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Economic Theory

by: April Jerde

Economic Theory ECON 712

April Jerde
GPA 3.6

Steven Durlauf

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This 75 page Class Notes was uploaded by April Jerde on Thursday September 17, 2015. The Class Notes belongs to ECON 712 at University of Wisconsin - Madison taught by Steven Durlauf in Fall. Since its upload, it has received 149 views. For similar materials see /class/205153/econ-712-university-of-wisconsin-madison in Economcs at University of Wisconsin - Madison.


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Economics 712 Steven N Durlauf Fall 2007 Lecture Notes 4 Univariate Forecasting 1 The Wiener Kolmogorov Prediction Formulas From previous notes he solution to the problem min Extk 2 41 MW is equal to the projection of xM onto Htx denoted as 9 Once again letting 14 39 Eat8M7 denote the fundamental MA representation of xi from the equivalence of 0 H2 x and H2 8 it is immediate that x may be written as tkt M ErrWet 42 0 The reason for this is simple at time I one can condition forecasts on 8 8H ie 1128 The component of x2 16 16 that 1s determ1ned by gtkgt1 1e EatEMF 1s j 0 orthogonal to H2 8 To achieve a more parsimonious expression of expressions such as 42 de ne the annihilation operator for lag polynomials De nition 41 Annihilation operator Let 7239L be a polynomial lag operator of the form 72214 721171 no 72391L 7239sz The annihilation operator denoted as eliminates all lag terms with negative exponents ie 7239L 07r1L7r2L2 43 Using the annihilation operator one can reexpress the optimal predictor as a L xtkt 82 44 When the MA polynomial is invertible aL71 x2 82 so that one can explicitly express the predictor as a weighted average of current and lagged x s 44 implies M aL 1x2 45 Equations 4 4 and 45 are known as the WienerKolmogorov prediction formulas Example 41 MA1 process Let x2 gt ng The AR WienerKolmogorov formula imply that optimal linear predictions of the process is 1 L x L5 1 pL1x 46 which equals 0 forj gt1 as one would expect since the process is unaffected by shocks more than one period in the past Example 42 AR1 process If x2 pr 82 then the AR WienerKolmogorov formula is kaIt p lkL l pLxt pkxt 47 10 The ARl thus has a very simple structure xmmlt prklt 2 The Hansen Sargent formula for geometrically discounted distributed leads The WienerKolmogorov formula has played a critical role in the development of modern empirical macroeconomics The reason for this is many dynamic aggregate models imply that one time series represents a combination of forecasts of others In a pair of seminal papers both of which are on the reading list Lars Hansen and Thomas Sargent developed the machinery to understand how such expectationsbased relationships lead to testable implications of macroeconomic theories This machinery has been of immense importance in the development of modern empirical macroeconomics I develop the basic HansenSargent formula in the context of the constant discount dividend stock price model This is an example of a geometric distributed leads model in that one variable is a geometrically weighted sum of expected values of future levels of another variable Defining the variables D2 dividends measured in real terms B stock price measured in real terms and fixed discount rate the constant discount dividend stock price asserts that the stock price series obeys BZ EDEB 48 where Pquot 2 ED 49 What is the economic interpretation of this model The model can be thought of as follows Suppose agents are risk neutral ie utility from consumption at t is linear in the level and discount utility geometrically at rate One can think of ownership of a unit of stock as providing a stream of real consumption via the dividends if held forever The opportunity cost of ownership is the consumption foregone today by buying a unit of stock The price formula expresses the price at which an expected utility maXimizer is indifferent between buying a unit or not which is what the price must do in equilibrium How can one test whether stock prices are consistent with this theory Suppose that dividends have the fundamental moving average representation DtaLgt 410 This representation is recoverable fromt the data One may use this representation to test the theory by computing the projection of B onto H t D that is implied by the theory 2 and comparing this projection to the direct projection P onto Ht Put differently the stock price model that has been described implies that PVOJ3913 IH DD 0 jPVOKDw H 2 DD 411 If one knows the process for the dividend series one can derive the right hand side expression and compare it to the left hand side formula In order to derive the projection on the right hand side of this expression one proceeds in two steps First one projects B onto 1128 to generate a time series 7239Lgt and second one inverts the 82 s to generate D s The derivation relies on the implication ofthe model that B may also be expressed as EB lHt D M3031 H DDt 412 which implies that nLg LLlgaLg 413 or 7239Lgt LL gtaLgt 414 Algebraic manipulation yields l L 1nLaL 7t01 415 To manipulate this expression in order to express 7239L in terms of aL notice that L then no a Substituting into this expression and rearranging yields ALFW 416 which leads to EOWHD aUJ M 81 y LrLJD min 2 l 1 Lrl l 1 Lrl l I This relationship between 7239L and aL is complicated For example if one writes a L EB lHt 13 2 g 418 then it is apparent that n i rm 419 3 Some properties of forecasts A number of tests of macroeconomic models may be constructed using properties of forecasts For example suppose that a variable x is if a certain model holds the optimal forecast of another variable y given an information set E ie mwmmm Mm Such a claim requires that the forecast error 772 y g is orthogonal to E This provides a simple way of testing such models Here are some examples of implications that have been exploited in the empirical literature i One of the elements of E is 9 so one implication of this is that forecasts must be orthogonal to forecast errors ii Suppose that y is an element of F 1 This will mean that the forecast errors 772 are uncorrelated iii Since varyt va1xt 772 varg var772when x2 is a forecast of y one can test this claimed relationship by evaluating the implication that the variances of realizations should be greater than the variance of rational forecasts of the realizations ie varyt gt varxt 421 The two are equal when forecasts errors are always equal to 0 I ignore this uninteresting case Evaluation of whether this inequality holds for a data series is known as an excess volatility test These tests were introduced by Stephen LeRoy and Robert Shiller One can construct many forms of excess volatility tests Suppose that 22 6E Then it is immediate that varyt z gt varxt z 422 Notice that varyt varxt 772 varg var772 2 covxt 772 so that varyt gt varxt implies that var772 gt 2 cov77 x which may be rewritten as cov x l lt 3 Hence an excess volat111ty test looks for negat1ve nonlocal correlations Var 772 between forecasts and forecast errors Excess volatility can thus be tested based on the regression 9 372134 423 Excess volat111ty requires that the coefficlent 1n th1s regress1on 1s less tha E ThlS regression has an unusual form as it regresses the forecast against the forecast errors rather than the forecast error against the forecast which is the regression suggested by the requirement that the forecast errors generated by rational forecasts are unpredictable Put differently excess volatility implies predictability of forecast errors although predictability of forecast errors does not imply excess volatility 4 Bubbles Returning to the dividend stock price model the basic equation 48 may be rewritten Pt Z EDt1DtZ lEDtx 10 11 D lEtDtx1 D2 EtZ lEm Dmu 43924 10 10 Dz Eth1 This means that the excess holding return Pz1Dz Pz 425 is unpredictable given all information available at time t as it is simply a particular forecast error A natural question is whether testing this unpredictability property is equivalent to testing the nonlinear restrictions embedded in the HansenSargent formula which relates the level of stock prices to the dividend process One way to answer this question is to ask whether 421 imposes fewer restrictions on the data that 48 It turns out the answer is yes Suppose that one appends to a B a process 3 with structure Bl 39lBH th 426 where 2 is unpredictable given information at t l This new process will violate 48 but not 421 Such a process is known as a bubble Tests of the unpredictability of excess holding returns cannot detect bubbles as they do not place any restrictions on the source of the returns Notice that Bl is an explosive process When present prices will not have a nite variance and in fact the assumptions required to construct a Hilbert space around current and lagged prices are violated Projections of prices onto current and lagged dividends will no longer be de ned This has implications for the construction of tests for the presence of bubbles Economics 712 Steven N Durlauf Fall 2006 Lecture Notes 5 Frequency Domain Approach to Time Series i spectral densities The spectral density of a time series is de ned as w i am je w E inn 51 7700 Note that since sin to 7 sin7w and am aw ij then iFim aw je W iFim aw jcos jw 03 igaw jcos jw 52 The spectral density is therefore the Fourier transform of the autocovariance function By the RieszFischer theorem this means that the spectral density function and the autocovariance function contain the same information about x2 One can recover the autocovariances Via the formula fl ltwgte dw a i 53 Notice a special case of the recovery formula is wdw 0w 0 54 which will prove to have a deep interpretation We now turn to some examples of spectral density functions Example 51 white noise process If mt 6t then the spectral density equals 27 55 The function is shaped as a rectangle for the interval 7239 7239 Each frequency produces the same value which is the origin of the term white noise Example 52 AR1 Process If mt pmH 6t then the spectral density equals 2 2 GE 7 as 27rltlipe wgtltlipeWgt i 27r12pcoswp2 56 When p gt 0 then the maximum of the function is w 0 If p lt 0 then u 0 minimum Notice that as p 3 l the spectral density function becomes arbitrarily large at w 0 Example 53 MA1 Process If mt 6t pEH then the spectral density equals 03lpe wlpew 03l2pcoswp2 57 27f 27f If p gt 0 then the function is maximal at u 0 wheras if p lt 0 the function is minimal Notice that the qualitative shape of the spectral density of an MAl is the same as for the AR1 Example 54 ARZ Process If mt plmH pg 6t then the spectral density equals 2 05 27r l 7 ple 7 pZe 2 l 7 pa 7 p262 2 of 58 27rlt1p12 p 2p1p2 72000qu 722 cos2w the maximum value of this function is difficult to compute except when p1 and p2 are both positive in which case the spectral density must have a maximum at u 0 ii spectral representation of a time series This section