III. Module 3. Empirical and Theoretical Techniques

Transcription

1 III. Module 3. Empirical and Theoreical Techniques Applied Saisical Techniques 3. Auocorrelaion Correcions Persisence affecs sandard errors. The radiional response is o rea he auocorrelaion as a echnical difficuly o be "correced," raher han evidence of possible model misspecificaion. The correcion is o ransform he daa such ha he error erm of he resuling modified model conforms o he OLS assumpion of no auocorrelaion. This generalized leas squares (GLS) ransformaion involves "generalized" or "quasi" differencing. This correcions have he following form:

2 Model: Y = β X + ε Persisence: ε = ρ ε + µ - Correcion: Y - ρy - = β( X - ρx - ) + ( ε- ρε - ), and, Y (- ρl) = βx (- ρl) + ε (- ρl) where L = ( - ). Or in Marix Form: β GLS = (X Ω - X) - X Ω - y, where : Ω can be represened as :

3 Ω = 2 ρ... ρ ρ ρ wih he " ρ" correcion: Ω - = - ρ ρ ρ ρ ρ 2

4 3.2 Error-in-Variables Regression Affec on Parameers (aenuaion and flucuaion) Consider he following: y = ˆ β x + u,, x = x + v *,, y = ˆ β x + u *,, where: ˆ β = Σx Σ, 2 x, y

5 = Σ x ( ˆ β x + u ) *,,, 2 Σx, Σx x Σ u = β + Σ Σ *,, x,, 2 2 x, x, (**)

6 * * * * 2 2 2, * 2,,,, The asympoic resuls: plim plim plim 0 Subsiue hese resuls ino (**): ˆ plim A regression now becomes: v x x x v x x x x x u y σ σ σ σ β β σ σ Σ = + Σ = Σ = = + * * 2,, 2 2 x v x x u σ β σ σ = + +

7 3.3 Two-Sage Leas Squares Simulaneiy leads o correlaion beween he error erm and independen variables. OLS coefficiens are biased and his can be manifesed in parameer insabiliy. Two-sage leas squares esimaes insrumens ha are independen of he error erm.

8 Social, Behavioral, and Economic Modeling Techniques 3.4 Condiional Expecaions In conras o a mahemaical expecaion, which is a summary measure (expeced value), a condiional expecaion is mahemaical expecaion wih a modified probabiliy disribuion ( informaion se ).

9 3.5 Forecasing Using Condiional Expecaions Forecas a random variable, y +, based on a se of observed variables. If we assume a quadraic loss funcion and le y + denoe he forecas. The loss funcion or MSE (mean square error) is: E y + y + 2 And we assume he forecas used is he one wih he smalles MSE: y + = E (y + x ) 3.6 Linear Projecion Forecass Use linear funcions of x o make forecass: by + = α 0 x, where α 0 is x m and x mx. If we assume "orhogonaliy" hen by + = P [y + x ]= α 0 x, which is he linear projecion of y + on x.

10 Secion 3.7 o 3.0 follow from Sargen (987) 3.7 Leas Squares Projecions If y + and x are ergodic and saionary hen he leas squares esimae is he sample esimae analogue for α. 3.8 Recursive Projecion Formula Recursive projecions allow one o sequenially esimae a process. A ime, we have daa only for a variable(s), bu as new daa (variables) comes along, we can updae using he following formula below. For wo variables (m = 2) he formula can be wrien: y = P [y,x,x 2 ]+ε, y = a 0 + a x + a 2 x 2 + ε If we assume ha orhogonaliy (i.e., E (ε) = 0,E(εx )=0,E(εx 2 )=0) hen he parameers a i are he leas squares parameer values. Recursive projecions can aid in sequenial learning in he following way. Assume ha daa only for x bu we wan o consider including new daa which in his case is x 2. 2

11 We would use he following recursive projecion which, assuming orhogonaliy, is akin o opimal leas squares learning (sequenial esimaion). The formula is: y = P [y,x ]+P [(y P [y,x ]) (x 2 P [x 2,x ])] + ε This formula ells us wheher he forecas error in predicing y can be improved upon by adding x 2. This formula can be generalized o mulivariae seings: P [y Ω,x]=P [y Ω]+P [(y P [y Ω]) (x P [x Ω])] 3.9 Law of Ieraed Projecions If we projec he random variable P [y Ω,x] agains Ω we ge he following relaion: Which is updaed via: P [P (y Ω,x) Ω] =P [y Ω] P [y Ω,x]=P [y Ω]+a(x P [x Ω]) Where he parameer a is a leas square parameer conaining he usual definiion. 3

12 3.0 Signal Exracion Using wha we have above now assume ha we wan o a leas squares esimae of variable S, bu only see observe he variable X. The wo variables are relaed in he following way: x = s + n From wha we know above we have: P [s,x]= a 0 + a x The parameer of ineres is a which (following leas squares) is definedinhefollowing way: a = Es 2 Es 2 + En 2 This resul is similar o he error in variables problem bu driven now by condiional expecaions and he various projecion formulas above. 4

13 3. Mehod of Undeermined Coefficiens and MSV Closing he model involves aking unknown variables (i.e., expecaions) and expressing hem in erms of oher known variables. The mehod of undeermined coefficiens is a paricular soluion process ha closes a model. A minimum sae variable (MSV) soluion is he simples soluion when using he mehod of undeermined coefficiens. 3.2 Adapive Learning Forecas funcions and he esimaion of heir parameers are explici. The expecaions and forecas funcions influence fuure daa poins. An adapive learning approach has he following feaures: o Find he raional expecaions equilibrium (REE). o Devise how agens forecas a variable of ineres --- he perceived law of moion (PLM).

