Study the bias (due to the nite dimensional approximation) and variance of the estimators
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- Herbert Shannon Thompson
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1 2 Series Methods 2. Geeral Approach A model has parameters (; ) where is ite-dimesioal ad is oparametric. (Sometimes, there is o :) We will focus o regressio. The fuctio is approximated by a series a ite dimesioal model which depeds o a iteger K ad a K dimesioal parameter : Let K () deote this approximatig fuctio. Typically, the parameters (; ) are estimated by a covetioal parametric techique (^; ^): The ^ = L (^) Tasks: To d a class of fuctios K () which are good approximatios to : Study the bias (due to the ite dimesioal approximatio) ad variace of the estimators Fid optimal rates for K to diverge to i ity Fid rules for selectio of K Show that ^; ^ are asymptotically ormal. Asymptotic variace computatio, ad stadard error calculatio. Data Trasformatio: Typically the methods are applied after trasformig the regressors X to lie i a speci c compact space, such as [0; ]. 2.2 Regressio ad Splies Take the uivariate regressio y i = g (X i ) + e i I this case, = g: Series Approximatios: power series (polyomial) works for low order polyomials ustable for high order polyomials trigoometric (si ad cos fuctios) bouded fuctios ca produce wiggly implausible oparametric fuctio estimates splies 89
2 piecewise polyomial of order r cotiuous derivatives up to r cubic splies popular joi poits (kots) ca be selected evely, or estimated 2.3 Splies It is useful to de e the positive part fuctio (a) + = max [0; a] = 8 >< >: 0 a < 0 a a 0 Liear, quadratic ad cubic splies with kots at t < t 2 < < t J are JX g K (x) = 0 + x + +j (x t j ) + j= JX g K (x) = 0 + x + 2 x j (x t j ) 2 + j= JX g K (x) = 0 + x + 2 x x j (x t j ) 3 + This model is set up so that it is everywhere a polyomial of order s; with cotiuous derivatives of order up to s, ad the s 0 th derivative chagig discotiuously at the kots. j= Cubic splies are smooth approximatig fuctios, exible, ad popular. The approximatio improves as the umber of kots icreases. The dimesio of is K = J + s: For a give set of kots the fuctio g K is liear i the parameters. De e 0 z = z(x) = x x 2 x 3 (x t ) 3 + (x t J ) 3 + ; the g K (x) = 0 Kz 2.4 B Splies Aother popular class of series approximatio are called B-splies. These are basis fuctios which are bouded, itegrable ad desity-shaped. They ca be costructed from a variety of basic shapes. Polyomials are commo. 90
3 Let X 2 [0; ] ad divide the support ito J equal subitervals, with kots are t j = j=j; j = 0; ; :::; J: We also eed kots outside of [0; ] so let t j = j=m for all itegers j: A r th order B-splie is a piecewise (r )-order polyomial. A liear (r = 2) B-splie base fuctios are liear o two adjacet subitervals, zero elsewhere. They take the form B 2 (x j t j ; t j+ ; t j+2 ) = (x t j ) + 2 (x t j+ ) + + (x t j+2 ) + : A quadratic (r = 3) B-splie base fuctio is piecewise quadratic over three subitervals B 3 (x j t j ; t j+ ; t j+2 ; t j+3 ) = (x t j ) + 3 (x t j+ ) (x t j+2 ) + (x t j+3 ) + : For geeral r B r (x j t j ; ; t j+r ) = rx ( ) s r (x t j+s ) s + : s=0 The B-splie is a liear combiatio of these basis fuctios. g K (x) = JX j= r = 0 Kz j B r (x j t j ; ; t j+r ) where z = z(x) is the vector of the basic fuctios. The dimesio of is K = J + r Estimatio For all of the examples, the fuctio g K is liear i the parameters (at least if the kots are xed). De e the vector Z i = z(x i ) as the sample base fuctio trasformatios. For example, i the case of a cubic splie Z i = X i X 2 i X 3 i (X i t ) 3 + (X i t J ) : From Z i ; costruct the regressor matrix Z: The LS estimate of K is ^ K = Z (Z 0 Z) Z 0 y: The estimate of g(x) is ^g(x) = z 0^K ; that of g(x i ) is ^g(x i ) = zi^ 0 K ad that of the vector g = (g(x ) ; :::; g(x )) 0 is ^g = Z^ K = P y where P = Z Z 0 Z Z 0 is a projectio matrix. 9
4 2.6 Bias Sice y = g + e the E ^K j X = Z 0 Z Z 0 E (y j X) = Z 0 Z Z 0 g = K the coe ciet from a regressio of g o Z: This is the e ective projectio or pseudo-true value. Similarly, E (^g j X) = P g = gk is the projectio of g o Z: The bias i estimatio of g is E (^g g j X) = g K g: If the series approximatio works well, the bias will decrease as K gets icreases. If g is -times di eretiable, the for splies ad power series sup jgk(x) g(x)j O K : x The itegrated squared bias is Z ISB K = (g K(x) g(x)) 2 df (x) O K 2 where F (x) is the margial distributio of X: This is approximately the same as the empirical average X (gk(x i ) g(x i )) 2 = (g K g) 0 (gk g) i= = g0 I P 0 (I P ) g = g0 (I P ) g 92