describes a decomposition of a time series into frequencyspecific components The decomposition will illustrate a deep relationship between a structure of atime series and its spectral density Theorem 51 Spectral representation theorem Cram r s theorem Let It be azero mean L2 process The art may be expressed as mt fwe mdszu 59 where dzw w is a complex valued random process such that i Ed2w an 0 m Ed2w mm 01 de m 510 M Eltd2w dezw my 0 u i w w 2m to is unique outside a set of measure zero The domain of integration is nal The use of the halfopen interval is a minor technicality which I will ignore What is dzw w This is an example of a random function This means that for each xed frequency 5 2w 5 is a random variable Suppose that for any xed frequenc1es ml lt LUZ S wS lt w Ezw r04 7 2 032 r02 7 2 w 0 511 then the process is said to possess orthogonal increments A standard example of such a process is Brownian motion A Brownian motion Bt t2 0 is random function such that Bt N0t What does this theorem tell us about the underlying structure of a time series Recall 54 which stated that the variance of a process is the integral of its spectral density Using 59 the variance can also be written as a 0 Em m2 mf mg m 512 The use of the complex conjugate is allowed since x2 is real Since dzw w is composed of orthogonal increments it must be the case that EU mg m f er2 m E f W m2 une mdzw w 513 dz momma The expected value operator can be moved inside the integral since the integral is a linear operator which means d2 WWW w Ef Eltdzw Lodz 01 f de m 514 This indicates that the spectral density reveals how each stochastic bit of mtconstributes to its overall variance Finally note that if m is real one can rewrite the spectral representation as follows First observe that in order for integral in 59 to be real with probability 1 it is necessary for dzw w Why Take a xed 5 and 7 J In order for the sum e mdzw J e mdzw 75 515 to be real it is necessary that e mdzw J e Wdzw 75 Of course 6quot 6 Since 2a is random dzw w dzw fin is needed to hold with probability 1 in order to ensure the integral is real Rewriting dzw w duw w 7 139de w the requirement that dzw w dzw flu is real means that duw w duw w and de w iduw 7w 516 Recall that by the spectral representation theorem Ed2w wd2w 7w 0 This can thus be rewritten Eduw unduw 71m 7 iEduw 11de 71m 7 517 iEduw 7w 11w 00 Ede 11de 710 0 Using 516 for 517 to hold at all frequencies it must be the case that E um w um 71 7 E Um to 11 00 0 implirs E duw 002 E 1 1102 We now rewrite the spectral representation of m as mt fir ewdzw w f1 cos wt isin wtduw w 7 de 00 w 4 w 4 w W 518 f cos wtduw m 7f cos wtde to if sin wtduw w if sin wtde to The first and fourth integrals in the third line of 518 are even functions meaning 97y 9y whereas the second and third are odd functions meaning that 9 9 9 Therefore as Cos wtduw w sinwtde w 7r 519 L cos wthw w sin wthw LAD where dUw w duw w duw 7w 2duw m 520 and de w de w 7 de 7w 2de m 521 Example 55 deterministic seasonal If m 6 cos Qt then the spectral density will not exist at J The reason for this is that the one frequency will contribute a non eligible amount of variance at J In this case de will still exist in this case 5 represents a jump point One way to think about the spectral density function is to de ne it as a 6w i a 522 w oz fw 27f where 60 7 E is the socalled Dirac delta function This is a function with the properties i 60 00 0 otherwise ii f1 50 7 EX w dw 90 This is a so called generalized function The Dirac function allows one to work with jumps in the spectral distribution function In general we can write the spectral density function as f w afuu Z mw 7 5 523 where f w is continuous iii filters Often one works with a time series that is a transformation of another ie y Lx2 524 In this case L is known as a filter The frequency domain allows for a number of insights into the effects off11ters The MA representation of a process may be thought of as describing a given process created by applying a lter to white noise The following theorem describes how this ltering applies to the spectral representation of the process Theorem 52 Construction of spectral representation from white noise spectral representation Suppose that 82 is a white noise process such that 82 e dz a Let x2 Lgt Then which means that in the spectral representation of xi Pf no 7 124 714 J e fe dz a2 no K wxm J e md a2 2 Z If e dzsw 525 526 527 528 where z 1 equot dz m 529 It is straightforward to verify that 51241 possesses all the necessary requirements for the spectral representation In particular 139 Edz equot Edzs 0 ii Edzx e39 quot e quot Edz lm W 5 0 530 we we gt25 iii Edz m1 alzX mj equot equot EdZ m1 dzsaj 0 if ml at m The relationship between the properties of dz a2 and dz a can be as follows Let 6 yael Observe that one can always do this by the standard properties of complex numbers We can first identify an effect of ya on the dz a term in the spectral 12 representation In particular Edzx a2dzxa yaEdz a2dz The first feature of any filter is that changes the length of dz a by ya This is referred to as the gain of the filter at a and illustrates how the filter alters the variance contributions of different frequencies Second consider the effect of em on 6 From the form of the spectral representation the effect is to alter the complex exponential in the sense that eltwewlw Wat 90 e This shifts the sine and cosine functions by This is called the phase shift There is nothing in this argument that does not immediately generalize to any lter Hence for y Lx2 y fire equot dzxa 531 and one can make the same sorts of arguments about the effects of the filter Example 55 differencing If y l Lx2 then fy a2 2 2cosa2 a2 Notice that for a differenced series fy 0 0 This makes intuitive sense Differencing eliminates the part of the process common to all observations Example 56 averaging Suppose that we define L such that 1Tj0T1 532 0 otherwise T71 This means that the filter averages the x2 process In this case e39 quot T39IZe39W 0 Note that equot 1 if a 0 rxTzo equot T11 87W a2 0 18 533 equot l a 0 Further W W 2 2 2cosTa 2sin2Ta2 e e T Z Zcosm j T sin2a2 53934 This is a function which converges to IN The spectral representation of an averaged series will therefore converge to 51X This makes sense when we average all the elements of 9 the part common to all elements is what remains this is what the zero frequency element dx 0 captures The spectral density function of the averaged series is sinz Tm2 fxwT sin2 wZ 535 a function whose integral over 7239 7239 will converge to zero This implies that a form of the law of large numbers holds for weakly stationary time series so long as the spectral density is bounded at 0 Example 57 band pass lter Take a time series x2 and suppose we wish to create a series y that removes the part of the Cramer representation that corresponds to those frequencies above some specified value 5 so that fya aifa a 3a0 if wgta 53936 This would imply that there eXists a filter L such that equot e quot 1 if lml S E 0 otherwise 537 What polynomial has this property We construct it as follows Let e39 quot Mm which requires that equot is symmetric and 2sided Hence in order to recover we can use the Fourier inversion formula 1 7 W 1 sin 5 zgj39i lldge de 01394g5c08mdw 538 This lter was proposed by Robert Engle to allow for band spectrum regression The idea was to allow one to analyze regressions based on the long run parts of various time series Notice that the lter does not preserve various temporal relationships in the data so that ltered data will not obey restrictions that economic theory for example places on the original series Economics 712 Steven N Durlauf Fall 2007 Lecture Notes 2 Hilbert Space Approach to Time Series The Hilbert space framework provides a very convenient language for discussing the relationship between various random variables Collections of random variables are called stochastic processes in common usage stochastic processes usually are indexed by time We focus on the case of a scalar stochastic process x2 where t is an integer We assume that this process is zeromean and second order stationary which means htat the autocovariances between x27 and 9 do not depend on t Formally E x 0 and Extxtij039jltoo For random variables such as the elements of the stochastic process x2 the natural notion of inner product between two elements is the covariance ltxxjgtExtxj 21 which means that the associated norm measures the standard deviation x2Exf 22 One can generate a Hilbert space around the sequence 99991ka What this means is that one forms a space by taking these elements adding all linear combinations of the elements all limits of the linear combinations etc We denote this Hilbert space as Htx The entire history of the stochastic process from oo to 00 generates Hoo x By construction HH x g H The general properties of Hilbert spaces described in Lecture 1 allow one to characterize the linear structure of H t x in ways that are very useful First observe that by the Hilbert space decomposition theorem one can decompose H t x so that HtxHHxCBG2 23 where G is another Hilbert space The dimension of this Hilbert space is either 0 or 1 This is so because the Hilbert space Gt must be spanned by the single random variable that is the difference between 99 projxtHHx where projxtHHx is the projection of x onto HHx To say the space G2 has dimension 0 means that 9 6 HH x ie varxt projxt lHH x 0 If one again applies the Hilbert space decomposition theorem one has H xHH 9063le G 24 Here GH is spanned by x271 projx271Ht2x One can of course repeat this decomposition any number of times The G2 spaces are by construction mutually orthogonal Notice that it is not the necessarily the case that the q s may be used to reconstruct H2 The reason for this is each space is constituted by elements that appear in the space HM x but not in the space H27Hx if there are elements that appear in every member of the sequence Ht xHH x they will not appear in any of the G s Elements that are common to all of the H2 x s form a Hilbert space as well Formally this space is defined as Hwx Hx 25 The Hilbert space generated by current and past xt s can therefore be decomposed as H x G 96 6611 x 26 This decomposition is the basis for two fundamental theorems in time series analysis each due to Herman Wold his 1948 article is still worth reading Rozanov 1967 is a deep treatment I nd Ash and Gardner s 1967 discussion to be especially useful Theorem 21 Wold decomposition theorem I Any weakly stationary zero mean weakly stationary process x2 may be decomposed as x x1 952 27 where x e Q EEGH EEGH GB 28 and x22 6 H o x 29 In this decomposition x is called the indeterministic component and x22 the deterministic component of 9 The terms refer to whether the components may be perfectly predicted from the past When a time series contains a nontrivial indeterministic component the time series itself is said to be indeterministic If the