14 o Subsiue he PLM ino he equaion of ineres and obain an acual law of moion (ALM). o This subsiuion yields a mapping from he PLM parameers ino he ALM parameers. o Only a he REE values do he ALM parameers equal he PLM parameers. Sabiliy condiions (E-Sabiliy) o Sabiliy condiions are needed o deermine wheher he REE is he sable oucome of a process in which PLM parameers adjus slowly oward he ALM parameers. o E-Sabiliy condiion is a condiion which governs wheher or no a given REE is sable. o Evans (989) defines he E-sabiliy condiion in erms of he ordinary differenial equaion (ODE): dθ = T dτ ( θ ) θ, where θ is a finie dimensional parameer specified in he PLM, T ( θ ) is a mapping (T-mapping) from he PLM o ALM and τ denoes noional ime. o E-sabiliy corresponds o local sabiliy of he REE under hese dynamics.

15 o Local sabiliy under adapive learning, or E-sabiliy, provides a selecion crierion in models wih muliple RE equilibria. 3.3 Leas Squares Learning Leas Squares Learning and Recursive Sochasic Algorihms Agens updae heir forecass using Recursive Leas Squares (RLS). The RLS formula is: ( ) ( ) ( ) ( ),,,,,,,,,,, ' ' + + = + + = i i i i i i i i i i i i i R z z T R R z y z R T ϕ ϕ ϕ

16 NOTE: To show he Convergence of he Esimaes, we need o pu he RLS in he form of a sandard Sochasic Recursive Algorihm (SRA): where: ( X ) θ = θ + γ Q θ, i,,, ' θ ( vec( ϕ ) vec( R )), X = ( z, z, η ) = i, ' i, + and = ( + ) Ti γ. i, i, i,

17 Module 3: Adapive Learning Basics The Model Equaion : Y = α Y +ρy +ζf +v. () Equaion 2: F = α F +γ A E A +λw +η. (2) Equaion 3: A = α A +δa +φf +βe A +ε. (3)

18 Raional Expecaion Soluion Agens use all available informaion up o ime (denoed E ): E A = α A +δa +φf +βe A. (4) The raional expecaions equilibrium (REE) is hen: E A =( β) α A +( β) δa +( β) φf. (5) 2

19 Specifying A Learning Mechanism Our unique raional expecaions soluion for (3.3) is: A = a RE A + b RE A A + c RE A F + ε, (6) where a RE A =( β) α A, b RE A =( β) δ and c RE A = ( β) φ. 0.. The Perceived Law of Moion Our nex sep is o specify how agens forecas hevariableofineres. The ideal case (he raional expecaions equilibrium) is when he agens forecasing equaion (he perceived law of moion) correcly predics he variable of ineres. We assume agens do no correcly specify heir likelihood funcions. They use an incorrec forecasing equaion. 3

20 Alhough he likelihood funcions do no correcly specify he process generaing he daa ha agens receive, in some cases he model ha agens use during he learning process includes he raional expecaions equilibrium. We assume ha agens updae heir forecass, in a way ha mimics leas squares, up o he period. Agens updae each period hereafer (Bray 982). Equaion (7) expresses he perceived law of moion (forecasing equaion): A = a A, + b A, A + c A, F + u (7) = θ 0 Z + u, 4

21 where θ = a A, b A, c A,, Z = A F, and u iid(0,σ 2 u). Subsiuing (7) ino (3) gives he acual law of moion: A = (α A + βa A, )+(δ + βb A, ) A +(φ + βc A, ) F + ε (8) = T (θ ) 0 Z + ε, where T (θ )=T α A + βa A, δ + βb A,, Z = A, and ε iid(0,σ 2 ε). φ + βc A, F 5

22 Condiions For Learning: E-Sabiliy DeCanio (979) and Evans (985, 989) devise a condiion under which (8) maps ino he raional expecaions equilibrium (6). Evans (989) defineshiscondiion,knownas expecaional sabiliy (or E-sabiliy), by he following ordinary differenial equaion: dθ dτ = T (θ) θ, (9) where θ is a finie dimension parameer specified in he perceived law of moion (7). T (θ) is a mapping (so-called T-mapping) from he perceived o acual laws of moion, and τ symbolizes eiher virual or arificial ime. The raional expecaions equilibrium, θ RE,corresponds o fixed poins of T (θ). 6