5 2.7 Itegrated Squared Error The itegrated squared error of ^g(x) for g(x) is Z ISE = (^g(x) g(x)) 2 df (x) ' X (^g(x i ) g(x i )) 2 i= = (^g g)0 (^g g) Sice ^g g = P (g + e) g = P e (I P ) g the ISE K = (P e (I P ) g)0 (P e (I P ) g) = e0 P P e + g0 I P 0 (I P ) g 2 e0 P (I P ) g ad whe P is a projectio matrix (as for LS estimatio) the this simpli es to ISE K = e0 P e + ISB K () The rst part represets estimatio variace, the secod is the itegrated squared bias. If the error is coditioally homoskedastic, the the coditioal expectatio of the rst part is E e0 P e j X = tr P E ee0 j X = tr (P ) 2 = K 2 I geeral, it ca be show that e0 P e = O p Put together with the aalyis of the ISB, we have ISE K O p K K + O K 2 : The optimal rate for K is K = =(2+) yieldig a MSE covergece 2=(2+) : This is the 93
6 same as the best rate attaied by kerel regressio usig higher-order kerels or local polyomials. 2.8 Asymptotic Normality The dimesio of ^ K grows with ; so we do ot discuss its asymptotic distributio. At ay x; the estimate of g (x) is ^g(x) = z 0^K ; a liear fuctio of the OLS estimator ^ K : Let ^V K be the covetioal (White) asymptoticcovariace matrix estimator for ^ K, so that for z 0^K is z 0 ^VK z: Applyig the CLT we ca d p (^g(x) g K (x)) p z 0 ^V K z! d N (0; ) Sice the estimator is oparametric, it is biased, so the estimator should be cetered at the projectio or pseudo-true value rather tha the true g(x): Alterative, if K is larger tha optimal, so the estimator is udersmoothed, the the squared bias will be of smaller order tha the variace ad it ca be omitted from the asymptotic expressio. The bottom lie is that for series estimatio, we calculate stadard errors usig the covetioal formula, as if the model were parametric. However, it is ot costructive to focus o stadard errors for idividual coe ciets, as they do ot have idividual meaig. Rather, stadard errors should be for ideti able parameters, such as the coditioal mea g(x). 2.9 Selectio of Series Terms The role of K is similar to that of the badwidth i kerel regressio. Automatic data-depedet procedures are ecessary for implemetatio. As we worked out before, the itegrated squared error is ISE K = e0 P e + ISB K The optimal K miimizes this expressio, but it is ukow. We ca estimate it usig the sum-of-squared residuals from a model. For a give K; there regressors de e a projectio matrix P; tted value ^g = P y ad residual vector ^e K = y P y: Note that ^e K = (I P ) y = (I P ) g + (I P ) e 94
7 Thus the SSE is Takig expectatios coditioal o X; ^e0 K ^e K = g0 (I P ) g + 2 g0 (I P ) e + e0 (I P ) e E ^e0 K ^e K j X = ISB K e0 P e + 2 g0 (I = ISE K 2 e0 P e + 2g0 (I P ) e + e0 e P ) e + e0 e 2 = E (ISE K j X) E e0 P e j X + 2 = E (ISE K j X) where the secod lie holds uder coditioal homoskedasticity. 2K Thus ^e0^e is biased for ISE K ; but this ca be corrected if we correct for the bias. This leads to Mallows (973) criteria C K = ^e 0 K ^e K + 2K ^ 2 where ^ 2 is a prelimiary estimate of 2 : The scale does t matter, so I have multiplied through by as is covetioal, ad the al 2 term does t matter, as it is idepedet of K: The Mallows estimate ^K is the value which miimizes C K : A method which does ot require homoskedasticity is cross-validatio. The CV criterio is CV K = X i= y i ^g K i (X i ) 2 where ^g K i is a K-th order series estimator omittig observatio i: The CV estimate ^K is the value which miimizes CV K : Li (987, Aals of Statistics) showed uder quite miimal coditios that Mallows, GCV, ad CV are asymptotically optimal for selectio of K; i the sese that ISE ^K if k ISE K! p Adrews (99, JoE) showed that this optimality oly exteds to the heteroskedastic case if CV is used for selectio. The reaso is that the Mallows criterio uses homoskedasticity to calculate the bias adjustmet, as we showed above, ad this is ot eeded uder CV. 2.0 Partially Liear ad Additive Models Suppose y i = W 0 i + g (X i ) + e i 95
8 with g oparametric. A series approximatio for g is z 0 K yieldig the model for estimatio y i = W 0 i + z 0 i K + error i which is estimated by least-squares. The estimate for is similar to that from the Robiso kerel estimator, which had a residual-regressio iterpretatio. The asymptotic distributio for ^ is the same as for the Robiso estimator, uder the coditio that the oparametric compoet has MSE covergig faster tha =2 ; e.g. if K= + K 2 = o =2. This is similar to the requiremet for the Robiso estimator. You ca easily geeralize this idea to multiple additive oparametric compoets y i = W 0 i + g (X i ) + g 2 (X 2i ) + e i I practice, the compoets X i ad X 2i are real-valued. As discussed i Li-Racie, W i ca cotai oliear iteractio e ects betwee X i ad X 2i, such as X i X 2i : The mai requiremet is that the compoets of W i caot be additively separable i X i ad X 2i : So i this sese the additive model ca allow for simple iteractio e ects. 96
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