process does not contain a deterministic component it is purely indeterministic What are examples of deterministic components One possibility is x22 cos wt t9where 9 is uniformly distributed on 7239 7239 From the perspective of prediction of a time series given its past the term x22 may be done from information in the arbitrarily distant past The second Wold Theorem characterizes the linear structure of the indeterministic part ofa time series Theorem 22 Wold decomposition theorem II If 9 is a purely indeterministic zeromean weakly stationary process then there eXists a representation such that x Zag 0101 210 13 no where 82 6G2 and 039 0 Vj 20182 1 is referred to as the fundamental moving 0 average MA representation of x and is unique Pf Since HtxHHxBG2 by construction G2 is a Hilbert space of maximum dimension 1 If the dimension is zero then HtxHHx and the process is not indeterministic which contradicts our assumption that the process is purely indeterministic Since the process is indeterministic one may find an element 82 in G such that the projection of xi onto G2 is 82 Since this element spans the space one can now treat the space as G For the spaces GM j gt 0 one can find an element in each of them denoted as 8H whose variance equals that of 82 each 8H spans its respective space This argument is cumbersome but allows one to think about the Gt 393 as generated by certain random variables Since x2 is purely indeterministic Ht x Gt8BGH8CB Letting projxt IGH 8 denote the projection of xi onto GM 8 by the Hilbert space projection theorem x Z prom IGH 8 Eat8 211 10 0 where the second equality follows from the construction of the 82 s This veri es the theorem except for uniqueness To prove uniqueness suppose that there existed another MA representation x2 Z j8 For this to be the case the variance of Z j8 Zaj8 must equal 10 10 0 zero since by assumption the parts of the expression are the same The variance of this expression equals 02aj j 2 which equals zero iff a 0 Vj 0 What is meant by the term fundamental in the description of the moving average representation Many different orthogonal processes may be used to generate a given time series Intuitively there is an in nity of ways to orthogonalize Htx The fundamental representation is based on one particular orthogonalization The associated8t s are unsurprisingly called fundamental innovations or errors as far as I know the term is taken from Rozanov 1967 There is an equivalence between the stochastic process x2 and the stochastic process 82 that are used to generate the fundamental moving average representation Theorem 23 Equivalence between the Hilbert space of a time series and its associated fundamental innovations Let Ht 8 denote the Hilbert space generated by 828271 the fundamental moving average components of a zeromean weakly stationary process x2 Then H t 8 Ht Pf This is left as an exercise Finally we consider the question of how to optimally predict a time series given its history Let x417 denote the projection of xi onto H 2 fx This projection is important in that it is also the solution to the linear prediction problem for 9 relative to the information set HM x 1 Theorem 24 Optimal linear predictor The projection x427j is the solution to min 55 HF x E x2 if Pf Let 5 solve the minimization problem The prediction error equals 82 x2l The variance of this term will equal 039 0 75 since 82 is orthogonal to x427j This 1 variance must exceed 039 unless x427j 5 is zero Uniqueness of the projection then veri es the result This theorem implies that 82 is the forecast error associated with the optimal in a minimum variance sense forecast of xi given the information set H Hx Hence one can think of a time series as a weighted average of current and past forecast errors This is intuitive since these forecast errors reveal aspects of the process that are realized each time period From the perspective of the Wold theorems the moving average representation of a time series is the natural way of thinking about its underlying linear structure much of time series analysis is based on this idea 1This theorem is actually an implication of the general result on the relationship between Hilbert space projections and certain minimization problems described in Lecture Notes 1 References Ash R and M Gardner 1975 Topics in Stochastic Processes New York Academic Press Rozanov Y 1967 Stationary Time Series San Francisco Holden Day Wold H 1948 On Prediction in Stationary Time Series Annals of Mathematical Statistics 19 4 558567 Economics 712 Steven N Durlauf Fall 2007 Lecture Notes 1 Linear Space Theory Much of time series analysis is based on the construction of linear decompositions of the processes under study These ideas are based on some results based that are taken from the mathematics of linear spaces Linear spaces are also sometimes called vector spaces These lecture notes describe some basic de nitions and results The results are all standard I follow Royden 1988 and Simmons 1963 closely in terms of presentation Ash 1976 chapter 3 is also useful 1 Basic de nitions De nition 111 Linear space Let F denote a nonempty set For each pair of elements y and y contained in F assume that V an operation called addition denoted as such that y y is an element of F Suppose that the addition operation obeys 117 7 717 ii 77j77717k iii F contains a unique element 0 such that y0 y Vy e F iv For each y e F V an element such that y y 0 Further for any element 0 of the set of complex numbers C assume that c may be combined with any yer to produce another element of F cy Suppose that this operation called scalar multiplication obeys v 60 yj cy cyj vi cdycyd vii cdycd viii lyy Then the set F and the addition and multiplication operations de ne a linear space If multiplication is de ned with respect to elements of the real line R rather than C then the set F and the addition and multiplication operations de ne a real linear space The properties which will developed for linear spaces will apply to real linear spaces I will switch between the two as appropriate In essence the de nition of a linear space requires that the space is closed with respect to addition of any pair of elements of the space and that multiplying any element by a scalar produces another element of the space This de nition does not require the space to be closed in any sense In order to discuss whether the space is closed one associates a linear space with a norm A norm is a function that de nes distances between elements of a space This de nition is required to fulfill certain properties De nition 112 Norm For a linear space F a norm denoted as I is a mapping from F to R such that i quotyIIZO yl0 gty0 13911 quot07quot c7 V c E C iii 71 3 ll7 zquotll7 The rst condition requires the norm to measure something interpretable as a length The second condition requires that the norm is linear The third condition is the triangle inequality This is a hint that intuitions concerning length that one has from elementary geometry will apply to more general spaces De nition 113 Normed linear space A normed linear space is a linear space equipped with a norm When a space is endowed with a norm one can discuss the convergence properties of various sequences of elements in the space An important type of sequence is a Cauchy sequence De nition 114 Cauchy sequence The sequence y is a Cauchy sequence if for any 8 gt 0 there exists an I which may depend on 8 such that for all ijgtI 71 7JIlt8 The notion of a Cauchy sequence naturally leads to the notion of a limit point for the sequence De nition 115 Limit of a Cauchy sequence The element 7 is the limit to Cauchy sequence y if for any 8 gt 0 there eXists an I which may depend on 8 such that for all 139 gt1 lt 8 7r It is not necessarily true that the limit point of Cauchy sequence is an element of space even if the all members of the Cauchy sequences are elements of the space For example consider the linear space 01 with the metric equal to the element ie y y 6 01 and the Cauchy sequence y in this case the limit of the sequence is 0 which is not 1 an element of the space Spaces where examples like this cannot occur ie which contain the limit points to all Cauchy sequences in the space are known as complete De nition 116 Completeness A linear space F is complete if the limit to every Cauchy sequence in the space is an element of the space Normed linear spaces that are complete form an important class of spaces De nition 117 Banach space A Banach space is a complete normed linear space The norm function provides an abstract notion of distance It is also important to be able to measure the way two elements of a space are related The key notion is the inner product of two elements of a space For the compleX number 0 5 denotes its compleX conjugate De nition 118 Inner product An inner product function is a mapping from F X F to C such that 139 lta7 b7 kgt alt77kgtblt7j7kgt ii lt7 71gt W iii yinl Z 0 1Vlt7171gt0 gt710 Our nal de nition describes spaces where the inner product is de nes the norm De nition 119 Hilbert space A Hilbert space is a Banach space with an inner product such that 21 y Hy ll2 Hilbert spaces are the foundations for the theory of time series 2 Properties of Hilbert spaces This section discusses a few properties of the general structure of Hilbert space Thes properties will depend critically on the notion of orthogonality between two elements of a Hilbert space Hilbert spaces possess a natural notion of orthogonality between two elements De nition 121 Orthogonality Two elements y and y are orthogonal if y 0 i Projections Orthogonality allows one to construct various decompositions of elements of Hilbert spaces and hence of the spaces themselves To see this take any two elements of a space y and y and consider the element lt77jgt lt7N7jgt j This element is also a member of the Hilbert space since the space is linear Further this element is orthogonal to y since mm lt7N7jgt lt71 77gtlt747jgtlt77gt0 This construction is suggestive of a fundamental property of Hilbert spaces namely that there exist ways to decompose these spaces into mutually orthogonal subspaces Theorem 121 Decomposability of Hilbert space into orthogonal subspaces Suppose that F and F1 are Hilbert spaces such that F1 g P Then there exists a unique Hilbert space F2 g P such that 139 rrlear2 n r1 LFZ The operator EB is called the direct sum The direct sum of two spaces produces athird space whose elements consist of all linear combinations of elements of the original spaces and the limits of all such combinations Corollary 121 Decomposition of an element of a Hilbert space into orthogonal components Suppose that F and F1 are Hilbert spaces such that F1 g F For a given element y e F there exist unique elements yl 6 F1 and yz 6 F2 where F2 is de ned in Theorem 121 such that i77172 whim In this corollary yl is called the projection of y onto F1 This projection has an interpretation in terms of an optimization problem Suppose one want to solve the