit a line "by eye," you would draw different lines We can use what is called a least-squares regression line to obtain the best fit line Consider the following diagram Each point of data is of the the form (x, y) and each point ofthe line of best fit using least-squares linear regression has the form (x, ŷ) The ŷ is read "y hat" and is the estimated value of y It is the value of y obtained using the regression line It is not generally equal to y from data The term y0 – ŷ0 = ε0 is called the "error" or residual It is not an error in the sense of a mistake The absolute value of a residual measures the vertical distance between the actual value of y and the estimated value of y In other words, it measures the vertical distance between the actual data point and the predicted point on the line If the observed data point lies above the line, the residual is positive, and the line underestimates the actual data value for y If the observed data point lies below the line, the residual is negative, and the line overestimates that actual data value for y In the diagram in [link], y0 – ŷ0 = ε0 is the residual for the point shown Here the point lies above the line and the residual is positive ε = the Greek letter epsilon For each data point, you can calculate the residuals or errors, yi - ŷi = εi for i = 1, 2, 3, , 11 Each |ε| is a vertical distance 3/11 The Regression Equation For the example about the third exam scores and the final exam scores for the 11 statistics students, there are 11 data points Therefore, there are 11 ε values If yousquare each ε and add, you get 11 (ε1)2 + (ε2)2 + + (ε11)2 = Σ ε2 i = This is called the Sum of Squared Errors (SSE) Using calculus, you can determine the values of a and b that make the SSE a minimum When you make the SSE a minimum, you have determined the points that are on the line of best fit It turns out that the line of best fit has the equation: ^ y = a + bx ¯ ¯ where a = y − bx and b = ¯ ¯ Σ(x − x)(y − y) ¯2 Σ(x − x) ¯ ¯ The sample means of the x values and the y values are x and y, respectively The best fit ¯ ¯ line always passes through the point (x, y) The slope b can be written as b = r ( ) where sy = the standard deviation of the y values sy sx and sx = the standard deviation of the x values r is the correlation coefficient, which is discussed in the next section Least Squares Criteria for Best Fit The process of fitting the best-fit line is called linear regression The idea behind finding the best-fit line is based on the assumption that the data are scattered about a straight line The criteria for the best fit line is that ... EXACT SMALL SAMPLE THEORY IN THE SIMULTANEOUS EQUATIONS MODEL Chapter 8 EXACT SMALL SAMPLE THEORY IN THE SIMULTANEOUS EQUATIONS MODEL P. C. B. PHILLIPS* Yale University Contents 1. Introduction 451 2. Simple mechanics of distribution theory 454 2. I. Primitive exact relations and useful inversion formulae 454 2.2. Approach via sample moments of the data 455 2.3. Asymptotic expansions and approximations 457 2.4. The Wishart distribution and related issues 459 3. Exact theory in the simultaneous equations model 463 3.1. 3.2. 3.3. 3.4. 3.5. 3.6. 3.1. 3.8. 3.9. 3.10. 3.11. 3.12. The model and notation Generic statistical forms of common single equation estimators The standardizing transformations The analysis of leading cases The exact distribution of the IV estimator in the general single equation case The case of two endogenous variables Structural variance estimators Test statistics Systems estimators and reduced-form coefficients Improved estimation of structural coefficients Supplementary results on moments Misspecification 463 464 467 469 472 478 482 484 490 497 499 501 *The present chapter is an abridgement of a longer work that contains inter nlia a fuller exposition and detailed proofs of results that are surveyed herein. Readers who may benefit from this greater degree of detail may wish to consult the longer work itself in Phillips (1982e). My warmest thanks go to Deborah Blood, Jerry Hausmann, Esfandiar Maasoumi, and Peter Reiss for their comments on a preliminary draft, to Glena Ames and Lydia Zimmerman for skill and effort in preparing the typescript under a tight schedule, and to the National Science Foundation for research support under grant number SES 800757 1. Handbook of Econometrics, Volume I, Edited by Z. Griliches and M.D. Intriligator 0 North-Holland Publishing Company, 1983 P. C. B. Phillips 4. A new approach to small sample theory 4.1 Intuitive ideas 4.2. Rational approximation 4.3. Curve fitting or constructive functional approximation? 5. Concluding remarks References 504 504 505 507 508 510 Ch. 8: Exact Small Sample Theoty 451 Little experience is sufficient to show that the traditional machinery of statistical processes is wholly unsuited to the needs of practical research. Not only does it take a cannon to shoot a sparrow, but it misses the sparrow! The elaborate mechanism built on the theory of infinitely large samples is not accurate enough for simple laboratory data. Only by systematically tackling small sample problems on their merits does it seem possible to apply accurate tests to practical data. Such at least has been the aim of this book. [From the Preface to the First Edition of R. A. Fisher (1925).] 1. Introduction Statistical procedures of estimation and inference are most frequently justified in econometric work on the basis of certain desirable asymptotic properties. One estimation procedure may, for example, be selected over another because it is known to provide consistent and asymptotically efficient parameter estimates under certain stochastic environments. Or, a statistical test may be preferred because it is known to be asymptotically most powerful for certain local alterna- tive hypotheses.’ Empirical investigators have, in particular, relied heavily on asymptotic theory to guide their choice of estimator, provide standard errors of their estimates and construct critical regions for their statistical tests. Such a heavy reliance on asymptotic theory can and does lead to serious problems of bias and Annals of Mathematics Propagation of singularities for the wave equation on manifolds with corners By Andr_as Vasy* Annals of Mathematics, 168 (2008), 749–812 Propagation of singularities for the wave equation on manifolds with corners By Andr ´ as Vasy* Abstract In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on C ∞ manifolds with corners M equipped with a Rie- mannian metric g. That is, for X = M ×R t , P = D 2 t −∆ M , and u ∈ H 1 loc (X) solving P u = 0 with homogeneous Dirichlet or Neumann boundary condi- tions, we show that WF b (u) is a union of maximally extended generalized broken bicharacteristics. This result is a C ∞ counterpart of Lebeau’s results for the propagation of analytic singularities on real analytic manifolds with appropriately stratified boundary, [11]. Our methods rely on b-microlocal pos- itive commutator estimates, thus providing a new proof for the propagation of singularities at hyperbolic points even if M has a smooth boundary (and no corners). 