23 In all cases, we base he es for learning, in he limi, on he sabiliy and almos sure convergence of he perceived and acual law of moion parameers o he raional expecaions parameers. Applying he E-sabiliy definiion above, we use (8) o see if (7) converges o (6). The resul is: T a A b A c A α A + βa A = δ + βb A. (0) φ + βc A To deermine he E-sabiliy condiion, we combine (9) and (0) and obain he associaed ordinary differenial equaion (differeniaed 7

24 wih respec o ime (τ)): d dτ a A b A = T a A b A a A b A, () c A c A c A and. a A = da A dτ = α A +(β ) a A, (2). b A = db A dτ = δ +(β ) b A, (3). c A = dc A dτ = φ +(β ) c A. (4) The E-sabiliy condiion is saisfied when β<. As long as his condiion holds agen s are able o learn (using leas squares) he REE in he long run. We summarize he E-sabiliy condiion in he following proposiion: 8

25 Proposiion. Consider (2), (3), and (4). If he E-sabiliy condiion is saisfied, β <, hen he laer erm vanishes in he limi: here is local convergence beween he perceived (7) and acual (8) laws of moion o he raional expecaions equilibrium (6). 9

26 Leas Squares Learning An added feaure of his resul is ha i is conneced wih leas squares learning. Leas squares learning, in he form of recursive leas squares (Hendry 995), can serve as a direc es for learning (Bray and Savin 986) and convergence o he seady sae. To demonsrae how agens can discern he campaign sraegis equaion, we sar wih he daa generaion process (DGP) or rue model (3): A = α A + δa + φf + βe A + ε. (5) Again, we assume ha he perceived law of moion for agens is: A = a A, + b A, A + c A, F + u, (6) 0

27 where u iid (0,σ 2 u). More compacly, A = ψ 0 z + u, (7) where ψ 0 µ a A, b A, c A,. If we subsiue (7) ino (6), he acual law of moion is: A =(α A + βa A, )+(δ + βb A, ) A +(φ + βc A, ) F + ε, (8) or in simpler erms: A = T (ψ ) 0 z + ε. (9) We assume he agen s use recursive leas squares o updae heir parameer esimaes of he (9):

28 ψ = ψ + R z A ψ 0 z (20) R = R + z z 0 R, for some appropriae values of ψ 0 and R 0. Our parameer updaing mechanism is a sochasic recursive algorihm: θ = θ + κ Q (θ,x,), (2) where θ 0 = vec (ψ ) 0,vec(R + ) 0, X 0 = z 0,z 0,ε, and κ =. 2

29 Sabiliy Condiions We esablish convergence properies of his algorihm by using he following ordinary differenial equaion and he E-sabiliy condiions esablished in (9) (4): and dθ dτ = f (θ), (22) f (θ) = lim EQ(θ, X,). (23) The associaed ordinary differenial equaion is: dθ dτ = T (θ) θ. (24) Now, le Ez z = Ez z = E A µ A F = F 3

30 Ω 2 Ω 3 Ω 2 Ω 22 Ω 23 M Ω 3 Ω 32 Ω 33 wih lim = and Ez + ε =0. The associaed ordinary differenial equaion in he model becomes: dψ dτ dr dτ = R M (T (ψ) ψ) (25) = M R. We have global sabiliy if R M (and if R is inverible), and R M I from any saring poin. Then: d dτ a A b A c A = T a A b A c A a A b A c A, (26) 4

31 or d dτ a A b A = α A + βa A a A δ + βb A b A. (27) c A φ + βc A c A 5

32 Model Illusraion Now ha we have esablished he condiions for expecaional sabiliy and convergence, a real or virual ime illusraion for learning is sraighforward. We examine convergence of he parameers o he raional expecaions equilibrium as. We are ineresed in wheher agens learn he raional expecaions equilibrium of A equaion (6). To demonsrae his we ake he self-referenial expecaions of (7): E A = a A, + b A, A + c A, F. (28) Implici is he assumpion ha agens have 6

33 he abiliy o learn equaion (3) parameers over ime equaion by using a A,, b A,,andc A,. 7

34 Resuls For he rue model (DGP) of (3), he illusraion conains he rue parameers: α A =5,δ = 0.5,φ = 0.5, and β = 0.5. ε is an unobservable normal whie noise process wih sandard deviaion one. The simulaion saring values for he perceived law of moion (7) are a A, =4,b A, =4,c A, = 4. We simulae equaions (7), (8), and (20) wih hese saring parameers. The simulaions have a virual ime period of 0,000. From (6) he raional expecaions equilibrium values a RE A,b RE A,c RE A are (3.33, 0.33, 0.33). If our model is correc, hen he perceived law of moion and he acual law of moion should have values ha converge o he raional expecaions values. 8

35 The resuls for he perceived law of moion show ha (a A,,b A,,c A, ) (3.28, 0.34, 0.32). Convergence o he raional expecaions equilibrium occurs for all parameers in roughly 200 ime periods. 9