problem linger quot7 5 In other words one wants to find that element of the subspace F1 that is closest to an element of F The solution to this problem is 5 y1 This follows immediately from recognizing that ll757172 571572 where the second equality follows immediately from the orthogonality of F1 and F2 ii Orthogonal Bases for Hilbert space De nition 123 Orthonormal set A subset S of a given Hilbert space is called an orthonormal set if 139 each element of S is orthogonal to every other element ii the norm of every element equals 1 De nition 124 Orthonormal basis An orthonormal set S is an orthonormal basis with respect to a given Hilbert space H if it is not a proper subset of any other orthonormal set in the space If a set of vectors span a space this means that if one takes all linear combinations and limits of the linear combination of elements of the set one reproduces the space Orthonormal bases have the important property that they span their respective spaces Theorem 122 Relationship between orthonormal basis and a given Hilbert space An orthonormal basis of a given Hilbert space F spans the space The next two theorems establish the existence of orthonormal bases Particularly clear proofs are found in Ash section 32 and Royden section 108 Theorem 123 Existence of an orthonormal basis Every Hilbert space contains an orthonormal basis Theorem 124 Existence of a countable orthonormal basis A Hilbert space contains an orthonormal basis iff the space is separable1 Countable orthonormal bases are of interest because they provide the axes for a Hilbert space that are analogous to the orthogonal unit vectors one uses to construct Rk When S is a countable orthonormal basis for F a typical element y can either be represented as or can be arbitrarily well represented this way The difference is that since the space is complete it contains limits of all such sums as well as the sums themselves The 1 A space is separable if it contains a countable dense subset The spaces we study will always be separable coefficients are implicitly determined by the projection of y onto each of the 1 dimensional subspaces spanned by the 81 s2 Using the construction of orthogonal decompositions developed above it is clear that a ysl Recall that swsl by assumption These are also known as Fourier coef cients 3 Examples and applications To provide an example of the abstract ideas we have discussed considering Hilbert space consider Rk kdimensional Euclidean space Letting x and y denote vectors in this space the inner product of the vectors is leyl with associated norm lexlfZ To see how the Hilbert space decomposition theorem applies consider a Hilbert space generated around the vector l000 Recall that generating a Hilbert space around some set of elements means adding all linear combinations and limits of linear combinations to elements to the set This will be a subspace of Rk What is the orthogonal complement to this space It is the Hilbert space spanned by the vectors 01000 001000000l Similarly a number of simple properties of Hilbert spaces are generalizations of results that one nds in a range of mathematical contexts Example 131 Pythagorean Theorem Suppose that some element of a Hilbert space call it x is the sum of two orthogonal elements y and Z In this case the square of the norm of x obeys ltxxgtltyzyzgt ltyygtltzzgt 2 Each 31 spans a separate ldimensional subspace by the fact that the s s form a basis since by orthogonality y z z y 0 But this is nothing more than the Pythagorean theorem for arbitrary Hilbert spaces as opposed to R2 Example 132 Cauchy Schwarz Inequality For any a the inner product of x ay with can be written as WWJHMWMMWHRampWHWJRO Suppose that a ltxaygt lty y Substituting this above ltxayxaygtltxxgt ltx ygtltyxgtlty xgtltxygt ltx ygt yygt bay 0 y Jay Mgto OW Jay 39 Rearranging terms this expression implies mmmwdmwr with equality iff x ay The iff claim is something you should verify This is the CauchySchwarz inequality for an arbitrary Hilbert space References Ash R 1976 Real Analysis anal Probability New York Academic Press Royden H 1988 Real Analysis Third Edition New York MacMillan Simmons G 1963 Introduction to Topology anal Modern Analysis New York Academic Press Econ 712 Discussion Section 9262008 Laura A Dague 1 Spectral Analysis We are now considering the value of I as a weighted sum of sines and cosinesi We use 503wt and sinwt and call w the frequency The goal is to determine the role of cycles of different frequencies in the behavior of the 1 process This is called frequencydomain analysis or spectral analysis1 Let7s quickly summarize what we already knowi Making the standard as sumptions on I 2700 71471 which has autocovariance function 075 j recall that we can represent the autocovariance function as its ztransform am V WM 700 or using the convolution formula aiwm where z is a complex scalar 6 Recall from the rst section that we can write this as 2 6080 7 isinw Maybe you wonder why we re taking Us and replacing them with z s and saying it s all the same and then saying that the z s are something else altogether and somehow we end up with sines and cosinesi Well this is what we7ve sort of been calling the ztransformi If you want more information you should refer speci cally to the discrete time Fourier transform which is a special type of ztransform on the unit circle But remember you already know this because of the RiezFischer Theorem and Lecture Notes 3 By de nition we write the spectral density of I as fxw Z canal F700 which from above we know is also am vzvz 1 It is useful to remember how to go back and forth between the spectrum and the time domain We have only done one way so far but you can go back by integrating the spectral density 1An encyclopedic reference for the time series material that includes not only theoretical exposition but estimation procedures is HaJnilton7s Time Series Analysis 11 Example Calculating the Spectral Density for an MA2 l7m sure you can do this on your own but just to be concrete let s do an example Let7s nd the spectral density of I 6 16271 26272 2 Examples from Past Exams The structure of exams in the past years has been 3 sections 0 815 TrueFalseUncertain short answer questions similar to HW ques tions wo models 0 l multipart short answer question involving a model similar to HW ques tions with multiple parts 0 l essay question which we will talk more about in future sections 2 1 2007 14 According to the pure expectations theory of the term structure of interest rates the spectral density of the 2 period rate must exceed the spectral density of the 1 period rate at the zero frequency 22 2007 15 If the zero frequency of the spectral density if positive this implies the mean of the time series is not zero 23 2007 18 A money supply feedback rule that reduces the overall variance of output com pared to a second rule must also reduce the variance at each frequency compared to the second rule 24 2006 16 For a strictly indeterministic secondorder stationary process the zero fre quency of the spectral density must equal zero 25 2006 17 The spectral density of a secondorder stationary process is nite at all frequen cies 26 2005 18 lfa time series is weakly stationary then the spectral density of its rst difference equals zero at the zero frequency 3 Solutions to HW3 31 HW3 Q1 Part i from Laura This is just saying that since shortperiod bonds can be combined to make longer period bonds the expected interest earned had better be the same for both if markets are complete and there is no arbitragei Here the LHS is the interest for the longerterm k period bond and the RHS is a weighted sum of expected interest rates of oneperiod bondsi art ii from Professor Durlauf and Laura Note we must have lpl lt 1 in order to satisfy stationarityi o a Ambiguousi Since i1 is AR1 i2z where p is the AR1 coefficient and we used that Eti1t1 pil Although if we consider the usual case of positive interest rates this would not seem to be possible by treating interest rates as AR1 the mean has been removed iiei negative observations are possible 0 bTruei We know from the assumptions of the problem that lt 1 so therefore f lt 1 and vari2t iffva im o c True because 1 p 2 vaTZ2t E171 150 var 1 1 2 21erEz21z vaT21rE17121z 32 HW3 Q2 0 i From Laura The Fisher equation is a rst order approximation for the relationship between real and nominal interest rates Suppose you want to buy a one dollar one period bond at time t Your return will be 1i1gt1 in the next period But if there is in ation it won t be worth that much instead your real return is 1 i1t 1 T T 1 Ez7rz1 where 7 is the real interest rate Solve for i1 and eliminate the TLEE n1 term because it is very smalli Name 7 1 c and we have the Fisher equa tion as stated here 0 ii from Professor Durlauf Since n1 7 in 7c 7rH1 7 Et7rt1 a projection of n1 7 i1 onto any Hilbert space based on information at time t assuming the space includes a constant will if the theory is correct produce a time series that is constant ie nothing can predict it except the constant iiifrom Laura lf i2 l2i1t Eti1t1 lt i1 then we have that Eti1t1 lt i1 Now use that i1gtt1 c Et17rt2 and take the expectation at time t getting Etc Et17rt2 c E7rz2 lt 11 C Ez7fz1 so we have the implication that Et7rt2 lt Et7rt1 or that in ation forecasts are strictly decreasingi If the term structure is strictly increasing longer term bonds have higher interest rates we have ipped inequality signs but remember we need to look at the 16 period bond case It seems to me that the kth forecast depends on an average of k l forecasts rather than the previous one You can make a counterexample with the rst three k 33 HW3 Q3 from Professor Durlauf The stock price is not stationary so we cannot directly use Hilbert space ideasi However if we consider the time series Pt1 7 Pt we can Under the random walk theory the projection of Pt1 7 Pt onto any Hilbert space HEZ must be zero for any vector Z so long as elements of the space indexed at time t and earlier are part of the information set of agents at time t Econ 712 Lecture Notes1 Laura Dague Steven Durlauf Second Half7 Fall 2008 1142008 Today Discussed ideas of what macroeconomic theory is Highlighted impor tance of welfare theorems 1 Macroeconomic Theory 11 Types 0 Suggestive or Weak theory recall y 7rLet is atheoretical purely statistical and model y 1 mt 7 Et1mt e is suggestive 0 Strong theory treats macro as a special case of micro general equilibrium theory Also called micro foundations 12 General Equilibrium Structure We need to specify the following 0 Agents consumers rms workers 0 Decision rules preferences constraints beliefs 0 Institutions 0 Adjudication how we will characterize equilibrium 2 Welfare Theorems 21 Context We consider a pure exchange economy with the following characteristics 0 I agents indexed by i lI 0 K commodities 0 Utility function U with 1 consumption bundle of agent i o Endowment ini De ne the set of prices and allocations pg as an equilibrium if a solves