1. Introduction In this paper we describe the propagation of C ∞ and Sobolev singularities for the wave equation on a manifold with corners M equipped with a smooth Riemannian metric g. We first recall the basic definitions from [12], and refer to [20, §2] as a more accessible reference. Thus, a tied (or t-) manifold with corners X of dimension n is a paracompact Hausdorff topological space with a C ∞ structure with corners. The latter simply means that the local coordi- nate charts map into [0, ∞) k × R n−k rather than into R n . Here k varies with the coordinate chart. We write ∂  X for the set of points p ∈ X such that in any local coordinates φ = (φ 1 , . . . , φ k , φ k+1 , . . . , φ n ) near p, with k as above, precisely  of the first k coordinate functions vanish at φ(p). We usually write such local coordinates as (x 1 , . . . , x k , y 1 , . . . , y n−k ). A boundary face of codi- mension  is the closure of a connected component of ∂  X. A boundary face of codimension 1 is called a boundary hypersurface. A manifold with corners is a tied manifold with corners such that all boundary hypersurfaces are embedded submanifolds. This implies the existence of global defining functions ρ H for *This work is partially supported by NSF grant #DMS-0201092, a fellowship from the Alfred P. Sloan Foundation and a Clay Research Fellowship. 750 ANDR ´ AS VASY each boundary hypersurface H (so that ρ H ∈ C ∞ (X), ρ H ≥ 0, ρ H vanishes exactly on H and dρ H = 0 on H); in each local coordinate chart intersecting H we may take one of the x j ’s (j = 1, . . . , k) to be ρ H . While our results are local, and hence hold for t-manifolds with corners, it is convenient to use the embeddedness occasionally to avoid overburdening the notation. Moreover, in a given coordinate system, we often write H j for the boundary hypersurface whose restriction to the given coordinate patch is given by x j = 0, so that the notation H j depends on a particular coordinate system having been chosen (but we usually ignore this point). If X is a manifold with corners, X ◦ denotes its interior, which is thus a C ∞ manifold (without boundary). Returning to the wave equation, let M be a manifold with corners equipped with a smooth Riemannian metric g. Let ∆ = ∆ g be the positive Laplacian of g, let X = M ×R t , P = D 2 t −∆, and consider Title: External Debt and Economic Growth Relationship Using the Simultaneous Equations JEL Classification: F34, C32, H63 List Of Keywords: Turkey, External Debt, Economic Growth Simultaneous Equations ERDAL KARAGOL UNIVERSITY OF BALIKESIR E mail: erdalkaragol@hotmail.com ABSTRACT This study will examine the interaction among economic growth, external debt service and capital inflow using time series data for Turkey and using a multi-equation model.The results show that the relationship between debt service and economic growth should be analysed with a simultaneous equation model, because there is a two-way relationship between debt service and growth. The rise in the debt-servicing ratio adversely affects economic growth whereas the decrease in the rate of growth, reduces the ability of an economy to service its debt. When Turkey is servicing its debt, debt servicing could impair economic growth. Servicing a heavy debt may exacerbate the debt problem. In order to service its debt, Turkey had to borrow more. The higher the lagged debt stock the higher the debt service. This result is consistent with the Turkish experience which shows the existence of two way relationships between total debt stock and debt service. External Debt and Economic Growth -1- Relationship Using the Simultaneous Equations 1.Introduction The relationship between external debt, economic growth and capital inflows can become complicated for several reasons. Firstly, there is a relationship between external debt servicing and economic growth. Secondly, government policies designed to influence the balance of payments, domestic interest rates and employment may affect the stock of foreign debt and hence, debt servicing and economic growth both directly and indirectly through their effects on exports, domestic savings and foreign capital inflows. Thirdly, there may be a two way relationship between debt stock and debt servicing. Finally, long term capital inflows, depending on its characteristics may also affect economic growth, investment and debt stock. Moreover, capital inflows could be affected by economic growth. Statistical methods for systems of simultaneous equations capture the mutual dependence among the variables in the model. Techniques in which equations are estimated one at a time are called limited information methods. Full information methods are those where all equations are estimated at the same time. Limited information methods do not take into account connections among variables from different equations within the system. Full information methods allow for these connections. Since all available information is incorparated, this produces more efficient parameter estimation. The three Stages-Least-Squares (3SLS) method used in this paper is a full information method. Variables in the system are categorized as endogenous and exogenous. Simultaneity within the model arises because some endogenous variables appear as explanatory variables in other equations. The set of exogenous variables often includes values of the explanatory variables. These predetermined variables impose the dynamic structure on the model. All -2- these complications imply that all possible links between debt service, capital inflow and economic growth can be analysed with a simultaneous equation ... distance 3/11 The Regression Equation For the example about the third exam scores and the final exam scores for the 11 statistics students, there are 11 data points Therefore, there are 11 ε... highlight Press the ZOOM key and then the number (for menu item "ZoomStat") ; the calculator will fit the window to the data To graph the best-fit line, press the "Y=" key and type the equation –173.5... linear regression The idea behind finding the best-fit line is based on the assumption that the data are scattered about a straight line The criteria for the best fit line is that the sum of the