mazmlqmlUi for al i b 1Please note that I have just typed up the lecture notes as I heard them and wrote them down Proofreading is minimal at this point If you nd a typo I would be happy to x it Email dague wiscvedu Use at your own risk 22 First Welfare Theorem Theorem Every competitive equilibrium is Pareto optimal Classic proof due to Arrow Suppose 31 such that S where the inequality is strict for at least one i Note that if M then the theorem is false It must be that p zf 2 p z in equilibrium so 10 gt 10 But since w 10 lt 10 If violates feasibility 1 23 Second Welfare Theorem Theorem Every Pareto optimal allocation is a competitive equilibrium under redistributious of initial endowment Proof Omitted You7ll get it in micro 1 Together the rst and second welfare theorems let us use the planners problem to solve for equilibria This is a big strategy in macro Hidden assumption 10 Ewi is bounded Important to recognize these types of assumptions Recall we considered pt pf BE which will violate this e are also assuming the commodity space is nite at this point We will be interested and consider cases where the assumptions are violated 24 Commodity Space time and uncertainty The commodity space we want to think about is an intertemporal one Think of commodities as typedate pairs We will generally work in terms of date zero prices ie pt pilompo Considering an in nite horizon makes the consumption bundle in nite e can also think about uncertainty in this context Consider the commod ity umbrella7 U and possible states for tomorrow rain R and sunshine 5 Suppose at time t you purchase U At time t 1 you own commodities U R and U S Point commodities can also be indexed by state Savage Axioms We observe people7s behavior under uncertainty and can assign probabilities to various events under certain conditions Sequence of trades can be de ned implicitly by behavior V Risk free assets Let7s index states of the world as 9 and de ne p9 as the price at time zero of a promise to pay 1 unit at time t given state Oj Called an Arrow security We could ask the question what is the price of a promise to pay 1 unit at time t regardless of 9 It must be that the riskfree price prf Ej 9 You re r 39 every t t V t quot so there is no risk 11608 Today Solow growth model weak theory and growth under general equilibrium theory Began a discussion of potential interest rate determination 3 Solow Growth Model The Solow model is what we would call weak theory that makes nice predictions in many cases 31 Context We will consider the model with no population growth no technological change and total depreciation so that the following equations describe the model 0 YLCLKL1 0 YzfKz o Kt1sYtsfKt Even in this simpli ed environment we still have a notion of dynamics Looking at the third equation it is clear that today7s state determines tomorrow s state Note that everything depends on the shape of We will make the following assumptions 0 gt 0 trivial o f 0 00 substantive o f lt 0 substantive in LucasRomer models opposite is truelRS o 0 substantive This allows the model to be fully analyzed with a diagram INSERT DIA GRAM Steady states If HKquot such that K sfK we call K the steady state level of capital stock Note that if 0 0 is a steady state Steady states themselves are not that interesting We want to know about the dynamics such as what happens with different initial values of capital stock K0 For small K0 K is increasing For large K0 K is decreasing In this context this implies that K 0 is an unstable steady state because if we started just a small distance away from it we would move away from this steady state Additionall it implies that K 0 is stable Cross country growth What can this model say about crosscountry growth differences Growth rates can be different Heterogeneity can be introduced through 0 Initial capital stock KW Even though 8 same countries with low K0 should grow faster 0 Savings rate 3 Higher more KHl 0 Production function Called new growth theory The new growth theory7 changes the meaning of It is no longer just a blueprint for production but can consider broader concepts like culture and geography Additionally it can broaden the concept of knowledge by considering things like openness to trade Gaps in Solow model In the Solow model economies should converge to a growth rate of 0 We observe wealthy economies converging to growth rates of 23 percent in practice This may be because the Solow model is driven by factor accumulation rather than innovation Additionally the Solow model predicts that countries with negative growth rates must have begun with a very high initial capital stock However the economies we observe with negative growth rates like those of Subsaharan Africa have surely not had this property Thus the Solow model cannot explain what is arguably one of the most serious problems in crosscountry growth So is it a bad model Thatls a bad question It explains some things very well and has little to say about others 4 General Equilibrium Approach 41 Context Maintain assumptions from beginning part Additionally consider 0 Representative consumer with preferences 20 jUCj j indexes date 0 Consumer budget constraint 20171395139 Y0 We where pt is the date zero price of the good at time t with p0 1 Y0 is the consumers ini tial endowment and We is the pro ts of the rm which is owned by the consumer 0 Representative rm with pro ts 7r0 Eilpj Kj 7 20 ijj1 De ne a competitive equilibrium as a sequence of prices 17 and allocations ct Kt with K0 given such that 0 Consumers maximize utility 0 Firms maximize pro ts 0 01 Kz1 YLW This results in the following rst order conditions Consumer UCt BU Ct1 Pt Pz1 Firm 0 Pz1fKz1 Rewrite as Pz1 UCt1 PL UCz and 1 fKz1pitl P If we solve these to get a joint understanding of consumer and producer equi librium decisions we get BU CH1 7 17 f Kz17UCt and nally UCz fKz1UCz1 If we rename fKHI l TKE where TKE denotes the net return to capital stock aka the real interest rate then Uct 61 TKUCt1 This gives us a theory of interest rate determination Capital is the only asset in this economy so the real interest rate is determined by the parameter 6 and the intertemporal ratio of marginal utility of consumption We can consider the implications of high and low marginal utilities have on the interest rate This is not that interesting since in models without uncertainty all assets will have the same rate of return We7ll get there If we rewrite the formula as UfUQ 7 K144 BU fKt1 7 Kt2gt we can interpret it as a law of motion for capital stock This is harder to analyze because it has three states Original Solow model had KHl sfKti Trick is to de ne Kt1 in equilibrium where gt 0 Either capital diverges or it will go to a steady state level You can prove this check Ljudgqvist and Sargent At Kquot 1 f so 6 determines the real return ofcapital in the long run Recall that B is just a parameter of consumer preferences describing patiencei What are implications of higher or lower 6 higher means more patient This setup allows us to apply the welfare theorems and talk about ef ciency 111108 Today Planner7s problem transversality conditions Bellman equations asset pricing and notion of risk 5 Ef cient solutions Planner s Problem Imagine that you are a benevolent social planner who wants everyone to be as well off as possible In the economy we are thinking about right now this amounts to him maximizing 00 2611461 j0 subject to 07 fUW C kn1 Consider the following thought experiment If you delay consumption by one period you give up uct and gain f kt1 u ct1i If you are on an optimal path it must be the case that we cannot improve in either direction so we will ave uCt fkt1 uCtl The social planner must directly consider the consequences of tradeoffs across time In the competitive equilibrium prices perform that function We can write the Euler Equation u kt ktlgt fkt1 ufkt1 7 kz2 If we know 160 and k1 we know the entire sequence How can we nd k1 Note that not every path consistent with the Euler Equation is optimal since it is just a rst order condition in the same way that even if we know the solution to g z 0 we do not know whether it is a maximum or minimumi We need something analogous to a second order conditioni Here it will be the Transversalz39ty Condition TVC tlim Btuct 0 Recall in the CE that the date zero price is a marginal utility The TVC is imposing that the price of goods in the arbitrarily distant future must be zero We need the TVC so that the First Welfare Theorem can hold and we do not get unbounded budget setsi Note that u ct would go to in nity as C went to zero so this is the case we are ruling out The TVC rules out bubbles in equilibrium models 6 Digression Socialist Calculation Debate Can socialist economies be ef cient We have been discussing the planners problem and we are getting the result that the planners problem yields the same as the competitive equilbiriumi Does this mean that socialist economies can reach the market economy solution The problem with this argument is the amount of information that the social planner needs to have in order to reach the CE He needs to know the preferences and their parameters technology initial capital stock etsi Market economies just use prices to aggregate all of this information No one needs to know a utility function if people want more or less of a good it shows up in the prices The resolution of the debate is that there is no way for a socialist economy to get at this information so they cannot reach the market solution 7 The Recursive Problem Bellman Equations Intro We want to solve the problem 00 2511451 j0 subject to 07 1 C kz1 Let7s consider an object V060 a value function which represents lifetime utility generated at the optimal solution VOW u00 5VK1 max Cok1fko ufko k1 5VK1 First Order Condition u ko k1 BVkl or in general if the path is optimal the following must be true 7060 kt1gtgt Vkwl We don7t know V but we can use the Envelope Theorem to get Vkz MICkt Combining these we end up with 7060 krlgtgt uCt1fktl which looks very familiar Thus the intertemporal decision problems we are trying to solve can be thought of recursively as well 8 Uncertainty and Asset Pricing We now consider a case in which the technology is described by fkt 1h where 1 is a productivity shocki Now the problem is mmZ jEluCjl The value function V061 now represents expected lifetime utility In the TVC the expected limit must be zero The math is a little more complicated but the economics is the same we are using tradeoffs across time to describe the optimal path FOC uCt Etu ct1fkz17 z1l Let7s de ne f kt1 t1 BK 1 TK where R is an expected gross real return and 7 is an expected net real returni Aside interpretation of ri If we buy a bond today and interest rate says 10 percent but we do not know in ation then the real return r is unknown although we can make expectations about it Consider any asset 239 name its return Rmi It must be that uCz EthituCtll7Vi since consumers must be indifferent between all assets in equilibriumi Rewrite this as 1 mm LC HH 1 WW We have restrictions on the behavior of rates of return De ne asset L as one in which EERLE BL so there is no uncertainty in the return Then The L asset has a high expected return when uCt1 is low and the reverse Now consider an asset S which is de ned so that Rs RLE1 415 where 41th 51 We price the S asset as 7 7 CH1 1 7 BEARS Wt which can be taken separately by independence so uCt1 1 EthStlEtWl so that 1 Eleszl BEA 65ml EleL BL u 5 The L asset was risk free The S asset has a higher variance but pays the same on average We have an intuition that riskier assets should have a higher rate of return But7 here we see a riskier asset with the same rate of return as a riskfree asset The relevant question what is the right way to measure risk Clearly in this problem7 assets L and S have the same amount of risk uppose you have access to stock in Honda and GMi The expected real returns are identical You work for Honda Which stock do you prefer to own 111308 Today Asset pricing and Consumption CAPM1 Model philosophy 9 Consumption Asset Pricing Recall EXY EXEY CO UX Y If we want to price any asset the Euler Equation tells us 1 warm WW which is equivalent using the above formula to 1 EARMEML SH comaMam u u 0 Use the risk free asset BL which has a rate R2 Ezwiulc uCz and multiplying both sides of the general equation by BL we get Rn Eleizl RLtCO39URit7 BM WW We can write this as 7 7 uCt1 Ethztl RLE1 COURztv uct or the Important Result Elma 7 BL iRLtcovRit u1LC1 Notice that what we are saying is that we need to know just two things in order to price any asset in this economy Even though we made no assumptions about the distributions of returns we can characterize them using just the rst 0 o si Additionally this equation gives us the Risk Premium which is the economi cally correct notion of risk It describes how risk interacts with uctuations in consumption Assets which covary negatively with consumption have a higher expected return This is sometimes called the Consumption Capital Asset Pric ing Model or Consumption CAPM1 The standard CAPM is not intertemporal and makes some assumptions on returns 10 Random Walk Theory with a little Model Philosophy Now let s think of an asset that pays a constant rate so that Rm 571 If such an asset exists it implies that uCt EtluCz If we suppose that the marginal utility of consumption is linear so that u ct kc then consumption becomes a random walk Ct EL CH1 This model is due to Robert Hall called the Random Walk Theory of Con sumptioni He proposed it in 1978 He found that taking the rst difference of the consumption series he got no correlation across time There was some evidence that changes in stock prices could explain it perhaps due to liquidity constraints The model does a remarkably good job in general Although it there are places it doesn7t they are not of rst order We could ask is it a good7 model Notice that we are missing alternative formulations that could explain this phenomena Consider for example the macro 102 model ctbYt Observed output is also a random walk so this model could explain consumption as a random walk as well The point here is that inferences are sensitive not only to the model and data but also to how well alternative formulations can explain phenomenal This is a big conceptual issue in empirical work How can we compare alternative models It also relates to identi cation looking for a dimension in which two different models have different implications All models are wrong but some are useful7 1 1 Stock Pricing Let7s price the stock s Different from the previous asset S Using our marginal cost marginal bene t analysis PstuCt dstuCt Etl uCEIP5E1l where d5 are dividends paid out in the same period Then uCt1 P d E 7P 5 5t B t um st1 lf agents are risk neutral we get our original constant discount dividend stock pricing mo e i 111808 Today Developing consumptionbased asset pricing with a consumer maximiz ing over a budget constraint a tax application and some theories on consump tion 12 The Model 11302515401 subject to 140 yr Az1 RAz y i 0 1n words a given initial asset endowment an income sequence with known stochastic process and a rule for asset holdings that depends on a constant rate of return Note that asset holdings are not required to be positive However we need to rule out the case of A 700 sometimes called Ponzi schemesi Possibilities are A 2 0 which would be a no borrowing constraint but it s a bit too strong for our taster Welll use the condition lianDo R TAT 0 This implies an intertemporal budget constraint of E0 213515 A0 E0 Zp yj j0 j0 A0 is called tangible wealth and the income sequence is intangible wealthi Think of it as potential future earnings Notice that B does not appear since the budget constraint depends only on market prices not subjective preferencesi R j is a date zero price for output at ji This lets us pin down the level of consumption 121 Application Taxes Suppose there exists a lump sum tax Tj which affects the consumers income so that total wealth is now A0 E0 20 R jyj 7 Tji Notice that income is exogenous at this point Assume that there are no transfers only government consumptioni We want to consider a change in taxes and how it affects con sumption Consider tax in two periods TE and Tt1i Suppose T is reduced by A and that government consumption will not change The government can nance this is three basic ways The rst two which we are not interested in are seignorage printing money and selling assets The third is to nance through borrowing from the public by selling bondsi Government offers 1 and TE1 must be such that the government will pay off the bond in one period What is the effect of the change on the consumer Nothingi He gains 6 in t and loses AR the amount by which taxes increase to nance the bond in t 1 1 It is exactly discounted so for the consumer it has no effect This is called Ricardian Equivalence Now we know7 it not to be true that people are unaffected by the timing of taxation right What is the problem with the model It could be that people face liquidity constraints or that people live nite lifetimes and changes in the taxdebt sequence affects who has to pay for it or it could be that people are not suf ciently rationali However the data suggests that people do consider the issues of the model resulting in a partial offseti7 The model gives insight into that behavior 122 Parenthetical Comments First Ricardian Equivalence says that the level of individual consumption is affected by government consumption just not the nance mechanismi Second this is related to the ModiglianiMiller Thoerem which says that the value of a corporation is unaffected by whether the corporation is nanced through debt or equity abstracting from tax code 13 Big Macro Ideas Idea 1 Keynes C a BY This leads to two important empirical facts in the UiSi the paradox of 57 crosssectionally but 9 intertemporally and the paradox of the dog that did not bark7 that there was no post WWH depression in the Uisi despite the huge increase in worker supply from men who had been in the army Idea 2 Friedman s Permanent lncome Hypothesis individuals consume a con stant part of their lifetime incomei Only changes in permanent income affect consumption Idea 3 Life cycle hypothesis Modigliani we observe individuals choosing dif ferent asset holdings over time negative positive draw down before retirement The modern form of the life cycle hypothesis is uct Et Ru CH1 and 00 00 E 2P ij A1 E 2ij jt jt which together de ne a unique consumption sequencer Recall our discussion of consumption as a random walk lf BR l and uct ken consumption is a random wa Now we know that still need the budget constraint to be satis ed in order to translate into consumption decisions consistent with utility maximization 112008 Today Finishing the consumption asset pricing in nite lifetime model intro to Overlapping Generations Models 14 Finishing ConsumptionBased Asset Pricing Recall To characterize the consumption sequence need both the Euler equation and the intertemporal budget constraint Letls assume With loss of generality that 1101 uo ulct 7 go We Will restrict ourselves to the part of the function Where the marginal utility is positive This functional form makes uct ul 7 ugct Then ui 1120 Et Rlul u2 z1l Rearrangin g EtcHl K0 510710 Where K is a constant we aren7t interested in This gives us a Markov process in consumption current information is suf cient to compute expected future consumptioni General form Ezcj1 Kj 6139715 Now take the budget constraint and compute E Z Pch E 2 Fri K BR 15 j0 j0 K Zlt R2ith j0 K 10 1 7 7 332 implying the Important Result 1 7 v Ct 1 A EthJLHjl This looks similar to what we were considering in the time series part Con sumption depends on a constant tangible and intangible wealth How could we test it Need the relevant data and the stochastic process for y Could use W K for prediction part and the HansenSargent formulation Suppose that BR 1 and call the whole of wealth wt Then wt 7 00R wt1 Suppose we wanted to consume so that wealth was unchanging so that wt 7 00R wt making 7 R 7 1 cl 7 th You could call this the modern version of the Permanent lncome Hypothesis Its the ip side of the random walk ideal The random walk theory was incomplete since it did not consider the budget constraint We get the result that trying to keep wealth constant implies trying to keep consumption constant when BR ll 15 Intro to Overlapping Generations Comment It can be tricky to write down monetary models Monetary equilibria tend to unravel at a termination point Money has value only because people think it does Demography of model One agent is born at each time period starting at t 71 01 00 Each agent lives for two periods including period born This makes the population constant at any time 2 Endowments for individuals are 0 so the agent only gets endowment in the rst period of life The endowment is perishable Denote Cy as consumption of young person at time t 00 as old person Total lifetime utility is ucyz ucot1 for an agent born at time t Assume u gt 0u lt 0 and note that the total supply of output is constant Consider the problem of agent li At period 0 no one will trade with him because he has nothing of value The same thing happens looking at any two agents at any time t So the allocation where Cy cot1 0 for all t is the unique competitive equilibrium with no trade uppose a social planner takes the endowments of everyone and splits them Guy 1 gets to consume 2 and the rest get 2 2i Since utility is concave u lt u 2 u 2 so as long as there is no termination date the planners allocation Pareto dominates the competitive equilibrium Why What is messing up the rst welfare theorem lt can7t be incomplete markets since agents are not disallowed from trade they just do not choose to make the trades Recall the rst welfare theorem needs equilibrium prices to not generate in nitely large budget sets Distant prices don7t have to get smaller in this model since the commodities today are consumed by a different person from those in the distant future Final comment This model is suggestive of how a social security system can be welfareenhancing In t e i i the Social Security program is pay as you go7 so that the current working population supports the current retired populationi You can think about What happens if the population growswsocial security can actually pay interest 112508 Today OLG and Money 16 Money Consider the same context from the last lecturer Is there a way to introduce the good equilibrium without the tax and transfer scheme Let7s consider M at money and endow it to the 1 guy M has no intrinsic value it s only a piece of paper Then i coil and i cop where the subscripts are 0 old y young date borni Changes in the price level will determine rates of return Suppose prices are at a steady state and E E or all t so pt 2M implying that Cy co 62 and the MUC for youth and age is the same and are equal to the price rationi This is the equilibrium solution If prices are nite this is the only possibilityi In nite prices are also a steady state Consider what happens when the money supply has an anticipated increase at some point in the future Contemporaneous in ation Suppose endowments change to 16 with e lt 1 so people still want to trans fer from youth to age Preferences are ucyt 00 M is constant The budget constraint of a young person is pt ploy M or 1 Cy Mpti The bud get constraint of an old person is epprl M 1014400 or e Mpt1 co The capacity to transfer consumption is determined by the relationship between prices in the two perio s As a thought experiment suppose we want to move epsilon consumption across t to tli We give up 610 and get Ept1 so pf is the gross real rate of return on money the only asset in this economy Another way to think about the rate of return on money is as the negative of the in ation rate This can help us think about liquidity traps which make money look like an asset worth hold ing and has consequences for the desire of people to hold other types of assets Additionally this is related to the zero bound problem nominal interest rates cannot be negative so it can be dif cult for entities to affect it once it gets too lowi Combining the budget constraints ploy 1014400 pt pt1e but the mech anism is money As an example consider ucyt coquot mfg all taking exponentials giving CobbDouglas formi Suppose B gt 6 so the individ ual wants to transfer consumption The functional form gives us constancy of budget shares 0 This can be transformed 1 1 7 5 1024460 7 15 P Pz1 101534 P PHIE and Market clearing says 634 COt71 1 6 Combining these i 16 1 This is a second order linear difference equation Any price sequence pt 2 0 compatible with it is an equilibrium price sequence The solution is pt k1 k2 is the solution In general pt klA MAE Can we get 161 and kg 161 is determined by the model Set k2 0 and solve for unique pt pt1 lt 00 But all we can say about the other is kg 2 0 so there are a continuum of solutions In summary there is one nite equilibrium and a continuum of solutions which imply that the economy demonitizesi If k2 gt 0 P Pz1 1071 105 1 5 prices are increasing over time so the equilibrium unravels at some point no one will trade This suggests there are limitations to the quantity theory of money which suggests a one to one tradeoff between money supply growth and in ation Historical example currency in the American coloniesi Colonial legislatures printed money and had different money supply rulesi Some would buy the money back in one year which resulted in no relation between changes in the money supply and prices and some would just print money which resulted in changes in the money supply associated with changes in the price level 120208 Today OLG With Production and Sunspots 17 Production Economy with Overlapping Gen erations Setup 0 Demography same as before 0 Preferences At time t get negative utility from work 7907 and at t l uct1 g g gt 0 and standard assumptions on u o Endowments Each generation has N ie labor individuals work when young Generation l has 1 unit money 0 9 production YE 77 constant returns to scale2 An equilibrium in this economy is a set ofprices and allocations p1 y 77 c fio such that 1 each agent maximizes utility subject to his budget 2 All markets clear We know it must be true that yt 77 CE in equilibrium Assume competi tive behavior among agents and perfect foresight of prices Consumers equate marginal cost and marginal bene t and we are using money to trade over time so M 7 M P Pt1 The budget constraint for individuals says that 1 Pth Pt1Ct1 so we can rewrite the MCMB condition as 9 yzyz u ytyt1 which is a nonlinear difference equation rede ne as 1 1 151yz 2yz1 This equation contains both the rst order conditions and the budget constraint ls any sequence of output levels that satis es this an equilibrium Almost Any sequence y 2 0 that is consistent with 11 is an equilibrium Suppose 3 3 such that g uE This would be a steady state Under the assumptions we have made the steady state is unique but the sequence y need not 2Side note on Returns to Scale We can ignore the case of decreasing returns to scale because it is always the result of a xed input or one left out ive entrepreneurship Mostly the CRS assumption is used here to avoid dealing with the redistribution of pro ts and because increasing returns would take us places we don7t want to go 19 171 Sunspots Are there stochastic equilibria Possibly Even though there is nothing in trinsically uncertain we cou ave extrinsic uncertainty leading to sunspot equilibrial William Stanley Jevons wrote the original 1878 paper that sug gested business cycles were related to sunspots literally dark and cool spot on sun due to something related to magnetic elds The idea of a sunspot in economics is that there is some random variable that affects outcomes but not L t likel f t and techhnlngv Where is there room for a sunspot here At time 0 suppose you are not sure whether or not the person following you will work the full amount you think there is some chance they might not By changing people s beliefs we can change their actions People can have ratio nal expectations and react to extrinsic variables although hasn t held up well empiricallyi Consider the problem of maximizing EJ757070 uCt1l which leads to 9 77 Etu0r1 Pt Pz1 We7ll assume all markets are spot marketsi Earlier identities still hold so Ezlyz1u yz1l Qty9 which is now a nonlinear stochastic difference equation Any stochastic process that is consistent with this such that y 2 0 is an equilibriumm This is similar to the idea of bubbles There is a particular type of random variable that can be introduced to the model that is still consistent with it Here output becomes stochastic because people are not perfectly predictable Example 2 possible output levels yH yL that follow a Markov chain described by An allocation that satis es yHgyH qHyHuyH1i qHyLuyL and ngyL quLuyL 1 7 qLyHuyH with yH yLqH 4L 2 0 is a sunspot equilibriumi Essentially every agent is ipping a coin with probabilities q to decide their effort Since everyone is doing it mixed strategies also make sense for everyone else Notice there are two equations four unknowns so it is within the realm of possiblityi 20 120408 Today Control Theory 18 De nitions De nition Neutrality of Money if ml and Amt produce the same real variables we say that money mt is neutral De nition Superneutrality of Money real variables are una ected by changes in the steadystate growth rate of money we say that money is superneutral 19 Control Theory State Equation 1 ALzt1 BLvt e e WLwt Where wt is White noise Feedback Rule 1 7FLzt Note that we are not trying to change steady state outcomes but considering relative to What process looks like how can the policymaker stabilize We Will be controlling the second moment 191 Example Single Input Single Output I axial bvtil w With U fIz Then I a bf1t71 wt and varzt a 7 bf2varzt varwt To minimize the variance of output f ab implying It w This rule removes dependence in the systemi Let7s contrast With the laissezfaire solution Where we don7t control f 0i rive axial wt We also suppose a gt 0 Then 0 fT NCW 27r1 7 2acosw a2 21 a la Friedman Under the optimal rule 2 011 w i fm gt 2 If you draw the picture you know that the area under the rectangle has less area since the integral of the spectral density is the variance and the optimal rule has smallest variancei But notice that the spectral density of optimal variance minimizing policy represents a tradeoff between the variance contributions of frequencies higher at some and lower at others Stabilization decreases the variance contributions of low frequencies and increases those of high frequencies Let7s think of some loss functions um mom and 7f Fwfxwdw where lquotw lquot7w 2 0 is increasing in Notice that lsteven gives differ ent weights to different frequencies Loss functions require preferences to justify them lneven can be justi ed using preferences of habit persistence uctcht1i This need for choosing a loss function can be a place to introduce model uncer tainty 192 Sensitivity Analysis Can we uniformly improve Pick a rule that does not have to trade off across frequencies No Call It If if FL 0 else rival Then W fzncw 2 1 7 eiiwAeiiw 2 and w 6440 2a2 1 161 l 2 w 4m 440 7m 4m 4w 2 llie Ae e Be Fe To compare them write 1 7 eiiwAeiiw 2 Alteiiweiiw eiineiiwFeiiw 2 namhmwm De ne 5a a sensitivity function as fxcw fxnclSWl2 We are looking at the control as shaping the spectral density of the non con trolled We7re assuming everything is nicely behaved of course so spectral den sities existi Can we choose 5w 0 No it isn7t feasible since wt is a shock to the state and the policy only works on lagi We can never take away all of the variance This suggests that not all functions S are admissablei 22 Theorem Bode Integral Constraint If INC is zeromean second order sta tionary fir longwl2dw 0 is required Implications Does there exist 5a such that lSwl2 lt 1 so that we could uniformly reduce variancel No It makes the log function negative all the time7 which could never integrate to zero Thus there is no uniform acrossfrequency variance reducing policy aka no free lunch All policies induce tradeoffsi Note that free lunches can become possible in a nonstationary environment7 or if the state equation is forward looking depends on future expectations rather than backward lookingi What is nonstandard here is using the frequency domain as a way to think about policy 12908 Today Finishing control theory relaxing assumptions Important Note Notes from last week have been corrected to adjust for a correction made in lecture today We began with a review from last lecture which I leave out 20 What Happens When We Relax Assump tions 201 What if alvc is nonstationary Suppose that rive is explosive or has a unit root What can the policymaker do Recall that we can factor the Fourier transform of the lag polynomial as 17 Ae i HQ 7 AJARe i quot lf AJAR lt 1 for all A then everything is stationary Nonstationarity matters for policy design since explosive or unit root processes have in nite variance The policy maker will then have to follow a twostep process rst eliminating the nonstationarity and second shaping the now stationary process Does the rst step affect the second Theorem General Integral Constraint For any process described by the model fj logl5wl2dw 4n E A leoglAJARl is required This implies that o it Ml 1 doesn7t matter only explosive roots do 0 iii The integral is 2 0 since it is summing over the log of things 2 1 Since we prefer the integral to be as small as possible to minimize variance we need to take care of the explosive parts rst The division of processes here is between those which are stationary or difference stationary and those which are explosive 202 What about expectations So far we have been considering only backwardlooking models Can we think of models that incorporate future expectations Let I EtIt1 AL1271 BLU271 6 This is called a hybrid model since it has both forward and backward looking parts Theorem Hybrid Model Integral Constraint For these processes if longwl2dw It must ho Points to make 0 ii kf may be negative This means we could achieve a uniform variance reductioni It works because we now can change how people think about the future instead of just changing the temporal dependence structure 0 iii Whether the st WLwt is the fundamental MA or not mattersi When incorporating expectations the information gap becomes important ie whether the w7s are structural or fundamental shoc s 203 Finally Why do we care Three reasons 0 1 1250 summarizes all effects 0 2 Remember 1 fir lquotwf Maybe there is more to the policy makers problem than minimizing variance Depends on preferences 0 3i Integral constraints are informative even if the goal is to minimize variance 204 Example Suppose five w existsi Temporarily name lswl2 pwi The problem is to choose the minimum pw of the lagrangian f powwow MKB 7 f logpwdw In this case KB 0 If we take the FCC fich APWVI 0 implying that under variance minimization pwf mw A for all w or in other words ff A so the general principle is that the spectral density of the controlled process should be white noise If we care about some frequencies more than others the problem changes and the spectral density of the controlled process may no longer be white noise Economics 712 Steven N Durlauf Fall 2007 Lecture Notes 3 Moving Average and Autoregressive Representations of Time Series In these notes some important ways of expressing linear time series relationships are developed 1 Lag Operators The representation and analysis of time series is greatly facilitated by lag operators De nition 31 Lag operator L is a linear operator which maps Ht x onto HHx such that Lxt x27 To see how concept simpli es time series notation using lag operator notation the moving averaging representation of a time series may be expressed as x aLgt where aL Zetij aL is an example ofa lag polynomial j0 Lag polynomials ful ll the basic rules of algebra in the sense that there is an isometric isomorphism1 between the Hilbert space of lag polynomials whose metric is the squared sum of coef cients and the space of algebraic polynomials 1Two spaces are isometrically isomorphic if there exists a one to one mapping between them that preserves the distances between elements so that the distance between any pair of elements in one space equals the distance between the mappings of the two elements in the other space 714 7124 02 00 016 cze a e 7239 As a result restrictions on operations with lag polynomials are straightforward to identify This equivalance between the properties of lag operators and complex polynomials is a consequence of the fact that the Hilbert space I2 which is de ned as the Hilbert space generated by all square summable sequences of compleX numbers m1 mH1 endowed with the inner product m n 0 mfg is 140 mm 171 isometrically isomorphic to the Hilbert space of L2 functions generated by compleX valued functions f m defined on the interval a e 7239 7239 2 and endowed with the inner product g fagada The key idea is that each element of 12 maps to an element of L2 in a way that preserves distances is formalized in the RieszFischer Theorem Notice that I state the theorem in terms of an orthonormal basis of L2 the normalized compleX eXponentials This orthonormal basis is the foundation of the theory of Fourier analysis which I will not develop further here Theorem 31 Riesz Fischer Theorem Define the sequence of orthogonal polynomials relative to the L2 norm Mag m 31 and denote c as any square summable sequence c1c0cl Then there eXists a function f m e L2 such that z a 1s sometrmes known as a frequency ii the in nite sum f w icy a2 arbitrarily well approximates fa in the 10 sensethat f fc 0 iii Any two series 0 and d are distinct if and only if the in nite sums f and f are also distinct ie f d 0 no iv f2ch2 1w As a result restrictions on operations with lag polynomials are straightforward to identify For example the product of aL and L corresponds to the product of ae39 quot and e39 quot Of particular importance one can take the inverse of aL if and only if ae is invertible When will this polynomial have an inverse To answer this 714 question recall that by the fundamental theorem of algebra ae can always be factored as the product of simple polynomials ie there exists a representation such that aequot Hl lequot A simple polynomiall te39W has an inverse if llt 1 Hence k ae39 quot possesses an inverse if Il7 ltl Vk I will return to the question of when lag polynomials may be inverted 2 z transforms Working with infinite sequences is cumbersome so it is valuable to be able work with functions that correspond to them For a given sequence one such function is its 2 transform The ztransform of any sequence 7239 j is defined as 7TZ i jzj 32 where z 6quot The Z transform is simply another way of describing elements of L2 but is particularly easy to work with Notice that the 2 s are orthogonal but not orthonormal In order to recover the original sequence of coefficients from the z transform one can employ the formula 63gt The z transform of the autocovariances 032 aj39z 34 jrno summarizes all second moment information in the time series Notice that this transform may not exist for all me 7r 7239 ie the function may be unbounded for some frequencies The z transform az similarly fully characterizes the Wold moving average representation The relationship between 03942 and az is important from the perspective of inference as it is only the autocovariances which are observable from the data Time series data are simply realizations of a stochastic process One can compute sample autocovariances which under mild regularity conditions will represent consistent estimates of the population autocovariances Our question is how to use these estimates to identify the moving average parameters of the process As a preliminary to establishing this relationship the following theorem establishes a relationship between any set of MA coefficients3 and the associated autocovariances of a time series 3This means that the theorem does not assume that the moving average representation is the fundamental one Theorem 32 Relationship between autocovariances and moving average coef cients If x Z 7177271 then 0162 ayzy27l 1w pf Clearly no E7L7727L7727k02 2 71M 35 jroo Therefore the z transform of the autocovariance function is 0212 7171716216 0 Z Z 712717162167 154 jroo kroo jroo 36 2 m2 i 721 027Z7Z 1 This factorization as noted before did not require that the 772 s were fundamental for the x2 process In general the autocovariance function of a stochastic process is consistent with an in nity of MA representations Intuitively this is obvious since there is no unique orthogonal basis for Htx For a speci c example suppose that the autocovariance function of x equals 42391 17 42 It is straightforward to verify that this autocovariance function could have been generated by a moving average representation of x described by 37 2 w1th oquot 16 or by a movmg average representatlon x2 772 477271 38 where 0311 Both are consistent with the autocovariances embedded in the z transform How can we tell whether either one is the fundamental representation 3 Fundamental moving average and autoregressive representations The relationship between the ztransform of the autocovariances and the fundamental MA coefficients can be best understood by considering the relationship between the moving average and autoregressive AR representation of 9 defined as Lxz 82 39 1 where o 1 If 0L is well de ned then the AR polynomial is immediately generated by the inversion of the MA polynomial whose existence and uniqueness is ensured by the second Wold theorem Consequently if we can identify the fundamental MA polynomial and it is invertible we can identify the AR polynomial This makes intuitive sense since if an AR representation exists this will define the projection 9cm1 Repeating an earlier argument the Fundamental Theorem of Algebra any MA polynomial 02 may be factored such that cz111 2z 310 which in turn means that the autocovariance generating function can always be written as azo 111 2z111 1kz 1 311 We assume that 1 Vk Now suppose that lt l Vk This means that the polynomial 12l lz is invertible Hence this must be the fundamental MA representation as it is the MA representation associated with the unique AR representation Conversely suppose that some of the roots are greater than one in magnitude Without loss of generality suppose that only 21 is greater then one Rewrite the 032 as a z a1 21 212g1 1zg1 1z 1 312 Now consider the following trick Since 1 21 2124 212 1 21lz1 11lz1 313 039X 2 may be rewritten as a 2 0 31 1 A z1 21451g1 1kzr11 1kzl 314 1 The polynomial l l39lzlllll lkznow possesses all roots inside the unit circle and therefore is invertible Hence this is the fundamental MA representation Notice that the nonfundamental innovation variance is 039 whereas the fundamental innovation variance 2 2 1s 03121 Therefore by 1pp1ng the roots of any MA representation ms1de the un1t c1rcle one can generate the fundamental polynomial structure Another way to interpret the transformation of the nonfundamental MA polynomial is that it is multiplied by terms of the form 1k1 152 PM 315 whenever I gt 1 What happens when llkl l This occurs for example when 9 gt 8H 316 In this case the representation is fundamental yet there does not exist an autoregressive representation with square summable coefficients Intuitively the projection of x ontoHH x lies in the closure of the linear combinations lagged x s used to generate the Hilbert space Therefore if the fundamental MA representation of a process is such that for at one l then an autoregressive representation does not eXist Taking these arguments together one can conclude that a given MA representation is fundamental if S l Vk


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